Subgroups — C₆₇×#20934 #30700185

Order 17152.

2066 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#518×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#418×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2309	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1879	C₂	1	8	yes	maximal, complemented
7	C₆₇×#1735	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1729	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×D₈×C₆₇	—	2	4	no	—
12	C₂²×D₈×C₆₇	—	2	4	no	—
13	C₂²×D₈×C₆₇	C₄	1	8	yes	complemented
14	C₂²×D₈×C₆₇	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	#518×C₆₇	—	2	4	no	—
18	#518×C₆₇	—	2	4	no	—
19	C₂×C₄×D₄×C₆₇	—	2	2	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	4	yes	complemented
21	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₂²⋊C₄×C₆₇	—	2	2	no	—
25	#418×C₆₇	—	2	4	no	—
26	#418×C₆₇	—	2	4	no	—
27	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
28	C₂×#102×C₆₇	C₂²	1	4	yes	complemented
29	#390×C₆₇	—	2	2	no	—
30	#390×C₆₇	C₂²	1	4	yes	complemented
31	#379×C₆₇	—	2	2	no	—
32	#340×C₆₇	C₂²	1	4	yes	complemented
33	#336×C₆₇	C₂²	1	4	yes	complemented
34	C₂⁵×C₆₇	—	2	1	no	—
35	C₂⁵×C₆₇	—	2	1	no	—
36	C₂²×D₄×C₆₇	—	2	1	no	—
37	C₂²×D₄×C₆₇	—	2	2	no	—
38	C₂²×D₄×C₆₇	—	2	2	no	—
39	C₂²×D₄×C₆₇	—	2	2	no	—
40	C₂²×D₄×C₆₇	—	2	2	no	—
41	C₂²×D₄×C₆₇	—	2	1	no	—
42	C₂²×D₄×C₆₇	—	2	2	no	—
43	C₂²×D₄×C₆₇	—	2	2	no	—
44	C₂²×D₄×C₆₇	—	2	2	no	—
45	C₂²×D₄×C₆₇	—	2	2	no	—
46	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	—
47	C₂²×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
48	C₂²×D₄×C₆₇	D₄	1	4	yes	—
49	C₂²×D₄×C₆₇	D₄	1	4	yes	—
50	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
51	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	—	2	4	no	—
53	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
54	C₂×D₈×C₆₇	—	2	32	no	—
55	C₂×D₈×C₆₇	—	2	8	no	—
56	C₂×D₈×C₆₇	—	2	8	no	—
57	C₂×D₈×C₆₇	C₂×C₄	1	16	yes	complemented
58	C₂×M₁₆×C₆₇	—	2	2	no	—
59	C₂×M₁₆×C₆₇	—	2	2	no	—
60	C₂×M₁₆×C₆₇	—	2	2	no	—
61	C₂²×C₈×C₆₇	—	2	2	no	—
62	C₂²×C₈×C₆₇	—	2	2	no	—
63	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	complemented
64	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
65	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
66	C₄×D₄×C₆₇	—	2	16	no	—
67	C₄×D₄×C₆₇	—	2	8	no	—
68	C₄×D₄×C₆₇	—	2	8	no	—
69	#117×C₆₇	—	2	2	no	—
70	#117×C₆₇	—	2	2	no	—
71	#117×C₆₇	—	2	2	no	—
72	#117×C₆₇	—	2	2	no	—
73	#117×C₆₇	—	2	2	no	—
74	#117×C₆₇	D₄	1	4	yes	—
75	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
81	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
82	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
83	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
84	C₂×C₄²×C₆₇	—	2	1	no	—
85	C₂×C₄²×C₆₇	—	2	1	no	—
86	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
87	#102×C₆₇	—	2	8	no	—
88	#102×C₆₇	—	2	8	no	—
89	C₂⁴×C₆₇	—	2	2	no	—
90	C₂⁴×C₆₇	—	2	4	no	—
91	C₂⁴×C₆₇	—	2	4	no	—
92	C₂⁴×C₆₇	—	2	4	no	—
93	C₂⁴×C₆₇	—	2	2	no	—
94	C₂⁴×C₆₇	—	2	4	no	—
95	C₂⁴×C₆₇	—	4	2	no	—
96	C₂⁴×C₆₇	—	4	2	no	—
97	C₂⁴×C₆₇	—	8	4	no	—
98	C₂⁴×C₆₇	—	8	4	no	—
99	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₆₇	—	2	8	no	—
101	C₂×D₄×C₆₇	—	2	8	no	—
102	C₂×D₄×C₆₇	—	2	4	no	—
