Subgroups — C₆₇×#26061 #30705312

Order 17152.

794 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#427×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2439	C₂	1	2	yes	maximal
4	C₆₇×#2085	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2057	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1941	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1941	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1854	C₂	1	2	yes	maximal
9	C₆₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1807	C₂	1	1	yes	maximal, characteristic
11	C₈×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×#117×C₆₇	C₂²	1	2	yes	—
15	#498×C₆₇	—	2	4	no	—
16	#427×C₆₇	—	2	4	no	—
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	C₂×#107×C₆₇	C₂²	1	2	yes	—
19	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#106×C₆₇	C₂²	1	2	yes	—
22	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#396×C₆₇	—	2	2	no	—
28	#393×C₆₇	—	2	2	no	—
29	#391×C₆₇	—	2	2	no	—
30	#389×C₆₇	C₂²	1	2	yes	—
31	#389×C₆₇	C₂²	1	2	yes	—
32	#387×C₆₇	—	2	2	no	—
33	#382×C₆₇	C₂²	1	2	yes	—
34	#380×C₆₇	C₂²	1	2	yes	—
35	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#336×C₆₇	C₂²	1	1	yes	characteristic
40	#336×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₆₇	—	2	2	no	—
42	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	—	2	2	no	—
45	C₂²×C₈×C₆₇	Q₈	1	4	yes	complemented
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	—	2	4	no	—
51	C₄⋊Q₈×C₆₇	—	4	2	no	—
52	#117×C₆₇	—	2	4	no	—
53	#117×C₆₇	—	4	2	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	C₂³	1	2	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
67	#107×C₆₇	—	2	4	no	—
68	#107×C₆₇	—	4	2	no	—
69	#106×C₆₇	—	2	2	no	—
70	#106×C₆₇	—	2	2	no	—
71	#106×C₆₇	—	2	2	no	—
72	#106×C₆₇	—	2	4	no	—
73	#106×C₆₇	D₄	1	4	yes	—
74	C₂²⋊C₈×C₆₇	Q₈	1	4	yes	complemented
75	C₄×C₈×C₆₇	—	2	2	no	—
76	C₄×C₈×C₆₇	—	2	2	no	—
77	#95×C₆₇	—	2	2	no	—
78	#95×C₆₇	—	2	2	no	—
79	#95×C₆₇	—	2	2	no	—
80	#95×C₆₇	—	2	2	no	—
81	#95×C₆₇	—	2	2	no	—
82	#95×C₆₇	D₄	1	2	yes	—
83	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂×Q₈×C₆₇	—	4	4	no	—
85	C₂×Q₈×C₆₇	—	4	4	no	—
86	C₂×Q₈×C₆₇	—	8	2	no	—
87	C₂²×C₄×C₆₇	—	2	1	no	—
88	C₂²×C₄×C₆₇	—	2	1	no	—
89	C₂²×C₄×C₆₇	—	2	1	no	—
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	—	2	2	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
100	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₈×C₆₇	—	2	1	no	—
102	C₂×C₈×C₆₇	—	2	1	no	—
103	C₂×C₈×C₆₇	—	2	1	no	—
104	C₂×C₈×C₆₇	—	2	1	no	—
105	C₂×C₈×C₆₇	—	2	1	no	—
106	C₂×C₈×C₆₇	—	2	1	no	—
107	C₂×C₈×C₆₇	—	2	8	no	—
108	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
110	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
111	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₆₇	#41	1	2	yes	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	2	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	2	no	—
118	C₄⋊C₄×C₆₇	—	2	2	no	—
119	C₄⋊C₄×C₆₇	—	2	2	no	—
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	4	4	no	—
122	C₄⋊C₄×C₆₇	—	4	4	no	—
123	C₄⋊C₄×C₆₇	—	8	2	no	—
124	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
125	C₂²⋊C₄×C₆₇	—	2	4	no	—
126	C₂²⋊C₄×C₆₇	—	4	4	no	—
127	C₂²⋊C₄×C₆₇	—	8	2	no	—
128	C₄²×C₆₇	—	2	1	no	—
129	C₄²×C₆₇	—	2	1	no	—
130	C₄²×C₆₇	—	2	1	no	—
131	C₄²×C₆₇	—	2	1	no	—
132	C₄²×C₆₇	—	4	4	no	—
133	C₂³×C₆₇	—	2	1	no	—
134	C₂³×C₆₇	—	2	1	no	—
135	C₂³×C₆₇	—	2	2	no	—
136	C₂³×C₆₇	—	2	2	no	—
137	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
138	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	Q₈×C₆₇	—	8	8	no	—
140	C₂×C₄×C₆₇	—	2	2	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	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	2	no	—
147	C₂×C₄×C₆₇	—	2	2	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	4	2	no	—
153	C₂×C₄×C₆₇	—	4	2	no	—
154	C₂×C₄×C₆₇	—	4	2	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	4	2	no	—
160	C₂×C₄×C₆₇	—	4	2	no	—
