Subgroups — #22559842

Order 8576.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863794	C₂	1	1	yes	maximal, characteristic
2	#16863794	C₂	1	2	yes	maximal
3	#16863776	C₂	1	1	yes	maximal, characteristic
4	#16863747	C₂	1	2	yes	maximal
5	#322×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₆₈	—	2	2	no	—
7	#14524043	C₂²	1	1	yes	characteristic
8	#14524043	C₂²	1	2	yes	—
9	#14524043	C₂²	1	1	yes	characteristic
10	#14523975	—	2	4	no	—
11	#14523975	—	2	2	no	—
12	#14523968	—	2	4	no	—
13	C₄×C₈×C₆₇	C₂²	1	2	yes	—
14	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₃₄	—	2	2	no	—
16	C₂×Dic₁₃₄	—	2	1	no	—
17	C₂×Dic₁₃₄	—	2	1	no	—
18	Dic₂₆₈	—	4	4	no	—
19	C₄⋊Dic₆₇	—	2	2	no	—
20	C₄⋊Dic₆₇	—	2	2	no	—
21	C₄⋊Dic₆₇	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₂×C₈×C₆₇	D₄	1	4	yes	—
24	C₂×C₈×C₆₇	D₄	1	2	yes	—
25	C₄²×C₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
28	C₂×Dic₆₇	—	4	1	no	—
29	C₂×Dic₆₇	—	4	1	no	—
30	C₂×Dic₆₇	—	4	2	no	—
31	Dic₁₃₄	—	4	2	no	—
32	Dic₁₃₄	—	4	2	no	—
33	Dic₁₃₄	—	4	4	no	—
34	C₈×C₆₇	—	2	4	no	—
35	C₈×C₆₇	SD₁₆	1	4	yes	—
36	C₂²×C₆₇	#127	1	1	yes	characteristic
37	C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
38	C₄×C₆₇	#137	1	4	yes	—
39	Dic₆₇	—	8	1	no	—
40	Dic₆₇	—	8	1	no	—
41	Dic₆₇	—	8	2	no	—
42	C₂×C₆₇	#492	1	1	yes	characteristic
43	C₂×C₆₇	#497	1	2	yes	—
44	#1716	—	67	1	no	maximal, Sylow
45	C₆₇	#1716	1	1	yes	characteristic, complemented, Sylow
46	#498	—	67	2	no	—
47	#498	—	67	1	no	—
48	#494	—	67	1	no	—
49	#487	—	67	2	no	—
50	#322	D₆₇	1	1	yes	characteristic
51	C₂×Q₁₆	—	134	2	no	—
52	C₄⋊Q₈	—	67	1	no	—
53	C₄⋊Q₈	—	67	2	no	—
54	C₄⋊Q₈	—	67	1	no	—
55	#106	—	134	4	no	—
56	#106	—	134	2	no	—
57	#103	—	134	4	no	—
58	C₄×C₈	C₂×D₆₇	1	2	yes	—
59	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
60	C₂×Q₈	—	134	2	no	—
61	C₂×Q₈	—	134	1	no	—
62	C₂×Q₈	—	134	1	no	—
63	Q₁₆	—	268	4	no	—
64	C₂×C₈	D₂₆₈	1	4	yes	—
65	C₂×C₈	D₂₆₈	1	2	yes	—
66	C₄⋊C₄	—	134	2	no	—
67	C₄⋊C₄	—	134	2	no	—
68	C₄⋊C₄	—	134	2	no	—
69	C₄⋊C₄	—	134	2	no	—
70	C₄²	C₂²×D₆₇	1	1	yes	characteristic
71	Q₈	—	268	2	no	—
72	Q₈	—	268	2	no	—
73	Q₈	—	268	4	no	—
74	C₂×C₄	—	268	1	no	—
75	C₂×C₄	—	268	1	no	—
76	C₂×C₄	—	268	2	no	—
77	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
78	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	#11761617	1	4	yes	—
81	C₂²	#14524040	1	1	yes	characteristic
82	C₄	—	536	1	no	—
83	C₄	—	536	1	no	—
84	C₄	—	536	2	no	—
85	C₄	C₂×#11761617	1	2	yes	—
86	C₄	#14524076	1	4	yes	—
87	C₂	#16863769	1	1	yes	characteristic
88	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	#1716
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
45. C₆₇	#1716	semidirect