Subgroups — C₂×C₆₇×#2396 #30713794

Order 17152.

2966 conjugacy classes of subgroups.

The subgroup lattice has 516 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×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2396	C₂	1	16	yes	maximal, direct product factor
4	C₂×#400×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#399×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#398×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×#396×C₆₇	C₂	1	2	yes	maximal
8	C₂×#395×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#393×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
13	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	complemented
17	C₂×#117×C₆₇	—	2	8	no	—
18	C₂×#117×C₆₇	C₂²	1	8	yes	—
19	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
26	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
27	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
33	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
34	#400×C₆₇	C₂²	1	8	yes	complemented
35	#399×C₆₇	C₂²	1	8	yes	—
36	#398×C₆₇	C₂²	1	16	yes	complemented
37	#396×C₆₇	C₂²	1	16	yes	—
38	#395×C₆₇	C₂²	1	8	yes	complemented
39	#393×C₆₇	C₂²	1	8	yes	—
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	16	yes	complemented
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	8	yes	complemented
42	C₄×C₂²⋊C₄×C₆₇	C₂²	1	16	yes	—
43	C₂×#95×C₆₇	C₂²	1	2	yes	—
44	C₂×#95×C₆₇	C₂²	1	2	yes	—
45	C₂×#95×C₆₇	C₂²	1	2	yes	—
46	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
47	C₂×#95×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×#95×C₆₇	C₂²	1	2	yes	—
49	C₂⁵×C₆₇	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₆₇	—	2	4	no	—
51	C₂²×Q₈×C₆₇	—	2	8	no	—
52	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
53	C₂³×C₄×C₆₇	—	2	4	no	—
54	C₂³×C₄×C₆₇	—	2	4	no	—
55	C₂³×C₄×C₆₇	Q₈	1	8	yes	complemented
56	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
59	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
60	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
62	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₇	C₂³	1	4	yes	—
64	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
65	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
66	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
67	C₄×Q₈×C₆₇	—	2	32	no	—
68	#117×C₆₇	—	2	32	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
74	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
75	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
79	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
81	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
84	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
85	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
86	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
89	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₆₇	C₂³	1	8	yes	—
91	C₂×C₄²×C₆₇	—	2	4	no	—
92	C₂×C₄²×C₆₇	—	2	8	no	—
93	C₂×C₄²×C₆₇	—	2	8	no	—
94	C₂×C₄²×C₆₇	D₄	1	8	yes	—
95	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
96	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
97	C₂×C₄²×C₆₇	C₂³	1	8	yes	—
98	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
99	#95×C₆₇	C₂³	1	8	yes	—
100	#95×C₆₇	C₂³	1	8	yes	—
101	#95×C₆₇	C₂³	1	8	yes	—
102	#95×C₆₇	C₂³	1	4	yes	—
103	#95×C₆₇	C₂³	1	4	yes	—
104	#95×C₆₇	C₂³	1	8	yes	—
105	C₂⁴×C₆₇	—	2	2	no	—
106	C₂⁴×C₆₇	—	2	4	no	—
107	C₂⁴×C₆₇	—	2	4	no	—
108	C₂⁴×C₆₇	—	2	2	no	—
109	C₂⁴×C₆₇	C₂×Q₈	1	4	yes	—
110	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
111	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
112	C₂×Q₈×C₆₇	—	2	32	no	—
113	C₂×Q₈×C₆₇	—	2	8	no	—
114	C₂×Q₈×C₆₇	—	2	16	no	—
115	C₂²×C₄×C₆₇	—	2	4	no	—
