Group #22568283 — subgroups

Subgroups — #22568283

Order 8576.

456 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×#14524037	C₂	1	2	yes	maximal, complemented
2	#16863326	C₂	1	3	yes	maximal, complemented
3	#16863318	C₂	1	6	yes	maximal, complemented
4	#16863313	C₂	1	3	yes	maximal
5	#384×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₆₈	C₂²	1	1	yes	characteristic
7	#14524037	—	2	8	no	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	3	yes	—
9	C₂×Dic₆₇⋊C₄	C₂²	1	6	yes	—
10	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
11	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	6	yes	—
12	C₆₇⋊#95	C₂²	1	3	yes	—
13	C₆₇⋊#95	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
15	C₂×C₄⋊C₄×C₆₇	C₂²	1	3	yes	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂³×D₆₇	C₂³	1	2	yes	—
18	C₂×D₂₆₈	—	4	3	no	—
19	C₂²×Dic₆₇	C₂³	1	6	yes	—
20	C₄⋊Dic₆₇	—	4	3	no	—
21	Dic₆₇⋊C₄	—	2	12	no	—
22	(C₂×D₆₇)⋊C₄	—	2	12	no	—
23	(C₂×D₆₇)⋊C₄	—	4	6	no	—
24	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
26	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
27	C₄⋊C₄×C₆₇	—	4	3	no	—
28	C₄⋊C₄×C₆₇	—	4	3	no	—
29	C₄²×C₆₇	D₄	1	4	yes	complemented
30	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
31	C₂²×D₆₇	—	2	2	no	—
32	C₂²×D₆₇	—	2	6	no	—
33	C₂²×D₆₇	—	2	6	no	—
34	C₂×C₄×C₆₇	—	2	3	no	—
35	C₂×C₄×C₆₇	—	2	3	no	—
36	C₂×C₄×C₆₇	—	2	6	no	—
37	C₂×C₄×C₆₇	—	2	3	no	—
38	C₂×C₄×C₆₇	—	2	3	no	—
39	C₂×C₄×C₆₇	C₂×D₄	1	6	yes	—
40	C₂×Dic₆₇	—	2	6	no	—
41	C₂×Dic₆₇	—	2	6	no	—
42	C₂×Dic₆₇	—	2	6	no	—
43	D₂₆₈	—	4	4	no	—
44	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
45	C₂²×C₆₇	H(5,𝔽₂)	1	3	yes	—
46	C₂²×C₆₇	2₋¹⁺⁴	1	3	yes	—
47	C₂×D₆₇	—	4	6	no	—
48	C₂×D₆₇	—	4	2	no	—
49	C₂×D₆₇	—	4	6	no	—
50	C₄×C₆₇	—	2	6	no	—
51	C₄×C₆₇	—	4	1	no	—
52	C₄×C₆₇	—	4	3	no	—
53	Dic₆₇	—	4	6	no	—
54	C₂×C₆₇	#537	1	3	yes	—
55	C₂×C₆₇	#560	1	1	yes	characteristic
56	C₂×C₆₇	#563	1	3	yes	—
57	D₆₇	—	8	2	no	—
58	#2823	—	67	1	no	maximal, Sylow
59	C₆₇	#2823	1	1	yes	characteristic, complemented, Sylow
60	C₂×#126	—	67	2	no	—
61	#398	—	67	3	no	—
62	#396	—	67	6	no	—
63	#394	—	67	3	no	—
64	#384	D₆₇	1	1	yes	characteristic, complemented
65	C₂²×D₄	—	67	1	no	—
66	#126	—	134	8	no	—
67	C₂×C₄⋊C₄	—	67	3	no	—
68	C₂×C₄⋊C₄	—	67	6	no	—
69	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
70	C₂×C₄⋊C₄	C₂×D₆₇	1	3	yes	—
71	C₂×C₂²⋊C₄	—	67	6	no	—
72	C₂×C₂²⋊C₄	—	67	6	no	—
73	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
74	#95	—	67	3	no	—
75	#95	—	67	3	no	—
76	C₂⁴	—	67	2	no	—
77	C₂×D₄	—	268	3	no	—
78	C₂²×C₄	—	67	6	no	—
79	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
80	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
81	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
82	C₄⋊C₄	—	4	3	no	—
83	C₄⋊C₄	—	4	3	no	—
84	C₄⋊C₄	—	268	3	no	—
85	C₄⋊C₄	—	134	12	no	—
86	C₂²⋊C₄	—	268	6	no	—
87	C₂²⋊C₄	—	134	12	no	—
88	C₄²	D₂₆₈	1	4	yes	complemented
89	C₂³	—	134	2	no	—
90	C₂³	—	134	6	no	—
91	C₂³	—	134	6	no	—
92	C₂³	C₂³×D₆₇	1	1	yes	characteristic
93	D₄	—	268	4	no	—
94	C₂×C₄	—	2	3	no	—
95	C₂×C₄	—	2	3	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	134	6	no	—
100	C₂×C₄	—	134	6	no	—
101	C₂×C₄	—	134	6	no	—
102	C₂×C₄	C₂×D₂₆₈	1	6	yes	—
103	C₂²	—	268	2	no	—
104	C₂²	—	268	6	no	—
105	C₂²	—	268	6	no	—
106	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
107	C₂²	#14524088	1	3	yes	—
108	C₂²	#14524091	1	3	yes	—
109	C₄	—	2	6	no	—
110	C₄	—	4	1	no	—
111	C₄	—	4	3	no	—
112	C₄	—	268	6	no	—
113	C₂	—	536	2	no	—
114	C₂	#16863977	1	3	yes	—
115	C₂	#16864133	1	1	yes	characteristic
116	C₂	#16864145	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524037

