Subgroups — C₆₇×#21886 #30701137

Order 17152.

1150 conjugacy classes of subgroups.

The subgroup lattice has 460 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	C₂²×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2283	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1989	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1983	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1893	C₂	1	2	yes	maximal
7	C₆₇×#1892	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	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
11	#529×C₆₇	—	2	4	no	—
12	#529×C₆₇	C₄	1	8	yes	complemented
13	C₂×#126×C₆₇	—	2	1	no	—
14	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×#123×C₆₇	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#121×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	—	2	1	no	—
21	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×#117×C₆₇	—	2	1	no	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	—	2	4	no	—
25	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	4	yes	—
28	C₂×#102×C₆₇	C₂²	1	4	yes	—
29	#383×C₆₇	—	2	1	no	—
30	#382×C₆₇	C₂²	1	2	yes	—
31	#381×C₆₇	—	2	1	no	—
32	#380×C₆₇	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
34	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
35	#376×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	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	—	2	4	no	—
46	C₂²×C₈×C₆₇	—	4	4	no	—
47	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
48	#126×C₆₇	—	2	8	no	—
49	#125×C₆₇	—	2	2	no	—
50	#125×C₆₇	C₂×C₄	1	4	yes	—
51	#124×C₆₇	—	2	2	no	—
52	#124×C₆₇	C₂×C₄	1	4	yes	—
53	#123×C₆₇	—	2	8	no	—
54	C₂²⋊Q₈×C₆₇	—	2	2	no	—
55	C₂²⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
56	#121×C₆₇	—	2	2	no	—
57	#121×C₆₇	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
60	C₄×D₄×C₆₇	—	2	8	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
63	#117×C₆₇	—	2	8	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	1	no	—
68	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
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	2	no	—
77	C₂×C₄²×C₆₇	—	2	1	no	—
78	C₂×C₄²×C₆₇	—	2	2	no	—
79	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
80	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
82	C₂×C₄²×C₆₇	D₄	1	4	yes	—
83	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₆₇	—	2	4	no	—
86	C₄⋊C₈×C₆₇	—	2	4	no	—
87	#103×C₆₇	—	4	4	no	—
88	#102×C₆₇	—	4	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	C₂²⋊C₄	1	4	yes	—
116	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
117	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
124	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
125	C₂×C₈×C₆₇	—	2	4	no	—
126	C₂×C₈×C₆₇	—	2	4	no	—
127	C₂×C₈×C₆₇	—	2	4	no	—
128	C₂×C₈×C₆₇	—	4	8	no	—
129	C₄⋊C₄×C₆₇	—	2	4	no	—
130	C₄⋊C₄×C₆₇	—	2	4	no	—
131	C₄⋊C₄×C₆₇	—	2	4	no	—
132	C₄⋊C₄×C₆₇	—	2	4	no	—
133	C₄⋊C₄×C₆₇	—	2	1	no	—
134	C₄⋊C₄×C₆₇	—	2	1	no	—
135	C₄⋊C₄×C₆₇	—	2	4	no	—
136	C₄⋊C₄×C₆₇	—	2	4	no	—
137	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
139	C₂²⋊C₄×C₆₇	—	2	4	no	—
140	C₂²⋊C₄×C₆₇	—	2	4	no	—
141	C₂²⋊C₄×C₆₇	—	2	4	no	—
142	C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂²⋊C₄×C₆₇	—	2	4	no	—
144	C₂²⋊C₄×C₆₇	—	4	4	no	—
145	C₄²×C₆₇	—	2	4	no	—
146	C₄²×C₆₇	—	4	4	no	—
147	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
148	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
149	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
150	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
151	C₄²×C₆₇	C₂×D₄	1	2	yes	—
152	C₄²×C₆₇	C₂×D₄	1	2	yes	—
153	C₂³×C₆₇	—	2	2	no	—
154	C₂³×C₆₇	—	2	2	no	—
155	C₂³×C₆₇	—	2	2	no	—
156	C₂³×C₆₇	—	4	2	no	—
157	C₂³×C₆₇	—	4	2	no	—
158	C₂³×C₆₇	—	4	2	no	—
159	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
160	C₂³×C₆₇	#135	1	4	yes	—
161	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₆₇	—	2	4	no	—
163	Q₈×C₆₇	—	4	1	no	—
164	Q₈×C₆₇	—	4	1	no	—
165	D₄×C₆₇	—	2	4	no	—
166	D₄×C₆₇	—	4	8	no	—
167	D₄×C₆₇	—	4	1	no	—
168	D₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	2	2	no	—
170	C₂×C₄×C₆₇	—	2	2	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	8	no	—
175	C₂×C₄×C₆₇	—	2	4	no	—
176	C₂×C₄×C₆₇	—	2	2	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	4	4	no	—
182	C₂×C₄×C₆₇	—	4	8	no	—
183	C₂×C₄×C₆₇	—	4	2	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	4	no	—
