Group #28417156 — subgroups

Subgroups — #28417156

Order 14912.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#21418534	C₂	1	1	yes	maximal, characteristic, complemented
2	#21418530	C₂	1	2	yes	maximal, complemented
3	#21418528	C₂	1	2	yes	maximal, complemented
4	#21418521	C₂	1	2	yes	maximal
5	#21418511	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#21418500	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊D₂₃₃	C₂	1	1	yes	maximal, characteristic
9	#21418494	C₂	1	1	yes	maximal, characteristic, complemented
10	#21418493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₂₃₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Dic₄₆₆	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₂₃₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₂₃₃	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₂₃₃	C₂²	1	2	yes	—
16	C₄⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
18	#16836573	C₂²	1	1	yes	characteristic
19	#16836573	C₂²	1	2	yes	—
20	#16836573	C₂²	1	2	yes	—
21	C₂²⋊Dic₂₃₃	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₂₃₃	C₂²	1	2	yes	—
23	#16836571	C₂²	1	1	yes	characteristic
24	#16836571	C₂²	1	2	yes	—
25	#16836571	C₂²	1	2	yes	—
26	C₄×Dic₂₃₃	C₂²	1	1	yes	characteristic
27	C₄×Dic₂₃₃	C₂²	1	2	yes	—
28	C₄×Dic₂₃₃	C₂²	1	2	yes	—
29	C₄×Dic₂₃₃	C₂²	1	2	yes	—
30	C₂×Q₈×C₂₃₃	C₂²	1	1	yes	characteristic
31	C₂²×C₄×C₂₃₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×C₂₃₃	C₂²	1	2	yes	—
33	C₄⋊C₄×C₂₃₃	C₂²	1	1	yes	characteristic
34	C₂²⋊C₄×C₂₃₃	C₂²	1	2	yes	—
35	C₂²×D₂₃₃	C₂³	1	1	yes	characteristic
36	Dic₄₆₆	—	2	4	no	—
37	C₂²⋊D₂₃₃	—	2	2	no	—
38	C₂×Dic₂₃₃	C₂³	1	2	yes	—
39	C₂×Dic₂₃₃	C₂³	1	1	yes	characteristic
40	C₂×Dic₂₃₃	C₂³	1	1	yes	characteristic
41	C₂×Dic₂₃₃	C₂³	1	2	yes	—
42	C₂×Dic₂₃₃	C₂³	1	1	yes	characteristic
43	C₄×D₂₃₃	—	2	2	no	—
44	C₂³×C₂₃₃	C₂³	1	1	yes	characteristic
45	Q₈×C₂₃₃	—	2	2	no	—
46	C₂×C₄×C₂₃₃	—	2	2	no	—
47	C₂×C₄×C₂₃₃	C₂³	1	2	yes	—
48	C₂×C₄×C₂₃₃	C₂³	1	2	yes	—
49	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
50	C₂×C₄×C₂₃₃	C₂³	1	1	yes	characteristic
51	C₂²×C₂₃₃	—	2	1	no	—
52	C₂²×C₂₃₃	—	2	1	no	—
53	C₂²×C₂₃₃	—	2	1	no	—
54	C₂²×C₂₃₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₂₃₃	—	2	1	no	—
56	C₂×D₂₃₃	—	2	1	no	—
57	C₂×D₂₃₃	—	2	1	no	—
58	C₄×C₂₃₃	—	2	1	no	—
59	C₄×C₂₃₃	—	2	2	no	—
