Group #22563315 — subgroups

Subgroups — #22563315

Order 8576.

448 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 221 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16863282	C₂	1	4	yes	maximal
2	#16863282	C₂	1	4	yes	maximal
3	C₄⋊C₄×Dic₆₇	C₂	1	4	yes	maximal
4	#16863242	C₂	1	2	yes	maximal
5	C₂×#125×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
7	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
8	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
9	C₆₇⋊#95	C₂²	1	2	yes	—
10	C₆₇⋊#95	C₂²	1	4	yes	—
11	C₆₇⋊#95	C₂²	1	2	yes	—
12	C₆₇⋊#95	C₂²	1	4	yes	—
13	C₆₇⋊#95	C₂²	1	4	yes	—
14	#125×C₆₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×Dic₆₇	C₂³	1	4	yes	—
19	C₂²×Dic₆₇	C₂³	1	4	yes	—
20	C₄⋊Dic₆₇	—	2	8	no	—
21	C₄×Dic₆₇	—	2	8	no	—
22	C₄×Dic₆₇	—	2	8	no	—
23	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
26	C₄⋊C₄×C₆₇	C₂×C₄	1	16	yes	—
27	C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
28	C₄²×C₆₇	C₂×C₄	1	4	yes	—
29	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₆₇	—	2	4	no	—
31	C₂×C₄×C₆₇	—	2	4	no	—
32	C₂×C₄×C₆₇	—	2	4	no	—
33	C₂×C₄×C₆₇	—	2	2	no	—
34	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
35	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₆₇	C₂²×C₄	1	8	yes	—
37	C₂×Dic₆₇	—	2	4	no	—
38	C₂×Dic₆₇	—	2	4	no	—
39	C₂×Dic₆₇	—	2	4	no	—
40	C₂×Dic₆₇	—	2	4	no	—
41	C₂×Dic₆₇	—	2	4	no	—
42	C₂×Dic₆₇	#41	1	8	yes	—
43	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
44	C₂²×C₆₇	C₂×#41	1	4	yes	—
45	C₂²×C₆₇	2₋¹⁺⁴	1	2	yes	—
46	C₄×C₆₇	—	2	4	no	—
47	C₄×C₆₇	—	2	8	no	—
48	C₄×C₆₇	—	2	2	no	—
49	Dic₆₇	—	2	8	no	—
50	Dic₆₇	—	4	4	no	—
51	C₂×C₆₇	C₄×#41	1	2	yes	—
52	C₂×C₆₇	#519	1	1	yes	characteristic
53	C₂×C₆₇	#552	1	4	yes	—
54	#2371	—	67	1	no	maximal, Sylow
55	C₆₇	#2371	1	1	yes	characteristic, complemented, Sylow
56	C₂×#125	D₆₇	1	1	yes	characteristic
57	#387	—	67	4	no	—
58	#387	—	67	4	no	—
59	C₄×C₄⋊C₄	—	67	4	no	—
60	#376	—	67	2	no	—
61	#125	Dic₆₇	1	8	yes	complemented
62	C₂×C₄⋊C₄	—	67	4	no	—
63	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
64	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
65	C₂×C₄²	—	67	4	no	—
66	C₂×C₄²	—	67	4	no	—
67	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
68	#95	—	67	2	no	—
69	#95	—	67	4	no	—
70	#95	—	67	2	no	—
71	#95	—	67	4	no	—
72	#95	—	67	4	no	—
73	C₂²×C₄	—	67	4	no	—
74	C₂²×C₄	—	67	4	no	—
75	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
76	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
77	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
78	C₄⋊C₄	—	134	8	no	—
79	C₄⋊C₄	C₂×Dic₆₇	1	16	yes	—
80	C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
81	C₄²	—	134	8	no	—
82	C₄²	—	134	8	no	—
83	C₄²	C₂×Dic₆₇	1	4	yes	—
84	C₂³	C₂³×D₆₇	1	1	yes	characteristic
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	67	8	no	—
90	C₂×C₄	—	134	4	no	—
91	C₂×C₄	—	134	4	no	—
92	C₂×C₄	—	134	4	no	—
93	C₂×C₄	—	134	4	no	—
94	C₂×C₄	—	134	4	no	—
95	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
96	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
97	C₂×C₄	C₂²×Dic₆₇	1	8	yes	—
98	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
99	C₂²	#41×D₆₇	1	4	yes	—
100	C₂²	#14524091	1	2	yes	—
101	C₄	—	2	4	no	—
102	C₄	—	2	8	no	—
103	C₄	—	2	2	no	—
104	C₄	—	268	4	no	—
105	C₄	—	134	8	no	—
106	C₂	#41×Dic₆₇	1	2	yes	—
107	C₂	#16863874	1	1	yes	characteristic
108	C₂	#16864078	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863282

C₂

yes

maximal

#16863282

C₂

yes

maximal

C₄⋊C₄×Dic₆₇

C₂

yes

maximal

#16863242

C₂

yes

maximal

C₂×#125×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

#125×C₆₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

#41

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₄×#41

yes

—

C₂×C₆₇

#519

yes

characteristic

C₂×C₆₇

#552

yes

—

#2371

—

maximal, Sylow

C₆₇

#2371

yes

characteristic, complemented, Sylow

C₂×#125

D₆₇

yes

characteristic

#387

—

#387

—

C₄×C₄⋊C₄

—

#376

—

#125

Dic₆₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

—

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

134

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄²

—

134

—

C₄²

—

134

—

C₄²

C₂×Dic₆₇

yes

—

C₂³

C₂³×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂×C₄

C₂²×Dic₆₇

yes

—

C₂²

C₂³×Dic₆₇

yes

characteristic

C₂²

#41×D₆₇

yes

—

100

C₂²

#14524091

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

268

—

105

C₄

—

134

—

106

C₂

#41×Dic₆₇

yes

—

107

C₂

#16863874

yes

characteristic

108

C₂

#16864078

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₄
Fitting	C₂×#125×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×#125	D₆₇
hypocenter	C₆₇	#2371
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator

C₂²×C₆₇

C₂³×C₄

Fitting

C₂×#125×C₆₇

C₂

socle

C₂³×C₆₇

C₂⁴

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
14. #125×C₆₇	C₄	semidirect
55. C₆₇	#2371	semidirect
61. #125	Dic₆₇	semidirect

H ≅ G/N

type

14. #125×C₆₇

C₄

semidirect

55. C₆₇

#2371

semidirect

61. #125

Dic₆₇

semidirect

Subgroups — #22563315

Conjugacy classes of subgroups

Complements