116	C₂²×C₄×C₆₇	—	2	2	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	4	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	4	no	—
122	C₂²×C₄×C₆₇	—	2	4	no	—
123	C₂²×C₄×C₆₇	—	2	32	no	—
124	C₂²×C₄×C₆₇	—	2	8	no	—
125	C₂²×C₄×C₆₇	—	2	8	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	4	no	—
130	C₂²×C₄×C₆₇	—	2	4	no	—
131	C₂²×C₄×C₆₇	—	2	4	no	—
132	C₂²×C₄×C₆₇	—	2	4	no	—
133	C₂²×C₄×C₆₇	—	2	4	no	—
134	C₂²×C₄×C₆₇	—	2	4	no	—
135	C₂²×C₄×C₆₇	—	2	4	no	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
138	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
140	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
141	C₂²×C₄×C₆₇	C₂×Q₈	1	16	yes	complemented
142	C₂²×C₄×C₆₇	#41	1	8	yes	—
143	C₂²×C₄×C₆₇	#41	1	8	yes	—
144	C₂²×C₄×C₆₇	#41	1	4	yes	—
145	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
146	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
149	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
152	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
153	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
154	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
155	C₄⋊C₄×C₆₇	—	2	8	no	—
156	C₄⋊C₄×C₆₇	—	2	8	no	—
157	C₄⋊C₄×C₆₇	—	2	16	no	—
158	C₄⋊C₄×C₆₇	—	2	8	no	—
159	C₄⋊C₄×C₆₇	—	2	16	no	—
160	C₄⋊C₄×C₆₇	—	2	16	no	—
161	C₄⋊C₄×C₆₇	—	2	16	no	—
162	C₂²⋊C₄×C₆₇	—	2	8	no	—
163	C₂²⋊C₄×C₆₇	—	2	8	no	—
164	C₂²⋊C₄×C₆₇	—	2	16	no	—
165	C₂²⋊C₄×C₆₇	—	2	16	no	—
166	C₄²×C₆₇	—	2	8	no	—
167	C₄²×C₆₇	—	2	16	no	—
168	C₄²×C₆₇	—	2	16	no	—
169	C₄²×C₆₇	C₂×D₄	1	16	yes	—
170	C₂³×C₆₇	—	2	8	no	—
171	C₂³×C₆₇	—	2	16	no	—
172	C₂³×C₆₇	—	2	2	no	—
173	C₂³×C₆₇	—	2	4	no	—
174	C₂³×C₆₇	—	2	4	no	—
175	C₂³×C₆₇	—	2	2	no	—
176	C₂³×C₆₇	—	4	4	no	—
177	C₂³×C₆₇	—	4	8	no	—
178	C₂³×C₆₇	—	4	4	no	—
179	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
180	C₂³×C₆₇	C₂²×Q₈	1	4	yes	—
181	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
182	C₂³×C₆₇	C₂×#41	1	2	yes	—
183	C₂³×C₆₇	C₂×#41	1	4	yes	—
184	C₂³×C₆₇	C₂×#41	1	4	yes	—
185	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
186	Q₈×C₆₇	—	2	32	no	—
187	C₂×C₄×C₆₇	—	2	8	no	—
188	C₂×C₄×C₆₇	—	2	8	no	—
189	C₂×C₄×C₆₇	—	2	16	no	—
190	C₂×C₄×C₆₇	—	2	16	no	—
191	C₂×C₄×C₆₇	—	2	8	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	4	no	—
195	C₂×C₄×C₆₇	—	2	4	no	—
196	C₂×C₄×C₆₇	—	2	4	no	—
197	C₂×C₄×C₆₇	—	2	2	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	4	no	—
200	C₂×C₄×C₆₇	—	2	32	no	—
201	C₂×C₄×C₆₇	—	2	2	no	—
202	C₂×C₄×C₆₇	—	2	2	no	—
203	C₂×C₄×C₆₇	—	2	4	no	—
204	C₂×C₄×C₆₇	—	2	4	no	—
205	C₂×C₄×C₆₇	—	2	4	no	—
206	C₂×C₄×C₆₇	—	2	4	no	—
207	C₂×C₄×C₆₇	—	2	4	no	—
208	C₂×C₄×C₆₇	—	2	4	no	—
209	C₂×C₄×C₆₇	—	2	4	no	—
210	C₂×C₄×C₆₇	—	2	4	no	—
211	C₂×C₄×C₆₇	—	4	4	no	—
212	C₂×C₄×C₆₇	—	4	8	no	—
213	C₂×C₄×C₆₇	—	4	4	no	—
214	C₂×C₄×C₆₇	—	4	8	no	—
215	C₂×C₄×C₆₇	C₂²⋊Q₈	1	16	yes	—
216	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
217	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
218	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
219	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
220	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
221	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
222	C₂×C₄×C₆₇	C₂×#41	1	8	yes	—
223	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
224	C₂²×C₆₇	—	2	4	no	—
225	C₂²×C₆₇	—	2	8	no	—
226	C₂²×C₆₇	—	4	16	no	—
227	C₂²×C₆₇	—	4	2	no	—
228	C₂²×C₆₇	—	4	4	no	—
229	C₂²×C₆₇	—	4	2	no	—
230	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	4	yes	—
231	C₂²×C₆₇	#525	1	4	yes	—
232	C₂²×C₆₇	#529	1	8	yes	—
233	C₂²×C₆₇	#532	1	4	yes	—
234	C₂²×C₆₇	#533	1	8	yes	—
235	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
236	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
237	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
238	C₂²×C₆₇	C₂²×#41	1	2	yes	—
