Subgroups — C₆₇×#20878 #30700129

Order 17152.

1458 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#520×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#428×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#417×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#416×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2316	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1865	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1864	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1862	C₂	1	2	yes	maximal, complemented
10	C₂×#340×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₆₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#520×C₆₇	—	2	4	no	—
16	#520×C₆₇	—	2	4	no	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₆₇	—	2	2	no	—
19	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
20	C₂×#117×C₆₇	—	2	1	no	—
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	#428×C₆₇	—	2	4	no	—
27	C₂×#107×C₆₇	C₂²	1	2	yes	—
28	C₂×#106×C₆₇	C₂²	1	2	yes	—
29	#417×C₆₇	C₄	1	8	yes	complemented
30	#416×C₆₇	—	2	4	no	—
31	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#103×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×#103×C₆₇	C₂²	1	2	yes	—
34	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×#102×C₆₇	C₂²	1	2	yes	—
37	#389×C₆₇	—	2	1	no	—
38	#389×C₆₇	—	2	1	no	—
39	#389×C₆₇	C₂²	1	2	yes	—
40	#389×C₆₇	C₂²	1	2	yes	—
41	#380×C₆₇	—	2	2	no	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	#336×C₆₇	C₂²	1	2	yes	—
46	C₂×#41×C₆₇	—	2	2	no	—
47	C₂×#41×C₆₇	—	2	2	no	—
48	C₂×#41×C₆₇	—	2	2	no	—
49	C₂×#41×C₆₇	C₂×C₄	1	4	yes	—
50	C₂×#41×C₆₇	D₄	1	4	yes	—
51	C₂×#41×C₆₇	Q₈	1	4	yes	—
52	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×D₄×C₆₇	—	2	1	no	—
54	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	—	2	1	no	—
56	C₂³×C₄×C₆₇	—	2	1	no	—
57	C₂³×C₄×C₆₇	—	2	1	no	—
58	C₂³×C₄×C₆₇	—	2	2	no	—
59	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂×M₁₆×C₆₇	—	2	2	no	—
62	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
63	C₂²×C₈×C₆₇	—	2	2	no	—
64	C₂²×C₈×C₆₇	—	2	2	no	—
65	C₂²×C₈×C₆₇	—	2	2	no	—
66	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
69	C₄×Q₈×C₆₇	—	2	4	no	—
70	C₄×Q₈×C₆₇	—	2	4	no	—
71	C₄×D₄×C₆₇	—	2	16	no	—
72	C₄×D₄×C₆₇	—	2	4	no	—
73	C₄×D₄×C₆₇	—	2	4	no	—
74	#117×C₆₇	—	2	8	no	—
75	#117×C₆₇	—	2	2	no	—
76	#117×C₆₇	—	2	2	no	—
77	#117×C₆₇	—	2	2	no	—
78	#117×C₆₇	—	2	2	no	—
79	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
80	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
87	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	complemented
88	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
89	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₆₇	D₄	1	4	yes	—
91	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
92	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
94	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
97	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
98	C₂×C₄²×C₆₇	—	2	2	no	—
99	C₂×C₄²×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
101	#107×C₆₇	—	2	4	no	—
102	#106×C₆₇	—	2	4	no	—
103	#103×C₆₇	—	2	2	no	—
104	#103×C₆₇	—	2	2	no	—
