Subgroups — C₆₇×#20915 #30700166

Order 17152.

1842 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#537×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#408×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1875	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1874	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1873	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1872	C₂	1	2	yes	maximal, complemented
10	C₆₇×#1736	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#340×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#537×C₆₇	—	2	4	no	—
17	#537×C₆₇	C₄	1	8	yes	complemented
18	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×#121×C₆₇	—	2	1	no	—
21	C₂×#121×C₆₇	—	2	2	no	—
22	C₂×#121×C₆₇	C₂²	1	2	yes	—
23	#520×C₆₇	—	2	4	no	—
24	#520×C₆₇	—	2	4	no	—
25	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
26	C₂×C₄×D₄×C₆₇	—	2	2	no	—
27	C₂×C₄×D₄×C₆₇	—	2	1	no	—
28	C₂×#117×C₆₇	C₂²	1	2	yes	—
29	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	#417×C₆₇	—	2	4	no	—
34	#416×C₆₇	—	2	4	no	—
35	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×#103×C₆₇	C₂²	1	2	yes	—
38	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#102×C₆₇	C₂²	1	2	yes	—
41	#408×C₆₇	—	2	4	no	—
42	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
43	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
44	#340×C₆₇	C₂²	1	2	yes	—
45	#340×C₆₇	C₂²	1	2	yes	—
46	#340×C₆₇	C₂²	1	2	yes	—
47	#340×C₆₇	C₂²	1	2	yes	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	—	2	2	no	—
50	C₂×#41×C₆₇	—	2	2	no	—
51	C₂×#41×C₆₇	—	2	2	no	—
52	C₂×#41×C₆₇	—	2	2	no	—
53	C₂×#41×C₆₇	—	2	2	no	—
54	C₂×#41×C₆₇	—	2	2	no	—
55	C₂×#41×C₆₇	—	2	2	no	—
56	C₂×#41×C₆₇	—	2	2	no	—
57	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
58	C₂×#41×C₆₇	C₂×C₄	1	4	yes	complemented
59	C₂×#41×C₆₇	D₄	1	4	yes	—
60	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
61	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₇	—	2	1	no	—
63	C₂²×D₄×C₆₇	—	2	2	no	—
64	C₂²×D₄×C₆₇	D₄	1	2	yes	—
65	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	—	2	2	no	—
68	C₂³×C₄×C₆₇	—	2	1	no	—
69	C₂³×C₄×C₆₇	—	2	1	no	—
70	C₂³×C₄×C₆₇	—	2	1	no	—
71	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
72	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₆₇	—	2	2	no	—
74	C₂×M₁₆×C₆₇	—	2	2	no	—
75	C₂×M₁₆×C₆₇	—	2	2	no	—
76	C₂²×C₈×C₆₇	—	2	2	no	—
77	C₂²×C₈×C₆₇	—	2	2	no	—
78	C₂²×C₈×C₆₇	—	2	2	no	—
79	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
82	C₂²⋊Q₈×C₆₇	—	2	8	no	—
83	C₂²⋊Q₈×C₆₇	—	2	4	no	—
84	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
85	#121×C₆₇	—	2	8	no	—
86	#121×C₆₇	—	2	16	no	—
87	#121×C₆₇	—	2	4	no	—
88	#121×C₆₇	C₂×C₄	1	8	yes	complemented
89	C₄×Q₈×C₆₇	—	2	8	no	—
90	C₄×D₄×C₆₇	—	2	16	no	—
91	C₄×D₄×C₆₇	—	2	8	no	—
92	#117×C₆₇	—	2	2	no	—
93	#117×C₆₇	—	2	4	no	—
94	#117×C₆₇	—	2	4	no	—
95	#117×C₆₇	D₄	1	4	yes	complemented