239	C₂²×C₆₇	C₂²×#41	1	2	yes	—
240	C₄×C₆₇	—	2	4	no	—
241	C₄×C₆₇	—	2	4	no	—
242	C₄×C₆₇	—	2	8	no	—
243	C₄×C₆₇	—	2	8	no	—
244	C₄×C₆₇	—	2	4	no	—
245	C₄×C₆₇	—	4	2	no	—
246	C₄×C₆₇	—	4	4	no	—
247	C₄×C₆₇	—	4	2	no	—
248	C₄×C₆₇	—	4	4	no	—
249	C₄×C₆₇	C₂×C₂²⋊Q₈	1	8	yes	—
250	C₂×#2396	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
251	C₂×C₆₇	—	4	4	no	—
252	C₂×C₆₇	#2396	1	8	yes	direct product factor
253	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
254	C₂×C₆₇	C₂×#525	1	1	yes	characteristic
255	C₂×C₆₇	C₂×#529	1	2	yes	—
256	C₂×C₆₇	C₂×#532	1	1	yes	characteristic
257	C₂×C₆₇	C₂×#533	1	2	yes	—
258	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
259	C₂²×#117	C₂×C₆₇	1	1	yes	characteristic
260	#2396	C₂×C₆₇	1	16	yes	direct product factor
261	C₂×#400	C₂×C₆₇	1	1	yes	characteristic, complemented
262	C₂×#399	C₂×C₆₇	1	1	yes	characteristic
263	C₂×#398	C₂×C₆₇	1	2	yes	complemented
264	C₂×#396	C₂×C₆₇	1	2	yes	—
265	C₂×#395	C₂×C₆₇	1	1	yes	characteristic, complemented
266	C₂×#393	C₂×C₆₇	1	1	yes	characteristic
267	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
268	C₂×C₄×C₄⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
269	C₂×C₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
270	C₆₇	C₂×#2396	1	1	yes	characteristic, direct product factor, Sylow
271	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
272	C₂⁴×C₄	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₄×Q₈	—	2	8	no	—
274	C₂×C₄×Q₈	C₂²×C₆₇	1	8	yes	complemented
275	C₂×#117	—	2	8	no	—
276	C₂×#117	C₂²×C₆₇	1	8	yes	—
277	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
278	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
279	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
280	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
281	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
282	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
283	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
284	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
285	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
286	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
287	C₂²×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
288	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
289	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
290	C₂²×C₄²	C₂²×C₆₇	1	2	yes	—
291	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
292	#400	C₂²×C₆₇	1	8	yes	complemented
293	#399	C₂²×C₆₇	1	8	yes	—
294	#398	C₂²×C₆₇	1	16	yes	complemented
295	#396	C₂²×C₆₇	1	16	yes	—
296	#395	C₂²×C₆₇	1	8	yes	complemented
297	#393	C₂²×C₆₇	1	8	yes	—
298	C₄×C₄⋊C₄	C₂²×C₆₇	1	16	yes	complemented
299	C₄×C₄⋊C₄	C₂²×C₆₇	1	8	yes	complemented
300	C₄×C₂²⋊C₄	C₂²×C₆₇	1	16	yes	—
301	C₂×#95	C₂²×C₆₇	1	2	yes	—
302	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
303	C₂×#95	C₂²×C₆₇	1	2	yes	—
304	C₂×#95	C₂²×C₆₇	1	1	yes	characteristic
305	C₂×#95	C₂²×C₆₇	1	2	yes	—
306	C₂×#95	C₂²×C₆₇	1	2	yes	—
307	C₂⁵	C₂³×C₆₇	1	1	yes	characteristic
308	C₂²×Q₈	—	2	4	no	—
309	C₂²×Q₈	—	2	8	no	—
310	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
311	C₂³×C₄	—	2	4	no	—
312	C₂³×C₄	—	2	4	no	—
313	C₂³×C₄	Q₈×C₆₇	1	8	yes	complemented
314	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
315	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
316	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
317	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
318	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
319	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
321	C₂³×C₄	C₂³×C₆₇	1	4	yes	—
322	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
323	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
324	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
