Subgroups — C₆₇×#11146 #30690397

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#421×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#104×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#413×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#344×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#339×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1728	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×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	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
15	#430×C₆₇	—	2	2	no	—
16	#430×C₆₇	—	2	2	no	—
17	C₂×#108×C₆₇	—	2	2	no	—
18	#421×C₆₇	—	2	2	no	—
19	#421×C₆₇	—	2	2	no	—
20	C₂×#104×C₆₇	—	2	2	no	—
21	C₂×#104×C₆₇	—	2	2	no	—
22	C₂×#104×C₆₇	—	2	2	no	—
23	C₂×#104×C₆₇	—	2	2	no	—
24	#413×C₆₇	—	2	2	no	—
25	#413×C₆₇	—	2	2	no	—
26	C₂×#101×C₆₇	C₄	1	2	yes	complemented
27	C₂×#100×C₆₇	C₄	1	2	yes	complemented
28	#344×C₆₇	—	2	2	no	—
29	#339×C₆₇	—	2	2	no	—
30	#336×C₆₇	—	2	2	no	—
31	C₂×#41×C₆₇	—	2	1	no	—
32	C₂×#41×C₆₇	—	2	1	no	—
33	C₂×#41×C₆₇	—	2	1	no	—
34	C₂×#41×C₆₇	—	2	1	no	—
35	C₂×#41×C₆₇	—	2	1	no	—
36	C₂×#41×C₆₇	—	2	1	no	—
37	C₂×#41×C₆₇	—	2	1	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	D₄	1	2	yes	—
42	C₂×#41×C₆₇	Q₈	1	2	yes	—
43	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₄×C₆₇	—	2	1	no	—
45	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₆₇	—	2	1	no	—
47	C₂³×C₄×C₆₇	—	2	1	no	—
48	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₆₇	—	2	1	no	—
50	C₂×M₁₆×C₆₇	—	2	1	no	—
51	C₂×M₁₆×C₆₇	—	2	1	no	—
52	C₂×M₁₆×C₆₇	—	2	1	no	—
53	C₂×M₁₆×C₆₇	—	2	1	no	—
54	C₂×M₁₆×C₆₇	—	2	1	no	—
55	C₂×M₁₆×C₆₇	—	2	1	no	—
56	C₂×M₁₆×C₆₇	—	2	1	no	—
57	C₂×M₁₆×C₆₇	—	2	1	no	—
58	C₂×M₁₆×C₆₇	—	2	1	no	—
59	C₂×M₁₆×C₆₇	—	2	1	no	—
60	C₂×M₁₆×C₆₇	—	2	1	no	—
61	C₂×M₁₆×C₆₇	—	2	1	no	—
62	C₂×M₁₆×C₆₇	—	2	1	no	—
63	C₂×M₁₆×C₆₇	—	2	1	no	—
64	C₂×M₁₆×C₆₇	—	2	1	no	—
65	C₂×M₁₆×C₆₇	—	2	1	no	—
66	C₂×M₁₆×C₆₇	—	2	1	no	—
67	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
69	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
70	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
71	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
72	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
73	C₂²×C₈×C₆₇	—	2	1	no	—
74	C₂²×C₈×C₆₇	—	2	1	no	—
75	C₂²×C₈×C₆₇	—	2	1	no	—
76	C₂²×C₈×C₆₇	—	2	1	no	—
77	#117×C₆₇	—	2	1	no	—
78	#117×C₆₇	—	2	1	no	—
79	#117×C₆₇	—	2	1	no	—
80	#117×C₆₇	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄²×C₆₇	—	2	1	no	—
84	C₂×C₄²×C₆₇	—	2	1	no	—
85	C₂×C₄²×C₆₇	—	2	1	no	—
86	C₂×C₄²×C₆₇	—	2	1	no	—
87	C₂×C₄²×C₆₇	—	2	1	no	—
88	C₂×C₄²×C₆₇	—	2	1	no	—
89	C₂×C₄²×C₆₇	D₄	1	2	yes	—
90	#108×C₆₇	—	2	8	no	—
91	#104×C₆₇	—	2	8	no	—
92	#104×C₆₇	—	2	8	no	—
93	#101×C₆₇	—	2	4	no	—
94	#100×C₆₇	—	2	4	no	—
95	C₂⁴×C₆₇	—	2	1	no	—
96	C₂⁴×C₆₇	#41	1	1	yes	characteristic
97	#41×C₆₇	—	2	2	no	—
98	#41×C₆₇	—	2	2	no	—
99	#41×C₆₇	—	2	2	no	—
100	#41×C₆₇	—	2	2	no	—
101	#41×C₆₇	—	2	2	no	—
102	#41×C₆₇	—	2	2	no	—
103	#41×C₆₇	—	2	2	no	—
104	#41×C₆₇	—	2	2	no	—