105	#103×C₆₇	C₂×C₄	1	8	yes	complemented
106	#102×C₆₇	—	2	2	no	—
107	#102×C₆₇	—	2	2	no	—
108	#102×C₆₇	C₂×C₄	1	8	yes	complemented
109	#95×C₆₇	—	2	2	no	—
110	#95×C₆₇	—	2	2	no	—
111	C₂⁴×C₆₇	—	2	1	no	—
112	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
113	#41×C₆₇	—	2	8	no	—
114	#41×C₆₇	—	2	8	no	—
115	#41×C₆₇	—	2	8	no	—
116	#41×C₆₇	C₄⋊C₄	1	16	yes	complemented
117	C₂×Q₈×C₆₇	—	2	1	no	—
118	C₂×Q₈×C₆₇	—	2	1	no	—
119	C₂×Q₈×C₆₇	—	2	1	no	—
120	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
122	C₂×Q₈×C₆₇	C₂×Q₈	1	2	yes	—
123	C₂×D₄×C₆₇	—	2	4	no	—
124	C₂×D₄×C₆₇	—	2	4	no	—
125	C₂×D₄×C₆₇	—	2	4	no	—
126	C₂×D₄×C₆₇	—	2	1	no	—
127	C₂×D₄×C₆₇	—	2	1	no	—
128	C₂×D₄×C₆₇	—	2	1	no	—
129	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
131	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	—	2	2	no	—
137	C₂²×C₄×C₆₇	—	2	2	no	—
138	C₂²×C₄×C₆₇	—	2	8	no	—
139	C₂²×C₄×C₆₇	—	2	4	no	—
140	C₂²×C₄×C₆₇	—	2	4	no	—
141	C₂²×C₄×C₆₇	—	2	1	no	—
142	C₂²×C₄×C₆₇	—	2	1	no	—
143	C₂²×C₄×C₆₇	—	8	2	no	—
144	C₂²×C₄×C₆₇	—	8	2	no	—
145	C₂²×C₄×C₆₇	—	8	1	no	—
146	C₂²×C₄×C₆₇	—	8	1	no	—
147	C₂²×C₄×C₆₇	—	8	1	no	—
148	C₂²×C₄×C₆₇	—	8	1	no	—
149	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
156	C₂²×C₄×C₆₇	#41	1	2	yes	—
157	C₂²×C₄×C₆₇	#41	1	2	yes	—
158	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
159	M₁₆×C₆₇	—	2	8	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	1	no	—
163	C₂×C₈×C₆₇	—	2	1	no	—
164	C₂×C₈×C₆₇	—	2	1	no	—
165	C₂×C₈×C₆₇	—	2	1	no	—
166	C₂×C₈×C₆₇	—	2	1	no	—
167	C₂×C₈×C₆₇	—	2	1	no	—
168	C₂×C₈×C₆₇	—	4	4	no	—
169	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
170	C₄⋊C₄×C₆₇	—	2	8	no	—
171	C₄⋊C₄×C₆₇	—	2	4	no	—
172	C₄⋊C₄×C₆₇	—	2	2	no	—
173	C₄⋊C₄×C₆₇	—	2	4	no	—
174	C₄⋊C₄×C₆₇	—	2	2	no	—
175	C₄⋊C₄×C₆₇	—	2	2	no	—
176	C₄⋊C₄×C₆₇	—	2	2	no	—
177	C₄⋊C₄×C₆₇	—	2	1	no	—
178	C₄⋊C₄×C₆₇	—	2	1	no	—
179	C₄⋊C₄×C₆₇	—	2	1	no	—
180	C₄⋊C₄×C₆₇	—	2	1	no	—
181	C₄⋊C₄×C₆₇	—	8	2	no	—
182	C₄⋊C₄×C₆₇	—	8	2	no	—
183	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
184	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
185	C₂²⋊C₄×C₆₇	—	2	8	no	—
186	C₂²⋊C₄×C₆₇	—	2	4	no	—
187	C₂²⋊C₄×C₆₇	—	4	2	no	—
188	C₄²×C₆₇	—	2	8	no	—
189	C₄²×C₆₇	—	2	2	no	—
190	C₄²×C₆₇	—	2	2	no	—
191	C₄²×C₆₇	—	4	2	no	—
192	C₂³×C₆₇	—	2	1	no	—
193	C₂³×C₆₇	—	2	1	no	—
194	C₂³×C₆₇	—	2	1	no	—
195	C₂³×C₆₇	—	2	1	no	—
196	C₂³×C₆₇	—	2	2	no	—
197	C₂³×C₆₇	—	2	2	no	—
198	C₂³×C₆₇	—	2	2	no	—
199	C₂³×C₆₇	—	4	1	no	—
200	C₂³×C₆₇	—	4	1	no	—
201	C₂³×C₆₇	—	4	1	no	—
202	C₂³×C₆₇	—	4	1	no	—
203	C₂³×C₆₇	C₄×D₄	1	2	yes	—
204	C₂³×C₆₇	#120	1	2	yes	—
205	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
206	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
207	Q₈×C₆₇	—	2	2	no	—
208	Q₈×C₆₇	—	2	2	no	—
209	Q₈×C₆₇	—	2	2	no	—
210	Q₈×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
211	D₄×C₆₇	—	2	16	no	—
212	D₄×C₆₇	—	2	2	no	—
213	D₄×C₆₇	—	2	2	no	—
214	D₄×C₆₇	—	2	2	no	—
215	D₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
216	C₂×C₄×C₆₇	—	2	2	no	—
217	C₂×C₄×C₆₇	—	2	2	no	—
218	C₂×C₄×C₆₇	—	2	2	no	—
219	C₂×C₄×C₆₇	—	2	16	no	—
220	C₂×C₄×C₆₇	—	2	4	no	—
221	C₂×C₄×C₆₇	—	2	2	no	—
222	C₂×C₄×C₆₇	—	2	2	no	—
223	C₂×C₄×C₆₇	—	2	4	no	—