96	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
98	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
101	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
102	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
103	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
104	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	complemented
105	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	complemented
106	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
110	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
111	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
114	C₂×C₄²×C₆₇	—	2	2	no	—
115	C₂×C₄²×C₆₇	—	2	1	no	—
116	#103×C₆₇	—	2	2	no	—
117	#103×C₆₇	—	2	2	no	—
118	#103×C₆₇	—	2	4	no	—
119	#102×C₆₇	—	2	2	no	—
120	#102×C₆₇	—	2	2	no	—
121	#102×C₆₇	—	2	4	no	—
122	C₂²⋊C₈×C₆₇	—	2	4	no	—
123	C₂²⋊C₈×C₆₇	—	2	4	no	—
124	C₂⁴×C₆₇	—	2	1	no	—
125	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
126	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
127	#41×C₆₇	—	2	8	no	—
128	#41×C₆₇	—	2	8	no	—
129	#41×C₆₇	—	2	8	no	—
130	#41×C₆₇	—	2	8	no	—
131	#41×C₆₇	—	2	8	no	—
132	#41×C₆₇	—	2	8	no	—
133	#41×C₆₇	—	4	4	no	—
134	#41×C₆₇	—	4	4	no	—
135	C₂×Q₈×C₆₇	—	2	2	no	—
136	C₂×Q₈×C₆₇	—	2	2	no	—
137	C₂×Q₈×C₆₇	—	2	1	no	—
138	C₂×Q₈×C₆₇	—	2	1	no	—
139	C₂×Q₈×C₆₇	—	2	1	no	—
140	C₂×Q₈×C₆₇	—	2	1	no	—
141	C₂×Q₈×C₆₇	—	2	1	no	—
142	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
143	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
144	C₂×D₄×C₆₇	—	2	4	no	—
145	C₂×D₄×C₆₇	—	2	4	no	—
146	C₂×D₄×C₆₇	—	2	4	no	—
147	C₂×D₄×C₆₇	—	2	8	no	—
148	C₂×D₄×C₆₇	—	2	2	no	—
149	C₂×D₄×C₆₇	—	2	2	no	—
150	C₂×D₄×C₆₇	—	2	1	no	—
151	C₂×D₄×C₆₇	—	2	1	no	—
152	C₂×D₄×C₆₇	—	2	1	no	—
153	C₂×D₄×C₆₇	—	2	1	no	—
154	C₂×D₄×C₆₇	—	2	1	no	—
155	C₂×D₄×C₆₇	—	4	2	no	—
156	C₂×D₄×C₆₇	—	4	2	no	—
157	C₂×D₄×C₆₇	—	8	2	no	—
158	C₂×D₄×C₆₇	—	8	2	no	—
159	C₂×D₄×C₆₇	C₂²⋊C₄	1	8	yes	—
160	C₂×D₄×C₆₇	C₂²⋊C₄	1	4	yes	—
161	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
162	C₂²×C₄×C₆₇	—	2	2	no	—
163	C₂²×C₄×C₆₇	—	2	2	no	—
164	C₂²×C₄×C₆₇	—	2	2	no	—
165	C₂²×C₄×C₆₇	—	2	2	no	—
166	C₂²×C₄×C₆₇	—	2	1	no	—
167	C₂²×C₄×C₆₇	—	2	2	no	—
168	C₂²×C₄×C₆₇	—	2	1	no	—
169	C₂²×C₄×C₆₇	—	2	2	no	—
170	C₂²×C₄×C₆₇	—	2	1	no	—
171	C₂²×C₄×C₆₇	—	2	8	no	—
172	C₂²×C₄×C₆₇	—	2	4	no	—
173	C₂²×C₄×C₆₇	—	2	4	no	—
174	C₂²×C₄×C₆₇	—	2	4	no	—
175	C₂²×C₄×C₆₇	—	4	2	no	—
176	C₂²×C₄×C₆₇	—	4	2	no	—
177	C₂²×C₄×C₆₇	—	8	1	no	—
178	C₂²×C₄×C₆₇	—	8	2	no	—
179	C₂²×C₄×C₆₇	—	8	1	no	—
180	C₂²×C₄×C₆₇	—	8	1	no	—
181	C₂²×C₄×C₆₇	—	8	2	no	—
182	C₂²×C₄×C₆₇	—	8	1	no	—
183	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
184	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
185	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
187	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
189	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
