Subgroups — #22559810

Order 8576.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863794	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863769	C₂	1	1	yes	maximal, characteristic
3	#16863747	C₂	1	2	yes	maximal, complemented
4	#16863743	C₂	1	2	yes	maximal
5	C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761617	—	2	2	no	—
7	#14524043	C₂²	1	1	yes	characteristic
8	#14524043	C₂²	1	2	yes	—
9	#14524040	C₂²	1	1	yes	characteristic
10	#14523975	—	2	2	no	—
11	#14523968	—	2	4	no	—
12	#14523965	—	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×D₂₆₈	—	2	2	no	—
18	C₂×D₂₆₈	—	2	1	no	—
19	#11761617	—	4	4	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	C₂²×D₆₇	—	4	1	no	—
27	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
29	C₂×Dic₆₇	—	4	1	no	—
30	C₂×Dic₆₇	—	4	2	no	—
31	D₂₆₈	—	4	2	no	—
32	D₂₆₈	—	4	4	no	—
33	Dic₁₃₄	—	4	2	no	—
34	Dic₁₃₄	—	4	4	no	—
35	C₈×C₆₇	—	2	4	no	—
36	C₈×C₆₇	SD₁₆	1	4	yes	—
37	C₂²×C₆₇	#127	1	1	yes	characteristic
38	C₂×D₆₇	—	8	1	no	—
39	C₂×D₆₇	—	8	2	no	—
40	C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
41	C₄×C₆₇	#135	1	4	yes	—
42	Dic₆₇	—	8	1	no	—
43	Dic₆₇	—	8	2	no	—
44	C₂×C₆₇	#492	1	1	yes	characteristic
45	C₂×C₆₇	#495	1	2	yes	—
46	D₆₇	—	16	1	no	—
47	#1708	—	67	1	no	maximal, Sylow
48	C₆₇	#1708	1	1	yes	characteristic, complemented, Sylow
49	#498	—	67	1	no	—
50	#492	—	67	1	no	—
51	#487	—	67	2	no	—
52	#486	—	67	2	no	—
53	C₈²	D₆₇	1	1	yes	characteristic, complemented
54	C₂×SD₁₆	—	134	2	no	—
55	C₄⋊Q₈	—	67	2	no	—
56	C₄⋊Q₈	—	67	1	no	—
57	#127	—	67	1	no	—
58	#106	—	134	2	no	—
59	#103	—	134	4	no	—
60	#102	—	134	4	no	—
61	C₄×C₈	C₂×D₆₇	1	2	yes	—
62	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
63	C₂×Q₈	—	134	2	no	—
64	C₂×Q₈	—	134	1	no	—
65	C₂×D₄	—	134	2	no	—
66	C₂×D₄	—	134	1	no	—
67	SD₁₆	—	268	4	no	—
68	C₂×C₈	D₂₆₈	1	4	yes	—
69	C₂×C₈	D₂₆₈	1	2	yes	—
70	C₄⋊C₄	—	134	2	no	—
71	C₄⋊C₄	—	134	2	no	—
72	C₄⋊C₄	—	134	2	no	—
73	C₄²	C₂²×D₆₇	1	1	yes	characteristic
74	C₂³	—	268	1	no	—
75	Q₈	—	268	2	no	—
76	Q₈	—	268	4	no	—
77	D₄	—	268	2	no	—
78	D₄	—	268	4	no	—
79	C₂×C₄	—	268	1	no	—
80	C₂×C₄	—	268	2	no	—
81	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
82	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	#11761617	1	4	yes	—
85	C₂²	—	536	1	no	—
86	C₂²	—	536	2	no	—
87	C₂²	#14524040	1	1	yes	characteristic
88	C₄	—	536	1	no	—
89	C₄	—	536	2	no	—
90	C₄	C₂×#11761617	1	2	yes	—
91	C₄	#14524065	1	4	yes	—
92	C₂	—	1072	1	no	—
93	C₂	#16863769	1	1	yes	characteristic
94	C₂	#16863780	1	2	yes	—

	subgroup	G/H
center	C₂²	#14524040
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	C₈²×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	C₈²	D₆₇
hypocenter	C₆₇	#1708
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #16863794	C₂	semidirect
3. #16863747	C₂	semidirect
5. C₈²×C₆₇	C₂	semidirect
48. C₆₇	#1708	semidirect
53. C₈²	D₆₇	semidirect