Group #11719347 — subgroups

Subgroups — #11719347

Order 832.

214 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#88200	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#88152	C₂	1	2	yes	maximal, complemented
7	#88146	C₂	1	1	yes	maximal, characteristic, complemented
8	#88144	C₂	1	2	yes	maximal, complemented
9	#88144	C₂	1	1	yes	maximal, characteristic, complemented
10	#88140	C₂	1	2	yes	maximal
11	#88131	C₂	1	2	yes	maximal, complemented
12	C₂×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₁₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃	C₂²	1	2	yes	—
16	C₂×C₄×D₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic
21	#6103	C₂²	1	2	yes	—
22	#6103	C₂²	1	2	yes	—
23	#6103	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₁₃	C₂²	1	2	yes	—
25	C₄⋊Dic₁₃	C₂²	1	2	yes	—
26	C₄⋊Dic₁₃	C₂²	1	2	yes	—
27	C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	#6101	C₂²	1	2	yes	—
30	#6101	C₂²	1	2	yes	—
31	#6101	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₃	C₂²	1	2	yes	—
33	C₄×Dic₁₃	C₂²	1	2	yes	—
34	C₄×Dic₁₃	C₂²	1	1	yes	characteristic
35	C₂³×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×D₁₃	C₂³	1	1	yes	characteristic
37	Q₈×C₁₃	—	2	2	no	—
38	C₂×C₄×C₁₃	—	2	2	no	—
39	C₂×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²⋊D₁₃	—	2	2	no	—
44	C₂×Dic₁₃	—	2	4	no	—
45	C₂×Dic₁₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₃	C₂³	1	2	yes	—
48	C₂×Dic₁₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₃	C₂³	1	2	yes	—
50	C₄×D₁₃	—	2	2	no	—
51	#552	—	13	1	no	maximal, Sylow
52	C₂²×C₁₃	—	2	1	no	—
53	C₂²×C₁₃	—	2	1	no	—
54	C₂²×C₁₃	#41	1	2	yes	—
55	C₂²×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₂×D₁₃	—	2	1	no	—
57	C₂×D₁₃	—	2	1	no	—
58	C₂×D₁₃	—	2	1	no	—
59	C₄×C₁₃	—	2	1	no	—
60	C₄×C₁₃	—	2	2	no	—
61	C₄×C₁₃	—	2	2	no	—
62	C₄×C₁₃	#41	1	2	yes	—
63	Dic₁₃	—	2	1	no	—
64	Dic₁₃	—	2	2	no	—
65	Dic₁₃	—	2	2	no	—
66	Dic₁₃	—	2	1	no	—
67	Dic₁₃	—	2	1	no	—
68	#126	—	13	2	no	—
69	#125	—	13	2	no	—
70	#125	—	13	1	no	—
71	#123	—	13	2	no	—
72	C₂²⋊Q₈	—	13	1	no	—
73	C₂²⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
74	C₄×Q₈	—	13	1	no	—
75	C₄×D₄	—	13	1	no	—
76	#117	—	13	1	no	—
77	#117	—	13	2	no	—
78	C₂×C₄⋊C₄	—	13	1	no	—
79	C₂×C₁₃	—	2	2	no	—
80	C₂×C₁₃	C₂×#41	1	1	yes	characteristic
81	C₂×C₁₃	C₂×#41	1	1	yes	characteristic
82	C₂×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
83	D₁₃	—	4	1	no	—
84	C₂×Q₈	C₂×D₁₃	1	1	yes	characteristic
85	C₂×D₄	—	13	1	no	—
86	C₂²×C₄	—	13	2	no	—
87	C₂²×C₄	—	13	1	no	—
88	C₂²×C₄	C₂×D₁₃	1	1	yes	characteristic
89	C₄⋊C₄	—	13	1	no	—
90	C₄⋊C₄	—	13	1	no	—
91	C₄⋊C₄	—	13	2	no	—
92	C₄⋊C₄	—	13	2	no	—
93	C₄⋊C₄	—	13	2	no	—
94	C₄⋊C₄	—	13	1	no	—
95	C₄⋊C₄	—	13	2	no	—
96	C₄⋊C₄	—	13	2	no	—
97	C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₁₃	1	2	yes	—
99	C₂²⋊C₄	—	13	1	no	—
100	C₂²⋊C₄	—	13	2	no	—
101	C₂²⋊C₄	—	13	1	no	—
102	C₂²⋊C₄	—	13	2	no	—
103	C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
104	C₄²	—	13	2	no	—
105	C₄²	—	13	2	no	—
106	C₄²	—	13	1	no	—
107	C₁₃	#552	1	1	yes	characteristic, complemented, Sylow
108	C₂³	—	13	1	no	—
109	C₂³	C₂²×D₁₃	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	D₄	—	26	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	13	1	no	—
114	C₂×C₄	—	13	1	no	—
115	C₂×C₄	—	13	2	no	—
116	C₂×C₄	—	13	1	no	—
117	C₂×C₄	—	13	2	no	—
118	C₂×C₄	—	26	4	no	—
119	C₂×C₄	—	26	2	no	—
120	C₂×C₄	C₂²×D₁₃	1	2	yes	—
121	C₂×C₄	C₂²×D₁₃	1	2	yes	—
122	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	26	1	no	—
127	C₂²	—	26	1	no	—
128	C₂²	—	26	1	no	—
129	C₂²	#6131	1	2	yes	—
130	C₂²	C₂³×D₁₃	1	1	yes	characteristic
131	C₄	—	2	1	no	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	26	1	no	—
135	C₄	—	26	2	no	—
136	C₄	—	26	2	no	—
137	C₄	—	26	1	no	—
138	C₄	—	26	1	no	—
139	C₄	#6129	1	2	yes	—
140	C₂	—	2	2	no	—
141	C₂	—	52	1	no	—
142	C₂	C₂×#6129	1	1	yes	characteristic
143	C₂	C₂×#6131	1	1	yes	characteristic
144	C₂	#88257	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₃×C₂²⋊Q₈