190	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₆₇	#41	1	2	yes	—
192	C₂²×C₄×C₆₇	#41	1	2	yes	—
193	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
194	M₁₆×C₆₇	—	4	4	no	—
195	C₂×C₈×C₆₇	—	2	1	no	—
196	C₂×C₈×C₆₇	—	2	1	no	—
197	C₂×C₈×C₆₇	—	2	2	no	—
198	C₂×C₈×C₆₇	—	2	1	no	—
199	C₂×C₈×C₆₇	—	2	1	no	—
200	C₂×C₈×C₆₇	—	2	2	no	—
201	C₂×C₈×C₆₇	—	2	1	no	—
202	C₂×C₈×C₆₇	—	2	1	no	—
203	C₂×C₈×C₆₇	—	2	2	no	—
204	C₂×C₈×C₆₇	—	4	4	no	—
205	C₄⋊C₄×C₆₇	—	2	8	no	—
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	2	8	no	—
208	C₄⋊C₄×C₆₇	—	2	4	no	—
209	C₄⋊C₄×C₆₇	—	2	2	no	—
210	C₄⋊C₄×C₆₇	—	2	1	no	—
211	C₄⋊C₄×C₆₇	—	2	2	no	—
212	C₄⋊C₄×C₆₇	—	2	1	no	—
213	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
214	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
215	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
216	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
217	C₂²⋊C₄×C₆₇	—	2	8	no	—
218	C₂²⋊C₄×C₆₇	—	2	8	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	2	8	no	—
221	C₄²×C₆₇	—	2	8	no	—
222	C₄²×C₆₇	—	2	4	no	—
223	C₂³×C₆₇	—	2	1	no	—
224	C₂³×C₆₇	—	2	1	no	—
225	C₂³×C₆₇	—	2	1	no	—
226	C₂³×C₆₇	—	2	1	no	—
227	C₂³×C₆₇	—	2	2	no	—
228	C₂³×C₆₇	—	2	2	no	—
229	C₂³×C₆₇	—	2	2	no	—
230	C₂³×C₆₇	—	2	2	no	—
231	C₂³×C₆₇	—	2	2	no	—
232	C₂³×C₆₇	—	2	2	no	—
233	C₂³×C₆₇	—	2	2	no	—
234	C₂³×C₆₇	—	4	1	no	—
235	C₂³×C₆₇	—	4	1	no	—
236	C₂³×C₆₇	—	4	1	no	—
237	C₂³×C₆₇	—	4	1	no	—
238	C₂³×C₆₇	—	4	1	no	—
239	C₂³×C₆₇	—	4	1	no	—
240	C₂³×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
241	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
242	C₂³×C₆₇	#120	1	2	yes	—
243	C₂³×C₆₇	#123	1	2	yes	—
244	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
245	Q₈×C₆₇	—	2	8	no	—
246	Q₈×C₆₇	—	2	2	no	—
247	Q₈×C₆₇	—	2	2	no	—
248	Q₈×C₆₇	—	4	1	no	—
249	Q₈×C₆₇	—	4	1	no	—
250	D₄×C₆₇	—	2	16	no	—
251	D₄×C₆₇	—	2	2	no	—
252	D₄×C₆₇	—	2	2	no	—
253	D₄×C₆₇	—	2	8	no	—
254	D₄×C₆₇	—	4	8	no	—
255	D₄×C₆₇	—	4	1	no	—
256	D₄×C₆₇	—	4	1	no	—
257	D₄×C₆₇	—	8	8	no	—
258	C₂×C₄×C₆₇	—	2	2	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	4	no	—
261	C₂×C₄×C₆₇	—	2	4	no	—
262	C₂×C₄×C₆₇	—	2	4	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	2	2	no	—
266	C₂×C₄×C₆₇	—	2	2	no	—
267	C₂×C₄×C₆₇	—	2	4	no	—
268	C₂×C₄×C₆₇	—	2	2	no	—
269	C₂×C₄×C₆₇	—	4	1	no	—
270	C₂×C₄×C₆₇	—	4	1	no	—
271	C₂×C₄×C₆₇	—	4	2	no	—
272	C₂×C₄×C₆₇	—	4	1	no	—
273	C₂×C₄×C₆₇	—	4	2	no	—
274	C₂×C₄×C₆₇	—	4	1	no	—
275	C₂×C₄×C₆₇	—	4	2	no	—
276	C₂×C₄×C₆₇	—	4	1	no	—
277	C₂×C₄×C₆₇	—	4	2	no	—
278	C₂×C₄×C₆₇	—	4	1	no	—
279	C₂×C₄×C₆₇	—	4	8	no	—
280	C₂×C₄×C₆₇	—	8	4	no	—
281	C₂×C₄×C₆₇	—	8	4	no	—
282	C₂×C₄×C₆₇	—	8	8	no	—
283	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
284	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
285	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
286	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
287	C₂×C₄×C₆₇	#120	1	2	yes	—
