Subgroups — #22562466

Order 8576.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523986	C₂	1	4	yes	maximal, complemented
2	#16863773	C₂	1	1	yes	maximal, characteristic
3	C₄×C₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₅₃₆	—	2	1	no	—
6	C₂×D₅₃₆	C₂²	1	2	yes	—
7	#14524040	—	2	1	no	—
8	#14523986	—	2	8	no	—
9	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₆₇	—	2	1	no	—
11	C₂×Q₁₆×C₆₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₆₇	—	2	1	no	—
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×D₂₆₈	—	2	2	no	—
15	C₂×D₂₆₈	—	4	2	no	—
16	D₅₃₆	—	2	4	no	—
17	D₅₃₆	—	4	2	no	—
18	C₆₇⋊C₁₆	D₄	1	4	yes	complemented
19	C₂×Q₈×C₆₇	—	2	2	no	—
20	Q₁₆×C₆₇	—	2	4	no	—
21	Q₁₆×C₆₇	—	4	2	no	—
22	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₆₇	—	4	2	no	—
25	C₄²×C₆₇	D₄	1	1	yes	characteristic
26	C₂²×D₆₇	—	4	2	no	—
27	Q₈×C₆₇	—	4	2	no	—
28	Q₈×C₆₇	—	4	2	no	—
29	C₂×C₄×C₆₇	—	4	2	no	—
30	C₂×C₄×C₆₇	D₈	1	2	yes	—
31	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
32	D₂₆₈	—	4	2	no	—
33	D₂₆₈	—	4	2	no	—
34	D₂₆₈	—	8	4	no	—
35	C₈×C₆₇	D₈	1	2	yes	—
36	C₈×C₆₇	C₂×D₄	1	2	yes	—
37	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
38	C₂×D₆₇	—	8	2	no	—
39	C₂×D₆₇	—	8	4	no	—
40	C₄×C₆₇	—	8	2	no	—
41	C₄×C₆₇	SD₃₂	1	4	yes	complemented
42	C₄×C₆₇	#127	1	1	yes	characteristic
43	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₆₇	#493	1	1	yes	characteristic
45	C₂×C₆₇	C₂×SD₃₂	1	2	yes	—
46	D₆₇	—	16	2	no	—
47	#2248	—	67	1	no	maximal, Sylow
48	C₆₇	#2248	1	1	yes	characteristic, complemented, Sylow
49	C₂×SD₃₂	—	67	4	no	—
50	#494	D₆₇	1	1	yes	characteristic, complemented
51	#493	—	67	1	no	—
52	C₄×C₁₆	—	67	1	no	—
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
55	C₂×D₈	—	67	2	no	—
56	C₂×D₈	—	134	1	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	134	1	no	—
59	SD₃₂	—	134	8	no	—
60	C₂×C₁₆	—	67	2	no	—
61	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
62	C₂×Q₈	—	2	2	no	—
63	C₂×D₄	—	268	2	no	—
64	C₂×D₄	—	134	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	268	2	no	—
68	D₈	—	134	4	no	—
69	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
73	C₁₆	—	67	4	no	—
74	C₂³	—	268	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	268	2	no	—
78	D₄	—	268	2	no	—
79	D₄	—	536	4	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	#11761613	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
83	C₈	#11761613	1	2	yes	—
84	C₈	D₄×D₆₇	1	2	yes	—
85	C₂²	—	536	2	no	—
86	C₂²	—	536	4	no	—
87	C₂²	C₂×#11761613	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#14523986	1	4	yes	complemented
90	C₄	#14524039	1	1	yes	characteristic
91	C₄	C₂×#11761613	1	1	yes	characteristic
92	C₂	—	1072	2	no	—
93	C₂	#16863772	1	1	yes	characteristic
94	C₂	C₂×#14523986	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₂×C₈×C₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#494	D₆₇
hypocenter	C₆₇	#2248
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₂×#14523986	C₂	semidirect
3. C₄×C₆₇⋊C₁₆	C₂	semidirect
4. #494×C₆₇	C₂	semidirect
18. C₆₇⋊C₁₆	D₄	semidirect
41. C₄×C₆₇	SD₃₂	semidirect
48. C₆₇	#2248	semidirect
50. #494	D₆₇	semidirect
89. C₄	#14523986	semidirect