161	C₂×C₄×C₆₇	—	4	2	no	—
162	C₂×C₄×C₆₇	—	4	2	no	—
163	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
164	C₂×C₄×C₆₇	#135	1	4	yes	—
165	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
168	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
171	C₈×C₆₇	—	2	2	no	—
172	C₈×C₆₇	—	2	2	no	—
173	C₈×C₆₇	—	2	2	no	—
174	C₈×C₆₇	C₂²⋊Q₈	1	4	yes	—
175	C₂²×C₆₇	—	2	2	no	—
176	C₂²×C₆₇	—	2	2	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	4	2	no	—
179	C₂²×C₆₇	—	4	2	no	—
180	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₂²×C₆₇	D₄²	1	1	yes	characteristic
182	C₂²×C₆₇	#547	1	1	yes	characteristic
183	C₂²×C₆₇	C₂×#135	1	2	yes	—
184	C₂²×C₆₇	#577	1	2	yes	—
185	C₄×C₆₇	—	2	2	no	—
186	C₄×C₆₇	—	2	4	no	—
187	C₄×C₆₇	—	4	2	no	—
188	C₄×C₆₇	—	4	4	no	—
189	C₄×C₆₇	—	8	2	no	—
190	C₄×C₆₇	—	8	2	no	—
191	C₄×C₆₇	—	8	2	no	—
192	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
193	C₄×C₆₇	#544	1	1	yes	characteristic
194	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
195	C₄×C₆₇	#554	1	1	yes	characteristic
196	#26061	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂×C₆₇	—	4	2	no	—
198	C₂×C₆₇	#2440	1	1	yes	characteristic
199	C₂×C₆₇	#3160	1	1	yes	characteristic
200	C₂×C₆₇	#3163	1	1	yes	characteristic
201	C₂×C₆₇	#3282	1	2	yes	—
202	C₂×C₆₇	#3340	1	2	yes	—
203	C₂×#498	C₂×C₆₇	1	1	yes	characteristic, complemented
204	C₂×#427	C₂×C₆₇	1	1	yes	characteristic
205	#2439	C₂×C₆₇	1	2	yes	—
206	#2085	C₂×C₆₇	1	2	yes	complemented
207	#2057	C₂×C₆₇	1	2	yes	complemented
208	#1941	C₂×C₆₇	1	1	yes	characteristic, complemented
209	#1941	C₂×C₆₇	1	1	yes	characteristic, complemented
210	#1854	C₂×C₆₇	1	2	yes	—
211	#1846	C₂×C₆₇	1	1	yes	characteristic, complemented
212	#1807	C₂×C₆₇	1	1	yes	characteristic
213	C₈×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic
214	C₆₇	#26061	1	1	yes	characteristic, direct product factor, Sylow
215	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
216	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
217	C₂×#117	C₂²×C₆₇	1	2	yes	—
218	#498	—	2	4	no	—
219	#427	—	2	4	no	—
220	C₂×#107	C₂²×C₆₇	1	2	yes	—
221	C₂×#107	C₂²×C₆₇	1	2	yes	—
222	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
223	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
224	C₂×#106	C₂²×C₆₇	1	2	yes	—
225	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
226	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
227	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
229	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
230	#396	—	2	2	no	—
231	#393	—	2	2	no	—
232	#391	—	2	2	no	—
233	#389	C₂²×C₆₇	1	2	yes	—
234	#389	C₂²×C₆₇	1	2	yes	—
235	#387	—	2	2	no	—
236	#382	C₂²×C₆₇	1	2	yes	—
237	#380	C₂²×C₆₇	1	2	yes	—
238	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
239	#340	C₂²×C₆₇	1	2	yes	—
240	#340	C₂²×C₆₇	1	2	yes	—
241	#340	C₂²×C₆₇	1	2	yes	—
242	#336	C₂²×C₆₇	1	1	yes	characteristic
243	#336	C₂²×C₆₇	1	1	yes	characteristic
244	C₂²×Q₈	—	2	2	no	—
245	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
246	C₂²×C₈	—	2	2	no	—
247	C₂²×C₈	—	2	2	no	—
248	C₂²×C₈	Q₈×C₆₇	1	4	yes	complemented
249	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
251	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
252	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
253	C₄⋊Q₈	—	2	4	no	—
254	C₄⋊Q₈	—	4	2	no	—
255	#117	—	2	4	no	—
256	#117	—	4	2	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	2	2	no	—
260	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
261	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
262	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
263	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
267	C₂×C₄²	—	2	2	no	—
268	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
269	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
270	#107	—	2	4	no	—
271	#107	—	4	2	no	—
272	#106	—	2	2	no	—
273	#106	—	2	2	no	—
274	#106	—	2	2	no	—
