Subgroups — C₆₇×#21861 #30701112

Order 17152.

1166 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#382×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1991	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1958	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1874	C₂	1	2	yes	maximal
7	C₆₇×#1872	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1805	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1804	C₂	1	2	yes	maximal, complemented
10	#529×C₆₇	—	2	4	no	—
11	#529×C₆₇	C₄	1	8	yes	complemented
12	C₂×#126×C₆₇	—	2	1	no	—
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#123×C₆₇	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	—	2	1	no	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	—	2	1	no	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	#431×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×#103×C₆₇	C₂²	1	4	yes	—
27	C₂×#102×C₆₇	C₂²	1	4	yes	—
28	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	#382×C₆₇	—	2	1	no	—
31	#382×C₆₇	—	2	1	no	—
32	#382×C₆₇	C₂²	1	2	yes	—
33	#382×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	—	2	1	no	—
35	C₂×#95×C₆₇	—	2	1	no	—
36	#340×C₆₇	C₂²	1	4	yes	—
37	#340×C₆₇	C₂²	1	4	yes	—
38	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₆₇	—	2	1	no	—
40	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	—	2	2	no	—
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
47	#126×C₆₇	—	2	8	no	—
48	#125×C₆₇	—	2	2	no	—
49	#125×C₆₇	C₂×C₄	1	4	yes	complemented
50	#124×C₆₇	—	2	2	no	—
51	#124×C₆₇	C₂×C₄	1	4	yes	complemented
52	#123×C₆₇	—	2	8	no	—
53	C₂²⋊Q₈×C₆₇	—	2	2	no	—
54	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
55	#121×C₆₇	—	2	2	no	—
56	#121×C₆₇	C₂×C₄	1	4	yes	—
57	C₄×Q₈×C₆₇	—	2	2	no	—
58	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	complemented
59	C₄×D₄×C₆₇	—	2	8	no	—
60	C₄×D₄×C₆₇	—	2	2	no	—
61	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	complemented
62	#117×C₆₇	—	2	8	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
75	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
76	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
79	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	D₄	1	4	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₄⋊C₈×C₆₇	—	2	4	no	—
85	#103×C₆₇	—	4	4	no	—
86	#102×C₆₇	—	4	4	no	—
87	C₂²⋊C₈×C₆₇	—	2	4	no	—
88	C₄×C₈×C₆₇	—	2	4	no	—
89	#95×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	2	no	—
92	#95×C₆₇	—	2	2	no	—
93	C₂⁴×C₆₇	—	2	1	no	—
94	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂×Q₈×C₆₇	—	2	1	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
99	C₂×D₄×C₆₇	—	2	4	no	—
100	C₂×D₄×C₆₇	—	2	1	no	—
101	C₂×D₄×C₆₇	—	2	2	no	—
102	C₂×D₄×C₆₇	—	2	2	no	—
103	C₂×D₄×C₆₇	—	4	4	no	—
104	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂²×C₄×C₆₇	—	2	4	no	—
106	C₂²×C₄×C₆₇	—	2	2	no	—
107	C₂²×C₄×C₆₇	—	2	2	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	1	no	—
111	C₂²×C₄×C₆₇	—	2	4	no	—
112	C₂²×C₄×C₆₇	—	2	4	no	—
113	C₂²×C₄×C₆₇	—	4	2	no	—
114	C₂²×C₄×C₆₇	—	4	4	no	—
115	C₂²×C₄×C₆₇	—	4	4	no	—
116	C₂²×C₄×C₆₇	—	4	4	no	—
117	C₂²×C₄×C₆₇	—	4	4	no	—
118	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
119	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
120	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
128	C₂×C₈×C₆₇	—	2	4	no	—
129	C₂×C₈×C₆₇	—	2	4	no	—
130	C₂×C₈×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	2	4	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	1	no	—
137	C₄⋊C₄×C₆₇	—	2	4	no	—
138	C₄⋊C₄×C₆₇	—	2	4	no	—
139	C₄⋊C₄×C₆₇	—	4	4	no	—
