Subgroups — C₆₇×#21864 #30701115

Order 17152.

1106 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2285	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1990	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1956	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1873	C₂	1	4	yes	maximal
7	C₆₇×#1805	C₂	1	4	yes	maximal, complemented
8	#533×C₆₇	—	2	4	no	—
9	#533×C₆₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂×#125×C₆₇	—	2	1	no	—
13	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₂×#117×C₆₇	—	2	1	no	—
18	C₂×#117×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#431×C₆₇	—	2	4	no	—
21	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
22	C₂×#103×C₆₇	C₂²	1	8	yes	—
23	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
25	#384×C₆₇	C₂²	1	2	yes	—
26	#382×C₆₇	C₂²	1	2	yes	—
27	#380×C₆₇	—	2	1	no	—
28	#380×C₆₇	—	2	1	no	—
29	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
30	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
31	#340×C₆₇	C₂²	1	8	yes	—
32	C₂²×Q₈×C₆₇	—	2	1	no	—
33	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
34	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₆₇	—	2	2	no	—
39	C₄⋊Q₈×C₆₇	—	2	2	no	—
40	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
41	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
42	#125×C₆₇	—	2	8	no	—
43	C₂²⋊Q₈×C₆₇	—	2	8	no	—
44	C₂²⋊Q₈×C₆₇	—	2	4	no	—
45	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
46	C₄×Q₈×C₆₇	—	2	8	no	—
47	C₄×Q₈×C₆₇	—	2	4	no	—
48	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
49	#117×C₆₇	—	2	8	no	—
50	#117×C₆₇	—	4	4	no	—
51	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₄²×C₆₇	—	2	2	no	—
63	C₂×C₄²×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
65	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	D₄	1	4	yes	—
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₆₇	—	2	4	no	—
71	#103×C₆₇	—	4	8	no	—
72	C₂²⋊C₈×C₆₇	—	2	4	no	—
73	C₄×C₈×C₆₇	—	2	4	no	—
74	#95×C₆₇	—	2	2	no	—
75	#95×C₆₇	—	2	2	no	—
76	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₂×Q₈×C₆₇	—	2	4	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	—	2	4	no	—
80	C₂×Q₈×C₆₇	—	2	4	no	—
81	C₂×Q₈×C₆₇	—	4	4	no	—
82	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
83	C₂²×C₄×C₆₇	—	2	4	no	—
84	C₂²×C₄×C₆₇	—	2	4	no	—
85	C₂²×C₄×C₆₇	—	2	2	no	—
86	C₂²×C₄×C₆₇	—	2	2	no	—
87	C₂²×C₄×C₆₇	—	2	2	no	—
88	C₂²×C₄×C₆₇	—	2	2	no	—
89	C₂²×C₄×C₆₇	—	2	4	no	—
90	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
91	C₂²×C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
92	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
100	C₂×C₈×C₆₇	—	2	4	no	—
101	C₂×C₈×C₆₇	—	2	4	no	—
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₄⋊C₄×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	2	4	no	—
105	C₄⋊C₄×C₆₇	—	2	4	no	—
106	C₄⋊C₄×C₆₇	—	2	8	no	—
107	C₄⋊C₄×C₆₇	—	2	8	no	—
108	C₄⋊C₄×C₆₇	—	2	2	no	—
109	C₄⋊C₄×C₆₇	—	2	8	no	—
110	C₄⋊C₄×C₆₇	—	4	4	no	—
111	C₄⋊C₄×C₆₇	—	4	4	no	—
112	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
113	C₂²⋊C₄×C₆₇	—	2	8	no	—
114	C₂²⋊C₄×C₆₇	—	4	4	no	—
115	C₄²×C₆₇	—	2	8	no	—
116	C₄²×C₆₇	—	4	4	no	—
117	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
118	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
119	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
120	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₄²×C₆₇	C₂×D₄	1	2	yes	—
123	C₂³×C₆₇	—	2	1	no	—
124	C₂³×C₆₇	—	2	1	no	—
125	C₂³×C₆₇	—	2	2	no	—
126	C₂³×C₆₇	—	2	2	no	—
127	C₂³×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₆₇	—	2	8	no	—
130	Q₈×C₆₇	—	4	4	no	—
131	Q₈×C₆₇	—	4	8	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	—	2	2	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	4	no	—
138	C₂×C₄×C₆₇	—	2	8	no	—
139	C₂×C₄×C₆₇	—	2	8	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	4	2	no	—
143	C₂×C₄×C₆₇	—	4	2	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	4	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	—	4	4	no	—
148	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