224	C₂×C₄×C₆₇	—	2	2	no	—
225	C₂×C₄×C₆₇	—	2	2	no	—
226	C₂×C₄×C₆₇	—	2	2	no	—
227	C₂×C₄×C₆₇	—	2	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	—	4	1	no	—
233	C₂×C₄×C₆₇	—	4	1	no	—
234	C₂×C₄×C₆₇	—	4	1	no	—
235	C₂×C₄×C₆₇	—	4	1	no	—
236	C₂×C₄×C₆₇	—	4	1	no	—
237	C₂×C₄×C₆₇	—	4	1	no	—
238	C₂×C₄×C₆₇	—	8	4	no	—
239	C₂×C₄×C₆₇	—	8	4	no	—
240	C₂×C₄×C₆₇	—	8	8	no	—
241	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
242	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
243	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
244	C₂×C₄×C₆₇	#120	1	2	yes	—
245	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
246	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
247	C₂×C₄×C₆₇	#123	1	4	yes	—
248	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
254	C₈×C₆₇	—	2	4	no	—
255	C₈×C₆₇	—	4	1	no	—
256	C₈×C₆₇	—	4	1	no	—
257	C₂²×C₆₇	—	2	2	no	—
258	C₂²×C₆₇	—	2	2	no	—
259	C₂²×C₆₇	—	2	2	no	—
260	C₂²×C₆₇	—	2	2	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	4	no	—
264	C₂²×C₆₇	—	4	2	no	—
265	C₂²×C₆₇	—	4	2	no	—
266	C₂²×C₆₇	—	4	2	no	—
267	C₂²×C₆₇	—	4	2	no	—
268	C₂²×C₆₇	—	4	2	no	—
269	C₂²×C₆₇	—	4	2	no	—
270	C₂²×C₆₇	#385	1	4	yes	—
271	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
272	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
273	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
274	C₂²×C₆₇	#576	1	1	yes	characteristic
275	C₂²×C₆₇	#577	1	1	yes	characteristic
276	C₂²×C₆₇	#577	1	1	yes	characteristic
277	C₂²×C₆₇	#578	1	1	yes	characteristic
278	C₄×C₆₇	—	2	4	no	—
279	C₄×C₆₇	—	4	4	no	—
280	C₄×C₆₇	—	4	2	no	—
281	C₄×C₆₇	—	4	4	no	—
282	C₄×C₆₇	—	4	2	no	—
283	C₄×C₆₇	—	8	1	no	—
284	C₄×C₆₇	—	8	1	no	—
285	C₄×C₆₇	#385	1	4	yes	—
286	C₄×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
287	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
288	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
289	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
290	#20878	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
291	C₂×C₆₇	—	2	4	no	—
292	C₂×C₆₇	—	4	4	no	—
293	C₂×C₆₇	C₂×#385	1	1	yes	characteristic
294	C₂×C₆₇	#2946	1	1	yes	characteristic
295	C₂×C₆₇	#2947	1	1	yes	characteristic
296	C₂×C₆₇	#3012	1	1	yes	characteristic
297	C₂×C₆₇	#3013	1	1	yes	characteristic
298	C₂×C₆₇	#3083	1	1	yes	characteristic
299	C₂×C₆₇	#3084	1	1	yes	characteristic
300	C₂×#520	C₂×C₆₇	1	1	yes	characteristic
301	C₂×#428	C₂×C₆₇	1	1	yes	characteristic, complemented
302	C₂×#417	C₂×C₆₇	1	1	yes	characteristic
303	C₂×#416	C₂×C₆₇	1	1	yes	characteristic
304	#2316	C₂×C₆₇	1	1	yes	characteristic, complemented
305	#1865	C₂×C₆₇	1	2	yes	complemented
306	#1864	C₂×C₆₇	1	2	yes	complemented
307	#1863	C₂×C₆₇	1	2	yes	complemented
308	#1862	C₂×C₆₇	1	2	yes	complemented
309	C₂×#340	C₂×C₆₇	1	1	yes	characteristic, complemented
310	#1729	C₂×C₆₇	1	1	yes	characteristic, complemented
311	C₆₇	#20878	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	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
315	#520	—	2	4	no	—
316	#520	—	2	4	no	—
317	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
318	C₂×C₄×D₄	—	2	2	no	—
319	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