140	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
141	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
142	C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂²⋊C₄×C₆₇	—	2	4	no	—
144	C₂²⋊C₄×C₆₇	—	2	4	no	—
145	C₂²⋊C₄×C₆₇	—	2	4	no	—
146	C₂²⋊C₄×C₆₇	—	2	4	no	—
147	C₄²×C₆₇	—	2	4	no	—
148	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
149	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
150	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
151	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
152	C₄²×C₆₇	C₂×D₄	1	2	yes	—
153	C₄²×C₆₇	C₂×D₄	1	2	yes	—
154	C₂³×C₆₇	—	2	1	no	—
155	C₂³×C₆₇	—	2	1	no	—
156	C₂³×C₆₇	—	2	2	no	—
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	4	1	no	—
159	C₂³×C₆₇	—	4	1	no	—
160	C₂³×C₆₇	—	4	2	no	—
161	C₂³×C₆₇	—	4	2	no	—
162	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
163	C₂³×C₆₇	#135	1	4	yes	—
164	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₆₇	—	2	4	no	—
166	Q₈×C₆₇	—	4	2	no	—
167	D₄×C₆₇	—	2	4	no	—
168	D₄×C₆₇	—	4	8	no	—
169	D₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	1	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	8	no	—
177	C₂×C₄×C₆₇	—	2	4	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	4	4	no	—
183	C₂×C₄×C₆₇	—	4	8	no	—
184	C₂×C₄×C₆₇	—	4	1	no	—
185	C₂×C₄×C₆₇	—	4	1	no	—
186	C₂×C₄×C₆₇	—	4	2	no	—
187	C₂×C₄×C₆₇	—	4	2	no	—
188	C₂×C₄×C₆₇	—	4	4	no	—
189	C₂×C₄×C₆₇	—	4	4	no	—
190	C₂×C₄×C₆₇	—	4	4	no	—
191	C₂×C₄×C₆₇	—	8	8	no	—
192	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
195	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
196	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
197	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
200	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
201	C₈×C₆₇	—	4	4	no	—
202	C₂²×C₆₇	—	2	2	no	—
203	C₂²×C₆₇	—	2	2	no	—
204	C₂²×C₆₇	—	2	4	no	—
205	C₂²×C₆₇	—	2	4	no	—
206	C₂²×C₆₇	—	4	2	no	—
207	C₂²×C₆₇	—	4	2	no	—
208	C₂²×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	8	2	no	—
210	C₂²×C₆₇	—	8	2	no	—
211	C₂²×C₆₇	#416	1	4	yes	—
212	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
213	C₂²×C₆₇	#535	1	1	yes	characteristic
214	C₂²×C₆₇	#536	1	1	yes	characteristic
215	C₂²×C₆₇	C₂×#135	1	2	yes	—
216	C₂²×C₆₇	#575	1	2	yes	—
217	C₄×C₆₇	—	2	2	no	—
218	C₄×C₆₇	—	2	2	no	—
219	C₄×C₆₇	—	2	2	no	—
220	C₄×C₆₇	—	2	2	no	—
221	C₄×C₆₇	—	2	2	no	—
222	C₄×C₆₇	—	4	2	no	—
223	C₄×C₆₇	—	4	2	no	—
224	C₄×C₆₇	—	4	2	no	—
225	C₄×C₆₇	—	8	4	no	—
226	C₄×C₆₇	#520	1	2	yes	—
227	C₄×C₆₇	#537	1	2	yes	—
228	#21861	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂×C₆₇	—	2	4	no	—
230	C₂×C₆₇	—	8	2	no	—
231	C₂×C₆₇	#2311	1	1	yes	characteristic
232	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
233	C₂×C₆₇	#2896	1	1	yes	characteristic
234	C₂×C₆₇	#3211	1	2	yes	—
235	C₂×C₆₇	#3232	1	2	yes	—
236	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
237	C₂×#431	C₂×C₆₇	1	1	yes	characteristic, complemented
238	C₂×#382	C₂×C₆₇	1	1	yes	characteristic, complemented
239	#1991	C₂×C₆₇	1	2	yes	complemented
240	#1958	C₂×C₆₇	1	2	yes	complemented
241	#1874	C₂×C₆₇	1	2	yes	—
242	#1872	C₂×C₆₇	1	2	yes	complemented
243	#1805	C₂×C₆₇	1	2	yes	complemented
244	#1804	C₂×C₆₇	1	2	yes	complemented
245	C₆₇	#21861	1	1	yes	characteristic, direct product factor, Sylow
246	#529	—	2	4	no	—
247	#529	C₄×C₆₇	1	8	yes	complemented
248	C₂×#126	—	2	1	no	—
249	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
251	C₂×#123	—	2	1	no	—
252	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
253	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
255	C₂×C₄×D₄	—	2	1	no	—