103	C₂×D₄×C₆₇	—	2	8	no	—
104	C₂×D₄×C₆₇	—	2	4	no	—
105	C₂×D₄×C₆₇	—	2	4	no	—
106	C₂×D₄×C₆₇	—	2	8	no	—
107	C₂×D₄×C₆₇	—	2	4	no	—
108	C₂×D₄×C₆₇	—	2	8	no	—
109	C₂×D₄×C₆₇	—	2	2	no	—
110	C₂×D₄×C₆₇	—	2	2	no	—
111	C₂×D₄×C₆₇	—	2	2	no	—
112	C₂×D₄×C₆₇	—	2	2	no	—
113	C₂×D₄×C₆₇	—	2	2	no	—
114	C₂×D₄×C₆₇	—	2	2	no	—
115	C₂×D₄×C₆₇	—	2	2	no	—
116	C₂×D₄×C₆₇	—	2	2	no	—
117	C₂×D₄×C₆₇	—	4	2	no	—
118	C₂×D₄×C₆₇	—	4	4	no	—
119	C₂×D₄×C₆₇	—	4	2	no	—
120	C₂×D₄×C₆₇	—	4	4	no	—
121	C₂×D₄×C₆₇	C₂²⋊C₄	1	16	yes	complemented
122	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
123	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
124	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
125	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	16	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	8	8	no	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
131	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
136	C₂²×C₄×C₆₇	#41	1	2	yes	—
137	C₂²×C₄×C₆₇	#41	1	2	yes	—
138	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
139	D₈×C₆₇	—	2	32	no	—
140	M₁₆×C₆₇	—	4	4	no	—
141	C₂×C₈×C₆₇	—	2	8	no	—
142	C₂×C₈×C₆₇	—	2	2	no	—
143	C₂×C₈×C₆₇	—	2	2	no	—
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
148	C₄⋊C₄×C₆₇	—	2	4	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	2	2	no	—
151	C₄⋊C₄×C₆₇	—	2	2	no	—
152	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
153	C₂²⋊C₄×C₆₇	—	2	16	no	—
154	C₂²⋊C₄×C₆₇	—	2	4	no	—
155	C₂²⋊C₄×C₆₇	—	2	4	no	—
156	C₂²⋊C₄×C₆₇	—	4	2	no	—
157	C₂²⋊C₄×C₆₇	—	8	8	no	—
158	C₄²×C₆₇	—	2	4	no	—
159	C₄²×C₆₇	—	2	2	no	—
160	C₄²×C₆₇	—	2	2	no	—
161	C₄²×C₆₇	—	4	2	no	—
162	C₂³×C₆₇	—	2	4	no	—
163	C₂³×C₆₇	—	2	4	no	—
164	C₂³×C₆₇	—	2	4	no	—
165	C₂³×C₆₇	—	2	4	no	—
166	C₂³×C₆₇	—	2	16	no	—
167	C₂³×C₆₇	—	2	4	no	—
168	C₂³×C₆₇	—	4	8	no	—
169	C₂³×C₆₇	—	4	2	no	—
170	C₂³×C₆₇	—	4	2	no	—
171	C₂³×C₆₇	—	4	8	no	—
172	C₂³×C₆₇	—	4	8	no	—
173	C₂³×C₆₇	—	8	8	no	—
174	C₂³×C₆₇	—	8	8	no	—
175	C₂³×C₆₇	—	8	8	no	—
176	C₂³×C₆₇	—	8	8	no	—
177	C₂³×C₆₇	—	8	8	no	—
178	C₂³×C₆₇	—	8	8	no	—
179	C₂³×C₆₇	C₄×D₄	1	2	yes	—
180	C₂³×C₆₇	#120	1	2	yes	—
181	C₂³×C₆₇	#121	1	2	yes	—
182	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
183	D₄×C₆₇	—	2	4	no	—
184	D₄×C₆₇	—	2	4	no	—
185	D₄×C₆₇	—	2	4	no	—
186	D₄×C₆₇	—	2	4	no	—
187	D₄×C₆₇	—	2	4	no	—
188	D₄×C₆₇	—	2	16	no	—
189	D₄×C₆₇	—	4	2	no	—
190	D₄×C₆₇	—	4	2	no	—
191	D₄×C₆₇	—	4	8	no	—
192	D₄×C₆₇	C₂×C₂²⋊C₄	1	8	yes	—
193	C₂×C₄×C₆₇	—	2	4	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	8	no	—
196	C₂×C₄×C₆₇	—	4	4	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	4	no	—
199	C₂×C₄×C₆₇	—	8	16	no	—
200	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
201	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
202	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
203	C₂×C₄×C₆₇	#120	1	2	yes	—
204	C₂×C₄×C₆₇	#121	1	4	yes	—
205	C₂×C₄×C₆₇	#121	1	2	yes	—
206	C₂×C₄×C₆₇	#123	1	4	yes	—
207	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
208	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
213	C₈×C₆₇	—	2	4	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	2	4	no	—
216	C₂²×C₆₇	—	2	4	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	2	8	no	—