320	C₂×#117	—	2	1	no	—
321	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
322	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
323	C₂²×C₄⋊C₄	—	2	1	no	—
324	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
325	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
326	#428	—	2	4	no	—
327	C₂×#107	C₂²×C₆₇	1	2	yes	—
328	C₂×#106	C₂²×C₆₇	1	2	yes	—
329	#417	C₄×C₆₇	1	8	yes	complemented
330	#416	—	2	4	no	—
331	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
332	C₂×#103	C₂²×C₆₇	1	1	yes	characteristic
333	C₂×#103	C₂²×C₆₇	1	2	yes	—
334	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
335	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
336	C₂×#102	C₂²×C₆₇	1	2	yes	—
337	#389	—	2	1	no	—
338	#389	—	2	1	no	—
339	#389	C₂²×C₆₇	1	2	yes	—
340	#389	C₂²×C₆₇	1	2	yes	—
341	#380	—	2	2	no	—
342	#340	C₂²×C₆₇	1	2	yes	—
343	#340	C₂²×C₆₇	1	2	yes	—
344	#336	C₂²×C₆₇	1	2	yes	—
345	#336	C₂²×C₆₇	1	2	yes	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	C₂×C₄×C₆₇	1	4	yes	—
350	C₂×#41	D₄×C₆₇	1	4	yes	—
351	C₂×#41	Q₈×C₆₇	1	4	yes	—
352	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
353	C₂²×D₄	—	2	1	no	—
354	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
355	C₂³×C₄	—	2	1	no	—
356	C₂³×C₄	—	2	1	no	—
357	C₂³×C₄	—	2	1	no	—
358	C₂³×C₄	—	2	2	no	—
359	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
360	C₂×M₁₆	—	2	2	no	—
361	C₂×M₁₆	—	2	2	no	—
362	C₂×M₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
363	C₂²×C₈	—	2	2	no	—
364	C₂²×C₈	—	2	2	no	—
365	C₂²×C₈	—	2	2	no	—
366	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
368	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
369	C₄×Q₈	—	2	4	no	—
370	C₄×Q₈	—	2	4	no	—
371	C₄×D₄	—	2	16	no	—
372	C₄×D₄	—	2	4	no	—
373	C₄×D₄	—	2	4	no	—
374	#117	—	2	8	no	—
375	#117	—	2	2	no	—
376	#117	—	2	2	no	—
377	#117	—	2	2	no	—
378	#117	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	8	no	—
380	C₂×C₄⋊C₄	—	2	1	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	2	no	—
383	C₂×C₄⋊C₄	—	2	1	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄⋊C₄	—	2	2	no	—
386	C₂×C₄⋊C₄	—	2	2	no	—
387	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	4	yes	complemented
388	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
389	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
390	C₂×C₄⋊C₄	D₄×C₆₇	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
392	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
393	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
394	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
395	C₂×C₂²⋊C₄	—	2	2	no	—
396	C₂×C₂²⋊C₄	—	2	1	no	—
397	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
398	C₂×C₄²	—	2	2	no	—
399	C₂×C₄²	—	2	1	no	—
400	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
401	#107	—	2	4	no	—
402	#106	—	2	4	no	—
403	#103	—	2	2	no	—
404	#103	—	2	2	no	—
405	#103	C₂×C₄×C₆₇	1	8	yes	complemented
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	#102	C₂×C₄×C₆₇	1	8	yes	complemented
409	#95	—	2	2	no	—
410	#95	—	2	2	no	—
411	C₂⁴	—	2	1	no	—
412	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
413	#41	—	2	8	no	—
414	#41	—	2	8	no	—