256	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
257	C₂×#117	—	2	1	no	—
258	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
259	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
260	#431	—	2	4	no	—
261	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
262	C₂×#103	C₂²×C₆₇	1	4	yes	—
263	C₂×#102	C₂²×C₆₇	1	4	yes	—
264	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
265	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
266	#382	—	2	1	no	—
267	#382	—	2	1	no	—
268	#382	C₂²×C₆₇	1	2	yes	—
269	#382	C₂²×C₆₇	1	2	yes	—
270	C₂×#95	—	2	1	no	—
271	C₂×#95	—	2	1	no	—
272	#340	C₂²×C₆₇	1	4	yes	—
273	#340	C₂²×C₆₇	1	4	yes	—
274	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
275	C₂²×D₄	—	2	1	no	—
276	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
277	C₂³×C₄	—	2	1	no	—
278	C₂³×C₄	—	2	2	no	—
279	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
280	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
282	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
283	#126	—	2	8	no	—
284	#125	—	2	2	no	—
285	#125	C₂×C₄×C₆₇	1	4	yes	complemented
286	#124	—	2	2	no	—
287	#124	C₂×C₄×C₆₇	1	4	yes	complemented
288	#123	—	2	8	no	—
289	C₂²⋊Q₈	—	2	2	no	—
290	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
291	#121	—	2	2	no	—
292	#121	C₂×C₄×C₆₇	1	4	yes	—
293	C₄×Q₈	—	2	2	no	—
294	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	complemented
295	C₄×D₄	—	2	8	no	—
296	C₄×D₄	—	2	2	no	—
297	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	complemented
298	#117	—	2	8	no	—
299	C₂×C₄⋊C₄	—	2	1	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	4	4	no	—
305	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
306	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
308	C₂×C₂²⋊C₄	—	2	1	no	—
309	C₂×C₂²⋊C₄	—	2	1	no	—
310	C₂×C₂²⋊C₄	—	2	1	no	—
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	1	no	—
313	C₂×C₄²	—	2	1	no	—
314	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
315	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
316	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
317	C₂×C₄²	D₄×C₆₇	1	4	yes	—
318	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
320	C₄⋊C₈	—	2	4	no	—
321	#103	—	4	4	no	—
322	#102	—	4	4	no	—
323	C₂²⋊C₈	—	2	4	no	—
324	C₄×C₈	—	2	4	no	—
325	#95	—	2	2	no	—
326	#95	—	2	2	no	—
327	#95	—	2	2	no	—
328	#95	—	2	2	no	—
329	C₂⁴	—	2	1	no	—
330	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	—	2	2	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	—	2	2	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	4	no	—
340	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	1	no	—
347	C₂²×C₄	—	2	4	no	—
348	C₂²×C₄	—	2	4	no	—
349	C₂²×C₄	—	4	2	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	—	4	4	no	—
352	C₂²×C₄	—	4	4	no	—
353	C₂²×C₄	—	4	4	no	—
354	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
355	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
356	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
357	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
362	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
363	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
364	C₂×C₈	—	2	4	no	—
365	C₂×C₈	—	2	4	no	—
366	C₂×C₈	—	2	4	no	—
367	C₄⋊C₄	—	2	4	no	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	4	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	4	4	no	—
376	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
377	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
378	C₂²⋊C₄	—	2	4	no	—