105	#41×C₆₇	—	4	1	no	—
106	#41×C₆₇	—	4	1	no	—
107	#41×C₆₇	—	4	1	no	—
108	#41×C₆₇	—	4	1	no	—
109	#41×C₆₇	—	4	1	no	—
110	#41×C₆₇	—	4	1	no	—
111	#41×C₆₇	—	4	1	no	—
112	#41×C₆₇	—	4	1	no	—
113	C₂×Q₈×C₆₇	—	2	1	no	—
114	C₂×Q₈×C₆₇	—	2	1	no	—
115	C₂×Q₈×C₆₇	—	2	1	no	—
116	C₂×Q₈×C₆₇	—	2	1	no	—
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
119	C₂×D₄×C₆₇	—	2	2	no	—
120	C₂×D₄×C₆₇	—	2	2	no	—
121	C₂×D₄×C₆₇	—	2	1	no	—
122	C₂×D₄×C₆₇	—	2	1	no	—
123	C₂×D₄×C₆₇	—	2	1	no	—
124	C₂×D₄×C₆₇	—	2	1	no	—
125	C₂×D₄×C₆₇	—	2	1	no	—
126	C₂×D₄×C₆₇	—	4	1	no	—
127	C₂×D₄×C₆₇	—	4	1	no	—
128	C₂×D₄×C₆₇	—	4	1	no	—
129	C₂×D₄×C₆₇	—	4	1	no	—
130	C₂×D₄×C₆₇	C₄⋊C₄	1	2	yes	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	2	no	—
133	C₂²×C₄×C₆₇	—	2	2	no	—
134	C₂²×C₄×C₆₇	—	2	2	no	—
135	C₂²×C₄×C₆₇	—	4	4	no	—
136	C₂²×C₄×C₆₇	—	4	1	no	—
137	C₂²×C₄×C₆₇	—	4	1	no	—
138	C₂²×C₄×C₆₇	—	4	1	no	—
139	C₂²×C₄×C₆₇	—	4	1	no	—
140	C₂²×C₄×C₆₇	—	4	1	no	—
141	C₂²×C₄×C₆₇	—	4	1	no	—
142	C₂²×C₄×C₆₇	—	4	1	no	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
150	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
151	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
152	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
153	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
155	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
157	M₁₆×C₆₇	—	2	2	no	—
158	M₁₆×C₆₇	—	2	2	no	—
159	M₁₆×C₆₇	—	2	2	no	—
160	M₁₆×C₆₇	—	2	2	no	—
161	M₁₆×C₆₇	—	2	2	no	—
162	M₁₆×C₆₇	—	2	2	no	—
163	M₁₆×C₆₇	—	2	2	no	—
164	M₁₆×C₆₇	—	2	2	no	—
165	M₁₆×C₆₇	—	2	2	no	—
166	M₁₆×C₆₇	—	4	1	no	—
167	M₁₆×C₆₇	—	4	1	no	—
168	M₁₆×C₆₇	—	4	1	no	—
169	M₁₆×C₆₇	—	4	1	no	—
170	M₁₆×C₆₇	—	4	1	no	—
171	M₁₆×C₆₇	—	4	1	no	—
172	M₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
173	C₂×C₈×C₆₇	—	2	2	no	—
174	C₂×C₈×C₆₇	—	2	2	no	—
175	C₂×C₈×C₆₇	—	2	2	no	—
176	C₂×C₈×C₆₇	—	2	2	no	—
177	C₂×C₈×C₆₇	—	2	2	no	—
178	C₂×C₈×C₆₇	—	2	2	no	—
179	C₂×C₈×C₆₇	—	4	1	no	—
180	C₂×C₈×C₆₇	—	4	1	no	—
181	C₂×C₈×C₆₇	—	4	1	no	—
182	C₂×C₈×C₆₇	—	4	1	no	—
183	C₂×C₈×C₆₇	—	4	1	no	—
184	C₂×C₈×C₆₇	—	4	1	no	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	4	1	no	—
187	C₂²⋊C₄×C₆₇	—	2	2	no	—
188	C₂²⋊C₄×C₆₇	—	4	1	no	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	—	2	2	no	—
192	C₄²×C₆₇	—	4	1	no	—
193	C₄²×C₆₇	—	4	1	no	—
194	C₄²×C₆₇	—	4	1	no	—
195	C₂³×C₆₇	—	2	2	no	—
196	C₂³×C₆₇	—	4	2	no	—
197	C₂³×C₆₇	—	4	1	no	—
198	C₂³×C₆₇	—	4	1	no	—
199	C₂³×C₆₇	—	8	2	no	—
200	C₂³×C₆₇	#117	1	1	yes	characteristic
201	C₂³×C₆₇	#121	1	1	yes	characteristic
202	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
203	C₂³×C₆₇	#123	1	1	yes	characteristic
204	C₂³×C₆₇	#123	1	1	yes	characteristic
205	C₂³×C₆₇	#124	1	1	yes	characteristic
206	C₂³×C₆₇	#124	1	1	yes	characteristic
207	Q₈×C₆₇	—	2	2	no	—
208	Q₈×C₆₇	—	2	2	no	—
209	Q₈×C₆₇	—	4	1	no	—
210	Q₈×C₆₇	—	4	1	no	—
211	D₄×C₆₇	—	2	2	no	—
212	D₄×C₆₇	—	2	2	no	—
213	D₄×C₆₇	—	4	2	no	—
214	D₄×C₆₇	—	4	2	no	—
215	D₄×C₆₇	—	4	2	no	—
216	D₄×C₆₇	—	4	2	no	—
217	D₄×C₆₇	—	4	1	no	—
218	D₄×C₆₇	—	4	1	no	—
219	C₂×C₄×C₆₇	—	2	4	no	—