187	C₂×C₄×C₆₇	—	4	4	no	—
188	C₂×C₄×C₆₇	—	4	4	no	—
189	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
190	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
191	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
192	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
194	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
195	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
197	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
198	C₈×C₆₇	—	4	4	no	—
199	C₂²×C₆₇	—	2	2	no	—
200	C₂²×C₆₇	—	2	2	no	—
201	C₂²×C₆₇	—	2	8	no	—
202	C₂²×C₆₇	—	4	2	no	—
203	C₂²×C₆₇	—	4	2	no	—
204	C₂²×C₆₇	—	4	2	no	—
205	C₂²×C₆₇	—	8	4	no	—
206	C₂²×C₆₇	#416	1	4	yes	—
207	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₂²×C₆₇	#537	1	2	yes	—
209	C₂²×C₆₇	C₂×#135	1	2	yes	—
210	C₂²×C₆₇	#575	1	2	yes	—
211	C₄×C₆₇	—	2	2	no	—
212	C₄×C₆₇	—	2	2	no	—
213	C₄×C₆₇	—	2	2	no	—
214	C₄×C₆₇	—	2	2	no	—
215	C₄×C₆₇	—	2	2	no	—
216	C₄×C₆₇	—	4	2	no	—
217	C₄×C₆₇	—	4	2	no	—
218	C₄×C₆₇	—	4	2	no	—
219	C₄×C₆₇	—	8	4	no	—
220	C₄×C₆₇	#520	1	2	yes	—
221	C₄×C₆₇	#535	1	1	yes	characteristic
222	C₄×C₆₇	#536	1	1	yes	characteristic
223	#21886	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₆₇	—	2	4	no	—
225	C₂×C₆₇	—	8	2	no	—
226	C₂×C₆₇	#2311	1	1	yes	characteristic
227	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
228	C₂×C₆₇	#2896	1	1	yes	characteristic
229	C₂×C₆₇	#3250	1	4	yes	—
230	C₂×#529	C₂×C₆₇	1	1	yes	characteristic
231	C₂²×C₄⋊C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
232	#2283	C₂×C₆₇	1	1	yes	characteristic, complemented
233	#1989	C₂×C₆₇	1	2	yes	complemented
234	#1983	C₂×C₆₇	1	2	yes	complemented
235	#1893	C₂×C₆₇	1	2	yes	—
236	#1892	C₂×C₆₇	1	2	yes	complemented
237	#1805	C₂×C₆₇	1	2	yes	complemented
238	#1804	C₂×C₆₇	1	2	yes	complemented
239	C₆₇	#21886	1	1	yes	characteristic, direct product factor, Sylow
240	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
241	#529	—	2	4	no	—
242	#529	C₄×C₆₇	1	8	yes	complemented
243	C₂×#126	—	2	1	no	—
244	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
245	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
246	C₂×#123	—	2	1	no	—
247	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
248	C₂×#121	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×C₄×D₄	—	2	1	no	—
251	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×#117	—	2	1	no	—
253	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
254	C₂×C₄⋊C₈	—	2	4	no	—
255	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
256	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
257	C₂×#103	C₂²×C₆₇	1	4	yes	—
258	C₂×#102	C₂²×C₆₇	1	4	yes	—
259	#383	—	2	1	no	—
260	#382	C₂²×C₆₇	1	2	yes	—
261	#381	—	2	1	no	—
262	#380	C₂²×C₆₇	1	2	yes	—
263	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
264	C₄×C₂²⋊C₄	—	2	1	no	—
265	#376	—	2	1	no	—
266	#340	C₂²×C₆₇	1	4	yes	—
267	#340	C₂²×C₆₇	1	4	yes	—
268	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
271	C₂³×C₄	—	2	1	no	—
272	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
273	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
274	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
275	C₂²×C₈	—	2	4	no	—
276	C₂²×C₈	—	4	4	no	—
277	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
278	#126	—	2	8	no	—
279	#125	—	2	2	no	—
280	#125	C₂×C₄×C₆₇	1	4	yes	—
281	#124	—	2	2	no	—
282	#124	C₂×C₄×C₆₇	1	4	yes	—
283	#123	—	2	8	no	—
284	C₂²⋊Q₈	—	2	2	no	—
285	C₂²⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
286	#121	—	2	2	no	—
287	#121	C₂×C₄×C₆₇	1	4	yes	—
288	C₄×Q₈	—	2	2	no	—
289	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
290	C₄×D₄	—	2	8	no	—
291	C₄×D₄	—	2	2	no	—
292	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
293	#117	—	2	8	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
299	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
300	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	1	no	—
302	C₂×C₂²⋊C₄	—	2	1	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	—	2	1	no	—
305	C₂×C₂²⋊C₄	—	2	1	no	—
306	C₂×C₂²⋊C₄	—	2	2	no	—
307	C₂×C₄²	—	2	1	no	—
308	C₂×C₄²	—	2	2	no	—
309	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
310	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
311	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
312	C₂×C₄²	D₄×C₆₇	1	4	yes	—
