Subgroups — C₆₇×#17390 #30696641

Order 17152.

940 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#542×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2983	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2982	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1473	C₂	1	4	yes	maximal, complemented
5	C₆₇×#1469	C₂	1	4	yes	maximal, complemented
6	C₆₇×#1460	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1458	C₂	1	2	yes	maximal, complemented
8	#542×C₆₇	—	2	4	no	—
9	#542×C₆₇	—	2	4	no	—
10	C₂×C₄⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
11	C₂×#126×C₆₇	C₂²	1	2	yes	—
12	C₂×#124×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#124×C₆₇	C₂²	1	2	yes	—
14	C₂×#123×C₆₇	C₄	1	2	yes	—
15	C₂×C₂²⋊Q₈×C₆₇	C₄	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
17	C₂×#121×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
21	#446×C₆₇	—	2	4	no	—
22	C₈×Q₈×C₆₇	—	2	4	no	—
23	#432×C₆₇	C₂²	1	4	yes	—
24	#431×C₆₇	C₂²	1	4	yes	—
25	#431×C₆₇	C₂²	1	2	yes	—
26	#431×C₆₇	C₂²	1	2	yes	—
27	#324×C₆₇	C₂²	1	4	yes	—
28	#323×C₆₇	C₂²	1	8	yes	—
29	#323×C₆₇	C₂²	1	4	yes	—
30	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
32	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×D₄×C₆₇	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
36	C₄⋊Q₈×C₆₇	—	2	4	no	—
37	#126×C₆₇	—	2	4	no	—
38	#126×C₆₇	—	2	4	no	—
39	#124×C₆₇	—	2	4	no	—
40	#124×C₆₇	—	2	4	no	—
41	#124×C₆₇	—	2	4	no	—
42	#123×C₆₇	—	2	8	no	—
43	C₂²⋊Q₈×C₆₇	—	2	4	no	—
44	C₂²⋊Q₈×C₆₇	—	2	8	no	—
45	#121×C₆₇	—	2	4	no	—
46	C₄×Q₈×C₆₇	D₄	1	4	yes	—
47	C₄×Q₈×C₆₇	D₄	1	4	yes	—
48	C₄×D₄×C₆₇	—	2	2	no	—
49	C₄×D₄×C₆₇	—	2	2	no	—
50	#117×C₆₇	—	2	2	no	—
51	#117×C₆₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
69	C₄⋊C₈×C₆₇	—	2	2	no	—
70	C₄⋊C₈×C₆₇	—	2	4	no	—
71	C₄⋊C₈×C₆₇	—	2	2	no	—
72	C₄⋊C₈×C₆₇	—	2	4	no	—
73	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
74	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
75	#97×C₆₇	—	2	4	no	—
76	C₄×C₈×C₆₇	—	2	2	no	—
77	C₄×C₈×C₆₇	—	2	4	no	—
78	C₄×C₈×C₆₇	—	2	2	no	—
79	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
80	C₂⁴×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂×Q₈×C₆₇	—	2	4	no	—
82	C₂×Q₈×C₆₇	—	2	4	no	—
83	C₂×Q₈×C₆₇	—	8	2	no	—
84	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
85	C₂×D₄×C₆₇	—	2	2	no	—
86	C₂×D₄×C₆₇	—	2	4	no	—
87	C₂×D₄×C₆₇	—	2	4	no	—
88	C₂×D₄×C₆₇	—	8	2	no	—
89	C₂²×C₄×C₆₇	—	2	2	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	8	2	no	—
93	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₆₇	—	2	4	no	—
103	C₂×C₈×C₆₇	—	2	4	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	2	4	no	—
108	C₄⋊C₄×C₆₇	—	2	1	no	—
109	C₄⋊C₄×C₆₇	—	2	1	no	—
110	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
111	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
112	C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂²⋊C₄×C₆₇	—	2	4	no	—
114	C₂²⋊C₄×C₆₇	—	2	4	no	—
115	C₂²⋊C₄×C₆₇	—	2	4	no	—
116	C₂²⋊C₄×C₆₇	—	2	4	no	—
117	C₂²⋊C₄×C₆₇	—	2	1	no	—
118	C₂²⋊C₄×C₆₇	—	2	1	no	—
119	C₂²⋊C₄×C₆₇	—	2	2	no	—
120	C₂²⋊C₄×C₆₇	—	2	2	no	—
121	C₄²×C₆₇	—	2	2	no	—
122	C₄²×C₆₇	C₂×D₄	1	2	yes	—
123	C₄²×C₆₇	C₂×D₄	1	2	yes	—
124	C₄²×C₆₇	C₂×D₄	1	4	yes	—
125	C₄²×C₆₇	C₂×D₄	1	4	yes	—
126	C₂³×C₆₇	—	2	1	no	—
127	C₂³×C₆₇	—	2	2	no	—
128	C₂³×C₆₇	—	2	2	no	—
129	C₂³×C₆₇	—	2	4	no	—
130	C₂³×C₆₇	—	4	2	no	—
131	C₂³×C₆₇	—	4	1	no	—
132	C₂³×C₆₇	—	4	1	no	—
133	C₂³×C₆₇	—	4	2	no	—
134	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
135	Q₈×C₆₇	—	2	8	no	—
136	Q₈×C₆₇	—	8	4	no	—
137	D₄×C₆₇	—	4	4	no	—
138	D₄×C₆₇	—	8	4	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	1	no	—
142	C₂×C₄×C₆₇	—	2	1	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	4	1	no	—
146	C₂×C₄×C₆₇	—	4	2	no	—
147	C₂×C₄×C₆₇	—	4	1	no	—
148	C₂×C₄×C₆₇	—	4	4	no	—
149	C₂×C₄×C₆₇	—	4	2	no	—
