Subgroups — C₂×C₆₇×#2398 #30713796

Order 17152.

3246 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2398	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂×#397×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#394×C₆₇	C₂	1	2	yes	maximal
7	C₂×#393×C₆₇	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	4	yes	maximal, complemented
9	C₂×#376×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₆₇	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
15	C₂×C₄×D₄×C₆₇	—	2	8	no	—
16	C₂×C₄×D₄×C₆₇	C₂²	1	8	yes	—
17	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
18	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
21	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
22	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
23	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
26	#400×C₆₇	C₂²	1	16	yes	complemented
27	#397×C₆₇	C₂²	1	16	yes	complemented
28	#394×C₆₇	C₂²	1	16	yes	—
29	#393×C₆₇	C₂²	1	16	yes	complemented
30	C₄×C₂²⋊C₄×C₆₇	C₂²	1	32	yes	complemented
31	#376×C₆₇	C₂²	1	8	yes	complemented
32	C₂×#95×C₆₇	C₂²	1	2	yes	—
33	C₂×#95×C₆₇	C₂²	1	2	yes	—
34	C₂×#95×C₆₇	C₂²	1	4	yes	—
35	C₂×#95×C₆₇	C₂²	1	4	yes	—
36	C₂⁵×C₆₇	C₂³	1	2	yes	—
37	C₂²×Q₈×C₆₇	—	2	4	no	—
38	C₂²×Q₈×C₆₇	—	2	8	no	—
39	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
40	C₂²×D₄×C₆₇	—	2	8	no	—
41	C₂²×D₄×C₆₇	—	2	4	no	—
42	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
43	C₂³×C₄×C₆₇	—	2	8	no	—
44	C₂³×C₄×C₆₇	—	2	16	no	—
45	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
47	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
48	C₂³×C₄×C₆₇	C₂³	1	8	yes	—
49	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
52	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₄×Q₈×C₆₇	—	2	32	no	—
54	C₄×D₄×C₆₇	—	2	32	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
58	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	16	no	—
64	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
65	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
66	C₂×C₂²⋊C₄×C₆₇	C₂³	1	16	yes	—
67	C₂×C₄²×C₆₇	—	2	8	no	—
68	C₂×C₄²×C₆₇	—	2	8	no	—
69	C₂×C₄²×C₆₇	D₄	1	8	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
73	#95×C₆₇	C₂³	1	8	yes	—
74	#95×C₆₇	C₂³	1	8	yes	—
75	#95×C₆₇	C₂³	1	16	yes	—
76	#95×C₆₇	C₂³	1	16	yes	—
77	C₂⁴×C₆₇	—	2	4	no	—
78	C₂⁴×C₆₇	—	2	8	no	—
79	C₂⁴×C₆₇	—	2	8	no	—
80	C₂⁴×C₆₇	—	2	4	no	—
81	C₂⁴×C₆₇	#41	1	8	yes	—
82	C₂⁴×C₆₇	C₂⁴	1	4	yes	—
83	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
84	C₂×Q₈×C₆₇	—	2	32	no	—
85	C₂×Q₈×C₆₇	—	2	8	no	—
86	C₂×Q₈×C₆₇	—	2	16	no	—
87	C₂×D₄×C₆₇	—	2	32	no	—
88	C₂×D₄×C₆₇	—	2	16	no	—
89	C₂×D₄×C₆₇	—	2	8	no	—
90	C₂²×C₄×C₆₇	—	2	4	no	—
91	C₂²×C₄×C₆₇	—	2	4	no	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	4	no	—
95	C₂²×C₄×C₆₇	—	2	4	no	—
96	C₂²×C₄×C₆₇	—	2	8	no	—
97	C₂²×C₄×C₆₇	—	2	8	no	—
