Subgroups — #22568820

Order 8576.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864163	C₂	1	2	yes	maximal
2	#16863445	C₂	1	6	yes	maximal, complemented
3	C₂×#14523979	C₂	1	6	yes	maximal, complemented
4	C₂×#131×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#14524050	—	2	8	no	—
6	C₂×#11761612	C₂²	1	12	yes	—
7	#14523979	C₂²	1	4	yes	—
8	#14523979	C₂²	1	12	yes	—
9	C₂×#41×C₆₇	C₂²	1	1	yes	characteristic
10	#131×C₆₇	C₄	1	8	yes	—
11	C₂×M₁₆×C₆₇	C₂²	1	3	yes	—
12	C₂²×C₈×C₆₇	C₂²	1	3	yes	—
13	#11761612	—	2	12	no	—
14	#11761612	C₂³	1	8	yes	—
15	C₂×C₆₇⋊C₈	—	2	12	no	—
16	#41×C₆₇	C₂×C₄	1	4	yes	—
17	#41×C₆₇	D₄	1	4	yes	—
18	C₂×Q₈×C₆₇	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₆₇	Q₈	1	3	yes	—
20	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
21	M₁₆×C₆₇	C₂×C₄	1	12	yes	—
22	C₂×C₈×C₆₇	C₂×C₄	1	12	yes	—
23	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₆₇	C₂³	1	3	yes	—
25	C₂³×C₆₇	C₂×Q₈	1	3	yes	—
26	Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
27	D₄×C₆₇	C₄⋊C₄	1	12	yes	—
28	C₂×C₄×C₆₇	C₂²×C₄	1	6	yes	—
29	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
30	C₂×C₄×C₆₇	C₂×Q₈	1	3	yes	—
31	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
32	C₈×C₆₇	C₂²×C₄	1	2	yes	—
33	C₈×C₆₇	C₂²×C₄	1	6	yes	—
34	C₆₇⋊C₈	—	2	8	no	—
35	C₂²×C₆₇	—	4	3	no	—
36	C₂²×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
37	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
38	C₄×C₆₇	C₂×C₄⋊C₄	1	6	yes	—
39	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
40	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₆₇	—	2	1	no	—
42	C₂×C₆₇	—	2	6	no	—
43	C₂×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
44	#2916	—	67	1	no	maximal, Sylow
45	C₆₇	#2916	1	1	yes	characteristic, complemented, Sylow
46	#568	—	67	2	no	—
47	C₂×#131	D₆₇	1	1	yes	characteristic
48	#430	—	67	6	no	—
49	C₂×#108	—	67	6	no	—
50	C₂×#41	C₂×D₆₇	1	1	yes	characteristic
51	#131	—	134	8	no	—
52	#131	Dic₆₇	1	8	yes	—
53	C₂×M₁₆	—	67	12	no	—
54	C₂×M₁₆	C₂×D₆₇	1	3	yes	—
55	C₂²×C₈	C₂×D₆₇	1	3	yes	—
56	#108	—	67	4	no	—
57	#108	—	67	12	no	—
58	#41	D₂₆₈	1	4	yes	—
59	#41	C₂×Dic₆₇	1	4	yes	—
60	C₂×Q₈	Dic₁₃₄	1	1	yes	characteristic
61	C₂×D₄	Dic₁₃₄	1	3	yes	—
62	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
63	M₁₆	—	67	8	no	—
64	M₁₆	—	134	12	no	—
65	M₁₆	C₂×Dic₆₇	1	12	yes	—
66	C₂×C₈	—	134	12	no	—
67	C₂×C₈	C₂×Dic₆₇	1	12	yes	—
68	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₆₇	1	3	yes	—
70	C₂³	C₂×Dic₁₃₄	1	3	yes	—
71	Q₈	C₄⋊Dic₆₇	1	4	yes	—
72	D₄	C₄⋊Dic₆₇	1	12	yes	—
73	C₂×C₄	C₂²×Dic₆₇	1	6	yes	—
74	C₂×C₄	C₂×D₂₆₈	1	6	yes	—
75	C₂×C₄	C₂×Dic₁₃₄	1	3	yes	—
76	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
77	C₈	—	134	8	no	—
78	C₈	C₂²×Dic₆₇	1	2	yes	—
79	C₈	C₂²×Dic₆₇	1	6	yes	—
80	C₂²	—	4	3	no	—
81	C₂²	C₂×C₄⋊Dic₆₇	1	6	yes	—
82	C₂²	C₂²×Dic₁₃₄	1	1	yes	characteristic
83	C₄	C₂×C₄⋊Dic₆₇	1	6	yes	—
84	C₄	C₂³×Dic₆₇	1	1	yes	characteristic
85	C₄	C₂²×D₂₆₈	1	1	yes	characteristic
86	C₂	—	2	1	no	—
87	C₂	—	2	6	no	—
88	C₂	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₂₆₈
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#131×C₆₇	C₂
socle	C₂×C₆₇	C₂²×C₄⋊C₄
hypercenter	C₂×#131	D₆₇
hypocenter	C₆₇	#2916
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
2. #16863445	C₂	semidirect
3. C₂×#14523979	C₂	semidirect
45. C₆₇	#2916	semidirect