Group #30682682 — subgroups

Subgroups — #30682682

Order 16832.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22554462	C₂	1	1	yes	maximal, characteristic, complemented
2	#22554458	C₂	1	1	yes	maximal, characteristic, complemented
3	#22554454	C₂	1	2	yes	maximal, complemented
4	#22554452	C₂	1	2	yes	maximal, complemented
5	#22554447	C₂	1	2	yes	maximal, complemented
6	#22554446	C₂	1	2	yes	maximal, complemented
7	#22554443	C₂	1	2	yes	maximal, complemented
8	#22554442	C₂	1	2	yes	maximal, complemented
9	#124×C₂₆₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₅₂₆	C₂²	1	1	yes	characteristic
11	C₂×D₁₀₅₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₂₆₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₂₆₃	C₂²	1	2	yes	—
14	C₂²×Dic₂₆₃	C₂²	1	2	yes	—
15	C₂×C₄×D₂₆₃	C₂²	1	2	yes	—
16	C₄⋊Dic₂₆₃	C₂²	1	2	yes	—
17	C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₂₆₃	C₂²	1	1	yes	characteristic
19	Dic₂₆₃⋊C₄	C₂²	1	2	yes	—
20	Dic₂₆₃⋊C₄	C₂²	1	2	yes	—
21	Dic₂₆₃⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₆₃	C₂²	1	2	yes	—
23	(C₂×D₂₆₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₂₆₃)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₂₆₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₂₆₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₂₆₃	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₂₆₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₆₃	C₂²	1	2	yes	—
30	C₄²×C₂₆₃	C₂²	1	1	yes	characteristic
31	C₂²×D₂₆₃	C₂³	1	2	yes	—
32	Dic₅₂₆	—	4	1	no	—
33	D₁₀₅₂	—	4	1	no	—
34	C₂²⋊D₂₆₃	—	4	2	no	—
35	C₂²⋊D₂₆₃	—	4	2	no	—
36	C₂×Dic₂₆₃	—	4	2	no	—
37	C₂×Dic₂₆₃	C₂³	1	2	yes	—
38	C₂×Dic₂₆₃	C₂³	1	2	yes	—
39	C₂×Dic₂₆₃	C₂³	1	2	yes	—
40	C₄×D₂₆₃	—	4	2	no	—
41	C₂³×C₂₆₃	C₂³	1	2	yes	—
42	Q₈×C₂₆₃	—	4	1	no	—
43	D₄×C₂₆₃	—	4	1	no	—
44	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₂₆₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₆₃	C₂³	1	2	yes	—
48	C₂²×C₂₆₃	—	2	2	no	—
49	C₂²×C₂₆₃	—	2	2	no	—
50	C₂²×C₂₆₃	—	2	2	no	—
51	C₂²×C₂₆₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₂₆₃	—	2	2	no	—
53	C₂×D₂₆₃	—	2	2	no	—
54	C₂×D₂₆₃	—	2	2	no	—
55	C₄×C₂₆₃	—	2	1	no	—
56	C₄×C₂₆₃	—	2	2	no	—
57	C₄×C₂₆₃	—	2	1	no	—
58	C₄×C₂₆₃	—	2	1	no	—
59	Dic₂₆₃	—	2	2	no	—
60	Dic₂₆₃	—	2	2	no	—
61	Dic₂₆₃	—	2	2	no	—
62	C₂×C₂₆₃	—	4	2	no	—
63	C₂×C₂₆₃	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₂₆₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₂₆₃	2₋¹⁺⁴	1	1	yes	characteristic
66	D₂₆₃	—	4	2	no	—
67	C₂₆₃	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	263	1	no	maximal, Sylow
69	#125	—	263	1	no	—
70	#124	—	263	1	no	—
71	#124	D₂₆₃	1	1	yes	characteristic, complemented
72	#123	—	263	2	no	—
73	#123	—	263	2	no	—
74	C₂²⋊Q₈	—	263	2	no	—
75	C₂²⋊Q₈	—	263	2	no	—
76	#121	—	263	2	no	—
77	#121	—	263	2	no	—
78	C₂×Q₈	—	263	1	no	—
79	C₂×Q₈	C₂×D₂₆₃	1	1	yes	characteristic
80	C₂×D₄	—	263	2	no	—
81	C₂×D₄	—	263	2	no	—
82	C₂×D₄	—	263	1	no	—
83	C₂×D₄	C₂×D₂₆₃	1	1	yes	characteristic
84	C₂²×C₄	—	263	2	no	—
85	C₂²×C₄	—	263	2	no	—
86	C₄⋊C₄	—	263	2	no	—
87	C₄⋊C₄	—	263	2	no	—
88	C₄⋊C₄	—	263	2	no	—
89	C₄⋊C₄	—	263	2	no	—
90	C₄⋊C₄	—	263	1	no	—
91	C₄⋊C₄	—	263	1	no	—
92	C₂²⋊C₄	—	263	2	no	—
93	C₂²⋊C₄	—	263	2	no	—
94	C₂²⋊C₄	—	263	2	no	—
95	C₂²⋊C₄	—	263	2	no	—
96	C₂²⋊C₄	C₂×D₂₆₃	1	2	yes	—
97	C₂²⋊C₄	C₂×D₂₆₃	1	2	yes	—
98	C₄²	C₂×D₂₆₃	1	1	yes	characteristic
99	C₂³	—	263	2	no	—
100	C₂³	C₂²×D₂₆₃	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	1052	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	1052	1	no	—
105	D₄	—	1052	2	no	—
106	D₄	—	1052	2	no	—
107	C₂×C₄	—	263	2	no	—
108	C₂×C₄	—	263	2	no	—
109	C₂×C₄	—	263	2	no	—
110	C₂×C₄	—	1052	2	no	—
111	C₂×C₄	—	1052	2	no	—
112	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₂₆₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₂₆₃	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	526	2	no	—
120	C₂²	—	526	2	no	—
121	C₂²	—	526	2	no	—
122	C₂²	C₂³×D₂₆₃	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	526	2	no	—
128	C₄	—	526	2	no	—
129	C₄	—	526	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	1052	2	no	—
132	C₂	#22554515	1	1	yes	characteristic
133	C₂	#22554516	1	1	yes	characteristic
134	C₂	#22554517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22554462