98	C₂²×C₄×C₆₇	—	2	8	no	—
99	C₂²×C₄×C₆₇	—	2	8	no	—
100	C₂²×C₄×C₆₇	—	2	8	no	—
101	C₂²×C₄×C₆₇	—	2	16	no	—
102	C₂²×C₄×C₆₇	—	2	32	no	—
103	C₂²×C₄×C₆₇	—	4	32	no	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
106	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
107	C₂²×C₄×C₆₇	#41	1	8	yes	—
108	C₂²×C₄×C₆₇	#41	1	8	yes	—
109	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
110	C₂²×C₄×C₆₇	C₂⁴	1	8	yes	—
111	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
112	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
113	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
115	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
116	C₄⋊C₄×C₆₇	—	2	8	no	—
117	C₄⋊C₄×C₆₇	—	2	16	no	—
118	C₄⋊C₄×C₆₇	—	2	8	no	—
119	C₂²⋊C₄×C₆₇	—	2	32	no	—
120	C₂²⋊C₄×C₆₇	—	2	16	no	—
121	C₂²⋊C₄×C₆₇	—	2	32	no	—
122	C₄²×C₆₇	—	2	16	no	—
123	C₄²×C₆₇	—	2	16	no	—
124	C₄²×C₆₇	C₂×D₄	1	16	yes	—
125	C₂³×C₆₇	—	2	16	no	—
126	C₂³×C₆₇	—	2	32	no	—
127	C₂³×C₆₇	—	2	4	no	—
128	C₂³×C₆₇	—	2	8	no	—
129	C₂³×C₆₇	—	2	8	no	—
130	C₂³×C₆₇	—	2	4	no	—
131	C₂³×C₆₇	—	4	8	no	—
132	C₂³×C₆₇	—	4	8	no	—
133	C₂³×C₆₇	—	4	16	no	—
134	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
135	C₂³×C₆₇	C₂×#41	1	8	yes	—
136	C₂³×C₆₇	C₂×#41	1	4	yes	—
137	C₂³×C₆₇	C₂×#41	1	4	yes	—
138	C₂³×C₆₇	C₂×#41	1	4	yes	—
139	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
140	Q₈×C₆₇	—	2	32	no	—
141	D₄×C₆₇	—	2	32	no	—
142	C₂×C₄×C₆₇	—	2	8	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	2	16	no	—
145	C₂×C₄×C₆₇	—	2	4	no	—
146	C₂×C₄×C₆₇	—	2	4	no	—
147	C₂×C₄×C₆₇	—	2	4	no	—
148	C₂×C₄×C₆₇	—	2	4	no	—
149	C₂×C₄×C₆₇	—	2	4	no	—
150	C₂×C₄×C₆₇	—	2	4	no	—
151	C₂×C₄×C₆₇	—	2	8	no	—
152	C₂×C₄×C₆₇	—	2	8	no	—
153	C₂×C₄×C₆₇	—	2	8	no	—
154	C₂×C₄×C₆₇	—	2	8	no	—
155	C₂×C₄×C₆₇	—	2	8	no	—
156	C₂×C₄×C₆₇	—	4	8	no	—
157	C₂×C₄×C₆₇	—	4	8	no	—
158	C₂×C₄×C₆₇	—	4	16	no	—
159	C₂×C₄×C₆₇	—	4	32	no	—
160	C₂×C₄×C₆₇	#124	1	16	yes	—
161	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
162	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
163	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
164	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
165	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
166	C₂²×C₆₇	—	2	8	no	—
167	C₂²×C₆₇	—	2	16	no	—
168	C₂²×C₆₇	—	4	32	no	—
169	C₂²×C₆₇	—	4	4	no	—
170	C₂²×C₆₇	—	4	4	no	—
171	C₂²×C₆₇	—	4	8	no	—
172	C₂²×C₆₇	#525	1	4	yes	—
173	C₂²×C₆₇	#525	1	4	yes	—
174	C₂²×C₆₇	C₂×#124	1	4	yes	—
175	C₂²×C₆₇	#529	1	16	yes	—
176	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
177	C₂²×C₆₇	C₂²×#41	1	2	yes	—
178	C₂²×C₆₇	C₂²×#41	1	2	yes	—
179	C₂²×C₆₇	C₂²×#41	1	2	yes	—
180	C₄×C₆₇	—	2	4	no	—
181	C₄×C₆₇	—	2	4	no	—
182	C₄×C₆₇	—	2	8	no	—
183	C₄×C₆₇	—	4	4	no	—
184	C₄×C₆₇	—	4	4	no	—
185	C₄×C₆₇	—	4	8	no	—
186	C₄×C₆₇	C₂×#124	1	8	yes	—
187	C₂×#2398	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₆₇	—	4	8	no	—