415	#41	—	2	8	no	—
416	#41	C₄⋊C₄×C₆₇	1	16	yes	complemented
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	—	2	1	no	—
420	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
421	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
422	C₂×Q₈	C₂×Q₈×C₆₇	1	2	yes	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
430	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
431	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	4	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	2	4	no	—
441	C₂²×C₄	—	2	1	no	—
442	C₂²×C₄	—	2	1	no	—
443	C₂²×C₄	—	8	2	no	—
444	C₂²×C₄	—	8	2	no	—
445	C₂²×C₄	—	8	1	no	—
446	C₂²×C₄	—	8	1	no	—
447	C₂²×C₄	—	8	1	no	—
448	C₂²×C₄	—	8	1	no	—
449	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
450	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
455	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
456	C₂²×C₄	#41×C₆₇	1	2	yes	—
457	C₂²×C₄	#41×C₆₇	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
459	M₁₆	—	2	8	no	—
460	C₂×C₈	—	2	2	no	—
461	C₂×C₈	—	2	2	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	2	1	no	—
464	C₂×C₈	—	2	1	no	—
465	C₂×C₈	—	2	1	no	—
466	C₂×C₈	—	2	1	no	—
467	C₂×C₈	—	2	1	no	—
468	C₂×C₈	—	4	4	no	—
469	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
470	C₄⋊C₄	—	2	8	no	—
471	C₄⋊C₄	—	2	4	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	2	no	—
475	C₄⋊C₄	—	2	2	no	—
476	C₄⋊C₄	—	2	2	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	2	1	no	—
481	C₄⋊C₄	—	8	2	no	—
482	C₄⋊C₄	—	8	2	no	—
483	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
484	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
485	C₂²⋊C₄	—	2	8	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	4	2	no	—
488	C₄²	—	2	8	no	—
489	C₄²	—	2	2	no	—
490	C₄²	—	2	2	no	—
491	C₄²	—	4	2	no	—
492	C₂³	—	2	1	no	—
493	C₂³	—	2	1	no	—
494	C₂³	—	2	1	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	2	no	—
497	C₂³	—	2	2	no	—
498	C₂³	—	2	2	no	—
499	C₂³	—	4	1	no	—
500	C₂³	—	4	1	no	—
501	C₂³	—	4	1	no	—
502	C₂³	—	4	1	no	—
503	C₂³	C₄×D₄×C₆₇	1	2	yes	—
504	C₂³	#120×C₆₇	1	2	yes	—
505	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
506	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
507	Q₈	—	2	2	no	—
508	Q₈	—	2	2	no	—
509	Q₈	—	2	2	no	—
510	Q₈	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
511	D₄	—	2	16	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	2	no	—
515	D₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	16	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	2	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	2	no	—
527	C₂×C₄	—	2	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	2	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	2	no	—
532	C₂×C₄	—	4	1	no	—
533	C₂×C₄	—	4	1	no	—
534	C₂×C₄	—	4	1	no	—
535	C₂×C₄	—	4	1	no	—
536	C₂×C₄	—	4	1	no	—
537	C₂×C₄	—	4	1	no	—
538	C₂×C₄	—	8	4	no	—
539	C₂×C₄	—	8	4	no	—
540	C₂×C₄	—	8	8	no	—
541	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	4	yes	—
542	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
543	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
544	C₂×C₄	#120×C₆₇	1	2	yes	—
545	C₂×C₄	C₂²⋊Q₈×C₆₇	1	4	yes	—
