Subgroups — #22559880

Order 8576.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863801	C₂	1	2	yes	maximal, complemented
2	#16863776	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863769	C₂	1	1	yes	maximal, characteristic
4	#16863743	C₂	1	2	yes	maximal
5	#322×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₆₈	—	2	2	no	—
7	C₂×#11761617	—	2	2	no	—
8	#14524043	C₂²	1	1	yes	characteristic
9	#14524043	C₂²	1	2	yes	—
10	#14524040	C₂²	1	1	yes	characteristic
11	#14523977	—	2	4	no	—
12	#14523965	—	2	4	no	—
13	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₇	C₂²	1	2	yes	—
15	C₂×Dic₁₃₄	—	2	1	no	—
16	C₂×Dic₁₃₄	—	2	2	no	—
17	C₂×D₂₆₈	—	2	2	no	—
18	C₂×D₂₆₈	—	2	1	no	—
19	Dic₂₆₈	—	4	4	no	—
20	#11761617	—	4	4	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	C₂×C₈×C₆₇	D₄	1	4	yes	—
25	C₂×C₈×C₆₇	D₄	1	2	yes	—
26	C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×D₆₇	—	4	1	no	—
28	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
30	C₂×Dic₆₇	—	4	2	no	—
31	C₂×Dic₆₇	—	4	1	no	—
32	D₂₆₈	—	4	2	no	—
33	D₂₆₈	—	4	4	no	—
34	Dic₁₃₄	—	4	2	no	—
35	Dic₁₃₄	—	4	4	no	—
36	C₈×C₆₇	—	2	4	no	—
37	C₈×C₆₇	D₈	1	4	yes	—
38	C₂²×C₆₇	#127	1	1	yes	characteristic
39	C₂×D₆₇	—	8	1	no	—
40	C₂×D₆₇	—	8	2	no	—
41	C₄×C₆₇	C₂×D₈	1	2	yes	—
42	C₄×C₆₇	#137	1	4	yes	—
43	Dic₆₇	—	8	1	no	—
44	Dic₆₇	—	8	2	no	—
45	C₂×C₆₇	#493	1	1	yes	characteristic
46	C₂×C₆₇	#497	1	2	yes	—
47	D₆₇	—	16	1	no	—
48	#1722	—	67	1	no	maximal, Sylow
49	C₆₇	#1722	1	1	yes	characteristic, complemented, Sylow
50	#500	—	67	2	no	—
51	#494	—	67	1	no	—
52	#492	—	67	1	no	—
53	#486	—	67	2	no	—
54	#322	D₆₇	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	134	2	no	—
56	C₂×SD₁₆	—	134	2	no	—
57	C₄⋊Q₈	—	67	2	no	—
58	C₄⋊Q₈	—	67	1	no	—
59	#127	—	67	1	no	—
60	#107	—	134	4	no	—
61	#102	—	134	4	no	—
62	C₄×C₈	C₂×D₆₇	1	2	yes	—
63	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
64	C₂×Q₈	—	134	1	no	—
65	C₂×Q₈	—	134	2	no	—
66	C₂×D₄	—	134	2	no	—
67	C₂×D₄	—	134	1	no	—
68	Q₁₆	—	268	4	no	—
69	SD₁₆	—	268	4	no	—
70	C₂×C₈	D₂₆₈	1	4	yes	—
71	C₂×C₈	D₂₆₈	1	2	yes	—
72	C₄⋊C₄	—	134	2	no	—
73	C₄⋊C₄	—	134	2	no	—
74	C₄⋊C₄	—	134	2	no	—
75	C₄²	C₂²×D₆₇	1	1	yes	characteristic
76	C₂³	—	268	1	no	—
77	Q₈	—	268	2	no	—
78	Q₈	—	268	4	no	—
79	D₄	—	268	2	no	—
80	D₄	—	268	4	no	—
81	C₂×C₄	—	268	2	no	—
82	C₂×C₄	—	268	1	no	—
83	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
84	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₅₃₆	1	4	yes	—
87	C₂²	—	536	1	no	—
88	C₂²	—	536	2	no	—
89	C₂²	#14524040	1	1	yes	characteristic
90	C₄	—	536	1	no	—
91	C₄	—	536	2	no	—
92	C₄	C₂×D₅₃₆	1	2	yes	—
93	C₄	#14524076	1	4	yes	—
94	C₂	—	1072	1	no	—
95	C₂	#16863773	1	1	yes	characteristic
96	C₂	#16863790	1	2	yes	—

	subgroup	G/H
center	C₂²	#14524040
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#322×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	#322	D₆₇
hypocenter	C₆₇	#1722
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #16863801	C₂	semidirect
2. #16863776	C₂	semidirect
5. #322×C₆₇	C₂	semidirect
49. C₆₇	#1722	semidirect
54. #322	D₆₇	semidirect