Subgroups — C₆₇×#10998 #30690249

Order 17152.

1350 conjugacy classes of subgroups.

The subgroup lattice has 776 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×#104×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#101×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#391×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1736	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#339×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	#536×C₆₇	—	2	2	no	—
13	#536×C₆₇	—	2	2	no	—
14	#536×C₆₇	—	2	2	no	—
15	#536×C₆₇	C₄	1	4	yes	complemented
16	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
17	C₂×#124×C₆₇	—	2	1	no	—
18	C₂×#123×C₆₇	—	2	2	no	—
19	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#104×C₆₇	—	2	2	no	—
24	C₂×#104×C₆₇	—	2	2	no	—
25	C₂×#104×C₆₇	—	2	2	no	—
26	#417×C₆₇	—	2	2	no	—
27	#417×C₆₇	—	2	2	no	—
28	C₂×#103×C₆₇	—	2	1	no	—
29	C₂×#102×C₆₇	—	2	1	no	—
30	C₂×#101×C₆₇	—	2	2	no	—
31	C₂×#101×C₆₇	—	2	2	no	—
32	C₂×#101×C₆₇	—	2	2	no	—
33	#391×C₆₇	—	2	2	no	—
34	#391×C₆₇	—	2	2	no	—
35	C₂×#95×C₆₇	—	2	2	no	—
36	#340×C₆₇	—	2	2	no	—
37	#339×C₆₇	—	2	2	no	—
38	C₂×#41×C₆₇	—	2	1	no	—
39	C₂×#41×C₆₇	—	2	1	no	—
40	C₂×#41×C₆₇	—	2	1	no	—
41	C₂×#41×C₆₇	—	2	1	no	—
42	C₂×#41×C₆₇	—	2	1	no	—
43	C₂×#41×C₆₇	—	2	1	no	—
44	C₂×#41×C₆₇	—	2	1	no	—
45	C₂×#41×C₆₇	—	2	1	no	—
46	C₂×#41×C₆₇	—	2	1	no	—
47	C₂×#41×C₆₇	—	2	1	no	—
48	C₂×#41×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×#41×C₆₇	D₄	1	2	yes	—
50	C₂²×Q₈×C₆₇	—	2	1	no	—
51	C₂²×Q₈×C₆₇	—	2	1	no	—
52	C₂²×Q₈×C₆₇	—	2	1	no	—
53	C₂²×Q₈×C₆₇	—	2	1	no	—
54	C₂²×Q₈×C₆₇	—	2	1	no	—
55	C₂²×Q₈×C₆₇	—	2	1	no	—
56	C₂²×Q₈×C₆₇	—	2	1	no	—
57	C₂²×Q₈×C₆₇	—	2	1	no	—
58	C₂²×Q₈×C₆₇	—	2	1	no	—
59	C₂²×Q₈×C₆₇	—	2	1	no	—
60	C₂²×Q₈×C₆₇	—	2	1	no	—
61	C₂²×Q₈×C₆₇	—	2	1	no	—
62	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
63	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
64	C₂²×Q₈×C₆₇	D₄	1	2	yes	—
65	C₂²×D₄×C₆₇	—	2	1	no	—
66	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	—	2	1	no	—
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂³×C₄×C₆₇	—	2	1	no	—
70	C₂³×C₄×C₆₇	D₄	1	2	yes	—
71	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₆₇	—	2	1	no	—
73	C₂×M₁₆×C₆₇	—	2	1	no	—
74	C₂×M₁₆×C₆₇	—	2	1	no	—
75	C₂×M₁₆×C₆₇	—	2	1	no	—
76	C₂×M₁₆×C₆₇	—	2	1	no	—
77	C₂×M₁₆×C₆₇	—	2	1	no	—
78	C₂×M₁₆×C₆₇	—	2	1	no	—
79	C₂×M₁₆×C₆₇	—	2	1	no	—
80	C₂×M₁₆×C₆₇	—	2	1	no	—
81	C₂×M₁₆×C₆₇	—	2	1	no	—
82	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
83	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
84	C₂²×C₈×C₆₇	—	2	1	no	—
85	C₂²×C₈×C₆₇	—	2	1	no	—
86	C₄⋊Q₈×C₆₇	—	2	4	no	—
87	C₄⋊Q₈×C₆₇	—	2	4	no	—
88	#124×C₆₇	—	2	4	no	—
89	#124×C₆₇	—	2	4	no	—
90	#123×C₆₇	—	2	8	no	—
91	#123×C₆₇	—	2	8	no	—
92	C₂²⋊Q₈×C₆₇	—	2	8	no	—
93	C₂²⋊Q₈×C₆₇	—	2	8	no	—
94	#117×C₆₇	—	2	1	no	—
95	#117×C₆₇	—	2	1	no	—
96	#117×C₆₇	C₂×C₄	1	2	yes	—
97	#117×C₆₇	D₄	1	2	yes	—
98	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
104	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
105	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