60	C₄×C₂₃₃	—	2	2	no	—
61	C₄×C₂₃₃	#41	1	2	yes	—
62	Dic₂₃₃	—	2	1	no	—
63	Dic₂₃₃	—	2	2	no	—
64	Dic₂₃₃	—	2	2	no	—
65	Dic₂₃₃	—	2	1	no	—
66	Dic₂₃₃	#41	1	2	yes	—
67	C₂×C₂₃₃	—	4	1	no	—
68	C₂×C₂₃₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₂₃₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₂₃₃	2₋¹⁺⁴	1	1	yes	characteristic
71	D₂₃₃	—	4	1	no	—
72	C₂₃₃	#556	1	1	yes	characteristic, complemented, Sylow
73	#556	—	233	1	no	maximal, Sylow
74	C₄⋊Q₈	—	233	1	no	—
75	#126	—	233	2	no	—
76	#126	—	233	2	no	—
77	#124	—	233	2	no	—
78	C₂²⋊Q₈	—	233	1	no	—
79	C₂²⋊Q₈	D₂₃₃	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	233	1	no	—
81	C₄×Q₈	—	233	2	no	—
82	C₄×D₄	—	233	1	no	—
83	#117	—	233	1	no	—
84	#117	—	233	1	no	—
85	C₂×Q₈	—	233	2	no	—
86	C₂×Q₈	C₂×D₂₃₃	1	1	yes	characteristic
87	C₂×D₄	—	233	1	no	—
88	C₂²×C₄	—	233	1	no	—
89	C₂²×C₄	C₂×D₂₃₃	1	1	yes	characteristic
90	C₄⋊C₄	—	233	1	no	—
91	C₄⋊C₄	—	233	1	no	—
92	C₄⋊C₄	—	233	1	no	—
93	C₄⋊C₄	—	233	2	no	—
94	C₄⋊C₄	—	233	2	no	—
95	C₄⋊C₄	—	233	2	no	—
96	C₄⋊C₄	C₂×D₂₃₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₂₃₃	1	2	yes	—
98	C₂²⋊C₄	—	233	2	no	—
99	C₂²⋊C₄	—	233	1	no	—
100	C₂²⋊C₄	—	233	2	no	—
101	C₂²⋊C₄	—	233	1	no	—
102	C₂²⋊C₄	—	233	2	no	—
103	C₂²⋊C₄	C₂×D₂₃₃	1	2	yes	—
104	C₄²	—	233	2	no	—
105	C₄²	—	233	2	no	—
106	C₄²	—	233	1	no	—
107	C₄²	—	233	2	no	—
108	C₂³	—	233	1	no	—
109	C₂³	C₂²×D₂₃₃	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	466	4	no	—
112	D₄	—	466	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	466	2	no	—
115	C₂×C₄	—	233	2	no	—
116	C₂×C₄	—	233	1	no	—
117	C₂×C₄	—	233	1	no	—
118	C₂×C₄	—	233	2	no	—
119	C₂×C₄	—	233	1	no	—
120	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₂₃₃	1	2	yes	—
122	C₂×C₄	C₂²×D₂₃₃	1	2	yes	—
123	C₂×C₄	C₂²×D₂₃₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	466	1	no	—
128	C₂²	—	466	1	no	—
129	C₂²	—	466	1	no	—
130	C₂²	C₂³×D₂₃₃	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	466	2	no	—
135	C₄	—	466	1	no	—
136	C₄	—	466	2	no	—
137	C₄	—	466	1	no	—
138	C₄	—	233	2	no	—
139	C₄	#16836593	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	932	1	no	—
142	C₂	#41×D₂₃₃	1	1	yes	characteristic
143	C₂	C₂×#16836593	1	1	yes	characteristic
144	C₂	#21418583	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21418534