379	C₂²⋊C₄	—	2	4	no	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	2	4	no	—
383	C₄²	—	2	4	no	—
384	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
385	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
386	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
387	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
388	C₄²	C₂×D₄×C₆₇	1	2	yes	—
389	C₄²	C₂×D₄×C₆₇	1	2	yes	—
390	C₂³	—	2	1	no	—
391	C₂³	—	2	1	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	2	no	—
394	C₂³	—	4	1	no	—
395	C₂³	—	4	1	no	—
396	C₂³	—	4	2	no	—
397	C₂³	—	4	2	no	—
398	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
399	C₂³	#135×C₆₇	1	4	yes	—
400	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
401	Q₈	—	2	4	no	—
402	Q₈	—	4	2	no	—
403	D₄	—	2	4	no	—
404	D₄	—	4	8	no	—
405	D₄	—	4	2	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	1	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	4	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	4	4	no	—
419	C₂×C₄	—	4	8	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	—	8	8	no	—
428	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
429	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
430	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
431	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
432	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
433	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
434	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
435	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
436	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
437	C₈	—	4	4	no	—
438	C₂²	—	2	2	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	2	4	no	—
441	C₂²	—	2	4	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	#416×C₆₇	1	4	yes	—
448	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
449	C₂²	#535×C₆₇	1	1	yes	characteristic
450	C₂²	#536×C₆₇	1	1	yes	characteristic
451	C₂²	C₂×#135×C₆₇	1	2	yes	—
452	C₂²	#575×C₆₇	1	2	yes	—
453	C₄	—	2	2	no	—
454	C₄	—	2	2	no	—
455	C₄	—	2	2	no	—
456	C₄	—	2	2	no	—
457	C₄	—	2	2	no	—
458	C₄	—	4	2	no	—
459	C₄	—	4	2	no	—
460	C₄	—	4	2	no	—
461	C₄	—	8	4	no	—
462	C₄	#520×C₆₇	1	2	yes	—
463	C₄	#537×C₆₇	1	2	yes	—
464	C₂	—	2	4	no	—
465	C₂	—	8	2	no	—
466	C₂	C₆₇×#2311	1	1	yes	characteristic
467	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
468	C₂	C₆₇×#2896	1	1	yes	characteristic
469	C₂	C₆₇×#3211	1	2	yes	—
470	C₂	C₆₇×#3232	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	C₂×#121×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#431×C₆₇	C₂	semidirect
3. C₂×#382×C₆₇	C₂	semidirect
4. C₆₇×#1991	C₂	semidirect
5. C₆₇×#1958	C₂	semidirect
7. C₆₇×#1872	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
9. C₆₇×#1804	C₂	semidirect
11. #529×C₆₇	C₄	semidirect
49. #125×C₆₇	C₂×C₄	semidirect
51. #124×C₆₇	C₂×C₄	semidirect
58. C₄×Q₈×C₆₇	C₂×C₄	semidirect
61. C₄×D₄×C₆₇	C₂×C₄	semidirect
228. #21861	C₆₇	direct (N × H)
237. C₂×#431	C₂×C₆₇	semidirect
238. C₂×#382	C₂×C₆₇	semidirect
239. #1991	C₂×C₆₇	semidirect
240. #1958	C₂×C₆₇	semidirect
242. #1872	C₂×C₆₇	semidirect
243. #1805	C₂×C₆₇	semidirect
244. #1804	C₂×C₆₇	semidirect
245. C₆₇	#21861	direct (N × H)
247. #529	C₄×C₆₇	semidirect
285. #125	C₂×C₄×C₆₇	semidirect
287. #124	C₂×C₄×C₆₇	semidirect
294. C₄×Q₈	C₂×C₄×C₆₇	semidirect
297. C₄×D₄	C₂×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×#529×C₆₇	2
C₂×#431×C₆₇	2
C₂×#382×C₆₇	2
C₆₇×#1991	2
C₆₇×#1958	2
C₆₇×#1874	2
C₆₇×#1872	2
C₆₇×#1805	2
C₆₇×#1804	2
#21861	67

Subgroups — C67×#21861 #30701112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21861 #30701112