288	C₂×C₄×C₆₇	#121	1	4	yes	—
289	C₂×C₄×C₆₇	#121	1	4	yes	—
290	C₂×C₄×C₆₇	#123	1	2	yes	—
291	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
292	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
293	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
294	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
295	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
296	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
297	C₈×C₆₇	—	4	1	no	—
298	C₈×C₆₇	—	4	1	no	—
299	C₈×C₆₇	—	4	2	no	—
300	C₂²×C₆₇	—	2	2	no	—
301	C₂²×C₆₇	—	2	2	no	—
302	C₂²×C₆₇	—	2	2	no	—
303	C₂²×C₆₇	—	2	2	no	—
304	C₂²×C₆₇	—	2	2	no	—
305	C₂²×C₆₇	—	2	2	no	—
306	C₂²×C₆₇	—	2	4	no	—
307	C₂²×C₆₇	—	2	4	no	—
308	C₂²×C₆₇	—	2	4	no	—
309	C₂²×C₆₇	—	4	2	no	—
310	C₂²×C₆₇	—	4	2	no	—
311	C₂²×C₆₇	—	4	2	no	—
312	C₂²×C₆₇	—	4	2	no	—
313	C₂²×C₆₇	—	4	2	no	—
314	C₂²×C₆₇	—	4	2	no	—
315	C₂²×C₆₇	—	4	2	no	—
316	C₂²×C₆₇	—	8	1	no	—
317	C₂²×C₆₇	—	8	1	no	—
318	C₂²×C₆₇	—	8	1	no	—
319	C₂²×C₆₇	—	8	1	no	—
320	C₂²×C₆₇	#386	1	4	yes	—
321	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
322	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
323	C₂²×C₆₇	C₂×#123	1	1	yes	characteristic
324	C₂²×C₆₇	#576	1	1	yes	characteristic
325	C₂²×C₆₇	#577	1	1	yes	characteristic
326	C₂²×C₆₇	#577	1	1	yes	characteristic
327	C₂²×C₆₇	#578	1	1	yes	characteristic
328	C₄×C₆₇	—	2	4	no	—
329	C₄×C₆₇	—	4	4	no	—
330	C₄×C₆₇	—	4	4	no	—
331	C₄×C₆₇	—	4	2	no	—
332	C₄×C₆₇	—	4	2	no	—
333	C₄×C₆₇	—	4	2	no	—
334	C₄×C₆₇	#386	1	4	yes	—
335	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
336	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
337	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
338	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
339	#20915	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
340	C₂×C₆₇	—	2	4	no	—
341	C₂×C₆₇	—	4	4	no	—
342	C₂×C₆₇	—	8	2	no	—
343	C₂×C₆₇	C₂×#386	1	1	yes	characteristic
344	C₂×C₆₇	#2897	1	1	yes	characteristic
345	C₂×C₆₇	#2898	1	1	yes	characteristic
346	C₂×C₆₇	#3012	1	1	yes	characteristic
347	C₂×C₆₇	#3013	1	1	yes	characteristic
348	C₂×C₆₇	#3098	1	1	yes	characteristic
349	C₂×C₆₇	#3099	1	1	yes	characteristic
350	C₂×#537	C₂×C₆₇	1	1	yes	characteristic
351	C₂×#520	C₂×C₆₇	1	1	yes	characteristic, complemented
352	C₂×#417	C₂×C₆₇	1	1	yes	characteristic, complemented
353	C₂×#416	C₂×C₆₇	1	1	yes	characteristic, complemented
354	C₂×#408	C₂×C₆₇	1	1	yes	characteristic, complemented
355	#1875	C₂×C₆₇	1	2	yes	complemented
356	#1874	C₂×C₆₇	1	2	yes	complemented
357	#1873	C₂×C₆₇	1	2	yes	complemented
358	#1872	C₂×C₆₇	1	2	yes	complemented
359	#1736	C₂×C₆₇	1	1	yes	characteristic, complemented
360	C₂×#340	C₂×C₆₇	1	1	yes	characteristic, complemented
361	C₆₇	#20915	1	1	yes	characteristic, direct product factor, Sylow
362	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
363	C₂²×#41	C₂²×C₆₇	1	1	yes	characteristic