C₂

yes

maximal, characteristic, complemented

#21418530

C₂

yes

maximal, complemented

#21418528

C₂

yes

maximal, complemented

#21418521

C₂

yes

maximal

#21418511

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₂₃₃

C₂

yes

maximal, characteristic, complemented

#21418500

C₂

yes

maximal, complemented

C₄×C₂²⋊D₂₃₃

C₂

yes

maximal, characteristic

#21418494

C₂

yes

maximal, characteristic, complemented

#21418493

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₂₃₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₆₆

C₂²

yes

—

C₂×C₂²⋊D₂₃₃

C₂²

yes

characteristic

C₂×C₄×D₂₃₃

C₂²

yes

characteristic

C₄⋊Dic₂₃₃

C₂²

yes

—

C₄⋊Dic₂₃₃

C₂²

yes

characteristic

C₄⋊Dic₂₃₃

C₂²

yes

characteristic

#16836573

C₂²

yes

characteristic

#16836573

C₂²

yes

—

#16836573

C₂²

yes

—

C₂²⋊Dic₂₃₃

C₂²

yes

characteristic

C₂²⋊Dic₂₃₃

C₂²

yes

—

#16836571

C₂²

yes

characteristic

#16836571

C₂²

yes

—

#16836571

C₂²

yes

—

C₄×Dic₂₃₃

C₂²

yes

characteristic

C₄×Dic₂₃₃

C₂²

yes

—

C₄×Dic₂₃₃

C₂²

yes

—

C₄×Dic₂₃₃

C₂²

yes

—

C₂×Q₈×C₂₃₃

C₂²

yes

characteristic

C₂²×C₄×C₂₃₃

C₂²

yes

characteristic

C₄⋊C₄×C₂₃₃

C₂²

yes

—

C₄⋊C₄×C₂₃₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₃₃

C₂²

yes

—

C₂²×D₂₃₃

C₂³

yes

characteristic

Dic₄₆₆

—

C₂²⋊D₂₃₃

—

C₂×Dic₂₃₃

C₂³

yes

—

C₂×Dic₂₃₃

C₂³

yes

characteristic

C₂×Dic₂₃₃

C₂³

yes

characteristic

C₂×Dic₂₃₃

C₂³

yes

—

C₂×Dic₂₃₃

C₂³

yes

characteristic

C₄×D₂₃₃

—

C₂³×C₂₃₃

C₂³

yes

characteristic

Q₈×C₂₃₃

—

C₂×C₄×C₂₃₃

—

C₂×C₄×C₂₃₃

C₂³

yes

—

C₂×C₄×C₂₃₃

C₂³

yes

—

C₂×C₄×C₂₃₃

C₂³

yes

characteristic

C₂×C₄×C₂₃₃

C₂³

yes

characteristic

C₂²×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

—

C₂²×C₂₃₃

C₂⁴

yes

characteristic

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₂×D₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