149	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
150	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
151	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
153	C₂×C₄×C₆₇	#135	1	4	yes	—
154	C₂×C₄×C₆₇	#137	1	4	yes	—
155	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
156	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
158	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
159	C₈×C₆₇	—	4	4	no	—
160	C₂²×C₆₇	—	2	2	no	—
161	C₂²×C₆₇	—	2	2	no	—
162	C₂²×C₆₇	—	2	2	no	—
163	C₂²×C₆₇	—	4	2	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
166	C₂²×C₆₇	#535	1	1	yes	characteristic
167	C₂²×C₆₇	#536	1	1	yes	characteristic
168	C₂²×C₆₇	C₂×#135	1	2	yes	—
169	C₂²×C₆₇	C₂×#137	1	2	yes	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	2	2	no	—
173	C₄×C₆₇	—	4	4	no	—
174	C₄×C₆₇	—	4	4	no	—
175	C₄×C₆₇	—	8	4	no	—
176	C₄×C₆₇	#416	1	4	yes	—
177	C₄×C₆₇	#419	1	4	yes	—
178	C₄×C₆₇	#520	1	2	yes	—
179	C₄×C₆₇	#537	1	2	yes	—
180	#21864	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₆₇	—	4	2	no	—
182	C₂×C₆₇	#2311	1	1	yes	characteristic
183	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
184	C₂×C₆₇	C₂×#419	1	1	yes	characteristic
185	C₂×C₆₇	#3210	1	2	yes	—
186	C₂×C₆₇	#3231	1	2	yes	—
187	C₂×#533	C₂×C₆₇	1	1	yes	characteristic
188	C₂×#431	C₂×C₆₇	1	1	yes	characteristic
189	#2285	C₂×C₆₇	1	1	yes	characteristic
190	#1990	C₂×C₆₇	1	2	yes	complemented
191	#1956	C₂×C₆₇	1	2	yes	complemented
192	#1873	C₂×C₆₇	1	4	yes	—
193	#1805	C₂×C₆₇	1	4	yes	complemented
194	C₆₇	#21864	1	1	yes	characteristic, direct product factor, Sylow
195	#533	—	2	4	no	—
196	#533	C₄×C₆₇	1	8	yes	complemented
197	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
198	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
199	C₂×#125	—	2	1	no	—
200	C₂×C₂²⋊Q₈	—	2	1	no	—
201	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
202	C₂×C₄×Q₈	—	2	1	no	—
203	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
204	C₂×#117	—	2	1	no	—
205	C₂×#117	C₂²×C₆₇	1	2	yes	—
206	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
207	#431	—	2	4	no	—
208	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
209	C₂×#103	C₂²×C₆₇	1	8	yes	—
210	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
211	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
212	#384	C₂²×C₆₇	1	2	yes	—
213	#382	C₂²×C₆₇	1	2	yes	—
214	#380	—	2	1	no	—
215	#380	—	2	1	no	—
216	C₄×C₄⋊C₄	—	2	1	no	—
217	C₄×C₄⋊C₄	—	2	1	no	—
218	#340	C₂²×C₆₇	1	8	yes	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
221	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
222	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
223	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
224	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
225	C₄⋊Q₈	—	2	2	no	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
228	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
229	#125	—	2	8	no	—
230	C₂²⋊Q₈	—	2	8	no	—
231	C₂²⋊Q₈	—	2	4	no	—
232	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
233	C₄×Q₈	—	2	8	no	—
234	C₄×Q₈	—	2	4	no	—
235	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	—
236	#117	—	2	8	no	—
237	#117	—	4	4	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	—	2	1	no	—
241	C₂×C₄⋊C₄	—	2	2	no	—
242	C₂×C₄⋊C₄	—	2	2	no	—
243	C₂×C₄⋊C₄	—	2	2	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
246	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
247	C₂×C₂²⋊C₄	—	2	2	no	—
248	C₂×C₂²⋊C₄	—	2	2	no	—
249	C₂×C₄²	—	2	2	no	—
250	C₂×C₄²	—	2	2	no	—
251	C₂×C₄²	C₂×C₄×C₆₇	1	4	yes	—
252	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
253	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
254	C₂×C₄²	D₄×C₆₇	1	4	yes	—
255	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
256	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
257	C₄⋊C₈	—	2	4	no	—
258	#103	—	4	8	no	—
259	C₂²⋊C₈	—	2	4	no	—
260	C₄×C₈	—	2	4	no	—
261	#95	—	2	2	no	—
262	#95	—	2	2	no	—
263	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