106	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
107	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
108	C₂×C₄²×C₆₇	—	2	1	no	—
109	C₂×C₄²×C₆₇	—	2	1	no	—
110	C₂×C₄²×C₆₇	—	2	1	no	—
111	C₂×C₄²×C₆₇	—	2	1	no	—
112	C₂×C₄²×C₆₇	—	2	1	no	—
113	C₂×C₄²×C₆₇	—	2	1	no	—
114	C₂×C₄²×C₆₇	D₄	1	2	yes	—
115	#104×C₆₇	—	2	8	no	—
116	#103×C₆₇	—	2	4	no	—
117	#102×C₆₇	—	2	4	no	—
118	#101×C₆₇	—	2	8	no	—
119	#95×C₆₇	—	2	8	no	—
120	C₂⁴×C₆₇	—	2	1	no	—
121	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
122	#41×C₆₇	—	2	2	no	—
123	#41×C₆₇	—	2	2	no	—
124	#41×C₆₇	—	2	2	no	—
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	—	2	2	no	—
127	#41×C₆₇	—	2	2	no	—
128	#41×C₆₇	—	2	2	no	—
129	#41×C₆₇	—	2	2	no	—
130	#41×C₆₇	—	4	1	no	—
131	#41×C₆₇	—	4	1	no	—
132	#41×C₆₇	—	4	1	no	—
133	#41×C₆₇	—	4	1	no	—
134	#41×C₆₇	—	4	1	no	—
135	#41×C₆₇	—	4	1	no	—
136	#41×C₆₇	—	4	1	no	—
137	#41×C₆₇	—	4	1	no	—
138	C₂×Q₈×C₆₇	—	2	2	no	—
139	C₂×Q₈×C₆₇	—	2	2	no	—
140	C₂×Q₈×C₆₇	—	2	2	no	—
141	C₂×Q₈×C₆₇	—	2	2	no	—
142	C₂×Q₈×C₆₇	—	2	2	no	—
143	C₂×Q₈×C₆₇	—	2	2	no	—
144	C₂×Q₈×C₆₇	—	2	2	no	—
145	C₂×Q₈×C₆₇	—	2	2	no	—
146	C₂×Q₈×C₆₇	—	2	2	no	—
147	C₂×Q₈×C₆₇	—	2	1	no	—
148	C₂×Q₈×C₆₇	—	2	2	no	—
149	C₂×Q₈×C₆₇	—	2	2	no	—
150	C₂×Q₈×C₆₇	—	2	2	no	—
151	C₂×Q₈×C₆₇	—	2	1	no	—
152	C₂×Q₈×C₆₇	—	2	1	no	—
153	C₂×Q₈×C₆₇	—	2	1	no	—
154	C₂×Q₈×C₆₇	—	2	1	no	—
155	C₂×Q₈×C₆₇	—	4	1	no	—
156	C₂×Q₈×C₆₇	—	4	1	no	—
157	C₂×Q₈×C₆₇	—	4	1	no	—
158	C₂×Q₈×C₆₇	—	4	1	no	—
159	C₂×Q₈×C₆₇	—	4	1	no	—
160	C₂×Q₈×C₆₇	—	4	1	no	—
161	C₂×Q₈×C₆₇	—	4	1	no	—
162	C₂×Q₈×C₆₇	—	4	1	no	—
163	C₂×Q₈×C₆₇	—	4	1	no	—
164	C₂×Q₈×C₆₇	—	4	1	no	—
165	C₂×Q₈×C₆₇	—	4	1	no	—
166	C₂×Q₈×C₆₇	—	4	1	no	—
167	C₂×Q₈×C₆₇	—	4	1	no	—
168	C₂×Q₈×C₆₇	—	4	1	no	—
169	C₂×Q₈×C₆₇	—	4	1	no	—
170	C₂×Q₈×C₆₇	—	4	1	no	—
171	C₂×Q₈×C₆₇	C₂²⋊C₄	1	2	yes	—
172	C₂×D₄×C₆₇	—	2	2	no	—
173	C₂×D₄×C₆₇	—	2	2	no	—
174	C₂×D₄×C₆₇	—	2	1	no	—
175	C₂×D₄×C₆₇	—	2	1	no	—
176	C₂×D₄×C₆₇	—	2	1	no	—
177	C₂×D₄×C₆₇	—	2	1	no	—
178	C₂×D₄×C₆₇	—	2	1	no	—
179	C₂×D₄×C₆₇	—	4	1	no	—
180	C₂×D₄×C₆₇	—	4	1	no	—
181	C₂×D₄×C₆₇	—	4	1	no	—
182	C₂×D₄×C₆₇	—	4	1	no	—
183	C₂×D₄×C₆₇	C₂²⋊C₄	1	2	yes	—
184	C₂²×C₄×C₆₇	—	2	4	no	—
185	C₂²×C₄×C₆₇	—	2	2	no	—
186	C₂²×C₄×C₆₇	—	2	2	no	—
187	C₂²×C₄×C₆₇	—	2	2	no	—
188	C₂²×C₄×C₆₇	—	2	2	no	—
189	C₂²×C₄×C₆₇	—	2	2	no	—
190	C₂²×C₄×C₆₇	—	2	1	no	—
191	C₂²×C₄×C₆₇	—	2	2	no	—
192	C₂²×C₄×C₆₇	—	2	2	no	—
193	C₂²×C₄×C₆₇	—	2	2	no	—
194	C₂²×C₄×C₆₇	—	2	2	no	—
195	C₂²×C₄×C₆₇	—	2	2	no	—
196	C₂²×C₄×C₆₇	—	2	2	no	—
197	C₂²×C₄×C₆₇	—	4	2	no	—
198	C₂²×C₄×C₆₇	—	4	1	no	—
199	C₂²×C₄×C₆₇	—	4	1	no	—
200	C₂²×C₄×C₆₇	—	4	1	no	—
201	C₂²×C₄×C₆₇	—	4	1	no	—
202	C₂²×C₄×C₆₇	—	4	1	no	—
203	C₂²×C₄×C₆₇	—	4	1	no	—
204	C₂²×C₄×C₆₇	—	4	4	no	—
205	C₂²×C₄×C₆₇	—	4	1	no	—
206	C₂²×C₄×C₆₇	—	4	1	no	—
207	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
208	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
209	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
211	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
212	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
