Group #11719340 — subgroups

Subgroups — #11719340

Order 832.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#88151	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₄×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#88143	C₂	1	2	yes	maximal, complemented
9	#88142	C₂	1	1	yes	maximal, characteristic, complemented
10	#88135	C₂	1	2	yes	maximal
11	#88132	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	C₂×C₄×D₁₃	C₂²	1	2	yes	—
18	C₂×Dic₂₆	C₂²	1	2	yes	—
19	C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₃	C₂²	1	1	yes	characteristic
23	#6103	C₂²	1	2	yes	—
24	#6103	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₁₃	C₂²	1	2	yes	—
26	C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₁₃	C₂²	1	2	yes	—
28	C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	#6101	C₂²	1	2	yes	—
30	#6101	C₂²	1	2	yes	—
31	#6101	C₂²	1	2	yes	—
32	#6101	C₂²	1	1	yes	characteristic
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	Q₈	1	4	yes	complemented
44	C₂×Dic₁₃	—	4	2	no	—
45	C₂×Dic₁₃	C₂³	1	2	yes	—
46	C₂×Dic₁₃	C₂³	1	1	yes	characteristic
47	C₂×Dic₁₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₁₃	C₂³	1	2	yes	—
49	C₂×Dic₁₃	C₂³	1	1	yes	characteristic
50	C₄×D₁₃	—	2	2	no	—
51	C₄×D₁₃	—	4	2	no	—
52	Dic₂₆	—	4	2	no	—
53	#554	—	13	1	no	maximal, Sylow
54	C₂²×C₁₃	—	2	1	no	—
55	C₂²×C₁₃	—	2	1	no	—
56	C₂²×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂²×C₁₃	C₂⁴	1	1	yes	characteristic
58	C₂×D₁₃	—	2	1	no	—
59	C₂×D₁₃	—	2	1	no	—
60	C₂×D₁₃	C₂×Q₈	1	2	yes	—
61	C₄×C₁₃	—	2	1	no	—
62	C₄×C₁₃	—	2	2	no	—
63	C₄×C₁₃	—	2	2	no	—
64	C₄×C₁₃	#41	1	2	yes	—
65	Dic₁₃	—	2	1	no	—
66	Dic₁₃	—	2	1	no	—
67	Dic₁₃	—	2	2	no	—
68	Dic₁₃	—	2	2	no	—
69	Dic₁₃	C₂×Q₈	1	2	yes	—
70	C₄⋊Q₈	—	13	1	no	—
71	#125	—	13	2	no	—
72	C₂²⋊Q₈	—	13	1	no	—
73	C₂²⋊Q₈	—	13	2	no	—
74	C₂²⋊Q₈	—	13	2	no	—
75	C₂²⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
76	C₄×Q₈	—	13	1	no	—
77	C₄×D₄	—	13	1	no	—
78	C₄×D₄	—	13	2	no	—
79	C₂×C₄⋊C₄	—	13	1	no	—
80	C₂×C₄⋊C₄	—	13	1	no	—
81	C₂×C₁₃	—	2	2	no	—
82	C₂×C₁₃	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₁₃	C₂×#41	1	1	yes	characteristic
84	C₂×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
85	D₁₃	—	2	2	no	—
86	C₂×Q₈	—	13	2	no	—
87	C₂×Q₈	C₂×D₁₃	1	1	yes	characteristic
88	C₂×D₄	—	13	1	no	—
89	C₂²×C₄	—	13	2	no	—
90	C₂²×C₄	—	13	1	no	—
91	C₂²×C₄	—	13	2	no	—
92	C₂²×C₄	C₂×D₁₃	1	1	yes	characteristic
93	C₄⋊C₄	—	13	1	no	—
94	C₄⋊C₄	—	13	2	no	—
95	C₄⋊C₄	—	13	2	no	—
96	C₄⋊C₄	—	13	1	no	—
97	C₄⋊C₄	—	13	1	no	—
98	C₄⋊C₄	—	13	2	no	—
99	C₄⋊C₄	—	13	2	no	—
100	C₄⋊C₄	—	13	2	no	—
101	C₄⋊C₄	C₂×D₁₃	1	1	yes	characteristic
102	C₄⋊C₄	C₂×D₁₃	1	2	yes	—
103	C₂²⋊C₄	—	13	1	no	—
104	C₂²⋊C₄	—	13	1	no	—
105	C₂²⋊C₄	—	13	2	no	—
106	C₂²⋊C₄	C₂×D₁₃	1	2	yes	—
107	C₄²	—	13	2	no	—
108	C₄²	—	13	1	no	—
109	C₁₃	#554	1	1	yes	characteristic, complemented, Sylow
110	C₂³	—	13	1	no	—
111	C₂³	C₂²×D₁₃	1	1	yes	characteristic
112	Q₈	—	2	2	no	—
113	Q₈	—	52	2	no	—
114	D₄	—	13	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	13	1	no	—
117	C₂×C₄	—	13	1	no	—
118	C₂×C₄	—	13	2	no	—
119	C₂×C₄	—	13	1	no	—
120	C₂×C₄	—	13	2	no	—
121	C₂×C₄	—	26	2	no	—
122	C₂×C₄	—	52	2	no	—
123	C₂×C₄	—	52	2	no	—
124	C₂×C₄	C₂²×D₁₃	1	2	yes	—
125	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₁₃	1	2	yes	—
127	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	13	2	no	—
131	C₂²	—	26	1	no	—
132	C₂²	—	26	1	no	—
133	C₂²	Q₈×D₁₃	1	2	yes	—
134	C₂²	C₂³×D₁₃	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	13	2	no	—
139	C₄	—	26	1	no	—
140	C₄	—	26	2	no	—
141	C₄	—	26	2	no	—
142	C₄	—	26	1	no	—
143	C₄	#6129	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	26	2	no	—
146	C₂	C₂×#6129	1	1	yes	characteristic
147	C₂	C₂×Q₈×D₁₃	1	1	yes	characteristic
148	C₂	#88256	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