546	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
547	C₂×C₄	#123×C₆₇	1	4	yes	—
548	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
549	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
550	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
551	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
552	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
553	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
554	C₈	—	2	4	no	—
555	C₈	—	4	1	no	—
556	C₈	—	4	1	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	2	no	—
560	C₂²	—	2	2	no	—
561	C₂²	—	2	2	no	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	4	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	2	no	—
569	C₂²	—	4	2	no	—
570	C₂²	#385×C₆₇	1	4	yes	—
571	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
572	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
573	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
574	C₂²	#576×C₆₇	1	1	yes	characteristic
575	C₂²	#577×C₆₇	1	1	yes	characteristic
576	C₂²	#577×C₆₇	1	1	yes	characteristic
577	C₂²	#578×C₆₇	1	1	yes	characteristic
578	C₄	—	2	4	no	—
579	C₄	—	4	2	no	—
580	C₄	—	4	4	no	—
581	C₄	—	4	2	no	—
582	C₄	—	4	4	no	—
583	C₄	—	8	1	no	—
584	C₄	—	8	1	no	—
585	C₄	#385×C₆₇	1	4	yes	—
586	C₄	C₂²×C₄⋊C₄×C₆₇	1	1	yes	characteristic
587	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
588	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
589	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
590	C₂	—	2	4	no	—
591	C₂	—	4	4	no	—
592	C₂	C₂×#385×C₆₇	1	1	yes	characteristic
593	C₂	C₆₇×#2946	1	1	yes	characteristic
594	C₂	C₆₇×#2947	1	1	yes	characteristic
595	C₂	C₆₇×#3012	1	1	yes	characteristic
596	C₂	C₆₇×#3013	1	1	yes	characteristic
597	C₂	C₆₇×#3083	1	1	yes	characteristic
598	C₂	C₆₇×#3084	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	C₂²×#41×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×#117×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#428×C₆₇	C₂	semidirect
5. C₆₇×#2316	C₂	semidirect
6. C₆₇×#1865	C₂	semidirect
7. C₆₇×#1864	C₂	semidirect
8. C₆₇×#1863	C₂	semidirect
9. C₆₇×#1862	C₂	semidirect
10. C₂×#340×C₆₇	C₂	semidirect
11. C₆₇×#1729	C₂	semidirect
29. #417×C₆₇	C₄	semidirect
62. C₂×M₁₆×C₆₇	C₂×C₄	semidirect
87. C₂×C₄⋊C₄×C₆₇	C₂×C₄	semidirect
105. #103×C₆₇	C₂×C₄	semidirect
108. #102×C₆₇	C₂×C₄	semidirect
116. #41×C₆₇	C₄⋊C₄	semidirect
290. #20878	C₆₇	direct (N × H)
301. C₂×#428	C₂×C₆₇	semidirect
304. #2316	C₂×C₆₇	semidirect
305. #1865	C₂×C₆₇	semidirect
306. #1864	C₂×C₆₇	semidirect
307. #1863	C₂×C₆₇	semidirect
308. #1862	C₂×C₆₇	semidirect
309. C₂×#340	C₂×C₆₇	semidirect
310. #1729	C₂×C₆₇	semidirect
311. C₆₇	#20878	direct (N × H)
329. #417	C₄×C₆₇	semidirect
362. C₂×M₁₆	C₂×C₄×C₆₇	semidirect
387. C₂×C₄⋊C₄	C₂×C₄×C₆₇	semidirect
405. #103	C₂×C₄×C₆₇	semidirect
408. #102	C₂×C₄×C₆₇	semidirect
416. #41	C₄⋊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×#520×C₆₇	2
C₂×#428×C₆₇	2
C₂×#417×C₆₇	2
C₂×#416×C₆₇	2
C₆₇×#2316	2
C₆₇×#1865	2
C₆₇×#1864	2
C₆₇×#1863	2
C₆₇×#1862	2
C₂×#340×C₆₇	2
C₆₇×#1729	2
#20878	67

Subgroups — C67×#20878 #30700129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20878 #30700129