313	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
315	C₄⋊C₈	—	2	4	no	—
316	C₄⋊C₈	—	2	4	no	—
317	#103	—	4	4	no	—
318	#102	—	4	4	no	—
319	#95	—	2	2	no	—
320	#95	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	—	2	2	no	—
323	C₂⁴	—	2	1	no	—
324	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
329	C₂×D₄	—	2	4	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
346	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
347	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
348	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
350	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
351	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
353	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
354	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
355	C₂×C₈	—	2	4	no	—
356	C₂×C₈	—	2	4	no	—
357	C₂×C₈	—	2	4	no	—
358	C₂×C₈	—	4	8	no	—
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	4	no	—
362	C₄⋊C₄	—	2	4	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	2	4	no	—
367	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
368	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
369	C₂²⋊C₄	—	2	4	no	—
370	C₂²⋊C₄	—	2	4	no	—
371	C₂²⋊C₄	—	2	4	no	—
372	C₂²⋊C₄	—	2	4	no	—
373	C₂²⋊C₄	—	2	4	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₄²	—	2	4	no	—
376	C₄²	—	4	4	no	—
377	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
378	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
379	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
380	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
381	C₄²	C₂×D₄×C₆₇	1	2	yes	—
382	C₄²	C₂×D₄×C₆₇	1	2	yes	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
390	C₂³	#135×C₆₇	1	4	yes	—
391	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
392	Q₈	—	2	4	no	—
393	Q₈	—	4	1	no	—
394	Q₈	—	4	1	no	—
395	D₄	—	2	4	no	—
396	D₄	—	4	8	no	—
397	D₄	—	4	1	no	—
398	D₄	—	4	1	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	4	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	4	4	no	—
412	C₂×C₄	—	4	8	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	4	no	—
417	C₂×C₄	—	4	4	no	—
418	C₂×C₄	—	4	4	no	—
419	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
420	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
421	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
422	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
423	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
424	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
425	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
426	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
427	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
428	C₈	—	4	4	no	—
429	C₂²	—	2	2	no	—
430	C₂²	—	2	2	no	—
431	C₂²	—	2	8	no	—
432	C₂²	—	4	2	no	—
433	C₂²	—	4	2	no	—
434	C₂²	—	4	2	no	—
435	C₂²	—	8	4	no	—
436	C₂²	#416×C₆₇	1	4	yes	—
437	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
438	C₂²	#537×C₆₇	1	2	yes	—
439	C₂²	C₂×#135×C₆₇	1	2	yes	—
440	C₂²	#575×C₆₇	1	2	yes	—
441	C₄	—	2	2	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	2	no	—
444	C₄	—	2	2	no	—
445	C₄	—	2	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	4	2	no	—
449	C₄	—	8	4	no	—
450	C₄	#520×C₆₇	1	2	yes	—
451	C₄	#535×C₆₇	1	1	yes	characteristic
452	C₄	#536×C₆₇	1	1	yes	characteristic
453	C₂	—	2	4	no	—
454	C₂	—	8	2	no	—
455	C₂	C₆₇×#2311	1	1	yes	characteristic
456	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
457	C₂	C₆₇×#2896	1	1	yes	characteristic
458	C₂	C₆₇×#3250	1	4	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. C₂²×C₄⋊C₈×C₆₇	C₂	semidirect
3. C₆₇×#2283	C₂	semidirect
4. C₆₇×#1989	C₂	semidirect
5. C₆₇×#1983	C₂	semidirect
7. C₆₇×#1892	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
9. C₆₇×#1804	C₂	semidirect
12. #529×C₆₇	C₄	semidirect
223. #21886	C₆₇	direct (N × H)
231. C₂²×C₄⋊C₈	C₂×C₆₇	semidirect
232. #2283	C₂×C₆₇	semidirect
233. #1989	C₂×C₆₇	semidirect
234. #1983	C₂×C₆₇	semidirect
236. #1892	C₂×C₆₇	semidirect
237. #1805	C₂×C₆₇	semidirect
238. #1804	C₂×C₆₇	semidirect
239. C₆₇	#21886	direct (N × H)
242. #529	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
C₂²×C₄⋊C₈×C₆₇	2
C₆₇×#2283	2
C₆₇×#1989	2
C₆₇×#1983	2
C₆₇×#1893	2
C₆₇×#1892	2
C₆₇×#1805	2
C₆₇×#1804	2
#21886	67

Subgroups — C67×#21886 #30701137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21886 #30701137