213	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
214	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
215	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
216	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
217	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
218	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
219	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
220	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
221	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
222	M₁₆×C₆₇	—	2	2	no	—
223	M₁₆×C₆₇	—	2	2	no	—
224	M₁₆×C₆₇	—	2	2	no	—
225	M₁₆×C₆₇	—	2	2	no	—
226	M₁₆×C₆₇	—	4	1	no	—
227	M₁₆×C₆₇	—	4	1	no	—
228	M₁₆×C₆₇	—	4	1	no	—
229	M₁₆×C₆₇	—	4	1	no	—
230	C₂×C₈×C₆₇	—	2	2	no	—
231	C₂×C₈×C₆₇	—	2	2	no	—
232	C₂×C₈×C₆₇	—	4	1	no	—
233	C₂×C₈×C₆₇	—	4	1	no	—
234	C₂×C₈×C₆₇	—	4	1	no	—
235	C₂×C₈×C₆₇	—	4	1	no	—
236	C₄⋊C₄×C₆₇	—	2	16	no	—
237	C₄⋊C₄×C₆₇	—	2	2	no	—
238	C₄⋊C₄×C₆₇	—	2	2	no	—
239	C₄⋊C₄×C₆₇	—	2	2	no	—
240	C₄⋊C₄×C₆₇	—	4	1	no	—
241	C₄⋊C₄×C₆₇	—	4	1	no	—
242	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
243	C₂²⋊C₄×C₆₇	—	2	16	no	—
244	C₂²⋊C₄×C₆₇	—	2	2	no	—
245	C₂²⋊C₄×C₆₇	—	2	2	no	—
246	C₄²×C₆₇	—	2	2	no	—
247	C₄²×C₆₇	—	2	2	no	—
248	C₄²×C₆₇	—	2	2	no	—
249	C₄²×C₆₇	—	2	2	no	—
250	C₄²×C₆₇	—	4	1	no	—
251	C₄²×C₆₇	—	4	1	no	—
252	C₂³×C₆₇	—	2	2	no	—
253	C₂³×C₆₇	—	2	2	no	—
254	C₂³×C₆₇	—	2	2	no	—
255	C₂³×C₆₇	—	4	1	no	—
256	C₂³×C₆₇	—	4	1	no	—
257	C₂³×C₆₇	—	4	2	no	—
258	C₂³×C₆₇	—	4	2	no	—
259	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
260	C₂³×C₆₇	#120	1	1	yes	characteristic
261	C₂³×C₆₇	#120	1	1	yes	characteristic
262	C₂³×C₆₇	#120	1	1	yes	characteristic
263	C₂³×C₆₇	#121	1	1	yes	characteristic
264	C₂³×C₆₇	#121	1	1	yes	characteristic
265	C₂³×C₆₇	#123	1	1	yes	characteristic
266	Q₈×C₆₇	—	2	2	no	—
267	Q₈×C₆₇	—	2	2	no	—
268	Q₈×C₆₇	—	4	2	no	—
269	Q₈×C₆₇	—	4	2	no	—
270	Q₈×C₆₇	—	4	2	no	—
271	Q₈×C₆₇	—	4	2	no	—
272	Q₈×C₆₇	—	4	2	no	—
273	Q₈×C₆₇	—	4	2	no	—
274	Q₈×C₆₇	—	4	1	no	—
275	Q₈×C₆₇	—	4	2	no	—
276	Q₈×C₆₇	—	4	2	no	—
277	Q₈×C₆₇	—	4	1	no	—
278	D₄×C₆₇	—	2	2	no	—
279	D₄×C₆₇	—	2	2	no	—
280	D₄×C₆₇	—	4	2	no	—
281	D₄×C₆₇	—	4	2	no	—
282	D₄×C₆₇	—	4	2	no	—
283	D₄×C₆₇	—	4	2	no	—
284	D₄×C₆₇	—	4	1	no	—
285	D₄×C₆₇	—	4	1	no	—
286	C₂×C₄×C₆₇	—	2	4	no	—
287	C₂×C₄×C₆₇	—	2	4	no	—
288	C₂×C₄×C₆₇	—	2	4	no	—
289	C₂×C₄×C₆₇	—	2	2	no	—
290	C₂×C₄×C₆₇	—	2	4	no	—
291	C₂×C₄×C₆₇	—	2	4	no	—
292	C₂×C₄×C₆₇	—	4	4	no	—
293	C₂×C₄×C₆₇	—	4	2	no	—
294	C₂×C₄×C₆₇	—	4	4	no	—
295	C₂×C₄×C₆₇	—	4	2	no	—
296	C₂×C₄×C₆₇	—	4	2	no	—
297	C₂×C₄×C₆₇	—	4	2	no	—
298	C₂×C₄×C₆₇	—	4	1	no	—
299	C₂×C₄×C₆₇	—	4	2	no	—
300	C₂×C₄×C₆₇	—	4	2	no	—
301	C₂×C₄×C₆₇	—	4	1	no	—
302	C₂×C₄×C₆₇	—	4	2	no	—
303	C₂×C₄×C₆₇	—	4	2	no	—
304	C₂×C₄×C₆₇	—	4	2	no	—
305	C₂×C₄×C₆₇	—	4	2	no	—
306	C₂×C₄×C₆₇	—	4	2	no	—
307	C₂×C₄×C₆₇	—	4	2	no	—
308	C₂×C₄×C₆₇	—	4	4	no	—
309	C₂×C₄×C₆₇	—	4	2	no	—
310	C₂×C₄×C₆₇	—	8	2	no	—
311	C₂×C₄×C₆₇	—	8	2	no	—
312	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
313	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
