Subgroups — #22561993

Order 8576.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863434	C₂	1	2	yes	maximal
2	#16863224	C₂	1	2	yes	maximal, complemented
3	#16863222	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#14524002	—	2	2	no	—
6	#14523977	C₂²	1	4	yes	—
7	#14523975	—	2	2	no	—
8	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₆₇	C₄	1	4	yes	complemented
10	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₆₇	—	2	4	no	—
13	C₂²⋊Dic₆₇	—	4	2	no	—
14	C₄×Dic₆₇	—	4	2	no	—
15	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₆₇	Q₈	1	2	yes	—
19	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₁₆×C₆₇	C₂×C₄	1	2	yes	—
21	C₁₆×C₆₇	C₂×C₄	1	2	yes	—
22	C₂³×C₆₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
24	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
25	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
26	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
27	C₂×Dic₆₇	—	4	4	no	—
28	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
29	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₆₇	—	2	2	no	—
32	C₂²×C₆₇	#107	1	2	yes	—
33	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
34	C₄×C₆₇	#107	1	2	yes	—
35	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
37	Dic₆₇	—	8	4	no	—
38	C₂×C₆₇	—	2	2	no	—
39	C₂×C₆₇	C₂×#107	1	1	yes	characteristic
40	C₂×C₆₇	#508	1	2	yes	—
41	#2158	—	67	1	no	maximal, Sylow
42	C₆₇	#2158	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₁₆	D₆₇	1	1	yes	characteristic
44	#427	—	67	2	no	—
45	#367	—	67	2	no	—
46	#366	—	67	2	no	—
47	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
48	#117	—	134	2	no	—
49	C₂×C₁₆	Dic₆₇	1	4	yes	complemented
50	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
51	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
52	#107	—	67	4	no	—
53	#106	—	134	2	no	—
54	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
55	C₂×C₈	Dic₁₃₄	1	2	yes	—
56	C₂×C₈	D₂₆₈	1	1	yes	characteristic
57	C₂×C₈	C₂×Dic₆₇	1	2	yes	—
58	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
59	C₄⋊C₄	—	134	4	no	—
60	C₂²⋊C₄	—	268	2	no	—
61	C₄²	—	268	2	no	—
62	C₁₆	C₂×Dic₆₇	1	2	yes	—
63	C₁₆	C₂×Dic₆₇	1	2	yes	—
64	C₂³	D₅₃₆	1	1	yes	characteristic
65	C₂×C₄	—	268	4	no	—
66	C₂×C₄	C₄⋊Dic₆₇	1	2	yes	—
67	C₂×C₄	D₅₃₆	1	1	yes	characteristic
68	C₂×C₄	Dic₂₆₈	1	2	yes	—
69	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
70	C₈	C₄⋊Dic₆₇	1	2	yes	—
71	C₈	C₂²×Dic₆₇	1	1	yes	characteristic
72	C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
73	C₂²	—	2	2	no	—
74	C₂²	#14523977	1	2	yes	—
75	C₂²	C₂×D₅₃₆	1	1	yes	characteristic
76	C₄	—	536	4	no	—
77	C₄	#14523977	1	2	yes	—
78	C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
79	C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	C₂×#14523977	1	1	yes	characteristic
82	C₂	#16863828	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₅₃₆
commutator	C₈×C₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	C₂²×C₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	C₂²×C₁₆	D₆₇
hypocenter	C₆₇	#2158
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. #16863224	C₂	semidirect
3. #16863222	C₂	semidirect
9. C₂×C₁₆×C₆₇	C₄	semidirect
42. C₆₇	#2158	semidirect
49. C₂×C₁₆	Dic₆₇	semidirect