150	C₂×C₄×C₆₇	—	4	1	no	—
151	C₂×C₄×C₆₇	—	4	1	no	—
152	C₂×C₄×C₆₇	—	8	4	no	—
153	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
154	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
156	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
157	C₂×C₄×C₆₇	#131	1	4	yes	—
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₆₇	—	4	4	no	—
161	C₈×C₆₇	—	4	4	no	—
162	C₂²×C₆₇	—	2	1	no	—
163	C₂²×C₆₇	—	2	1	no	—
164	C₂²×C₆₇	—	4	1	no	—
165	C₂²×C₆₇	—	4	2	no	—
166	C₂²×C₆₇	—	4	1	no	—
167	C₂²×C₆₇	—	4	2	no	—
168	C₂²×C₆₇	—	4	2	no	—
169	C₂²×C₆₇	—	4	1	no	—
170	C₂²×C₆₇	—	8	2	no	—
171	C₂²×C₆₇	—	8	4	no	—
172	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
173	C₂²×C₆₇	C₂×#131	1	2	yes	—
174	C₄×C₆₇	—	2	4	no	—
175	C₄×C₆₇	—	2	1	no	—
176	C₄×C₆₇	—	2	1	no	—
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	4	no	—
179	C₄×C₆₇	—	4	1	no	—
180	C₄×C₆₇	—	4	1	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	—	4	1	no	—
183	C₄×C₆₇	#408	1	4	yes	—
184	#17390	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
185	C₂×C₆₇	—	2	2	no	—
186	C₂×C₆₇	—	8	1	no	—
187	C₂×C₆₇	—	8	2	no	—
188	C₂×C₆₇	C₂×#408	1	1	yes	characteristic
189	C₂×C₆₇	#2884	1	1	yes	characteristic
190	C₂×C₆₇	#2888	1	1	yes	characteristic
191	C₂×#542	C₂×C₆₇	1	1	yes	characteristic
192	#2983	C₂×C₆₇	1	1	yes	characteristic, complemented
193	#2982	C₂×C₆₇	1	1	yes	characteristic, complemented
194	#1473	C₂×C₆₇	1	4	yes	complemented
195	#1469	C₂×C₆₇	1	4	yes	complemented
196	#1460	C₂×C₆₇	1	2	yes	complemented
197	#1458	C₂×C₆₇	1	2	yes	complemented
198	C₆₇	#17390	1	1	yes	characteristic, direct product factor, Sylow
199	#542	—	2	4	no	—
200	#542	—	2	4	no	—
201	C₂×C₄⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
202	C₂×#126	C₂²×C₆₇	1	2	yes	—
203	C₂×#124	C₄×C₆₇	1	1	yes	characteristic
204	C₂×#124	C₂²×C₆₇	1	2	yes	—
205	C₂×#123	C₄×C₆₇	1	2	yes	—
206	C₂×C₂²⋊Q₈	C₄×C₆₇	1	2	yes	—
207	C₂×C₂²⋊Q₈	C₄×C₆₇	1	1	yes	characteristic
208	C₂×#121	C₄×C₆₇	1	1	yes	characteristic
209	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
210	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
211	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
212	#446	—	2	4	no	—
213	C₈×Q₈	—	2	4	no	—
214	#432	C₂²×C₆₇	1	4	yes	—
215	#431	C₂²×C₆₇	1	2	yes	—
216	#431	C₂²×C₆₇	1	4	yes	—
217	#431	C₂²×C₆₇	1	2	yes	—
218	#324	C₂²×C₆₇	1	4	yes	—
219	#323	C₂²×C₆₇	1	4	yes	—
220	#323	C₂²×C₆₇	1	8	yes	—
221	C₂²×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
222	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
223	C₂²×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
224	C₂²×D₄	C₂×C₄×C₆₇	1	2	yes	—
225	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
226	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
227	C₄⋊Q₈	—	2	4	no	—
228	#126	—	2	4	no	—
229	#126	—	2	4	no	—
230	#124	—	2	4	no	—
231	#124	—	2	4	no	—
232	#124	—	2	4	no	—
233	#123	—	2	8	no	—
234	C₂²⋊Q₈	—	2	4	no	—
235	C₂²⋊Q₈	—	2	8	no	—
236	#121	—	2	4	no	—
237	C₄×Q₈	D₄×C₆₇	1	4	yes	—
238	C₄×Q₈	D₄×C₆₇	1	4	yes	—
239	C₄×D₄	—	2	2	no	—
240	C₄×D₄	—	2	2	no	—
241	#117	—	2	2	no	—
242	#117	—	2	2	no	—
243	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
244	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
246	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
247	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
248	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
249	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
250	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
251	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
252	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
253	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
254	C₂×C₂²⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
255	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