314	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
315	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
316	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
317	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
318	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
319	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
320	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
321	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
322	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
323	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
324	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
325	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
326	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
327	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
328	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
329	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
330	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
331	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
332	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
333	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
334	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
335	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
336	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
337	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
338	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
339	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
340	C₈×C₆₇	—	4	2	no	—
341	C₈×C₆₇	—	4	2	no	—
342	C₂²×C₆₇	—	2	2	no	—
343	C₂²×C₆₇	—	2	2	no	—
344	C₂²×C₆₇	—	2	1	no	—
345	C₂²×C₆₇	—	2	1	no	—
346	C₂²×C₆₇	—	2	1	no	—
347	C₂²×C₆₇	—	2	1	no	—
348	C₂²×C₆₇	—	2	2	no	—
349	C₂²×C₆₇	—	2	2	no	—
350	C₂²×C₆₇	—	4	4	no	—
351	C₂²×C₆₇	—	4	2	no	—
352	C₂²×C₆₇	—	8	1	no	—
353	C₂²×C₆₇	—	8	1	no	—
354	C₂²×C₆₇	—	8	2	no	—
355	C₂²×C₆₇	#379	1	1	yes	characteristic
356	C₂²×C₆₇	#386	1	1	yes	characteristic
357	C₂²×C₆₇	#386	1	1	yes	characteristic
358	C₂²×C₆₇	#392	1	1	yes	characteristic
359	C₂²×C₆₇	#392	1	1	yes	characteristic
360	C₂²×C₆₇	#394	1	1	yes	characteristic
361	C₂²×C₆₇	#394	1	1	yes	characteristic
362	C₂²×C₆₇	#454	1	2	yes	—
363	C₂²×C₆₇	#456	1	2	yes	—
364	C₄×C₆₇	—	4	4	no	—
365	C₄×C₆₇	—	4	4	no	—
366	C₄×C₆₇	—	4	4	no	—
367	C₄×C₆₇	—	4	2	no	—
368	C₄×C₆₇	—	8	2	no	—
369	C₄×C₆₇	#386	1	1	yes	characteristic
370	C₄×C₆₇	#386	1	1	yes	characteristic
371	C₄×C₆₇	#390	1	1	yes	characteristic
372	C₄×C₆₇	#390	1	1	yes	characteristic
373	C₄×C₆₇	#392	1	1	yes	characteristic
374	C₄×C₆₇	#392	1	1	yes	characteristic
375	C₄×C₆₇	#394	1	1	yes	characteristic
376	C₄×C₆₇	#394	1	1	yes	characteristic
377	#10998	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
378	C₂×C₆₇	—	2	1	no	—
379	C₂×C₆₇	—	2	1	no	—
380	C₂×C₆₇	—	2	1	no	—
381	C₂×C₆₇	—	2	1	no	—
382	C₂×C₆₇	—	8	2	no	—
383	C₂×C₆₇	#1438	1	1	yes	characteristic
384	C₂×C₆₇	#1793	1	2	yes	—
385	C₂×C₆₇	#2003	1	1	yes	characteristic
386	C₂×C₆₇	#2005	1	1	yes	characteristic
387	C₂×C₆₇	#2038	1	2	yes	—
