Group #22569320 — subgroups

Subgroups — #22569320

Order 8576.

432 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 297 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	Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
2	D₄×C₆₇⋊C₈	C₂	1	6	yes	maximal, complemented
3	#16863449	C₂	1	3	yes	maximal, complemented
4	C₂×C₄⋊(C₆₇⋊C₈)	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₈	C₂²	1	6	yes	—
7	C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
8	C₄⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
9	C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂²⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
11	C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
12	C₂×#41×C₆₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₆₇	C₄	1	3	yes	—
16	C₄×D₄×C₆₇	C₂²	1	3	yes	—
17	#117×C₆₇	C₄	1	3	yes	—
18	C₂×C₄²×C₆₇	C₂²	1	3	yes	—
19	C₂×C₆₇⋊C₈	—	4	6	no	—
20	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
21	C₂×C₆₇⋊C₈	C₂³	1	6	yes	—
22	#41×C₆₇	C₈	1	8	yes	complemented
23	C₂×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₆₇	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₆₇	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₆₇	C₂×C₄	1	3	yes	—
27	C₂²×C₄×C₆₇	C₂³	1	3	yes	—
28	C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
31	C₄²×C₆₇	C₂×C₄	1	3	yes	—
32	C₄²×C₆₇	C₂×C₄	1	3	yes	—
33	C₄²×C₆₇	C₂³	1	1	yes	characteristic
34	C₄²×C₆₇	C₂³	1	3	yes	—
35	C₂³×C₆₇	C₂²×C₄	1	3	yes	—
36	Q₈×C₆₇	C₂×C₈	1	4	yes	complemented
37	D₄×C₆₇	C₂×C₈	1	12	yes	complemented
38	C₂×C₄×C₆₇	—	2	3	no	—
39	C₂×C₄×C₆₇	—	2	3	no	—
40	C₂×C₄×C₆₇	—	4	3	no	—
41	C₂×C₄×C₆₇	C₂×C₈	1	12	yes	complemented
42	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂²×C₄	1	3	yes	—
47	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
48	C₆₇⋊C₈	—	2	6	no	—
49	C₆₇⋊C₈	—	2	2	no	—
50	C₂²×C₆₇	—	2	3	no	—
51	C₂²×C₆₇	—	2	3	no	—
52	C₂²×C₆₇	C₂²×C₈	1	6	yes	—
53	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₆₇	—	2	3	no	—
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	—	2	3	no	—
57	C₄×C₆₇	C₂²×C₈	1	2	yes	—
58	C₄×C₆₇	C₂²×C₈	1	6	yes	—
59	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₆₇	—	2	6	no	—
62	C₂×C₆₇	#520	1	1	yes	characteristic