C₂

yes

maximal, complemented

#16863326

C₂

yes

maximal, complemented

#16863318

C₂

yes

maximal, complemented

#16863313

C₂

yes

maximal

#384×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

C₂²

yes

characteristic

#14524037

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₆₇)⋊C₄

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂³×D₆₇

C₂³

yes

—

C₂×D₂₆₈

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

complemented

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

C₂×C₆₇

#537

yes

—

C₂×C₆₇

#560

yes

characteristic

C₂×C₆₇

#563

yes

—

D₆₇

—

#2823

—

maximal, Sylow

C₆₇

#2823

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂²×D₄

—

#126

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#95

—

#95

—

C₂⁴

—

C₂×D₄

—

268

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

268

—

C₂²⋊C₄

—

134

—

C₄²

D₂₆₈

yes

complemented

C₂³

—

134

—

C₂³

—

134

—

C₂³

—

134

—

C₂³

C₂³×D₆₇

yes

characteristic

D₄

—

268

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

134

—

100

C₂×C₄

—

134

—

101

C₂×C₄

—

134

—

102

C₂×C₄

C₂×D₂₆₈

yes

—

103

C₂²

—

268

—

104

C₂²

—

268

—

105

C₂²

—

268

—

106

C₂²

C₂²×D₂₆₈

yes

characteristic

107

C₂²

#14524088

yes

—

108

C₂²

#14524091

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

268

—

113

C₂

—

536

—

114

C₂

#16863977

yes

—

115

C₂

#16864133

yes

characteristic

116

C₂

#16864145

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator, socle	C₂³×C₆₇	C₂⁴
Fitting	#384×C₆₇	C₂
hypercenter	#384	D₆₇
hypocenter	C₆₇	#2823
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator, socle

C₂³×C₆₇

C₂⁴

Fitting

#384×C₆₇

C₂

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#14524037	C₂	semidirect
2. #16863326	C₂	semidirect
3. #16863318	C₂	semidirect
5. #384×C₆₇	C₂	semidirect
29. C₄²×C₆₇	D₄	semidirect
59. C₆₇	#2823	semidirect
64. #384	D₆₇	semidirect
88. C₄²	D₂₆₈	semidirect

H ≅ G/N

type

1. C₂×#14524037

C₂

semidirect

2. #16863326

C₂

semidirect

3. #16863318

C₂

semidirect

5. #384×C₆₇

C₂

semidirect

29. C₄²×C₆₇

D₄

semidirect

59. C₆₇

#2823

semidirect

64. #384

D₆₇

semidirect

88. C₄²

D₂₆₈

semidirect

Subgroups — #22568283

Conjugacy classes of subgroups

Complements