219	C₂²×C₆₇	—	4	8	no	—
220	C₂²×C₆₇	—	4	8	no	—
221	C₂²×C₆₇	—	4	8	no	—
222	C₂²×C₆₇	—	4	8	no	—
223	C₂²×C₆₇	—	4	4	no	—
224	C₂²×C₆₇	—	8	16	no	—
225	C₂²×C₆₇	—	8	16	no	—
226	C₂²×C₆₇	—	8	4	no	—
227	C₂²×C₆₇	—	8	4	no	—
228	C₂²×C₆₇	#386	1	4	yes	—
229	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
230	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
231	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
232	C₂²×C₆₇	#576	1	4	yes	—
233	C₄×C₆₇	—	4	4	no	—
234	C₄×C₆₇	—	4	4	no	—
235	C₄×C₆₇	—	8	2	no	—
236	C₄×C₆₇	#386	1	4	yes	—
237	C₄×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
238	C₄×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
239	C₄×C₆₇	C₂×#121	1	1	yes	characteristic
240	C₄×C₆₇	C₂×#123	1	1	yes	characteristic
241	#20934	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂×C₆₇	—	2	4	no	—
243	C₂×C₆₇	—	4	8	no	—
244	C₂×C₆₇	—	8	4	no	—
245	C₂×C₆₇	C₂×#386	1	1	yes	characteristic
246	C₂×C₆₇	#2948	1	2	yes	—
247	C₂×C₆₇	#3008	1	2	yes	—
248	C₂×C₆₇	#3041	1	2	yes	—
249	C₂³×D₈	C₂×C₆₇	1	1	yes	characteristic
250	C₂×#518	C₂×C₆₇	1	1	yes	characteristic, complemented
251	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
252	C₂×#418	C₂×C₆₇	1	1	yes	characteristic, complemented
253	#2309	C₂×C₆₇	1	1	yes	characteristic, complemented
254	#1879	C₂×C₆₇	1	8	yes	complemented
255	#1735	C₂×C₆₇	1	1	yes	characteristic, complemented
256	#1729	C₂×C₆₇	1	1	yes	characteristic, complemented
257	C₆₇	#20934	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
259	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
260	C₂²×D₈	—	2	4	no	—
261	C₂²×D₈	—	2	4	no	—
262	C₂²×D₈	C₄×C₆₇	1	8	yes	complemented
263	C₂²×D₈	C₂²×C₆₇	1	4	yes	—
264	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
265	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
266	#518	—	2	4	no	—
267	#518	—	2	4	no	—
268	C₂×C₄×D₄	—	2	2	no	—
269	C₂×C₄×D₄	C₂²×C₆₇	1	4	yes	complemented
270	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
271	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
273	C₂²×C₂²⋊C₄	—	2	2	no	—
274	#418	—	2	4	no	—
275	#418	—	2	4	no	—
276	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
277	C₂×#102	C₂²×C₆₇	1	4	yes	complemented
278	#390	—	2	2	no	—
279	#390	C₂²×C₆₇	1	4	yes	complemented
280	#379	—	2	2	no	—
281	#340	C₂²×C₆₇	1	4	yes	complemented
282	#336	C₂²×C₆₇	1	4	yes	complemented
283	C₂⁵	—	2	1	no	—
284	C₂⁵	—	2	1	no	—
285	C₂²×D₄	—	2	1	no	—
286	C₂²×D₄	—	2	2	no	—
287	C₂²×D₄	—	2	2	no	—
288	C₂²×D₄	—	2	2	no	—
289	C₂²×D₄	—	2	2	no	—
290	C₂²×D₄	—	2	1	no	—
291	C₂²×D₄	—	2	2	no	—
292	C₂²×D₄	—	2	2	no	—
293	C₂²×D₄	—	2	2	no	—
294	C₂²×D₄	—	2	2	no	—
295	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	—
296	C₂²×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
297	C₂²×D₄	D₄×C₆₇	1	4	yes	—
298	C₂²×D₄	D₄×C₆₇	1	4	yes	—
299	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
300	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
301	C₂³×C₄	—	2	4	no	—
302	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
303	C₂×D₈	—	2	32	no	—
304	C₂×D₈	—	2	8	no	—
305	C₂×D₈	—	2	8	no	—
306	C₂×D₈	C₂×C₄×C₆₇	1	16	yes	complemented
307	C₂×M₁₆	—	2	2	no	—
308	C₂×M₁₆	—	2	2	no	—
309	C₂×M₁₆	—	2	2	no	—
310	C₂²×C₈	—	2	2	no	—