388	C₂×#536	C₂×C₆₇	1	1	yes	characteristic
389	C₂²×#104	C₂×C₆₇	1	1	yes	characteristic
390	C₂×#417	C₂×C₆₇	1	1	yes	characteristic
391	C₂²×#101	C₂×C₆₇	1	1	yes	characteristic, complemented
392	C₂×#391	C₂×C₆₇	1	1	yes	characteristic, complemented
393	#1736	C₂×C₆₇	1	1	yes	characteristic, complemented
394	C₂×#339	C₂×C₆₇	1	1	yes	characteristic, complemented
395	C₆₇	#10998	1	1	yes	characteristic, direct product factor, Sylow
396	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
397	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
398	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
399	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
400	#536	—	2	2	no	—
401	#536	—	2	2	no	—
402	#536	—	2	2	no	—
403	#536	C₄×C₆₇	1	4	yes	complemented
404	C₂×C₄⋊Q₈	—	2	1	no	—
405	C₂×#124	—	2	1	no	—
406	C₂×#123	—	2	2	no	—
407	C₂×C₂²⋊Q₈	—	2	2	no	—
408	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
409	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
410	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
411	C₂×#104	—	2	2	no	—
412	C₂×#104	—	2	2	no	—
413	C₂×#104	—	2	2	no	—
414	#417	—	2	2	no	—
415	#417	—	2	2	no	—
416	C₂×#103	—	2	1	no	—
417	C₂×#102	—	2	1	no	—
418	C₂×#101	—	2	2	no	—
419	C₂×#101	—	2	2	no	—
420	C₂×#101	—	2	2	no	—
421	#391	—	2	2	no	—
422	#391	—	2	2	no	—
423	C₂×#95	—	2	2	no	—
424	#340	—	2	2	no	—
425	#339	—	2	2	no	—
426	C₂×#41	—	2	1	no	—
427	C₂×#41	—	2	1	no	—
428	C₂×#41	—	2	1	no	—
429	C₂×#41	—	2	1	no	—
430	C₂×#41	—	2	1	no	—
431	C₂×#41	—	2	1	no	—
432	C₂×#41	—	2	1	no	—
433	C₂×#41	—	2	1	no	—
434	C₂×#41	—	2	1	no	—
435	C₂×#41	—	2	1	no	—
436	C₂×#41	C₂×C₄×C₆₇	1	2	yes	—
437	C₂×#41	D₄×C₆₇	1	2	yes	—
438	C₂²×Q₈	—	2	1	no	—
439	C₂²×Q₈	—	2	1	no	—
440	C₂²×Q₈	—	2	1	no	—
441	C₂²×Q₈	—	2	1	no	—
442	C₂²×Q₈	—	2	1	no	—
443	C₂²×Q₈	—	2	1	no	—
444	C₂²×Q₈	—	2	1	no	—
445	C₂²×Q₈	—	2	1	no	—
446	C₂²×Q₈	—	2	1	no	—
447	C₂²×Q₈	—	2	1	no	—
448	C₂²×Q₈	—	2	1	no	—
449	C₂²×Q₈	—	2	1	no	—
450	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
451	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
452	C₂²×Q₈	D₄×C₆₇	1	2	yes	—
453	C₂²×D₄	—	2	1	no	—
454	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
455	C₂³×C₄	—	2	1	no	—
456	C₂³×C₄	—	2	1	no	—
457	C₂³×C₄	—	2	1	no	—
458	C₂³×C₄	D₄×C₆₇	1	2	yes	—
459	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
460	C₂×M₁₆	—	2	1	no	—
461	C₂×M₁₆	—	2	1	no	—
462	C₂×M₁₆	—	2	1	no	—
463	C₂×M₁₆	—	2	1	no	—
464	C₂×M₁₆	—	2	1	no	—
465	C₂×M₁₆	—	2	1	no	—
466	C₂×M₁₆	—	2	1	no	—
467	C₂×M₁₆	—	2	1	no	—
468	C₂×M₁₆	—	2	1	no	—
469	C₂×M₁₆	—	2	1	no	—
470	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
471	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
472	C₂²×C₈	—	2	1	no	—
473	C₂²×C₈	—	2	1	no	—
474	C₄⋊Q₈	—	2	4	no	—
475	C₄⋊Q₈	—	2	4	no	—
476	#124	—	2	4	no	—
477	#124	—	2	4	no	—
478	#123	—	2	8	no	—
479	#123	—	2	8	no	—
480	C₂²⋊Q₈	—	2	8	no	—
481	C₂²⋊Q₈	—	2	8	no	—
482	#117	—	2	1	no	—
483	#117	—	2	1	no	—
484	#117	C₂×C₄×C₆₇	1	2	yes	—
485	#117	D₄×C₆₇	1	2	yes	—
486	C₂×C₄⋊C₄	—	2	4	no	—
487	C₂×C₄⋊C₄	—	2	1	no	—
