Subgroups — #22558079

Order 8576.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863453	C₂	1	2	yes	maximal
2	C₂×C₄⋊C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
5	#14523954	C₄	1	4	yes	—
6	C₂²×C₈×C₆₇	C₄	1	2	yes	—
7	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₆₇	—	2	2	no	—
9	C₂×C₆₇⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
12	C₂²×C₄×C₆₇	Q₈	1	1	yes	characteristic
13	C₂×C₈×C₆₇	—	2	2	no	—
14	C₂×C₈×C₆₇	—	4	2	no	—
15	C₄²×C₆₇	C₂×C₄	1	2	yes	—
16	C₄²×C₆₇	D₄	1	2	yes	—
17	C₂³×C₆₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₄×C₆₇	—	2	2	no	—
19	C₂×C₄×C₆₇	—	2	1	no	—
20	C₂×C₄×C₆₇	—	2	1	no	—
21	C₂×C₄×C₆₇	—	4	1	no	—
22	C₂×C₄×C₆₇	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
26	C₈×C₆₇	—	4	2	no	—
27	C₆₇⋊C₈	—	4	4	no	—
28	C₂²×C₆₇	—	2	1	no	—
29	C₂²×C₆₇	—	2	1	no	—
30	C₂²×C₆₇	#95	1	1	yes	characteristic
31	C₂²×C₆₇	#108	1	2	yes	—
32	C₄×C₆₇	—	2	2	no	—
33	C₄×C₆₇	—	2	2	no	—
34	C₄×C₆₇	—	2	1	no	—
35	C₄×C₆₇	#100	1	1	yes	characteristic
36	C₄×C₆₇	#101	1	1	yes	characteristic
37	C₂×C₆₇	—	2	2	no	—
38	C₂×C₆₇	H(3,ℤ/4)	1	1	yes	characteristic
39	C₂×C₆₇	#341	1	1	yes	characteristic
40	C₂×C₆₇	#343	1	1	yes	characteristic
41	#1290	—	67	1	no	maximal, Sylow
42	C₆₇	#1290	1	1	yes	characteristic, complemented, Sylow
43	#432	—	67	2	no	—
44	C₂×C₄⋊C₈	D₆₇	1	1	yes	characteristic
45	C₂²×C₈	Dic₆₇	1	2	yes	—
46	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
47	C₄⋊C₈	—	2	2	no	—
48	C₄⋊C₈	—	134	2	no	—
49	C₂²⋊C₈	—	134	2	no	—
50	#97	—	67	4	no	—
51	C₂²×C₄	Dic₁₃₄	1	1	yes	characteristic
52	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
53	C₂²×C₄	C₂×Dic₆₇	1	1	yes	characteristic
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	4	2	no	—
56	C₂×C₈	—	134	4	no	—
57	C₄²	C₄×D₆₇	1	2	yes	—
58	C₄²	C₂²⋊D₆₇	1	2	yes	—
59	C₂³	C₄⋊Dic₆₇	1	1	yes	characteristic
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	1	no	—
63	C₂×C₄	—	4	1	no	—
64	C₂×C₄	C₄×Dic₆₇	1	1	yes	characteristic
65	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
66	C₂×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
67	C₂×C₄	Dic₆₇⋊C₄	1	2	yes	—
68	C₈	—	4	2	no	—
69	C₈	—	268	4	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	2	1	no	—
72	C₂²	C₆₇⋊#95	1	1	yes	characteristic
73	C₂²	#14523979	1	2	yes	—
74	C₄	—	2	2	no	—
75	C₄	—	2	2	no	—
76	C₄	—	2	1	no	—
77	C₄	#14523960	1	1	yes	characteristic
78	C₄	#14523962	1	1	yes	characteristic
79	C₂	—	2	2	no	—
80	C₂	#16863153	1	1	yes	characteristic
81	C₂	#16863161	1	1	yes	characteristic
82	C₂	#16863165	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₆₇⋊#95
commutator	C₂×C₄×C₆₇	C₄²
Frattini	C₂×C₄²	C₂×D₆₇
Fitting	C₂×C₄⋊C₈×C₆₇	C₂
socle	C₂²×C₆₇	#95
hypercenter	C₂×C₄⋊C₈	D₆₇
hypocenter	C₆₇	#1290
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
42. C₆₇	#1290	semidirect