Group #22568395 — subgroups

Subgroups — #22568395

Order 8576.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₆₇⋊#401	C₂	1	2	yes	maximal, complemented
2	#16863329	C₂	1	6	yes	maximal, complemented
3	#16863321	C₂	1	3	yes	maximal
4	#16863268	C₂	1	3	yes	maximal
5	#399×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂³×Dic₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₇	C₂²	1	3	yes	—
8	C₂×Dic₆₇⋊C₄	C₂²	1	6	yes	—
9	C₂×C₂²⋊Dic₆₇	C₂²	1	3	yes	—
10	C₆₇⋊#95	C₂²	1	6	yes	—
11	C₆₇⋊#95	C₂²	1	3	yes	—
12	C₆₇⋊#95	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
14	C₂×C₂²⋊C₄×C₆₇	C₂²	1	3	yes	—
15	#95×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×Dic₆₇	—	4	3	no	—
17	C₂²×Dic₆₇	C₂³	1	6	yes	—
18	C₂²×Dic₆₇	C₂³	1	2	yes	—
19	C₄⋊Dic₆₇	—	4	3	no	—
20	Dic₆₇⋊C₄	—	4	6	no	—
21	C₂²⋊Dic₆₇	—	2	6	no	—
22	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
24	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
25	C₄⋊C₄×C₆₇	—	4	3	no	—
26	C₂²⋊C₄×C₆₇	—	4	3	no	—
27	C₂³×C₆₇	—	2	3	no	—
28	C₂³×C₆₇	—	2	3	no	—
29	C₂³×C₆₇	#41	1	2	yes	—
30	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₆₇	—	2	3	no	—
32	C₂×C₄×C₆₇	—	2	3	no	—
33	C₂×C₄×C₆₇	—	2	6	no	—
34	C₂×C₄×C₆₇	—	2	6	no	—
35	C₂×Dic₆₇	—	2	6	no	—
36	C₂×Dic₆₇	—	2	6	no	—
37	C₂×Dic₆₇	—	2	6	no	—
38	C₂×Dic₆₇	—	2	6	no	—
39	C₂×Dic₆₇	—	4	4	no	—
40	C₂²×C₆₇	—	2	6	no	—
41	C₂²×C₆₇	—	4	1	no	—
42	C₂²×C₆₇	—	4	3	no	—
43	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
44	C₂²×C₆₇	H(5,𝔽₂)	1	3	yes	—
45	C₂²×C₆₇	2₋¹⁺⁴	1	3	yes	—
46	C₄×C₆₇	—	4	3	no	—
47	C₄×C₆₇	—	4	3	no	—
48	Dic₆₇	—	4	6	no	—
49	Dic₆₇	—	4	2	no	—
50	C₂×C₆₇	—	4	2	no	—
51	C₂×C₆₇	#539	1	3	yes	—
52	C₂×C₆₇	#560	1	1	yes	characteristic
53	C₂×C₆₇	#563	1	3	yes	—
54	#2834	—	67	1	no	maximal, Sylow
55	C₆₇	#2834	1	1	yes	characteristic, complemented, Sylow
56	#401	—	67	2	no	—
57	#399	D₆₇	1	1	yes	characteristic
58	#398	—	67	6	no	—
59	#397	—	67	3	no	—
60	#385	—	67	3	no	—
61	C₂³×C₄	—	67	1	no	—
62	C₂×C₄⋊C₄	—	67	6	no	—
63	C₂×C₄⋊C₄	—	67	3	no	—
64	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
65	C₂×C₂²⋊C₄	—	67	3	no	—
66	C₂×C₂²⋊C₄	C₂×D₆₇	1	3	yes	—
67	#95	—	67	6	no	—
68	#95	—	67	3	no	—
69	#95	—	67	6	no	—
70	#95	C₂×D₆₇	1	1	yes	characteristic
71	C₂⁴	C₂²×D₆₇	1	1	yes	characteristic
72	C₂²×C₄	—	268	3	no	—
73	C₂²×C₄	—	67	6	no	—
74	C₂²×C₄	—	67	2	no	—
75	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
76	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
77	C₄⋊C₄	—	4	3	no	—
78	C₄⋊C₄	—	268	6	no	—
79	C₄⋊C₄	—	268	3	no	—
80	C₂²⋊C₄	—	4	3	no	—
81	C₂²⋊C₄	—	134	6	no	—
82	C₂³	—	2	3	no	—
83	C₂³	—	2	3	no	—
84	C₂³	#11761629	1	2	yes	—
85	C₂³	C₂³×D₆₇	1	1	yes	characteristic
86	C₂×C₄	—	2	3	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	2	6	no	—
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	268	4	no	—
91	C₂×C₄	—	134	6	no	—
92	C₂×C₄	—	134	6	no	—
93	C₂×C₄	—	134	6	no	—
94	C₂×C₄	—	134	6	no	—
95	C₂²	—	2	6	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	3	no	—
98	C₂²	C₂×#11761629	1	1	yes	characteristic
99	C₂²	#14524088	1	3	yes	—
100	C₂²	#14524090	1	3	yes	—
101	C₄	—	4	3	no	—
102	C₄	—	4	3	no	—
103	C₄	—	268	2	no	—
104	C₄	—	268	6	no	—
105	C₂	—	4	2	no	—
106	C₂	#16863995	1	3	yes	—
107	C₂	#16864133	1	1	yes	characteristic
108	C₂	#16864149	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇⋊#401

C₂

yes

maximal, complemented

#16863329

C₂

yes

maximal, complemented

#16863321

C₂

yes

maximal

#16863268

C₂

yes

maximal

#399×C₆₇

C₂

yes

maximal, characteristic

C₂³×Dic₆₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×C₂²⋊Dic₆₇

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₂⁴×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#41

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#539

yes

—

C₂×C₆₇

#560

yes

characteristic

C₂×C₆₇

#563

yes

—

#2834

—

maximal, Sylow

C₆₇

#2834

yes

characteristic, complemented, Sylow

#401

—

#399

D₆₇

yes

—

—

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₆₇

yes

—

—

—

—

C₂×D₆₇

yes

characteristic

C₂⁴

C₂²×D₆₇

yes

characteristic

C₂²×C₄

—

268

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

268

—

C₂²⋊C₄

—

C₂²⋊C₄

—

134

—

C₂³

—

C₂³

—

C₂³

#11761629

yes

—

C₂³

C₂³×D₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

268

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂×C₄

—

134

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×#11761629

yes

characteristic

C₂²

#14524088

yes

—

100

C₂²

#14524090

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

268

—

104

C₄

—

268

—

105

C₂

—

106

C₂

#16863995

yes

—

107

C₂

#16864133

yes

characteristic

108

C₂

#16864149

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#399×C₆₇	C₂
hypercenter	#399	D₆₇
hypocenter	C₆₇	#2834
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#399×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. C₆₇⋊#401	C₂	semidirect
2. #16863329	C₂	semidirect
55. C₆₇	#2834	semidirect

H ≅ G/N

type

1. C₆₇⋊#401

C₂

semidirect

2. #16863329

C₂

semidirect

55. C₆₇

#2834

semidirect

Subgroups — #22568395

Conjugacy classes of subgroups

Complements