C₂

yes

maximal, characteristic, complemented

#88200

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₁₃

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₃

C₂

yes

maximal, characteristic, complemented

#88152

C₂

yes

maximal, complemented

#88146

C₂

yes

maximal, characteristic, complemented

#88144

C₂

yes

maximal, complemented

#88144

C₂

yes

maximal, characteristic, complemented

#88140

C₂

yes

maximal

#88131

C₂

yes

maximal, complemented

C₂×Q₈×C₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃

C₂²

yes

characteristic

C₂²×Dic₁₃

C₂²

yes

—

C₂×C₄×D₁₃

C₂²

yes

characteristic

C₁₃×C₄⋊C₄

C₂²

yes

characteristic

C₁₃×C₄⋊C₄

C₂²

yes

—

C₁₃×C₂²⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₃

C₂²

yes

characteristic

#6103

C₂²

yes

—

#6103

C₂²

yes

—

#6103

C₂²

yes

characteristic

C₄⋊Dic₁₃

C₂²

yes

—

C₄⋊Dic₁₃

C₂²

yes

—

C₄⋊Dic₁₃

C₂²

yes

—

C₄⋊Dic₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₃

C₂²

yes

characteristic

#6101

C₂²

yes

—

#6101

C₂²

yes

—

#6101

C₂²

yes

characteristic

C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₁₃

C₂²

yes

—

C₄×Dic₁₃

C₂²

yes

characteristic

C₂³×C₁₃

C₂³

yes

characteristic

C₂²×D₁₃

C₂³

yes

characteristic

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂³

yes

—

C₂×C₄×C₁₃

C₂³

yes

—

C₂×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₃

C₂³

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₁₃

C₂³

yes

—

C₂×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₁₃

C₂³

yes

—

C₄×D₁₃

—

#552

—

maximal, Sylow

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

#41

yes

—

C₂²×C₁₃

C₂⁴

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#41

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₁₃

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₁₃

—

C₂×C₁₃

C₂×#41

yes

characteristic

C₂×C₁₃

C₂×#41

yes

characteristic

C₂×C₁₃

2₋¹⁺⁴

yes

characteristic

D₁₃

—

C₂×Q₈

C₂×D₁₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₃

yes

characteristic

C₄⋊C₄

C₂×D₁₃

yes

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₁₃

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₁₃

#552

yes

characteristic, complemented, Sylow

108

C₂³

—

109

C₂³

C₂²×D₁₃

yes

characteristic

110

Q₈

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×D₁₃

yes

—

121

C₂×C₄

C₂²×D₁₃

yes

—

122

C₂×C₄

C₂²×D₁₃

yes

characteristic

123

C₂×C₄

C₂²×D₁₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#6131

yes

—

130

C₂²

C₂³×D₁₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#6129

yes

—

140

C₂

—

141

C₂

—

142

C₂

C₂×#6129

yes

characteristic

143

C₂

C₂×#6131

yes

characteristic

144

C₂

#88257

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁴
Fitting	C₁₃×C₂²⋊Q₈	C₂
prime radical	C₂×C₂²⋊D₁₃	C₂²
hypercenter	C₂²⋊Q₈	D₁₃
hypocenter	C₁₃	#552
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃

commutator, socle, cosocle

C₂²×C₁₃

C₂⁴

Fitting

C₁₃×C₂²⋊Q₈

C₂

prime radical

C₂×C₂²⋊D₁₃

C₂²

hypercenter

C₂²⋊Q₈

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. C₁₃×C₂²⋊Q₈	C₂	semidirect
2. #88200	C₂	semidirect
3. Q₈×Dic₁₃	C₂	semidirect
5. C₂×C₄⋊Dic₁₃	C₂	semidirect
6. #88152	C₂	semidirect
7. #88146	C₂	semidirect
8. #88144	C₂	semidirect
9. #88144	C₂	semidirect
11. #88131	C₂	semidirect
73. C₂²⋊Q₈	D₁₃	semidirect
107. C₁₃	#552	semidirect

H ≅ G/N

type

1. C₁₃×C₂²⋊Q₈

C₂

semidirect

2. #88200

C₂

semidirect

3. Q₈×Dic₁₃

C₂

semidirect

5. C₂×C₄⋊Dic₁₃

C₂

semidirect

6. #88152

C₂

semidirect

7. #88146

C₂

semidirect

8. #88144

C₂

semidirect

9. #88144

C₂

semidirect

11. #88131

C₂

semidirect

73. C₂²⋊Q₈

D₁₃

semidirect

107. C₁₃

#552

semidirect

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

subgroup	index
C₁₃×C₂²⋊Q₈	2
#88200	2
Q₈×Dic₁₃	2
C₄×C₂²⋊D₁₃	2
C₂×C₄⋊Dic₁₃	2
#88152	2
#88146	2
#88144	2
#88140	2
#88131	2
#552	13

subgroup

index

C₁₃×C₂²⋊Q₈

#88200

Q₈×Dic₁₃

C₄×C₂²⋊D₁₃

C₂×C₄⋊Dic₁₃

Subgroups — #11719347

Conjugacy classes of subgroups

Complements

Maximal subgroups