63	C₂×C₆₇	C₂³×C₈	1	1	yes	characteristic
64	C₂×C₆₇	#568	1	1	yes	characteristic
65	#2988	—	67	1	no	maximal, Sylow
66	C₆₇	#2988	1	1	yes	characteristic, complemented, Sylow
67	C₄×#41	D₆₇	1	1	yes	characteristic
68	C₈×Q₈	—	67	2	no	—
69	C₈×D₄	—	67	6	no	—
70	#431	—	67	3	no	—
71	C₂×C₄⋊C₈	—	67	3	no	—
72	C₂×#41	Dic₆₇	1	1	yes	characteristic
73	C₂²×C₈	—	67	6	no	—
74	C₄×Q₈	Dic₆₇	1	1	yes	characteristic
75	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
76	C₄×D₄	Dic₆₇	1	3	yes	—
77	C₄×D₄	C₂×D₆₇	1	3	yes	—
78	#117	Dic₆₇	1	3	yes	—
79	C₂×C₄²	C₂×D₆₇	1	3	yes	—
80	C₄⋊C₈	—	67	3	no	—
81	C₄⋊C₈	—	67	6	no	—
82	C₄⋊C₈	—	67	1	no	—
83	C₂²⋊C₈	—	67	6	no	—
84	C₄×C₈	—	67	6	no	—
85	#41	C₆₇⋊C₈	1	8	yes	complemented
86	C₂×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
87	C₂×D₄	C₂×Dic₆₇	1	3	yes	—
88	C₂²×C₄	C₂×Dic₆₇	1	3	yes	—
89	C₂²×C₄	C₂×Dic₆₇	1	3	yes	—
90	C₂²×C₄	C₂²×D₆₇	1	3	yes	—
91	C₂×C₈	—	268	6	no	—
92	C₂×C₈	—	67	2	no	—
93	C₂×C₈	—	67	6	no	—
94	C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
95	C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
96	C₂²⋊C₄	C₂×Dic₆₇	1	6	yes	—
97	C₄²	C₂×Dic₆₇	1	3	yes	—
98	C₄²	C₂×Dic₆₇	1	3	yes	—
99	C₄²	C₂²×D₆₇	1	3	yes	—
100	C₄²	C₂²×D₆₇	1	1	yes	characteristic
101	C₂³	C₂²×Dic₆₇	1	3	yes	—
102	Q₈	C₂×C₆₇⋊C₈	1	4	yes	complemented
103	D₄	C₂×C₆₇⋊C₈	1	12	yes	complemented
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	4	3	no	—
107	C₂×C₄	C₂×C₆₇⋊C₈	1	12	yes	complemented
108	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
110	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
111	C₂×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₆₇	1	3	yes	—
113	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
114	C₈	—	134	6	no	—
115	C₈	—	134	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	C₂²×C₆₇⋊C₈	1	6	yes	—
119	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
124	C₄	C₂²×C₆₇⋊C₈	1	6	yes	—
125	C₄	#14524089	1	1	yes	characteristic
126	C₄	#14524090	1	1	yes	characteristic
127	C₂	—	2	6	no	—
128	C₂	#16863879	1	1	yes	characteristic
129	C₂	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
130	C₂	#16864163	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₆₇⋊C₈