220	C₂×C₄×C₆₇	—	2	2	no	—
221	C₂×C₄×C₆₇	—	4	2	no	—
222	C₂×C₄×C₆₇	—	4	4	no	—
223	C₂×C₄×C₆₇	—	4	4	no	—
224	C₂×C₄×C₆₇	—	4	1	no	—
225	C₂×C₄×C₆₇	—	4	1	no	—
226	C₂×C₄×C₆₇	—	4	2	no	—
227	C₂×C₄×C₆₇	—	4	2	no	—
228	C₂×C₄×C₆₇	—	4	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	—	4	2	no	—
231	C₂×C₄×C₆₇	—	4	2	no	—
232	C₂×C₄×C₆₇	—	8	2	no	—
233	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
239	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
240	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
241	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
242	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
243	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
244	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
245	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
246	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
247	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
248	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
255	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
256	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
257	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
258	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
259	C₂×C₄×C₆₇	#126	1	1	yes	characteristic
260	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
261	C₈×C₆₇	—	2	4	no	—
262	C₈×C₆₇	—	4	2	no	—
263	C₈×C₆₇	—	4	2	no	—
264	C₈×C₆₇	—	4	2	no	—
265	C₂²×C₆₇	—	2	2	no	—
266	C₂²×C₆₇	—	2	2	no	—
267	C₂²×C₆₇	—	2	2	no	—
268	C₂²×C₆₇	—	4	1	no	—
269	C₂²×C₆₇	—	4	1	no	—
270	C₂²×C₆₇	—	4	1	no	—
271	C₂²×C₆₇	—	4	1	no	—
272	C₂²×C₆₇	—	4	2	no	—
273	C₂²×C₆₇	—	8	2	no	—
274	C₂²×C₆₇	—	8	2	no	—
275	C₂²×C₆₇	—	8	2	no	—
276	C₂²×C₆₇	#380	1	1	yes	characteristic
277	C₂²×C₆₇	#388	1	1	yes	characteristic
278	C₂²×C₆₇	#388	1	1	yes	characteristic
279	C₂²×C₆₇	#393	1	1	yes	characteristic
280	C₂²×C₆₇	#394	1	1	yes	characteristic
281	C₂²×C₆₇	#394	1	1	yes	characteristic
282	C₂²×C₆₇	#399	1	1	yes	characteristic
283	C₄×C₆₇	—	4	4	no	—
284	C₄×C₆₇	—	4	2	no	—
285	C₄×C₆₇	—	8	2	no	—
286	C₄×C₆₇	#385	1	1	yes	characteristic
287	C₄×C₆₇	#385	1	1	yes	characteristic
288	C₄×C₆₇	#386	1	1	yes	characteristic
289	C₄×C₆₇	#390	1	1	yes	characteristic
290	C₄×C₆₇	#392	1	1	yes	characteristic
291	C₄×C₆₇	#396	1	1	yes	characteristic
292	C₄×C₆₇	#396	1	1	yes	characteristic
293	C₄×C₆₇	#397	1	1	yes	characteristic
294	#11146	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
295	C₂×C₆₇	—	2	1	no	—
296	C₂×C₆₇	—	2	1	no	—
297	C₂×C₆₇	—	2	1	no	—
298	C₂×C₆₇	—	2	1	no	—
299	C₂×C₆₇	—	2	1	no	—
300	C₂×C₆₇	—	2	1	no	—
301	C₂×C₆₇	—	8	2	no	—
302	C₂×C₆₇	#1439	1	1	yes	characteristic
303	C₂×C₆₇	#1932	1	2	yes	—
304	C₂×#430	C₂×C₆₇	1	1	yes	characteristic, complemented
305	C₂×#421	C₂×C₆₇	1	1	yes	characteristic
306	C₂²×#104	C₂×C₆₇	1	1	yes	characteristic
307	C₂×#413	C₂×C₆₇	1	1	yes	characteristic
308	C₂×#344	C₂×C₆₇	1	1	yes	characteristic, complemented
309	C₂×#339	C₂×C₆₇	1	1	yes	characteristic, complemented