189	C₂×C₆₇	#2398	1	8	yes	direct product factor
190	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
191	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
192	C₂×C₆₇	C₂²×#124	1	1	yes	characteristic
193	C₂×C₆₇	C₂×#529	1	4	yes	—
194	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
195	C₂²×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
196	#2398	C₂×C₆₇	1	16	yes	direct product factor
197	C₂×#400	C₂×C₆₇	1	2	yes	complemented
198	C₂×#397	C₂×C₆₇	1	2	yes	complemented
199	C₂×#394	C₂×C₆₇	1	2	yes	—
200	C₂×#393	C₂×C₆₇	1	2	yes	complemented
201	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	4	yes	complemented
202	C₂×#376	C₂×C₆₇	1	1	yes	characteristic, complemented
203	C₆₇	C₂×#2398	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
205	C₂³×D₄	C₂²×C₆₇	1	1	yes	characteristic
206	C₂⁴×C₄	C₂²×C₆₇	1	2	yes	—
207	C₂×C₄×Q₈	—	2	8	no	—
208	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
209	C₂×C₄×D₄	—	2	8	no	—
210	C₂×C₄×D₄	C₂²×C₆₇	1	8	yes	—
211	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
212	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
213	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
214	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
215	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
216	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
217	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
218	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
219	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
220	#400	C₂²×C₆₇	1	16	yes	complemented
221	#397	C₂²×C₆₇	1	16	yes	complemented
222	#394	C₂²×C₆₇	1	16	yes	—
223	#393	C₂²×C₆₇	1	16	yes	complemented
224	C₄×C₂²⋊C₄	C₂²×C₆₇	1	32	yes	complemented
225	#376	C₂²×C₆₇	1	8	yes	complemented
226	C₂×#95	C₂²×C₆₇	1	2	yes	—
227	C₂×#95	C₂²×C₆₇	1	4	yes	—
228	C₂×#95	C₂²×C₆₇	1	2	yes	—
229	C₂×#95	C₂²×C₆₇	1	4	yes	—
230	C₂⁵	C₂³×C₆₇	1	2	yes	—
231	C₂²×Q₈	—	2	4	no	—
232	C₂²×Q₈	—	2	8	no	—
233	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
234	C₂²×D₄	—	2	8	no	—
235	C₂²×D₄	—	2	4	no	—
236	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
237	C₂³×C₄	—	2	8	no	—
238	C₂³×C₄	—	2	16	no	—
239	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
240	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
241	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
242	C₂³×C₄	C₂³×C₆₇	1	8	yes	—
243	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
244	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
245	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
246	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
247	C₄×Q₈	—	2	32	no	—
248	C₄×D₄	—	2	32	no	—
249	C₂×C₄⋊C₄	—	2	4	no	—
250	C₂×C₄⋊C₄	—	2	8	no	—
251	C₂×C₄⋊C₄	—	2	4	no	—
252	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