C₂

yes

maximal, complemented

D₄×C₆₇⋊C₈

C₂

yes

maximal, complemented

#16863449

C₂

yes

maximal, complemented

C₂×C₄⋊(C₆₇⋊C₈)

C₂

yes

maximal, complemented

C₄×#41×C₆₇

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₈

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

characteristic

C₂²⋊(C₆₇⋊C₈)

C₂²

yes

—

C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×#41×C₆₇

C₄

yes

characteristic

C₄×Q₈×C₆₇

C₄

yes

characteristic

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄×D₄×C₆₇

C₄

yes

—

C₄×D₄×C₆₇

C₂²

yes

—

#117×C₆₇

C₄

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₂×C₆₇⋊C₈

C₂³

yes

—

#41×C₆₇

C₈

yes

complemented

C₂×Q₈×C₆₇

C₂×C₄

yes

characteristic

C₂×D₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂³

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂²×C₄

yes

—

Q₈×C₆₇

C₂×C₈

yes

complemented

D₄×C₆₇

C₂×C₈

yes

complemented

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₈

yes

complemented

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×C₈

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²×C₈

yes

—

C₄×C₆₇

C₂²×C₈

yes

—

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

#520

yes

characteristic

C₂×C₆₇

C₂³×C₈

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₄×#41

D₆₇

yes

characteristic

C₈×Q₈

—

C₈×D₄

—

#431

—

C₂×C₄⋊C₈

—

C₂×#41

Dic₆₇

yes

characteristic

C₂²×C₈

—

C₄×Q₈

Dic₆₇

yes

characteristic

C₄×Q₈

C₂×D₆₇

yes

characteristic

C₄×D₄

Dic₆₇

yes

—

C₄×D₄

C₂×D₆₇

yes

—

#117

Dic₆₇

yes

—

C₂×C₄²

C₂×D₆₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

C₆₇⋊C₈

yes

complemented

C₂×Q₈

C₂×Dic₆₇

yes

characteristic

C₂×D₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂×C₈

—

268

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₂²⋊C₄

C₂×Dic₆₇

yes

—

C₄²

C₂×Dic₆₇

yes

—

C₄²

C₂×Dic₆₇

yes

—

C₄²

C₂²×D₆₇

yes

—

100

C₄²

C₂²×D₆₇

yes

characteristic

101

C₂³

C₂²×Dic₆₇

yes

—

102

Q₈

C₂×C₆₇⋊C₈

yes

complemented

103

D₄

C₂×C₆₇⋊C₈

yes

complemented

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₆₇⋊C₈

yes

complemented

108

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

109

C₂×C₄

C₂²×Dic₆₇

yes

—

110

C₂×C₄

C₂²×Dic₆₇

yes

—

111

C₂×C₄

C₂²×Dic₆₇

yes

characteristic

112

C₂×C₄

C₂²×Dic₆₇

yes

—

113

C₂×C₄

C₂³×D₆₇

yes

characteristic

114

C₈

—

134

—

115

C₈

—

134

—

116

C₂²

—

117

C₂²

—

118

C₂²

C₂²×C₆₇⋊C₈

yes

—

119

C₂²

C₂³×Dic₆₇

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂²×C₆₇⋊C₈

yes

—

124

C₄

C₂²×C₆₇⋊C₈

yes

—

125

C₄

#14524089

yes

characteristic

126

C₄

#14524090

yes

characteristic

127

C₂

—

128

C₂

#16863879

yes

characteristic

129

C₂

C₂³×C₆₇⋊C₈

yes

characteristic

130

C₂

#16864163

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₇
commutator	C₂×C₆₇	C₂³×C₈
Fitting	C₄×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂³×C₄
hypercenter	C₄×#41	D₆₇
hypocenter	C₆₇	#2988
LCS	G ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₇

commutator

C₂×C₆₇

C₂³×C₈

Fitting

C₄×#41×C₆₇

C₂

socle

C₂²×C₆₇

C₂³×C₄

hypercenter

C₄×#41

D₆₇

C₆₇

G ▷ C₂×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×C₆₇⋊C₈	C₂	semidirect
2. D₄×C₆₇⋊C₈	C₂	semidirect
3. #16863449	C₂	semidirect
4. C₂×C₄⋊(C₆₇⋊C₈)	C₂	semidirect
22. #41×C₆₇	C₈	semidirect
36. Q₈×C₆₇	C₂×C₈	semidirect
37. D₄×C₆₇	C₂×C₈	semidirect
41. C₂×C₄×C₆₇	C₂×C₈	semidirect
66. C₆₇	#2988	semidirect
85. #41	C₆₇⋊C₈	semidirect
102. Q₈	C₂×C₆₇⋊C₈	semidirect
103. D₄	C₂×C₆₇⋊C₈	semidirect
107. C₂×C₄	C₂×C₆₇⋊C₈	semidirect

H ≅ G/N

type

1. Q₈×C₆₇⋊C₈

C₂

semidirect

2. D₄×C₆₇⋊C₈

C₂

semidirect

3. #16863449

C₂

semidirect

4. C₂×C₄⋊(C₆₇⋊C₈)

C₂

semidirect

22. #41×C₆₇

C₈

semidirect

36. Q₈×C₆₇

C₂×C₈

semidirect

37. D₄×C₆₇

C₂×C₈

semidirect

41. C₂×C₄×C₆₇

C₂×C₈

semidirect

66. C₆₇

#2988

semidirect

85. #41

C₆₇⋊C₈

semidirect

102. Q₈

C₂×C₆₇⋊C₈

semidirect

103. D₄

C₂×C₆₇⋊C₈

semidirect

107. C₂×C₄

C₂×C₆₇⋊C₈

semidirect

Subgroups — #22569320

Conjugacy classes of subgroups

Complements