310	#1728	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₆₇	#11146	1	1	yes	characteristic, direct product factor, Sylow
312	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
313	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
314	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
315	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
316	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
317	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
318	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
319	#430	—	2	2	no	—
320	#430	—	2	2	no	—
321	C₂×#108	—	2	2	no	—
322	#421	—	2	2	no	—
323	#421	—	2	2	no	—
324	C₂×#104	—	2	2	no	—
325	C₂×#104	—	2	2	no	—
326	C₂×#104	—	2	2	no	—
327	C₂×#104	—	2	2	no	—
328	#413	—	2	2	no	—
329	#413	—	2	2	no	—
330	C₂×#101	C₄×C₆₇	1	2	yes	complemented
331	C₂×#100	C₄×C₆₇	1	2	yes	complemented
332	#344	—	2	2	no	—
333	#339	—	2	2	no	—
334	#336	—	2	2	no	—
335	C₂×#41	—	2	1	no	—
336	C₂×#41	—	2	1	no	—
337	C₂×#41	—	2	1	no	—
338	C₂×#41	—	2	1	no	—
339	C₂×#41	—	2	1	no	—
340	C₂×#41	—	2	1	no	—
341	C₂×#41	—	2	1	no	—
342	C₂×#41	—	2	1	no	—
343	C₂×#41	—	2	1	no	—
344	C₂×#41	—	2	1	no	—
345	C₂×#41	D₄×C₆₇	1	2	yes	—
346	C₂×#41	Q₈×C₆₇	1	2	yes	—
347	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
348	C₂²×D₄	—	2	1	no	—
349	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
350	C₂³×C₄	—	2	1	no	—
351	C₂³×C₄	—	2	1	no	—
352	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
353	C₂×M₁₆	—	2	1	no	—
354	C₂×M₁₆	—	2	1	no	—
355	C₂×M₁₆	—	2	1	no	—
356	C₂×M₁₆	—	2	1	no	—
357	C₂×M₁₆	—	2	1	no	—
358	C₂×M₁₆	—	2	1	no	—
359	C₂×M₁₆	—	2	1	no	—
360	C₂×M₁₆	—	2	1	no	—
361	C₂×M₁₆	—	2	1	no	—
362	C₂×M₁₆	—	2	1	no	—
363	C₂×M₁₆	—	2	1	no	—
364	C₂×M₁₆	—	2	1	no	—
365	C₂×M₁₆	—	2	1	no	—
366	C₂×M₁₆	—	2	1	no	—
367	C₂×M₁₆	—	2	1	no	—
368	C₂×M₁₆	—	2	1	no	—
369	C₂×M₁₆	—	2	1	no	—
370	C₂×M₁₆	—	2	1	no	—
371	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
372	C₂×M₁₆	C₂×C₄×C₆₇	1	2	yes	—
373	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
374	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
375	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
376	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
377	C₂²×C₈	—	2	1	no	—
378	C₂²×C₈	—	2	1	no	—
379	C₂²×C₈	—	2	1	no	—
380	C₂²×C₈	—	2	1	no	—
381	#117	—	2	1	no	—
382	#117	—	2	1	no	—
383	#117	—	2	1	no	—
384	#117	D₄×C₆₇	1	2	yes	—
385	C₂×C₄⋊C₄	—	2	1	no	—
386	C₂×C₂²⋊C₄	—	2	1	no	—
387	C₂×C₄²	—	2	1	no	—
388	C₂×C₄²	—	2	1	no	—
389	C₂×C₄²	—	2	1	no	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	—	2	1	no	—
392	C₂×C₄²	—	2	1	no	—
393	C₂×C₄²	D₄×C₆₇	1	2	yes	—
394	#108	—	2	8	no	—
395	#104	—	2	8	no	—
396	#104	—	2	8	no	—
397	#101	—	2	4	no	—
398	#100	—	2	4	no	—
399	C₂⁴	—	2	1	no	—
400	C₂⁴	#41×C₆₇	1	1	yes	characteristic
401	#41	—	2	2	no	—
402	#41	—	2	2	no	—
403	#41	—	2	2	no	—
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	—	2	2	no	—
408	#41	—	2	2	no	—
409	#41	—	4	1	no	—
410	#41	—	4	1	no	—
411	#41	—	4	1	no	—
412	#41	—	4	1	no	—
413	#41	—	4	1	no	—
414	#41	—	4	1	no	—
415	#41	—	4	1	no	—