311	C₂²×C₈	—	2	2	no	—
312	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	complemented
313	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
314	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
315	C₄×D₄	—	2	16	no	—
316	C₄×D₄	—	2	8	no	—
317	C₄×D₄	—	2	8	no	—
318	#117	—	2	2	no	—
319	#117	—	2	2	no	—
320	#117	—	2	2	no	—
321	#117	—	2	2	no	—
322	#117	—	2	2	no	—
323	#117	D₄×C₆₇	1	4	yes	—
324	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
327	C₂×C₂²⋊C₄	—	2	16	no	—
328	C₂×C₂²⋊C₄	—	2	1	no	—
329	C₂×C₂²⋊C₄	—	2	4	no	—
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₂²⋊C₄	—	2	8	no	—
332	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
336	#102	—	2	8	no	—
337	#102	—	2	8	no	—
338	C₂⁴	—	2	2	no	—
339	C₂⁴	—	2	4	no	—
340	C₂⁴	—	2	4	no	—
341	C₂⁴	—	2	4	no	—
342	C₂⁴	—	2	2	no	—
343	C₂⁴	—	2	4	no	—
344	C₂⁴	—	4	2	no	—
345	C₂⁴	—	4	2	no	—
346	C₂⁴	—	8	4	no	—
347	C₂⁴	—	8	4	no	—
348	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂×D₄	—	2	8	no	—
350	C₂×D₄	—	2	8	no	—
351	C₂×D₄	—	2	4	no	—
352	C₂×D₄	—	2	8	no	—
353	C₂×D₄	—	2	4	no	—
354	C₂×D₄	—	2	4	no	—
355	C₂×D₄	—	2	8	no	—
356	C₂×D₄	—	2	4	no	—
357	C₂×D₄	—	2	8	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	4	2	no	—
367	C₂×D₄	—	4	4	no	—
368	C₂×D₄	—	4	2	no	—
369	C₂×D₄	—	4	4	no	—
370	C₂×D₄	C₂²⋊C₄×C₆₇	1	16	yes	complemented
371	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
372	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
373	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
374	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	16	no	—
377	C₂²×C₄	—	2	2	no	—
378	C₂²×C₄	—	8	8	no	—
379	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
380	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
381	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
382	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
383	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
385	C₂²×C₄	#41×C₆₇	1	2	yes	—
386	C₂²×C₄	#41×C₆₇	1	2	yes	—
387	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
388	D₈	—	2	32	no	—
389	M₁₆	—	4	4	no	—
390	C₂×C₈	—	2	8	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	2	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	2	no	—
401	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
402	C₂²⋊C₄	—	2	16	no	—
403	C₂²⋊C₄	—	2	4	no	—
404	C₂²⋊C₄	—	2	4	no	—
405	C₂²⋊C₄	—	4	2	no	—
406	C₂²⋊C₄	—	8	8	no	—
407	C₄²	—	2	4	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	—	4	2	no	—
411	C₂³	—	2	4	no	—
412	C₂³	—	2	4	no	—
413	C₂³	—	2	4	no	—
414	C₂³	—	2	4	no	—
415	C₂³	—	2	16	no	—
416	C₂³	—	2	4	no	—
417	C₂³	—	4	8	no	—
418	C₂³	—	4	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	8	no	—
421	C₂³	—	4	8	no	—
422	C₂³	—	8	8	no	—
423	C₂³	—	8	8	no	—
424	C₂³	—	8	8	no	—
425	C₂³	—	8	8	no	—
426	C₂³	—	8	8	no	—
427	C₂³	—	8	8	no	—
428	C₂³	C₄×D₄×C₆₇	1	2	yes	—
429	C₂³	#120×C₆₇	1	2	yes	—
430	C₂³	#121×C₆₇	1	2	yes	—
431	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
432	D₄	—	2	4	no	—
433	D₄	—	2	4	no	—
434	D₄	—	2	4	no	—
435	D₄	—	2	4	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	16	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	8	no	—
441	D₄	C₂×C₂²⋊C₄×C₆₇	1	8	yes	—