253	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
254	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
255	C₂×C₂²⋊C₄	—	2	16	no	—
256	C₂×C₂²⋊C₄	—	2	8	no	—
257	C₂×C₂²⋊C₄	—	2	16	no	—
258	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
259	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
260	C₂×C₂²⋊C₄	C₂³×C₆₇	1	16	yes	—
261	C₂×C₄²	—	2	8	no	—
262	C₂×C₄²	—	2	8	no	—
263	C₂×C₄²	D₄×C₆₇	1	8	yes	—
264	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
265	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
266	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
267	#95	C₂³×C₆₇	1	8	yes	—
268	#95	C₂³×C₆₇	1	8	yes	—
269	#95	C₂³×C₆₇	1	16	yes	—
270	#95	C₂³×C₆₇	1	16	yes	—
271	C₂⁴	—	2	4	no	—
272	C₂⁴	—	2	8	no	—
273	C₂⁴	—	2	8	no	—
274	C₂⁴	—	2	4	no	—
275	C₂⁴	#41×C₆₇	1	8	yes	—
276	C₂⁴	C₂⁴×C₆₇	1	4	yes	—
277	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
278	C₂×Q₈	—	2	32	no	—
279	C₂×Q₈	—	2	8	no	—
280	C₂×Q₈	—	2	16	no	—
281	C₂×D₄	—	2	32	no	—
282	C₂×D₄	—	2	16	no	—
283	C₂×D₄	—	2	8	no	—
284	C₂²×C₄	—	2	4	no	—
285	C₂²×C₄	—	2	4	no	—
286	C₂²×C₄	—	2	4	no	—
287	C₂²×C₄	—	2	4	no	—
288	C₂²×C₄	—	2	4	no	—
289	C₂²×C₄	—	2	4	no	—
290	C₂²×C₄	—	2	8	no	—
291	C₂²×C₄	—	2	8	no	—
292	C₂²×C₄	—	2	8	no	—
293	C₂²×C₄	—	2	8	no	—
294	C₂²×C₄	—	2	8	no	—
295	C₂²×C₄	—	2	16	no	—
296	C₂²×C₄	—	2	32	no	—
297	C₂²×C₄	—	4	32	no	—
298	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
299	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
300	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
301	C₂²×C₄	#41×C₆₇	1	8	yes	—
302	C₂²×C₄	#41×C₆₇	1	8	yes	—
303	C₂²×C₄	C₂⁴×C₆₇	1	8	yes	—
304	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
305	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
306	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
307	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
308	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
309	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
310	C₄⋊C₄	—	2	8	no	—
311	C₄⋊C₄	—	2	16	no	—
312	C₄⋊C₄	—	2	8	no	—
313	C₂²⋊C₄	—	2	32	no	—
314	C₂²⋊C₄	—	2	16	no	—
315	C₂²⋊C₄	—	2	32	no	—
316	C₄²	—	2	16	no	—
317	C₄²	—	2	16	no	—
318	C₄²	C₂×D₄×C₆₇	1	16	yes	—
319	C₂³	—	2	16	no	—
320	C₂³	—	2	32	no	—
321	C₂³	—	2	4	no	—
322	C₂³	—	2	8	no	—
323	C₂³	—	2	8	no	—
324	C₂³	—	2	4	no	—
325	C₂³	—	4	8	no	—
326	C₂³	—	4	8	no	—
327	C₂³	—	4	16	no	—
328	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
329	C₂³	C₂×#41×C₆₇	1	8	yes	—
330	C₂³	C₂×#41×C₆₇	1	4	yes	—
331	C₂³	C₂×#41×C₆₇	1	4	yes	—
332	C₂³	C₂×#41×C₆₇	1	4	yes	—
333	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
334	Q₈	—	2	32	no	—
335	D₄	—	2	32	no	—
336	C₂×C₄	—	2	8	no	—
337	C₂×C₄	—	2	8	no	—
338	C₂×C₄	—	2	16	no	—
339	C₂×C₄	—	2	4	no	—
340	C₂×C₄	—	2	4	no	—
341	C₂×C₄	—	2	4	no	—
342	C₂×C₄	—	2	4	no	—
343	C₂×C₄	—	2	4	no	—