325	C₄×Q₈	—	2	32	no	—
326	#117	—	2	32	no	—
327	C₂×C₄⋊C₄	—	2	4	no	—
328	C₂×C₄⋊C₄	—	2	4	no	—
329	C₂×C₄⋊C₄	—	2	8	no	—
330	C₂×C₄⋊C₄	—	2	4	no	—
331	C₂×C₄⋊C₄	—	2	8	no	—
332	C₂×C₄⋊C₄	—	2	8	no	—
333	C₂×C₄⋊C₄	—	2	8	no	—
334	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
335	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
336	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
337	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
338	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
339	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
341	C₂×C₂²⋊C₄	—	2	4	no	—
342	C₂×C₂²⋊C₄	—	2	4	no	—
343	C₂×C₂²⋊C₄	—	2	8	no	—
344	C₂×C₂²⋊C₄	—	2	8	no	—
345	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₆₇	1	4	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₆₇	1	8	yes	—
349	C₂×C₄²	—	2	4	no	—
350	C₂×C₄²	—	2	8	no	—
351	C₂×C₄²	—	2	8	no	—
352	C₂×C₄²	D₄×C₆₇	1	8	yes	—
353	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
354	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
355	C₂×C₄²	C₂³×C₆₇	1	8	yes	—
356	C₂×C₄²	C₂³×C₆₇	1	4	yes	—
357	#95	C₂³×C₆₇	1	8	yes	—
358	#95	C₂³×C₆₇	1	8	yes	—
359	#95	C₂³×C₆₇	1	8	yes	—
360	#95	C₂³×C₆₇	1	4	yes	—
361	#95	C₂³×C₆₇	1	4	yes	—
362	#95	C₂³×C₆₇	1	8	yes	—
363	C₂⁴	—	2	2	no	—
364	C₂⁴	—	2	4	no	—
365	C₂⁴	—	2	4	no	—
366	C₂⁴	—	2	2	no	—
367	C₂⁴	C₂×Q₈×C₆₇	1	4	yes	—
368	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
369	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
370	C₂×Q₈	—	2	32	no	—
371	C₂×Q₈	—	2	8	no	—
372	C₂×Q₈	—	2	16	no	—
373	C₂²×C₄	—	2	4	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	4	no	—
376	C₂²×C₄	—	2	4	no	—
377	C₂²×C₄	—	2	4	no	—
378	C₂²×C₄	—	2	2	no	—
379	C₂²×C₄	—	2	4	no	—
380	C₂²×C₄	—	2	4	no	—
381	C₂²×C₄	—	2	32	no	—
382	C₂²×C₄	—	2	8	no	—
383	C₂²×C₄	—	2	8	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	4	no	—
389	C₂²×C₄	—	2	4	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	2	4	no	—
393	C₂²×C₄	—	2	4	no	—
394	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
395	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
396	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
397	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
398	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
399	C₂²×C₄	C₂×Q₈×C₆₇	1	16	yes	complemented
400	C₂²×C₄	#41×C₆₇	1	8	yes	—
401	C₂²×C₄	#41×C₆₇	1	8	yes	—
402	C₂²×C₄	#41×C₆₇	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
404	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
405	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
406	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
407	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
409	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
410	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
411	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
412	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
413	C₄⋊C₄	—	2	8	no	—
414	C₄⋊C₄	—	2	8	no	—
415	C₄⋊C₄	—	2	16	no	—
416	C₄⋊C₄	—	2	8	no	—
417	C₄⋊C₄	—	2	16	no	—
418	C₄⋊C₄	—	2	16	no	—
419	C₄⋊C₄	—	2	16	no	—
420	C₂²⋊C₄	—	2	8	no	—
421	C₂²⋊C₄	—	2	8	no	—
422	C₂²⋊C₄	—	2	16	no	—
423	C₂²⋊C₄	—	2	16	no	—
424	C₄²	—	2	8	no	—
425	C₄²	—	2	16	no	—
426	C₄²	—	2	16	no	—
427	C₄²	C₂×D₄×C₆₇	1	16	yes	—
428	C₂³	—	2	8	no	—
429	C₂³	—	2	16	no	—
430	C₂³	—	2	2	no	—
431	C₂³	—	2	4	no	—
432	C₂³	—	2	4	no	—
433	C₂³	—	2	2	no	—
434	C₂³	—	4	4	no	—
435	C₂³	—	4	8	no	—
436	C₂³	—	4	4	no	—
437	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
438	C₂³	C₂²×Q₈×C₆₇	1	4	yes	—
439	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
440	C₂³	C₂×#41×C₆₇	1	2	yes	—
441	C₂³	C₂×#41×C₆₇	1	4	yes	—
442	C₂³	C₂×#41×C₆₇	1	4	yes	—
443	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