C₂

yes

maximal, characteristic, complemented

#22554458

C₂

yes

maximal, characteristic, complemented

#22554454

C₂

yes

maximal, complemented

#22554452

C₂

yes

maximal, complemented

#22554447

C₂

yes

maximal, complemented

#22554446

C₂

yes

maximal, complemented

#22554443

C₂

yes

maximal, complemented

#22554442

C₂

yes

maximal, complemented

#124×C₂₆₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₂₆

C₂²

yes

characteristic

C₂×D₁₀₅₂

C₂²

yes

characteristic

C₂×C₂²⋊D₂₆₃

C₂²

yes

—

C₂×C₂²⋊D₂₆₃

C₂²

yes

—

C₂²×Dic₂₆₃

C₂²

yes

—

C₂×C₄×D₂₆₃

C₂²

yes

—

C₄⋊Dic₂₆₃

C₂²

yes

—

C₄⋊Dic₂₆₃

C₂²

yes

characteristic

C₄⋊Dic₂₆₃

C₂²

yes

characteristic

Dic₂₆₃⋊C₄

C₂²

yes

—

Dic₂₆₃⋊C₄

C₂²

yes

—

Dic₂₆₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₂₆₃

C₂²

yes

—

(C₂×D₂₆₃)⋊C₄

C₂²

yes

—

(C₂×D₂₆₃)⋊C₄

C₂²

yes

—

(C₂×D₂₆₃)⋊C₄

C₂²

yes

—

C₂×Q₈×C₂₆₃

C₂²

yes

characteristic

C₂×D₄×C₂₆₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₆₃

C₂²

yes

—

C₂²⋊C₄×C₂₆₃

C₂²

yes

—

C₄²×C₂₆₃

C₂²

yes

characteristic

C₂²×D₂₆₃

C₂³

yes

—

Dic₅₂₆

—

D₁₀₅₂

—

C₂²⋊D₂₆₃

—

C₂²⋊D₂₆₃

—

C₂×Dic₂₆₃

—

C₂×Dic₂₆₃

C₂³

yes

—

C₂×Dic₂₆₃

C₂³

yes

—

C₂×Dic₂₆₃

C₂³

yes

—

C₄×D₂₆₃

—

C₂³×C₂₆₃

C₂³

yes

—

Q₈×C₂₆₃

—

D₄×C₂₆₃

—

C₂×C₄×C₂₆₃

C₂³

yes

characteristic

C₂×C₄×C₂₆₃

C₂³

yes

characteristic

C₂×C₄×C₂₆₃

C₂³

yes

characteristic

C₂×C₄×C₂₆₃

C₂³

yes

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

—

C₂²×C₂₆₃

C₂⁴

yes

characteristic

C₂×D₂₆₃

—

C₂×D₂₆₃

—

C₂×D₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

C₄×C₂₆₃

—

Dic₂₆₃

—

Dic₂₆₃

—

Dic₂₆₃

—

C₂×C₂₆₃

—

C₂×C₂₆₃

yes

C₂×C₂₆₃

yes

C₂×C₂₆₃

2₋¹⁺⁴

yes

characteristic

D₂₆₃

—

C₂₆₃

#562

yes

characteristic, complemented, Sylow

#562

—

263

maximal, Sylow

#125

—

263

—

#124

—

263

—

#124

D₂₆₃

yes

characteristic, complemented

#123

—

263

—

#123

—

263

—

C₂²⋊Q₈

—

263

—

C₂²⋊Q₈

—

263

—

#121

—

263

—

#121

—

263

—

C₂×Q₈

—

263

—

C₂×Q₈

C₂×D₂₆₃

yes

characteristic

C₂×D₄

—

263

—

C₂×D₄

—

263

—

C₂×D₄

—

263

—

C₂×D₄

C₂×D₂₆₃

yes

characteristic

C₂²×C₄

—

263

—

C₂²×C₄

—

263

—

C₄⋊C₄

—

263

—

C₄⋊C₄

—

263

—

C₄⋊C₄

—

263

—

C₄⋊C₄

—

263

—