416	#41	—	4	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	—	2	1	no	—
421	C₂×Q₈	—	2	1	no	—
422	C₂×Q₈	C₄⋊C₄×C₆₇	1	2	yes	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	1	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	4	1	no	—
431	C₂×D₄	—	4	1	no	—
432	C₂×D₄	—	4	1	no	—
433	C₂×D₄	—	4	1	no	—
434	C₂×D₄	C₄⋊C₄×C₆₇	1	2	yes	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	4	4	no	—
440	C₂²×C₄	—	4	1	no	—
441	C₂²×C₄	—	4	1	no	—
442	C₂²×C₄	—	4	1	no	—
443	C₂²×C₄	—	4	1	no	—
444	C₂²×C₄	—	4	1	no	—
445	C₂²×C₄	—	4	1	no	—
446	C₂²×C₄	—	4	1	no	—
447	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
455	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
457	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
458	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
459	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
460	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
461	M₁₆	—	2	2	no	—
462	M₁₆	—	2	2	no	—
463	M₁₆	—	2	2	no	—
464	M₁₆	—	2	2	no	—
465	M₁₆	—	2	2	no	—
466	M₁₆	—	2	2	no	—
467	M₁₆	—	2	2	no	—
468	M₁₆	—	2	2	no	—
469	M₁₆	—	2	2	no	—
470	M₁₆	—	4	1	no	—
471	M₁₆	—	4	1	no	—
472	M₁₆	—	4	1	no	—
473	M₁₆	—	4	1	no	—
474	M₁₆	—	4	1	no	—
475	M₁₆	—	4	1	no	—
476	M₁₆	C₂²⋊C₄×C₆₇	1	4	yes	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	2	no	—
479	C₂×C₈	—	2	2	no	—
480	C₂×C₈	—	2	2	no	—
481	C₂×C₈	—	2	2	no	—
482	C₂×C₈	—	2	2	no	—
483	C₂×C₈	—	4	1	no	—
484	C₂×C₈	—	4	1	no	—
485	C₂×C₈	—	4	1	no	—
486	C₂×C₈	—	4	1	no	—
487	C₂×C₈	—	4	1	no	—
488	C₂×C₈	—	4	1	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	4	1	no	—
491	C₂²⋊C₄	—	2	2	no	—
492	C₂²⋊C₄	—	4	1	no	—
493	C₄²	—	2	2	no	—
494	C₄²	—	2	2	no	—
495	C₄²	—	2	2	no	—
496	C₄²	—	4	1	no	—
497	C₄²	—	4	1	no	—
498	C₄²	—	4	1	no	—
499	C₂³	—	2	2	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	—	8	2	no	—
504	C₂³	#117×C₆₇	1	1	yes	characteristic
505	C₂³	#121×C₆₇	1	1	yes	characteristic
506	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
507	C₂³	#123×C₆₇	1	1	yes	characteristic
508	C₂³	#123×C₆₇	1	1	yes	characteristic
509	C₂³	#124×C₆₇	1	1	yes	characteristic
510	C₂³	#124×C₆₇	1	1	yes	characteristic
511	Q₈	—	2	2	no	—
512	Q₈	—	2	2	no	—
513	Q₈	—	4	1	no	—
514	Q₈	—	4	1	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	2	no	—
517	D₄	—	4	2	no	—
518	D₄	—	4	2	no	—
519	D₄	—	4	2	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	1	no	—
522	D₄	—	4	1	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	2	no	—
533	C₂×C₄	—	4	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	8	2	no	—
537	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
538	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
539	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
540	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
541	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
542	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
543	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
544	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
545	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