344	C₂×C₄	—	2	4	no	—
345	C₂×C₄	—	2	8	no	—
346	C₂×C₄	—	2	8	no	—
347	C₂×C₄	—	2	8	no	—
348	C₂×C₄	—	2	8	no	—
349	C₂×C₄	—	2	8	no	—
350	C₂×C₄	—	4	8	no	—
351	C₂×C₄	—	4	8	no	—
352	C₂×C₄	—	4	16	no	—
353	C₂×C₄	—	4	32	no	—
354	C₂×C₄	#124×C₆₇	1	16	yes	—
355	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
356	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
357	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
358	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
359	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
360	C₂²	—	2	8	no	—
361	C₂²	—	2	16	no	—
362	C₂²	—	4	32	no	—
363	C₂²	—	4	4	no	—
364	C₂²	—	4	4	no	—
365	C₂²	—	4	8	no	—
366	C₂²	#525×C₆₇	1	4	yes	—
367	C₂²	#525×C₆₇	1	4	yes	—
368	C₂²	C₂×#124×C₆₇	1	4	yes	—
369	C₂²	#529×C₆₇	1	16	yes	—
370	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
371	C₂²	C₂²×#41×C₆₇	1	2	yes	—
372	C₂²	C₂²×#41×C₆₇	1	2	yes	—
373	C₂²	C₂²×#41×C₆₇	1	2	yes	—
374	C₄	—	2	4	no	—
375	C₄	—	2	4	no	—
376	C₄	—	2	8	no	—
377	C₄	—	4	4	no	—
378	C₄	—	4	4	no	—
379	C₄	—	4	8	no	—
380	C₄	C₂×#124×C₆₇	1	8	yes	—
381	C₂	—	4	8	no	—
382	C₂	C₆₇×#2398	1	8	yes	direct product factor
383	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
384	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
385	C₂	C₂²×#124×C₆₇	1	1	yes	characteristic
386	C₂	C₂×#529×C₆₇	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂⁴×C₆₇	C₂⁴
prime radical	C₂³×D₄×C₆₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2398	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
5. C₂×#397×C₆₇	C₂	semidirect
7. C₂×#393×C₆₇	C₂	semidirect
8. C₂×C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
9. C₂×#376×C₆₇	C₂	semidirect
14. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
26. #400×C₆₇	C₂²	semidirect
27. #397×C₆₇	C₂²	semidirect
29. #393×C₆₇	C₂²	semidirect
30. C₄×C₂²⋊C₄×C₆₇	C₂²	semidirect
31. #376×C₆₇	C₂²	semidirect
187. C₂×#2398	C₆₇	direct (N × H)
189. C₂×C₆₇	#2398	direct (N × H)
194. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
196. #2398	C₂×C₆₇	direct (N × H)
197. C₂×#400	C₂×C₆₇	semidirect
198. C₂×#397	C₂×C₆₇	semidirect
200. C₂×#393	C₂×C₆₇	semidirect
201. C₂×C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
202. C₂×#376	C₂×C₆₇	semidirect
203. C₆₇	C₂×#2398	direct (N × H)
208. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
220. #400	C₂²×C₆₇	semidirect
221. #397	C₂²×C₆₇	semidirect
223. #393	C₂²×C₆₇	semidirect
224. C₄×C₂²⋊C₄	C₂²×C₆₇	semidirect
225. #376	C₂²×C₆₇	semidirect
382. C₂	C₆₇×#2398	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂²×C₄×D₄×C₆₇	2
C₆₇×#2398	2
C₂×#400×C₆₇	2
C₂×#397×C₆₇	2
C₂×#394×C₆₇	2
C₂×#393×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#376×C₆₇	2
C₂×#2398	67

Subgroups — C2×C67×#2398 #30713796

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2398 #30713796