—

C₄×C₂₃₃

#41

yes

—

Dic₂₃₃

—

Dic₂₃₃

—

Dic₂₃₃

—

Dic₂₃₃

—

Dic₂₃₃

#41

yes

—

C₂×C₂₃₃

—

C₂×C₂₃₃

C₂×#41

yes

characteristic

C₂×C₂₃₃

C₂×#41

yes

characteristic

C₂×C₂₃₃

2₋¹⁺⁴

yes

characteristic

D₂₃₃

—

C₂₃₃

#556

yes

characteristic, complemented, Sylow

#556

—

233

maximal, Sylow

C₄⋊Q₈

—

233

—

#126

—

233

—

#126

—

233

—

#124

—

233

—

C₂²⋊Q₈

—

233

—

C₂²⋊Q₈

D₂₃₃

yes

characteristic, complemented

C₄×Q₈

—

233

—

C₄×Q₈

—

233

—

C₄×D₄

—

233

—

#117

—

233

—

#117

—

233

—

C₂×Q₈

—

233

—

C₂×Q₈

C₂×D₂₃₃

yes

characteristic

C₂×D₄

—

233

—

C₂²×C₄

—

233

—

C₂²×C₄

C₂×D₂₃₃

yes

characteristic

C₄⋊C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

—

233

—

C₄⋊C₄

C₂×D₂₃₃

yes

characteristic

C₄⋊C₄

C₂×D₂₃₃

yes

—

C₂²⋊C₄

—

233

—

C₂²⋊C₄

—

233

—

100

C₂²⋊C₄

—

233

—

101

C₂²⋊C₄

—

233

—

102

C₂²⋊C₄

—

233

—

103

C₂²⋊C₄

C₂×D₂₃₃

yes

—

104

C₄²

—

233

—

105

C₄²

—

233

—

106

C₄²

—

233

—

107

C₄²

—

233

—

108

C₂³

—

233

—

109

C₂³

C₂²×D₂₃₃

yes

characteristic

110

Q₈

—

111

Q₈

—

466

—

112

D₄

—

466

—

113

C₂×C₄

—

114

C₂×C₄

—

466

—

115

C₂×C₄

—

233

—

116

C₂×C₄

—

233

—

117

C₂×C₄

—

233

—

118

C₂×C₄

—

233

—

119

C₂×C₄

—

233

—

120

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

121

C₂×C₄

C₂²×D₂₃₃

yes

—

122

C₂×C₄

C₂²×D₂₃₃

yes

—

123

C₂×C₄

C₂²×D₂₃₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

466

—

128

C₂²

—

466

—

129

C₂²

—

466

—

130

C₂²

C₂³×D₂₃₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

466

—

135

C₄

—

466

—

136

C₄

—

466

—

137

C₄

—

466

—

138

C₄

—

233

—

139

C₄

#16836593

yes

—

140

C₂

—

141

C₂

—

932

—

142

C₂

#41×D₂₃₃

yes

characteristic

143

C₂

C₂×#16836593

yes

characteristic

144

C₂

#21418583

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₃₃
commutator, socle, cosocle	C₂²×C₂₃₃	C₂⁴
Fitting	C₂²⋊Q₈×C₂₃₃	C₂
prime radical	C₂×C₂²⋊D₂₃₃	C₂²
hypercenter	C₂²⋊Q₈	D₂₃₃
hypocenter	C₂₃₃	#556
LCS	G ▷ C₂²×C₂₃₃ ▷ C₂₃₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₃₃

commutator, socle, cosocle

C₂²×C₂₃₃

C₂⁴

Fitting

C₂²⋊Q₈×C₂₃₃

C₂

prime radical

C₂×C₂²⋊D₂₃₃

C₂²

hypercenter

C₂²⋊Q₈

D₂₃₃

C₂₃₃

G ▷ C₂²×C₂₃₃ ▷ C₂₃₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #21418534	C₂	semidirect
2. #21418530	C₂	semidirect
3. #21418528	C₂	semidirect
5. #21418511	C₂	semidirect
6. Q₈×Dic₂₃₃	C₂	semidirect
7. #21418500	C₂	semidirect
9. #21418494	C₂	semidirect
10. #21418493	C₂	semidirect
11. C₂²⋊Q₈×C₂₃₃	C₂	semidirect
72. C₂₃₃	#556	semidirect
79. C₂²⋊Q₈	D₂₃₃	semidirect

H ≅ G/N

type

1. #21418534

C₂

semidirect

2. #21418530

C₂

semidirect

3. #21418528

C₂

semidirect

5. #21418511

C₂

semidirect

6. Q₈×Dic₂₃₃

C₂

semidirect

7. #21418500

C₂

semidirect

9. #21418494

C₂

semidirect

10. #21418493

C₂

semidirect

11. C₂²⋊Q₈×C₂₃₃

C₂

semidirect

72. C₂₃₃

#556

semidirect

79. C₂²⋊Q₈

D₂₃₃

semidirect

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

subgroup	index
#21418534	2
#21418530	2
#21418528	2
#21418521	2
#21418511	2
Q₈×Dic₂₃₃	2
#21418500	2
C₄×C₂²⋊D₂₃₃	2
#21418494	2
#21418493	2
C₂²⋊Q₈×C₂₃₃	2
#556	233

subgroup

index

Q₈×Dic₂₃₃

C₄×C₂²⋊D₂₃₃

C₂²⋊Q₈×C₂₃₃

233

Subgroups — #28417156

Conjugacy classes of subgroups

Complements

Maximal subgroups