364	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
365	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
366	#537	—	2	4	no	—
367	#537	C₄×C₆₇	1	8	yes	complemented
368	C₂×C₂²⋊Q₈	—	2	1	no	—
369	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
370	C₂×#121	—	2	1	no	—
371	C₂×#121	—	2	2	no	—
372	C₂×#121	C₂²×C₆₇	1	2	yes	—
373	#520	—	2	4	no	—
374	#520	—	2	4	no	—
375	C₂×C₄×Q₈	—	2	1	no	—
376	C₂×C₄×D₄	—	2	2	no	—
377	C₂×C₄×D₄	—	2	1	no	—
378	C₂×#117	C₂²×C₆₇	1	2	yes	—
379	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
380	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
381	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
382	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
383	#417	—	2	4	no	—
384	#416	—	2	4	no	—
385	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
386	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
387	C₂×#103	C₂²×C₆₇	1	2	yes	—
388	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
389	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
390	C₂×#102	C₂²×C₆₇	1	2	yes	—
391	#408	—	2	4	no	—
392	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
393	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
394	#340	C₂²×C₆₇	1	2	yes	—
395	#340	C₂²×C₆₇	1	2	yes	—
396	#340	C₂²×C₆₇	1	2	yes	—
397	#340	C₂²×C₆₇	1	2	yes	—
398	C₂×#41	—	2	2	no	—
399	C₂×#41	—	2	2	no	—
400	C₂×#41	—	2	2	no	—
401	C₂×#41	—	2	2	no	—
402	C₂×#41	—	2	2	no	—
403	C₂×#41	—	2	2	no	—
404	C₂×#41	—	2	2	no	—
405	C₂×#41	—	2	2	no	—
406	C₂×#41	—	2	2	no	—
407	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
408	C₂×#41	C₂×C₄×C₆₇	1	4	yes	complemented
409	C₂×#41	D₄×C₆₇	1	4	yes	—
410	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
411	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
412	C₂²×D₄	—	2	1	no	—
413	C₂²×D₄	—	2	2	no	—
414	C₂²×D₄	D₄×C₆₇	1	2	yes	—
415	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
416	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
417	C₂³×C₄	—	2	2	no	—
418	C₂³×C₄	—	2	1	no	—
419	C₂³×C₄	—	2	1	no	—
420	C₂³×C₄	—	2	1	no	—
421	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
422	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
423	C₂×M₁₆	—	2	2	no	—
424	C₂×M₁₆	—	2	2	no	—
425	C₂×M₁₆	—	2	2	no	—
426	C₂²×C₈	—	2	2	no	—
427	C₂²×C₈	—	2	2	no	—
428	C₂²×C₈	—	2	2	no	—
429	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
430	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
432	C₂²⋊Q₈	—	2	8	no	—
433	C₂²⋊Q₈	—	2	4	no	—
434	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
435	#121	—	2	8	no	—
436	#121	—	2	16	no	—
437	#121	—	2	4	no	—
438	#121	C₂×C₄×C₆₇	1	8	yes	complemented
439	C₄×Q₈	—	2	8	no	—
440	C₄×D₄	—	2	16	no	—
441	C₄×D₄	—	2	8	no	—
442	#117	—	2	2	no	—
443	#117	—	2	4	no	—
444	#117	—	2	4	no	—
445	#117	D₄×C₆₇	1	4	yes	complemented
446	C₂×C₄⋊C₄	—	2	2	no	—
447	C₂×C₄⋊C₄	—	2	1	no	—