488	C₂×C₄⋊C₄	—	2	1	no	—
489	C₂×C₄⋊C₄	—	2	1	no	—
490	C₂×C₄⋊C₄	—	2	1	no	—
491	C₂×C₄⋊C₄	—	2	1	no	—
492	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
493	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
494	C₂×C₂²⋊C₄	—	2	4	no	—
495	C₂×C₂²⋊C₄	—	2	1	no	—
496	C₂×C₄²	—	2	1	no	—
497	C₂×C₄²	—	2	1	no	—
498	C₂×C₄²	—	2	1	no	—
499	C₂×C₄²	—	2	1	no	—
500	C₂×C₄²	—	2	1	no	—
501	C₂×C₄²	—	2	1	no	—
502	C₂×C₄²	D₄×C₆₇	1	2	yes	—
503	#104	—	2	8	no	—
504	#103	—	2	4	no	—
505	#102	—	2	4	no	—
506	#101	—	2	8	no	—
507	#95	—	2	8	no	—
508	C₂⁴	—	2	1	no	—
509	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
510	#41	—	2	2	no	—
511	#41	—	2	2	no	—
512	#41	—	2	2	no	—
513	#41	—	2	2	no	—
514	#41	—	2	2	no	—
515	#41	—	2	2	no	—
516	#41	—	2	2	no	—
517	#41	—	2	2	no	—
518	#41	—	4	1	no	—
519	#41	—	4	1	no	—
520	#41	—	4	1	no	—
521	#41	—	4	1	no	—
522	#41	—	4	1	no	—
523	#41	—	4	1	no	—
524	#41	—	4	1	no	—
525	#41	—	4	1	no	—
526	C₂×Q₈	—	2	2	no	—
527	C₂×Q₈	—	2	2	no	—
528	C₂×Q₈	—	2	2	no	—
529	C₂×Q₈	—	2	2	no	—
530	C₂×Q₈	—	2	2	no	—
531	C₂×Q₈	—	2	2	no	—
532	C₂×Q₈	—	2	2	no	—
533	C₂×Q₈	—	2	2	no	—
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	2	1	no	—
536	C₂×Q₈	—	2	2	no	—
537	C₂×Q₈	—	2	2	no	—
538	C₂×Q₈	—	2	2	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	2	1	no	—
542	C₂×Q₈	—	2	1	no	—
543	C₂×Q₈	—	4	1	no	—
544	C₂×Q₈	—	4	1	no	—
545	C₂×Q₈	—	4	1	no	—
546	C₂×Q₈	—	4	1	no	—
547	C₂×Q₈	—	4	1	no	—
548	C₂×Q₈	—	4	1	no	—
549	C₂×Q₈	—	4	1	no	—
550	C₂×Q₈	—	4	1	no	—
551	C₂×Q₈	—	4	1	no	—
552	C₂×Q₈	—	4	1	no	—
553	C₂×Q₈	—	4	1	no	—
554	C₂×Q₈	—	4	1	no	—
555	C₂×Q₈	—	4	1	no	—
556	C₂×Q₈	—	4	1	no	—
557	C₂×Q₈	—	4	1	no	—
558	C₂×Q₈	—	4	1	no	—
559	C₂×Q₈	C₂²⋊C₄×C₆₇	1	2	yes	—
560	C₂×D₄	—	2	2	no	—
561	C₂×D₄	—	2	2	no	—
562	C₂×D₄	—	2	1	no	—
563	C₂×D₄	—	2	1	no	—
564	C₂×D₄	—	2	1	no	—
565	C₂×D₄	—	2	1	no	—
566	C₂×D₄	—	2	1	no	—
567	C₂×D₄	—	4	1	no	—
568	C₂×D₄	—	4	1	no	—
569	C₂×D₄	—	4	1	no	—
570	C₂×D₄	—	4	1	no	—
571	C₂×D₄	C₂²⋊C₄×C₆₇	1	2	yes	—
572	C₂²×C₄	—	2	4	no	—
573	C₂²×C₄	—	2	2	no	—
574	C₂²×C₄	—	2	2	no	—
575	C₂²×C₄	—	2	2	no	—
576	C₂²×C₄	—	2	2	no	—
577	C₂²×C₄	—	2	1	no	—
578	C₂²×C₄	—	2	2	no	—
579	C₂²×C₄	—	2	2	no	—
580	C₂²×C₄	—	2	2	no	—
581	C₂²×C₄	—	2	2	no	—
582	C₂²×C₄	—	2	2	no	—
583	C₂²×C₄	—	2	2	no	—
584	C₂²×C₄	—	2	2	no	—
585	C₂²×C₄	—	4	2	no	—
586	C₂²×C₄	—	4	1	no	—
587	C₂²×C₄	—	4	1	no	—
588	C₂²×C₄	—	4	1	no	—
589	C₂²×C₄	—	4	1	no	—
590	C₂²×C₄	—	4	1	no	—
591	C₂²×C₄	—	4	1	no	—
592	C₂²×C₄	—	4	4	no	—
593	C₂²×C₄	—	4	1	no	—
594	C₂²×C₄	—	4	1	no	—
595	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
596	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
597	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
598	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
599	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
600	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
601	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
602	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
603	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