264	C₂×Q₈	—	2	4	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	4	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	4	4	no	—
269	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
270	C₂²×C₄	—	2	4	no	—
271	C₂²×C₄	—	2	4	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	2	2	no	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	4	no	—
277	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
278	C₂²×C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
279	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
280	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
281	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
284	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
286	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
287	C₂×C₈	—	2	4	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	2	4	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₄⋊C₄	—	2	4	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	8	no	—
294	C₄⋊C₄	—	2	8	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	8	no	—
297	C₄⋊C₄	—	4	4	no	—
298	C₄⋊C₄	—	4	4	no	—
299	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
300	C₂²⋊C₄	—	2	8	no	—
301	C₂²⋊C₄	—	4	4	no	—
302	C₄²	—	2	8	no	—
303	C₄²	—	4	4	no	—
304	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
305	C₄²	C₂²⋊C₄×C₆₇	1	4	yes	—
306	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
307	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
308	C₄²	C₂×D₄×C₆₇	1	2	yes	—
309	C₄²	C₂×D₄×C₆₇	1	2	yes	—
310	C₂³	—	2	1	no	—
311	C₂³	—	2	1	no	—
312	C₂³	—	2	2	no	—
313	C₂³	—	2	2	no	—
314	C₂³	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
315	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
316	Q₈	—	2	8	no	—
317	Q₈	—	4	4	no	—
318	Q₈	—	4	8	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	4	no	—
325	C₂×C₄	—	2	8	no	—
326	C₂×C₄	—	2	8	no	—
327	C₂×C₄	—	2	4	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	4	no	—
332	C₂×C₄	—	4	4	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	4	4	no	—
335	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
336	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
337	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
338	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
339	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
340	C₂×C₄	#135×C₆₇	1	4	yes	—
341	C₂×C₄	#137×C₆₇	1	4	yes	—
342	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
343	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
344	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
345	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
346	C₈	—	4	4	no	—
347	C₂²	—	2	2	no	—
348	C₂²	—	2	2	no	—
349	C₂²	—	2	2	no	—
350	C₂²	—	4	2	no	—
351	C₂²	—	4	2	no	—
352	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
353	C₂²	#535×C₆₇	1	1	yes	characteristic
354	C₂²	#536×C₆₇	1	1	yes	characteristic
355	C₂²	C₂×#135×C₆₇	1	2	yes	—
356	C₂²	C₂×#137×C₆₇	1	2	yes	—
357	C₄	—	2	2	no	—
358	C₄	—	2	2	no	—
359	C₄	—	2	2	no	—
360	C₄	—	4	4	no	—
361	C₄	—	4	4	no	—
362	C₄	—	8	4	no	—
363	C₄	#416×C₆₇	1	4	yes	—
364	C₄	#419×C₆₇	1	4	yes	—
365	C₄	#520×C₆₇	1	2	yes	—
366	C₄	#537×C₆₇	1	2	yes	—
367	C₂	—	4	2	no	—
368	C₂	C₆₇×#2311	1	1	yes	characteristic
369	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
370	C₂	C₂×#419×C₆₇	1	1	yes	characteristic
371	C₂	C₆₇×#3210	1	2	yes	—
372	C₂	C₆₇×#3231	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	C₂⁴×C₆₇	C₂×D₄
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
4. C₆₇×#1990	C₂	semidirect
5. C₆₇×#1956	C₂	semidirect
7. C₆₇×#1805	C₂	semidirect
9. #533×C₆₇	C₄	semidirect
180. #21864	C₆₇	direct (N × H)
190. #1990	C₂×C₆₇	semidirect
191. #1956	C₂×C₆₇	semidirect
193. #1805	C₂×C₆₇	semidirect
194. C₆₇	#21864	direct (N × H)
196. #533	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×#533×C₆₇	2
C₂×#431×C₆₇	2
C₆₇×#2285	2
C₆₇×#1990	2
C₆₇×#1956	2
C₆₇×#1873	2
C₆₇×#1805	2
#21864	67

Subgroups — C67×#21864 #30701115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#21864 #30701115