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	—	2	2	no	—
451	C₂×C₄⋊C₄	—	2	2	no	—
452	C₂×C₄⋊C₄	—	2	4	no	—
453	C₂×C₄⋊C₄	—	2	4	no	—
454	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	complemented
455	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	complemented
456	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
457	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
458	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
459	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
460	C₂×C₂²⋊C₄	—	2	2	no	—
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	—	2	2	no	—
463	C₂×C₂²⋊C₄	—	2	2	no	—
464	C₂×C₄²	—	2	2	no	—
465	C₂×C₄²	—	2	1	no	—
466	#103	—	2	2	no	—
467	#103	—	2	2	no	—
468	#103	—	2	4	no	—
469	#102	—	2	2	no	—
470	#102	—	2	2	no	—
471	#102	—	2	4	no	—
472	C₂²⋊C₈	—	2	4	no	—
473	C₂²⋊C₈	—	2	4	no	—
474	C₂⁴	—	2	1	no	—
475	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
476	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
477	#41	—	2	8	no	—
478	#41	—	2	8	no	—
479	#41	—	2	8	no	—
480	#41	—	2	8	no	—
481	#41	—	2	8	no	—
482	#41	—	2	8	no	—
483	#41	—	4	4	no	—
484	#41	—	4	4	no	—
485	C₂×Q₈	—	2	2	no	—
486	C₂×Q₈	—	2	2	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
493	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	2	8	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	1	no	—
504	C₂×D₄	—	2	1	no	—
505	C₂×D₄	—	4	2	no	—
506	C₂×D₄	—	4	2	no	—
507	C₂×D₄	—	8	2	no	—
508	C₂×D₄	—	8	2	no	—
509	C₂×D₄	C₂²⋊C₄×C₆₇	1	8	yes	—
510	C₂×D₄	C₂²⋊C₄×C₆₇	1	4	yes	—
511	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	2	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	2	no	—
516	C₂²×C₄	—	2	1	no	—
517	C₂²×C₄	—	2	2	no	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	—	2	2	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	8	no	—
522	C₂²×C₄	—	2	4	no	—
523	C₂²×C₄	—	2	4	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	4	2	no	—
526	C₂²×C₄	—	4	2	no	—
527	C₂²×C₄	—	8	1	no	—
528	C₂²×C₄	—	8	2	no	—
529	C₂²×C₄	—	8	1	no	—
530	C₂²×C₄	—	8	1	no	—
531	C₂²×C₄	—	8	2	no	—
532	C₂²×C₄	—	8	1	no	—
533	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
534	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
535	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
536	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
537	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
538	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
539	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
540	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
541	C₂²×C₄	#41×C₆₇	1	2	yes	—
542	C₂²×C₄	#41×C₆₇	1	2	yes	—
543	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
544	M₁₆	—	4	4	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	2	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	2	no	—
551	C₂×C₈	—	2	1	no	—
552	C₂×C₈	—	2	1	no	—
553	C₂×C₈	—	2	2	no	—
554	C₂×C₈	—	4	4	no	—