444	Q₈	—	2	32	no	—
445	C₂×C₄	—	2	8	no	—
446	C₂×C₄	—	2	8	no	—
447	C₂×C₄	—	2	16	no	—
448	C₂×C₄	—	2	16	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	4	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	4	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	32	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	2	no	—
461	C₂×C₄	—	2	4	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	4	4	no	—
470	C₂×C₄	—	4	8	no	—
471	C₂×C₄	—	4	4	no	—
472	C₂×C₄	—	4	8	no	—
473	C₂×C₄	C₂²⋊Q₈×C₆₇	1	16	yes	—
474	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
475	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
476	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
477	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
478	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
479	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
480	C₂×C₄	C₂×#41×C₆₇	1	8	yes	—
481	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
482	C₂²	—	2	4	no	—
483	C₂²	—	2	8	no	—
484	C₂²	—	4	16	no	—
485	C₂²	—	4	2	no	—
486	C₂²	—	4	4	no	—
487	C₂²	—	4	2	no	—
488	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	4	yes	—
489	C₂²	#525×C₆₇	1	4	yes	—
490	C₂²	#529×C₆₇	1	8	yes	—
491	C₂²	#532×C₆₇	1	4	yes	—
492	C₂²	#533×C₆₇	1	8	yes	—
493	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
494	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
495	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
496	C₂²	C₂²×#41×C₆₇	1	2	yes	—
497	C₂²	C₂²×#41×C₆₇	1	2	yes	—
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	8	no	—
501	C₄	—	2	8	no	—
502	C₄	—	2	4	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	4	no	—
505	C₄	—	4	2	no	—
506	C₄	—	4	4	no	—
507	C₄	C₂×C₂²⋊Q₈×C₆₇	1	8	yes	—
508	C₂	—	4	4	no	—
509	C₂	C₆₇×#2396	1	8	yes	direct product factor
510	C₂	C₂²×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
511	C₂	C₂×#525×C₆₇	1	1	yes	characteristic
512	C₂	C₂×#529×C₆₇	1	2	yes	—
513	C₂	C₂×#532×C₆₇	1	1	yes	characteristic
514	C₂	C₂×#533×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₆₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₆₇
socle	C₂⁴×C₆₇	C₂⁴
prime radical	C₂⁵×C₆₇	C₂³
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×#2396	C₂	direct (N × H)
4. C₂×#400×C₆₇	C₂	semidirect
6. C₂×#398×C₆₇	C₂	semidirect
8. C₂×#395×C₆₇	C₂	semidirect
10. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
11. C₂×C₄×C₄⋊C₄×C₆₇	C₂	semidirect
16. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
34. #400×C₆₇	C₂²	semidirect
36. #398×C₆₇	C₂²	semidirect
38. #395×C₆₇	C₂²	semidirect
40. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
41. C₄×C₄⋊C₄×C₆₇	C₂²	semidirect
55. C₂³×C₄×C₆₇	Q₈	semidirect
141. C₂²×C₄×C₆₇	C₂×Q₈	semidirect
250. C₂×#2396	C₆₇	direct (N × H)
252. C₂×C₆₇	#2396	direct (N × H)
258. C₂²×C₄×Q₈	C₂×C₆₇	semidirect
260. #2396	C₂×C₆₇	direct (N × H)
261. C₂×#400	C₂×C₆₇	semidirect
263. C₂×#398	C₂×C₆₇	semidirect
265. C₂×#395	C₂×C₆₇	semidirect
267. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
268. C₂×C₄×C₄⋊C₄	C₂×C₆₇	semidirect
270. C₆₇	C₂×#2396	direct (N × H)
274. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
292. #400	C₂²×C₆₇	semidirect
294. #398	C₂²×C₆₇	semidirect
296. #395	C₂²×C₆₇	semidirect
298. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
299. C₄×C₄⋊C₄	C₂²×C₆₇	semidirect
313. C₂³×C₄	Q₈×C₆₇	semidirect
399. C₂²×C₄	C₂×Q₈×C₆₇	semidirect
509. C₂	C₆₇×#2396	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×#117×C₆₇	2
C₆₇×#2396	2
C₂×#400×C₆₇	2
C₂×#399×C₆₇	2
C₂×#398×C₆₇	2
C₂×#396×C₆₇	2
C₂×#395×C₆₇	2
C₂×#393×C₆₇	2
C₂×C₄×C₄⋊C₄×C₆₇	2
C₂×C₄×C₂²⋊C₄×C₆₇	2
C₂×#2396	67

Subgroups — C2×C67×#2396 #30713794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#2396 #30713794