604	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
605	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
606	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
607	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
608	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
609	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
610	M₁₆	—	2	2	no	—
611	M₁₆	—	2	2	no	—
612	M₁₆	—	2	2	no	—
613	M₁₆	—	2	2	no	—
614	M₁₆	—	4	1	no	—
615	M₁₆	—	4	1	no	—
616	M₁₆	—	4	1	no	—
617	M₁₆	—	4	1	no	—
618	C₂×C₈	—	2	2	no	—
619	C₂×C₈	—	2	2	no	—
620	C₂×C₈	—	4	1	no	—
621	C₂×C₈	—	4	1	no	—
622	C₂×C₈	—	4	1	no	—
623	C₂×C₈	—	4	1	no	—
624	C₄⋊C₄	—	2	16	no	—
625	C₄⋊C₄	—	2	2	no	—
626	C₄⋊C₄	—	2	2	no	—
627	C₄⋊C₄	—	2	2	no	—
628	C₄⋊C₄	—	4	1	no	—
629	C₄⋊C₄	—	4	1	no	—
630	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
631	C₂²⋊C₄	—	2	16	no	—
632	C₂²⋊C₄	—	2	2	no	—
633	C₂²⋊C₄	—	2	2	no	—
634	C₄²	—	2	2	no	—
635	C₄²	—	2	2	no	—
636	C₄²	—	2	2	no	—
637	C₄²	—	2	2	no	—
638	C₄²	—	4	1	no	—
639	C₄²	—	4	1	no	—
640	C₂³	—	2	2	no	—
641	C₂³	—	2	2	no	—
642	C₂³	—	2	2	no	—
643	C₂³	—	4	2	no	—
644	C₂³	—	4	2	no	—
645	C₂³	—	4	1	no	—
646	C₂³	—	4	1	no	—
647	C₂³	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
648	C₂³	#120×C₆₇	1	1	yes	characteristic
649	C₂³	#120×C₆₇	1	1	yes	characteristic
650	C₂³	#120×C₆₇	1	1	yes	characteristic
651	C₂³	#121×C₆₇	1	1	yes	characteristic
652	C₂³	#121×C₆₇	1	1	yes	characteristic
653	C₂³	#123×C₆₇	1	1	yes	characteristic
654	Q₈	—	2	2	no	—
655	Q₈	—	2	2	no	—
656	Q₈	—	4	2	no	—
657	Q₈	—	4	2	no	—
658	Q₈	—	4	2	no	—
659	Q₈	—	4	2	no	—
660	Q₈	—	4	2	no	—
661	Q₈	—	4	2	no	—
662	Q₈	—	4	1	no	—
663	Q₈	—	4	2	no	—
664	Q₈	—	4	2	no	—
665	Q₈	—	4	1	no	—
666	D₄	—	2	2	no	—
667	D₄	—	2	2	no	—
668	D₄	—	4	2	no	—
669	D₄	—	4	2	no	—
670	D₄	—	4	2	no	—
671	D₄	—	4	2	no	—
672	D₄	—	4	1	no	—
673	D₄	—	4	1	no	—
674	C₂×C₄	—	2	4	no	—
675	C₂×C₄	—	2	4	no	—
676	C₂×C₄	—	2	4	no	—
677	C₂×C₄	—	2	2	no	—
678	C₂×C₄	—	2	4	no	—
679	C₂×C₄	—	2	4	no	—
680	C₂×C₄	—	4	2	no	—
681	C₂×C₄	—	4	4	no	—
682	C₂×C₄	—	4	4	no	—
683	C₂×C₄	—	4	2	no	—
684	C₂×C₄	—	4	2	no	—
685	C₂×C₄	—	4	2	no	—
686	C₂×C₄	—	4	2	no	—
687	C₂×C₄	—	4	2	no	—
688	C₂×C₄	—	4	1	no	—
689	C₂×C₄	—	4	2	no	—
690	C₂×C₄	—	4	2	no	—
691	C₂×C₄	—	4	1	no	—
692	C₂×C₄	—	4	2	no	—
693	C₂×C₄	—	4	2	no	—
694	C₂×C₄	—	4	2	no	—
695	C₂×C₄	—	4	2	no	—
696	C₂×C₄	—	4	4	no	—
697	C₂×C₄	—	4	2	no	—
698	C₂×C₄	—	8	2	no	—
699	C₂×C₄	—	8	2	no	—
700	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
701	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
702	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
703	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
704	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
705	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
706	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
707	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
708	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
709	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
710	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