442	C₂×C₄	—	2	4	no	—
443	C₂×C₄	—	2	4	no	—
444	C₂×C₄	—	2	8	no	—
445	C₂×C₄	—	4	4	no	—
446	C₂×C₄	—	4	2	no	—
447	C₂×C₄	—	4	4	no	—
448	C₂×C₄	—	8	16	no	—
449	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
450	C₂×C₄	C₄×D₄×C₆₇	1	4	yes	—
451	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
452	C₂×C₄	#120×C₆₇	1	2	yes	—
453	C₂×C₄	#121×C₆₇	1	4	yes	—
454	C₂×C₄	#121×C₆₇	1	2	yes	—
455	C₂×C₄	#123×C₆₇	1	4	yes	—
456	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
457	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
458	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
460	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
462	C₈	—	2	4	no	—
463	C₈	—	4	2	no	—
464	C₂²	—	2	4	no	—
465	C₂²	—	2	4	no	—
466	C₂²	—	2	4	no	—
467	C₂²	—	2	8	no	—
468	C₂²	—	4	8	no	—
469	C₂²	—	4	8	no	—
470	C₂²	—	4	8	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	8	no	—
473	C₂²	—	8	16	no	—
474	C₂²	—	8	16	no	—
475	C₂²	—	8	4	no	—
476	C₂²	—	8	4	no	—
477	C₂²	#386×C₆₇	1	4	yes	—
478	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
479	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
480	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
481	C₂²	#576×C₆₇	1	4	yes	—
482	C₄	—	4	4	no	—
483	C₄	—	4	4	no	—
484	C₄	—	8	2	no	—
485	C₄	#386×C₆₇	1	4	yes	—
486	C₄	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
487	C₄	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
488	C₄	C₂×#121×C₆₇	1	1	yes	characteristic
489	C₄	C₂×#123×C₆₇	1	1	yes	characteristic
490	C₂	—	2	4	no	—
491	C₂	—	4	8	no	—
492	C₂	—	8	4	no	—
493	C₂	C₂×#386×C₆₇	1	1	yes	characteristic
494	C₂	C₆₇×#2948	1	2	yes	—
495	C₂	C₆₇×#3008	1	2	yes	—
496	C₂	C₆₇×#3041	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂³×D₈×C₆₇	C₂
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×#518×C₆₇	C₂	semidirect
3. C₂×#418×C₆₇	C₂	semidirect
4. C₂×#418×C₆₇	C₂	semidirect
5. C₆₇×#2309	C₂	semidirect
6. C₆₇×#1879	C₂	semidirect
7. C₆₇×#1735	C₂	semidirect
8. C₆₇×#1729	C₂	semidirect
13. C₂²×D₈×C₆₇	C₄	semidirect
20. C₂×C₄×D₄×C₆₇	C₂²	semidirect
27. C₂×#102×C₆₇	C₂²	semidirect
28. C₂×#102×C₆₇	C₂²	semidirect
30. #390×C₆₇	C₂²	semidirect
32. #340×C₆₇	C₂²	semidirect
33. #336×C₆₇	C₂²	semidirect
47. C₂²×D₄×C₆₇	C₂×C₄	semidirect
57. C₂×D₈×C₆₇	C₂×C₄	semidirect
63. C₂²×C₈×C₆₇	C₂×C₄	semidirect
121. C₂×D₄×C₆₇	C₂²⋊C₄	semidirect
241. #20934	C₆₇	direct (N × H)
250. C₂×#518	C₂×C₆₇	semidirect
251. C₂×#418	C₂×C₆₇	semidirect
252. C₂×#418	C₂×C₆₇	semidirect
253. #2309	C₂×C₆₇	semidirect
254. #1879	C₂×C₆₇	semidirect
255. #1735	C₂×C₆₇	semidirect
256. #1729	C₂×C₆₇	semidirect
257. C₆₇	#20934	direct (N × H)
262. C₂²×D₈	C₄×C₆₇	semidirect
269. C₂×C₄×D₄	C₂²×C₆₇	semidirect
276. C₂×#102	C₂²×C₆₇	semidirect
277. C₂×#102	C₂²×C₆₇	semidirect
279. #390	C₂²×C₆₇	semidirect
281. #340	C₂²×C₆₇	semidirect
282. #336	C₂²×C₆₇	semidirect
296. C₂²×D₄	C₂×C₄×C₆₇	semidirect
306. C₂×D₈	C₂×C₄×C₆₇	semidirect
312. C₂²×C₈	C₂×C₄×C₆₇	semidirect
370. C₂×D₄	C₂²⋊C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₆₇	2
C₂×#518×C₆₇	2
C₂×#418×C₆₇	2
C₆₇×#2309	2
C₆₇×#1879	2
C₆₇×#1735	2
C₆₇×#1729	2
#20934	67

Subgroups — C67×#20934 #30700185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#20934 #30700185