Subgroups — #22559878

Order 8576.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863799	C₂	1	2	yes	maximal, complemented
2	#16863776	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863742	C₂	1	2	yes	maximal
4	#16863117	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	—	2	2	no	—
7	#14524043	C₂²	1	2	yes	—
8	#14523976	—	2	4	no	—
9	#14523964	—	2	4	no	—
10	C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₆₇	—	2	2	no	—
12	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#127×C₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₃₄	—	2	2	no	—
16	Dic₂₆₈	—	4	4	no	—
17	C₂×C₆₇⋊C₈	D₄	1	4	yes	—
18	C₄⋊Dic₆₇	—	2	2	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₂×Q₈×C₆₇	—	2	1	no	—
21	C₂×D₄×C₆₇	—	2	1	no	—
22	C₂×D₄×C₆₇	—	2	2	no	—
23	SD₁₆×C₆₇	—	4	4	no	—
24	C₂×C₈×C₆₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₆₇	—	2	2	no	—
26	C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₂³×C₆₇	—	4	1	no	—
28	Q₈×C₆₇	—	4	2	no	—
29	D₄×C₆₇	—	4	2	no	—
30	D₄×C₆₇	—	4	4	no	—
31	C₂×C₄×C₆₇	—	4	1	no	—
32	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₆₇	—	4	2	no	—
35	Dic₁₃₄	—	4	4	no	—
36	C₈×C₆₇	D₈	1	4	yes	—
37	C₆₇⋊C₈	—	2	4	no	—
38	C₂²×C₆₇	—	8	1	no	—
39	C₂²×C₆₇	—	8	2	no	—
40	C₂²×C₆₇	#127	1	1	yes	characteristic
41	C₄×C₆₇	—	8	1	no	—
42	C₄×C₆₇	C₂×D₈	1	2	yes	—
43	C₄×C₆₇	#137	1	4	yes	—
44	Dic₆₇	—	8	2	no	—
45	C₂×C₆₇	—	16	1	no	—
46	C₂×C₆₇	#493	1	1	yes	characteristic
47	C₂×C₆₇	#497	1	2	yes	—
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	D₆₇	1	1	yes	characteristic
53	#486	—	67	2	no	—
54	#322	—	67	1	no	—
55	C₂×Q₁₆	—	134	2	no	—
56	C₂×SD₁₆	—	2	2	no	—
57	C₄⋊Q₈	—	67	2	no	—
58	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
59	#127	C₂×D₆₇	1	1	yes	characteristic
60	#107	—	134	4	no	—
61	#102	—	134	4	no	—
62	C₄×C₈	—	67	2	no	—
63	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	134	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	1	no	—
68	Q₁₆	—	268	4	no	—
69	SD₁₆	—	4	4	no	—
70	C₂×C₈	—	67	4	no	—
71	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	134	2	no	—
74	C₄⋊C₄	—	134	2	no	—
75	C₄²	C₂²×D₆₇	1	1	yes	characteristic
76	C₂³	—	4	1	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	268	4	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	4	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	268	2	no	—
83	C₂×C₄	D₄×D₆₇	1	2	yes	—
84	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
85	C₈	—	134	4	no	—
86	C₈	#11761613	1	4	yes	—
87	C₂²	—	8	1	no	—
88	C₂²	—	8	2	no	—
89	C₂²	#14524039	1	1	yes	characteristic
90	C₄	—	8	1	no	—
91	C₄	—	536	2	no	—
92	C₄	C₂×#11761613	1	2	yes	—
93	C₄	#14524072	1	4	yes	—
94	C₂	—	16	1	no	—
95	C₂	#16863772	1	1	yes	characteristic
96	C₂	#16863787	1	2	yes	—

	subgroup	G/H
center	C₂²	#14524039
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#492×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	#492	D₆₇
hypocenter	C₆₇	#1722
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. #16863799	C₂	semidirect
2. #16863776	C₂	semidirect
4. #16863117	C₂	semidirect
49. C₆₇	#1722	semidirect