275	#106	—	2	4	no	—
276	#106	D₄×C₆₇	1	4	yes	—
277	C₂²⋊C₈	Q₈×C₆₇	1	4	yes	complemented
278	C₄×C₈	—	2	2	no	—
279	C₄×C₈	—	2	2	no	—
280	#95	—	2	2	no	—
281	#95	—	2	2	no	—
282	#95	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	D₄×C₆₇	1	2	yes	—
286	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
287	C₂×Q₈	—	4	4	no	—
288	C₂×Q₈	—	4	4	no	—
289	C₂×Q₈	—	8	2	no	—
290	C₂²×C₄	—	2	1	no	—
291	C₂²×C₄	—	2	1	no	—
292	C₂²×C₄	—	2	1	no	—
293	C₂²×C₄	—	2	1	no	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	—	2	2	no	—
296	C₂²×C₄	—	2	2	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
299	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
301	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
302	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
303	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	—	2	8	no	—
311	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
312	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
313	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
314	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
315	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
316	C₂×C₈	#41×C₆₇	1	2	yes	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	4	4	no	—
325	C₄⋊C₄	—	4	4	no	—
326	C₄⋊C₄	—	8	2	no	—
327	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
328	C₂²⋊C₄	—	2	4	no	—
329	C₂²⋊C₄	—	4	4	no	—
330	C₂²⋊C₄	—	8	2	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	4	4	no	—
336	C₂³	—	2	1	no	—
337	C₂³	—	2	1	no	—
338	C₂³	—	2	2	no	—
339	C₂³	—	2	2	no	—
340	C₂³	C₄⋊Q₈×C₆₇	1	2	yes	—
341	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
342	Q₈	—	8	8	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	4	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	C₄⋊Q₈×C₆₇	1	2	yes	—
367	C₂×C₄	#135×C₆₇	1	4	yes	—
368	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
369	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
370	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
372	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
373	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	2	2	no	—
376	C₈	—	2	2	no	—
377	C₈	C₂²⋊Q₈×C₆₇	1	4	yes	—
378	C₂²	—	2	2	no	—
379	C₂²	—	2	2	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	4	2	no	—
382	C₂²	—	4	2	no	—
383	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
384	C₂²	D₄²×C₆₇	1	1	yes	characteristic
385	C₂²	#547×C₆₇	1	1	yes	characteristic
386	C₂²	C₂×#135×C₆₇	1	2	yes	—
387	C₂²	#577×C₆₇	1	2	yes	—
388	C₄	—	2	4	no	—
389	C₄	—	2	2	no	—
390	C₄	—	4	4	no	—
391	C₄	—	4	2	no	—
392	C₄	—	8	2	no	—
393	C₄	—	8	2	no	—
394	C₄	—	8	2	no	—
395	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
396	C₄	#544×C₆₇	1	1	yes	characteristic
397	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
398	C₄	#554×C₆₇	1	1	yes	characteristic
399	C₂	—	4	2	no	—
400	C₂	C₆₇×#2440	1	1	yes	characteristic
401	C₂	C₆₇×#3160	1	1	yes	characteristic
402	C₂	C₆₇×#3163	1	1	yes	characteristic
403	C₂	C₆₇×#3282	1	2	yes	—
404	C₂	C₆₇×#3340	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄×C₂²⋊C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#498×C₆₇	C₂	semidirect
4. C₆₇×#2085	C₂	semidirect
5. C₆₇×#2057	C₂	semidirect
6. C₆₇×#1941	C₂	semidirect
7. C₆₇×#1941	C₂	semidirect
9. C₆₇×#1846	C₂	semidirect
45. C₂²×C₈×C₆₇	Q₈	semidirect
74. C₂²⋊C₈×C₆₇	Q₈	semidirect
196. #26061	C₆₇	direct (N × H)
203. C₂×#498	C₂×C₆₇	semidirect
206. #2085	C₂×C₆₇	semidirect
207. #2057	C₂×C₆₇	semidirect
208. #1941	C₂×C₆₇	semidirect
209. #1941	C₂×C₆₇	semidirect
211. #1846	C₂×C₆₇	semidirect
214. C₆₇	#26061	direct (N × H)
248. C₂²×C₈	Q₈×C₆₇	semidirect
277. C₂²⋊C₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#498×C₆₇	2
C₂×#427×C₆₇	2
C₆₇×#2439	2
C₆₇×#2085	2
C₆₇×#2057	2
C₆₇×#1941	2
C₆₇×#1854	2
C₆₇×#1846	2
C₆₇×#1807	2
C₈×C₂²⋊C₄×C₆₇	2
#26061	67

Subgroups — C67×#26061 #30705312

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#26061 #30705312