555	C₄⋊C₄	—	2	8	no	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	8	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	2	no	—
560	C₄⋊C₄	—	2	1	no	—
561	C₄⋊C₄	—	2	2	no	—
562	C₄⋊C₄	—	2	1	no	—
563	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
564	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
565	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
566	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
567	C₂²⋊C₄	—	2	8	no	—
568	C₂²⋊C₄	—	2	8	no	—
569	C₂²⋊C₄	—	2	4	no	—
570	C₂²⋊C₄	—	2	8	no	—
571	C₄²	—	2	8	no	—
572	C₄²	—	2	4	no	—
573	C₂³	—	2	1	no	—
574	C₂³	—	2	1	no	—
575	C₂³	—	2	1	no	—
576	C₂³	—	2	1	no	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	2	2	no	—
581	C₂³	—	2	2	no	—
582	C₂³	—	2	2	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	4	1	no	—
585	C₂³	—	4	1	no	—
586	C₂³	—	4	1	no	—
587	C₂³	—	4	1	no	—
588	C₂³	—	4	1	no	—
589	C₂³	—	4	1	no	—
590	C₂³	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
591	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
592	C₂³	#120×C₆₇	1	2	yes	—
593	C₂³	#123×C₆₇	1	2	yes	—
594	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
595	Q₈	—	2	8	no	—
596	Q₈	—	2	2	no	—
597	Q₈	—	2	2	no	—
598	Q₈	—	4	1	no	—
599	Q₈	—	4	1	no	—
600	D₄	—	2	16	no	—
601	D₄	—	2	2	no	—
602	D₄	—	2	2	no	—
603	D₄	—	2	8	no	—
604	D₄	—	4	8	no	—
605	D₄	—	4	1	no	—
606	D₄	—	4	1	no	—
607	D₄	—	8	8	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	4	no	—
612	C₂×C₄	—	2	4	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	4	no	—
615	C₂×C₄	—	2	2	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	2	2	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	4	2	no	—
622	C₂×C₄	—	4	1	no	—
623	C₂×C₄	—	4	2	no	—
624	C₂×C₄	—	4	2	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	4	1	no	—
628	C₂×C₄	—	4	1	no	—
629	C₂×C₄	—	4	8	no	—
630	C₂×C₄	—	8	4	no	—
631	C₂×C₄	—	8	4	no	—
632	C₂×C₄	—	8	8	no	—
633	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
634	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
635	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
636	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
637	C₂×C₄	#120×C₆₇	1	2	yes	—
638	C₂×C₄	#121×C₆₇	1	4	yes	—
639	C₂×C₄	#121×C₆₇	1	4	yes	—
640	C₂×C₄	#123×C₆₇	1	2	yes	—
641	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
642	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
643	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
644	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
645	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
646	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
647	C₈	—	4	1	no	—
648	C₈	—	4	1	no	—
649	C₈	—	4	2	no	—
650	C₂²	—	2	2	no	—
651	C₂²	—	2	2	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	2	2	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	2	4	no	—
657	C₂²	—	2	4	no	—
658	C₂²	—	2	4	no	—