711	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
712	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
713	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
714	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
715	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
716	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
717	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
718	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
719	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
720	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
721	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
722	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
723	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
724	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
725	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
726	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
727	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
728	C₈	—	4	2	no	—
729	C₈	—	4	2	no	—
730	C₂²	—	2	1	no	—
731	C₂²	—	2	1	no	—
732	C₂²	—	2	1	no	—
733	C₂²	—	2	1	no	—
734	C₂²	—	2	2	no	—
735	C₂²	—	2	2	no	—
736	C₂²	—	2	2	no	—
737	C₂²	—	2	2	no	—
738	C₂²	—	4	4	no	—
739	C₂²	—	4	2	no	—
740	C₂²	—	8	1	no	—
741	C₂²	—	8	1	no	—
742	C₂²	—	8	2	no	—
743	C₂²	#379×C₆₇	1	1	yes	characteristic
744	C₂²	#386×C₆₇	1	1	yes	characteristic
745	C₂²	#386×C₆₇	1	1	yes	characteristic
746	C₂²	#392×C₆₇	1	1	yes	characteristic
747	C₂²	#392×C₆₇	1	1	yes	characteristic
748	C₂²	#394×C₆₇	1	1	yes	characteristic
749	C₂²	#394×C₆₇	1	1	yes	characteristic
750	C₂²	#454×C₆₇	1	2	yes	—
751	C₂²	#456×C₆₇	1	2	yes	—
752	C₄	—	4	4	no	—
753	C₄	—	4	4	no	—
754	C₄	—	4	4	no	—
755	C₄	—	4	2	no	—
756	C₄	—	8	2	no	—
757	C₄	#386×C₆₇	1	1	yes	characteristic
758	C₄	#386×C₆₇	1	1	yes	characteristic
759	C₄	#390×C₆₇	1	1	yes	characteristic
760	C₄	#390×C₆₇	1	1	yes	characteristic
761	C₄	#392×C₆₇	1	1	yes	characteristic
762	C₄	#392×C₆₇	1	1	yes	characteristic
763	C₄	#394×C₆₇	1	1	yes	characteristic
764	C₄	#394×C₆₇	1	1	yes	characteristic
765	C₂	—	2	1	no	—
766	C₂	—	2	1	no	—
767	C₂	—	2	1	no	—
768	C₂	—	2	1	no	—
769	C₂	—	8	2	no	—
770	C₂	C₆₇×#1438	1	1	yes	characteristic
771	C₂	C₆₇×#1793	1	2	yes	—
772	C₂	C₆₇×#2003	1	1	yes	characteristic
773	C₂	C₆₇×#2005	1	1	yes	characteristic
774	C₂	C₆₇×#2038	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	#120
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₂²×#101×C₆₇	C₂	semidirect
5. C₂×#391×C₆₇	C₂	semidirect
6. C₆₇×#1736	C₂	semidirect
7. C₂×#339×C₆₇	C₂	semidirect
15. #536×C₆₇	C₄	semidirect
377. #10998	C₆₇	direct (N × H)
391. C₂²×#101	C₂×C₆₇	semidirect
392. C₂×#391	C₂×C₆₇	semidirect
393. #1736	C₂×C₆₇	semidirect
394. C₂×#339	C₂×C₆₇	semidirect
395. C₆₇	#10998	direct (N × H)
403. #536	C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#536×C₆₇	2
C₂²×#104×C₆₇	2
C₂×#417×C₆₇	2
C₂²×#101×C₆₇	2
C₂×#391×C₆₇	2
C₆₇×#1736	2
C₂×#339×C₆₇	2
#10998	67

Subgroups — C67×#10998 #30690249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10998 #30690249