C₄⋊C₄

—

263

—

C₄⋊C₄

—

263

—

C₂²⋊C₄

—

263

—

C₂²⋊C₄

—

263

—

C₂²⋊C₄

—

263

—

C₂²⋊C₄

—

263

—

C₂²⋊C₄

C₂×D₂₆₃

yes

—

C₂²⋊C₄

C₂×D₂₆₃

yes

—

C₄²

C₂×D₂₆₃

yes

characteristic

C₂³

—

263

—

100

C₂³

C₂²×D₂₆₃

yes

—

101

Q₈

—

102

Q₈

—

1052

—

103

D₄

—

104

D₄

—

1052

—

105

D₄

—

1052

—

106

D₄

—

1052

—

107

C₂×C₄

—

263

—

108

C₂×C₄

—

263

—

109

C₂×C₄

—

263

—

110

C₂×C₄

—

1052

—

111

C₂×C₄

—

1052

—

112

C₂×C₄

C₂²×D₂₆₃

yes

characteristic

113

C₂×C₄

C₂²×D₂₆₃

yes

characteristic

114

C₂×C₄

C₂²×D₂₆₃

yes

characteristic

115

C₂×C₄

C₂²×D₂₆₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

526

—

120

C₂²

—

526

—

121

C₂²

—

526

—

122

C₂²

C₂³×D₂₆₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

526

—

128

C₄

—

526

—

129

C₄

—

526

—

130

C₂

—

131

C₂

—

1052

—

132

C₂

#22554515

yes

characteristic

133

C₂

#22554516

yes

characteristic

134

C₂

#22554517

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₆₃
commutator, socle, cosocle	C₂²×C₂₆₃	C₂⁴
Fitting	#124×C₂₆₃	C₂
hypercenter	#124	D₂₆₃
hypocenter	C₂₆₃	#562
LCS	G ▷ C₂²×C₂₆₃ ▷ C₂₆₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₆₃

commutator, socle, cosocle

C₂²×C₂₆₃

C₂⁴

Fitting

#124×C₂₆₃

C₂

D₂₆₃

C₂₆₃

G ▷ C₂²×C₂₆₃ ▷ C₂₆₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #22554462	C₂	semidirect
2. #22554458	C₂	semidirect
3. #22554454	C₂	semidirect
4. #22554452	C₂	semidirect
5. #22554447	C₂	semidirect
6. #22554446	C₂	semidirect
7. #22554443	C₂	semidirect
8. #22554442	C₂	semidirect
9. #124×C₂₆₃	C₂	semidirect
67. C₂₆₃	#562	semidirect
71. #124	D₂₆₃	semidirect

H ≅ G/N

type

1. #22554462

C₂

semidirect

2. #22554458

C₂

semidirect

3. #22554454

C₂

semidirect

4. #22554452

C₂

semidirect

5. #22554447

C₂

semidirect

6. #22554446

C₂

semidirect

7. #22554443

C₂

semidirect

8. #22554442

C₂

semidirect

9. #124×C₂₆₃

C₂

semidirect

67. C₂₆₃

#562

semidirect

71. #124

D₂₆₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22554462	2
#22554458	2
#22554454	2
#22554452	2
#22554447	2
#22554446	2
#22554443	2
#22554442	2
#124×C₂₆₃	2
#562	263

subgroup

index

#124×C₂₆₃

263

Subgroups — #30682682

Conjugacy classes of subgroups

Complements

Maximal subgroups