659	C₂²	—	4	2	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	2	no	—
662	C₂²	—	4	2	no	—
663	C₂²	—	4	2	no	—
664	C₂²	—	4	2	no	—
665	C₂²	—	4	2	no	—
666	C₂²	—	8	1	no	—
667	C₂²	—	8	1	no	—
668	C₂²	—	8	1	no	—
669	C₂²	—	8	1	no	—
670	C₂²	#386×C₆₇	1	4	yes	—
671	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
672	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
673	C₂²	C₂×#123×C₆₇	1	1	yes	characteristic
674	C₂²	#576×C₆₇	1	1	yes	characteristic
675	C₂²	#577×C₆₇	1	1	yes	characteristic
676	C₂²	#577×C₆₇	1	1	yes	characteristic
677	C₂²	#578×C₆₇	1	1	yes	characteristic
678	C₄	—	2	4	no	—
679	C₄	—	4	4	no	—
680	C₄	—	4	2	no	—
681	C₄	—	4	2	no	—
682	C₄	—	4	2	no	—
683	C₄	—	4	4	no	—
684	C₄	#386×C₆₇	1	4	yes	—
685	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
686	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
687	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
688	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
689	C₂	—	2	4	no	—
690	C₂	—	4	4	no	—
691	C₂	—	8	2	no	—
692	C₂	C₂×#386×C₆₇	1	1	yes	characteristic
693	C₂	C₆₇×#2897	1	1	yes	characteristic
694	C₂	C₆₇×#2898	1	1	yes	characteristic
695	C₂	C₆₇×#3012	1	1	yes	characteristic
696	C₂	C₆₇×#3013	1	1	yes	characteristic
697	C₂	C₆₇×#3098	1	1	yes	characteristic
698	C₂	C₆₇×#3099	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#537×C₆₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#520×C₆₇	C₂	semidirect
3. C₂×#417×C₆₇	C₂	semidirect
4. C₂×#416×C₆₇	C₂	semidirect
5. C₂×#408×C₆₇	C₂	semidirect
6. C₆₇×#1875	C₂	semidirect
7. C₆₇×#1874	C₂	semidirect
8. C₆₇×#1873	C₂	semidirect
9. C₆₇×#1872	C₂	semidirect
10. C₆₇×#1736	C₂	semidirect
11. C₂×#340×C₆₇	C₂	semidirect
17. #537×C₆₇	C₄	semidirect
58. C₂×#41×C₆₇	C₂×C₄	semidirect
84. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
88. #121×C₆₇	C₂×C₄	semidirect
95. #117×C₆₇	D₄	semidirect
104. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
105. C₂×C₄⋊C₄×C₆₇	D₄	semidirect
339. #20915	C₆₇	direct (N × H)
351. C₂×#520	C₂×C₆₇	semidirect
352. C₂×#417	C₂×C₆₇	semidirect
353. C₂×#416	C₂×C₆₇	semidirect
354. C₂×#408	C₂×C₆₇	semidirect
355. #1875	C₂×C₆₇	semidirect
356. #1874	C₂×C₆₇	semidirect
357. #1873	C₂×C₆₇	semidirect
358. #1872	C₂×C₆₇	semidirect
359. #1736	C₂×C₆₇	semidirect
360. C₂×#340	C₂×C₆₇	semidirect
361. C₆₇	#20915	direct (N × H)
367. #537	C₄×C₆₇	semidirect
408. C₂×#41	C₂×C₄×C₆₇	semidirect
434. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
438. #121	C₂×C₄×C₆₇	semidirect
445. #117	D₄×C₆₇	semidirect
454. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
455. C₂×C₄⋊C₄	D₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#537×C₆₇	2
C₂×#520×C₆₇	2
C₂×#417×C₆₇	2
C₂×#416×C₆₇	2
C₂×#408×C₆₇	2
C₆₇×#1875	2
C₆₇×#1874	2
C₆₇×#1873	2
C₆₇×#1872	2
C₆₇×#1736	2
C₂×#340×C₆₇	2
#20915	67

Subgroups — C67×#20915 #30700166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20915 #30700166