Subgroups — #22569369

Order 8576.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863527	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863474	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863468	C₂	1	2	yes	maximal, complemented
5	D₄×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	D₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#16863457	C₂	1	2	yes	maximal, complemented
8	#16863449	C₂	1	1	yes	maximal, characteristic, complemented
9	#16863423	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#529×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×#11761612	C₂²	1	2	yes	—
13	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
14	C₂²×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
16	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
17	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
20	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
22	C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
25	C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
26	#14523954	C₂²	1	2	yes	—
27	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
29	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	#126×C₆₇	C₄	1	2	yes	—
31	#125×C₆₇	C₄	1	1	yes	characteristic
32	#124×C₆₇	C₄	1	1	yes	characteristic
33	#123×C₆₇	C₄	1	2	yes	—
34	C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
35	#121×C₆₇	C₄	1	1	yes	characteristic
36	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₆₇	C₂²	1	2	yes	—
38	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
39	#117×C₆₇	C₂²	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
41	#11761612	—	4	2	no	—
42	C₂×C₆₇⋊C₈	—	4	2	no	—
43	C₂×C₆₇⋊C₈	—	4	2	no	—
44	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
45	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
46	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
47	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×D₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
57	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
58	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
68	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂³	1	1	yes	characteristic
71	C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₄²×C₆₇	C₂³	1	2	yes	—
73	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
74	C₂³×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₆₇	—	2	2	no	—
76	D₄×C₆₇	—	2	4	no	—
77	D₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	1	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
93	C₆₇⋊C₈	—	2	1	no	—
94	C₆₇⋊C₈	—	2	2	no	—
95	C₆₇⋊C₈	—	2	2	no	—
96	C₆₇⋊C₈	—	2	1	no	—
97	C₆₇⋊C₈	—	2	1	no	—
98	C₆₇⋊C₈	—	2	1	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	2	2	no	—
104	C₂²×C₆₇	#131	1	2	yes	—
105	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	2	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	#131	1	2	yes	—
114	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₆₇	—	2	2	no	—
117	C₂×C₆₇	—	4	2	no	—
118	C₂×C₆₇	#520	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
120	C₂×C₆₇	C₂×#131	1	1	yes	characteristic
121	#2994	—	67	1	no	maximal, Sylow
122	C₆₇	#2994	1	1	yes	characteristic, complemented, Sylow
123	#529	D₆₇	1	1	yes	characteristic
124	#446	—	67	1	no	—
125	C₈×Q₈	—	67	1	no	—
126	#436	—	67	1	no	—
127	#435	—	67	2	no	—
128	C₈×D₄	—	67	1	no	—
129	C₈×D₄	—	67	2	no	—
130	#433	—	67	2	no	—
131	#431	—	67	1	no	—
132	#424	—	67	2	no	—
133	C₂×C₄⋊C₈	—	67	1	no	—
134	C₂×M₁₆	—	67	2	no	—
135	C₂²×C₈	—	67	2	no	—
136	C₂²×C₈	—	67	2	no	—
137	#126	Dic₆₇	1	2	yes	—
138	#125	Dic₆₇	1	1	yes	characteristic
139	#124	Dic₆₇	1	1	yes	characteristic
140	#123	Dic₆₇	1	2	yes	—
141	C₂²⋊Q₈	Dic₆₇	1	1	yes	characteristic
142	#121	Dic₆₇	1	1	yes	characteristic
143	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
144	C₄×D₄	C₂×D₆₇	1	2	yes	—
145	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
146	#117	C₂×D₆₇	1	2	yes	—
147	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
148	C₄⋊C₈	—	67	2	no	—
149	C₄⋊C₈	—	67	1	no	—
150	C₄⋊C₈	—	67	1	no	—
151	C₄⋊C₈	—	67	1	no	—
152	C₄⋊C₈	—	67	2	no	—
153	C₄⋊C₈	—	67	2	no	—
154	C₄⋊C₈	—	67	1	no	—
155	C₂²⋊C₈	—	67	1	no	—
156	C₂²⋊C₈	—	67	1	no	—
157	C₂²⋊C₈	—	67	2	no	—
158	C₂²⋊C₈	—	67	2	no	—
159	#97	—	67	2	no	—
160	C₄×C₈	—	67	1	no	—
161	C₄×C₈	—	67	1	no	—
162	C₄×C₈	—	67	2	no	—
163	C₂×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
164	C₂×D₄	C₂×Dic₆₇	1	2	yes	—
165	C₂×D₄	C₂×Dic₆₇	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
170	M₁₆	—	268	2	no	—
171	C₂×C₈	—	268	2	no	—
172	C₂×C₈	—	268	2	no	—
173	C₂×C₈	—	67	1	no	—
174	C₂×C₈	—	67	1	no	—
175	C₂×C₈	—	67	1	no	—
176	C₂×C₈	—	67	2	no	—
177	C₂×C₈	—	67	1	no	—
178	C₂×C₈	—	67	2	no	—
179	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
181	C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
182	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
186	C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
190	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
191	C₄²	C₂×Dic₆₇	1	1	yes	characteristic
192	C₄²	C₂²×D₆₇	1	1	yes	characteristic
193	C₄²	C₂²×D₆₇	1	2	yes	—
194	C₄²	C₂²×D₆₇	1	1	yes	characteristic
195	C₂³	C₂²×Dic₆₇	1	2	yes	—
196	C₂³	C₂²×Dic₆₇	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
211	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
212	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
213	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
215	C₈	—	134	1	no	—
216	C₈	—	134	2	no	—
217	C₈	—	134	2	no	—
218	C₈	—	134	1	no	—
219	C₈	—	134	1	no	—
220	C₈	—	134	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	#14524050	1	2	yes	—
227	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	#14524050	1	2	yes	—
236	C₄	#14524089	1	1	yes	characteristic
237	C₄	#14524090	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	#16863879	1	1	yes	characteristic
241	C₂	C₂×#14524050	1	1	yes	characteristic
242	C₂	C₂×#14524050	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator, socle	C₂²×C₆₇	C₂³×C₄
Fitting	#529×C₆₇	C₂
hypercenter	#529	D₆₇
hypocenter	C₆₇	#2994
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #16863527	C₂	semidirect
2. Q₈×C₆₇⋊C₈	C₂	semidirect
3. #16863474	C₂	semidirect
4. #16863468	C₂	semidirect
5. D₄×C₆₇⋊C₈	C₂	semidirect
6. D₄×C₆₇⋊C₈	C₂	semidirect
7. #16863457	C₂	semidirect
8. #16863449	C₂	semidirect
9. #16863423	C₂	semidirect
10. C₂×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
122. C₆₇	#2994	semidirect