546	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
547	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
548	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
549	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
550	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
551	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
552	C₂×C₄	#121×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
554	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
555	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
556	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
557	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
558	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
559	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
560	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
561	C₂×C₄	#124×C₆₇	1	1	yes	characteristic
562	C₂×C₄	#126×C₆₇	1	1	yes	characteristic
563	C₂×C₄	#126×C₆₇	1	1	yes	characteristic
564	C₂×C₄	#127×C₆₇	1	1	yes	characteristic
565	C₈	—	2	4	no	—
566	C₈	—	4	2	no	—
567	C₈	—	4	2	no	—
568	C₈	—	4	2	no	—
569	C₂²	—	2	2	no	—
570	C₂²	—	2	2	no	—
571	C₂²	—	2	2	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	8	2	no	—
578	C₂²	—	8	2	no	—
579	C₂²	—	8	2	no	—
580	C₂²	#380×C₆₇	1	1	yes	characteristic
581	C₂²	#388×C₆₇	1	1	yes	characteristic
582	C₂²	#388×C₆₇	1	1	yes	characteristic
583	C₂²	#393×C₆₇	1	1	yes	characteristic
584	C₂²	#394×C₆₇	1	1	yes	characteristic
585	C₂²	#394×C₆₇	1	1	yes	characteristic
586	C₂²	#399×C₆₇	1	1	yes	characteristic
587	C₄	—	4	4	no	—
588	C₄	—	4	2	no	—
589	C₄	—	8	2	no	—
590	C₄	#385×C₆₇	1	1	yes	characteristic
591	C₄	#385×C₆₇	1	1	yes	characteristic
592	C₄	#386×C₆₇	1	1	yes	characteristic
593	C₄	#390×C₆₇	1	1	yes	characteristic
594	C₄	#392×C₆₇	1	1	yes	characteristic
595	C₄	#396×C₆₇	1	1	yes	characteristic
596	C₄	#396×C₆₇	1	1	yes	characteristic
597	C₄	#397×C₆₇	1	1	yes	characteristic
598	C₂	—	2	1	no	—
599	C₂	—	2	1	no	—
600	C₂	—	2	1	no	—
601	C₂	—	2	1	no	—
602	C₂	—	2	1	no	—
603	C₂	—	2	1	no	—
604	C₂	—	8	2	no	—
605	C₂	C₆₇×#1439	1	1	yes	characteristic
606	C₂	C₆₇×#1932	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	#121
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂³×C₄	C₂³×C₆₇
socle	C₂²×C₆₇	#388
prime radical	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#430×C₆₇	C₂	semidirect
5. C₂×#344×C₆₇	C₂	semidirect
6. C₂×#339×C₆₇	C₂	semidirect
7. C₆₇×#1728	C₂	semidirect
26. C₂×#101×C₆₇	C₄	semidirect
27. C₂×#100×C₆₇	C₄	semidirect
294. #11146	C₆₇	direct (N × H)
304. C₂×#430	C₂×C₆₇	semidirect
308. C₂×#344	C₂×C₆₇	semidirect
309. C₂×#339	C₂×C₆₇	semidirect
310. #1728	C₂×C₆₇	semidirect
311. C₆₇	#11146	direct (N × H)
330. C₂×#101	C₄×C₆₇	semidirect
331. C₂×#100	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×#430×C₆₇	2
C₂×#421×C₆₇	2
C₂²×#104×C₆₇	2
C₂×#413×C₆₇	2
C₂×#344×C₆₇	2
C₂×#339×C₆₇	2
C₆₇×#1728	2
#11146	67

Subgroups — C67×#11146 #30690397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#11146 #30690397