#88151

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₃

C₂

yes

maximal, characteristic, complemented

#88143

C₂

yes

maximal, complemented

#88142

C₂

yes

maximal, characteristic, complemented

#88135

C₂

yes

maximal

#88132

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

C₂×C₄×D₁₃

C₂²

yes

—

C₂×Dic₂₆

C₂²

yes

—

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

characteristic

C₄⋊Dic₁₃

C₂²

yes

—

C₄⋊Dic₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₃

C₂²

yes

—

C₄⋊Dic₁₃

C₂²

yes

characteristic

#6101

C₂²

yes

—

#6101

C₂²

yes

—

#6101

C₂²

yes

—

#6101

C₂²

yes

characteristic

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

Q₈

yes

complemented

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂³

yes

—

C₂×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₁₃

C₂³

yes

characteristic

C₂×Dic₁₃

C₂³

yes

—

C₂×Dic₁₃

C₂³

yes

characteristic

C₄×D₁₃

—

C₄×D₁₃

—

Dic₂₆

—

#554

—

maximal, Sylow

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂×Q₈

yes

—

C₂²×C₁₃

C₂⁴

yes

characteristic

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

C₂×Q₈

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#41

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

C₂×Q₈

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₁₃

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₁₃

—

C₂×C₁₃

C₂²×Q₈

yes

characteristic

C₂×C₁₃

C₂×#41

yes

characteristic

C₂×C₁₃

H(5,𝔽₂)

yes

characteristic

D₁₃

—

C₂×Q₈

—

C₂×Q₈

C₂×D₁₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

C₂×D₁₃

yes

characteristic

102

C₄⋊C₄

C₂×D₁₃

yes

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

C₂×D₁₃

yes

—

107

C₄²

—

108

C₄²

—

109

C₁₃

#554

yes

characteristic, complemented, Sylow

110

C₂³

—

111

C₂³

C₂²×D₁₃

yes

characteristic

112

Q₈

—

113

Q₈

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂²×D₁₃

yes

—

125

C₂×C₄

C₂²×D₁₃

yes

characteristic

126

C₂×C₄

C₂²×D₁₃

yes

—

127

C₂×C₄

C₂²×D₁₃

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

Q₈×D₁₃

yes

—

134

C₂²

C₂³×D₁₃

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#6129

yes

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#6129

yes

characteristic

147

C₂

C₂×Q₈×D₁₃

yes

characteristic

148

C₂

#88256

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	#554
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. #88151	C₂	semidirect
7. C₄⋊C₄×D₁₃	C₂	semidirect
8. #88143	C₂	semidirect
9. #88142	C₂	semidirect
11. #88132	C₂	semidirect
43. C₂²⋊D₁₃	Q₈	semidirect
75. C₂²⋊Q₈	D₁₃	semidirect
109. C₁₃	#554	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. #88151

C₂

semidirect

7. C₄⋊C₄×D₁₃

C₂

semidirect

8. #88143

C₂

semidirect

9. #88142

C₂

semidirect

11. #88132

C₂

semidirect

43. C₂²⋊D₁₃

Q₈

semidirect

75. C₂²⋊Q₈

D₁₃

semidirect

109. C₁₃

#554

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
#88151	2
C₄⋊C₄×D₁₃	2
#88143	2
#88142	2
#88135	2
#88132	2
#554	13

subgroup

index

C₁₃×C₂²⋊Q₈

#88200

Q₈×Dic₁₃

C₄×C₂²⋊D₁₃

C₂×C₄⋊Dic₁₃

C₄⋊C₄×D₁₃

Subgroups — #11719340

Conjugacy classes of subgroups

Complements

Maximal subgroups