256	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
257	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
258	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
259	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	4	no	—
262	C₄⋊C₈	—	2	2	no	—
263	C₄⋊C₈	—	2	4	no	—
264	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
265	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
266	#97	—	2	4	no	—
267	C₄×C₈	—	2	2	no	—
268	C₄×C₈	—	2	4	no	—
269	C₄×C₈	—	2	2	no	—
270	C₂⁴	C₂²×C₄×C₆₇	1	1	yes	characteristic
271	C₂⁴	C₂²×C₄×C₆₇	1	2	yes	—
272	C₂×Q₈	—	2	4	no	—
273	C₂×Q₈	—	2	4	no	—
274	C₂×Q₈	—	8	2	no	—
275	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	4	no	—
278	C₂×D₄	—	2	4	no	—
279	C₂×D₄	—	8	2	no	—
280	C₂²×C₄	—	2	2	no	—
281	C₂²×C₄	—	2	4	no	—
282	C₂²×C₄	—	2	4	no	—
283	C₂²×C₄	—	8	2	no	—
284	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
286	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
289	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
291	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
292	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
293	C₂×C₈	—	2	4	no	—
294	C₂×C₈	—	2	4	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
302	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	4	no	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₂²⋊C₄	—	2	4	no	—
307	C₂²⋊C₄	—	2	4	no	—
308	C₂²⋊C₄	—	2	1	no	—
309	C₂²⋊C₄	—	2	1	no	—
310	C₂²⋊C₄	—	2	2	no	—
311	C₂²⋊C₄	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	C₂×D₄×C₆₇	1	2	yes	—
314	C₄²	C₂×D₄×C₆₇	1	2	yes	—
315	C₄²	C₂×D₄×C₆₇	1	4	yes	—
316	C₄²	C₂×D₄×C₆₇	1	4	yes	—
317	C₂³	—	2	1	no	—
318	C₂³	—	2	2	no	—
319	C₂³	—	2	2	no	—
320	C₂³	—	2	4	no	—
321	C₂³	—	4	2	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	4	1	no	—
324	C₂³	—	4	2	no	—
325	C₂³	C₂³×C₄×C₆₇	1	1	yes	characteristic
326	Q₈	—	2	8	no	—
327	Q₈	—	8	4	no	—
328	D₄	—	4	4	no	—
329	D₄	—	8	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	8	4	no	—
344	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
345	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
346	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
347	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
348	C₂×C₄	#131×C₆₇	1	4	yes	—
349	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
350	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
351	C₈	—	4	4	no	—
352	C₈	—	4	4	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	4	1	no	—
356	C₂²	—	4	2	no	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	2	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	8	4	no	—
363	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
364	C₂²	C₂×#131×C₆₇	1	2	yes	—
365	C₄	—	2	4	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	4	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	2	no	—
373	C₄	—	4	1	no	—
374	C₄	#408×C₆₇	1	4	yes	—
375	C₂	—	2	2	no	—
376	C₂	—	8	1	no	—
377	C₂	—	8	2	no	—
378	C₂	C₂×#408×C₆₇	1	1	yes	characteristic
379	C₂	C₆₇×#2884	1	1	yes	characteristic
380	C₂	C₆₇×#2888	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#121×C₆₇	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#2983	C₂	semidirect
3. C₆₇×#2982	C₂	semidirect
4. C₆₇×#1473	C₂	semidirect
5. C₆₇×#1469	C₂	semidirect
6. C₆₇×#1460	C₂	semidirect
7. C₆₇×#1458	C₂	semidirect
184. #17390	C₆₇	direct (N × H)
192. #2983	C₂×C₆₇	semidirect
193. #2982	C₂×C₆₇	semidirect
194. #1473	C₂×C₆₇	semidirect
195. #1469	C₂×C₆₇	semidirect
196. #1460	C₂×C₆₇	semidirect
197. #1458	C₂×C₆₇	semidirect
198. C₆₇	#17390	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#542×C₆₇	2
C₆₇×#2983	2
C₆₇×#2982	2
C₆₇×#1473	2
C₆₇×#1469	2
C₆₇×#1460	2
C₆₇×#1458	2
#17390	67

Subgroups — C67×#17390 #30696641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#17390 #30696641