Group #13120948 — subgroups

Subgroups — #13120948

Order 1664.

612 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88201	C₂	1	2	yes	maximal
3	C₂×#88198	C₂	1	2	yes	maximal, complemented
4	C₂×#88129	C₂	1	1	yes	maximal, characteristic
5	#11718653	C₂	1	8	yes	maximal, complemented
6	#11718593	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
8	C₂²×D₅₂	C₂²	1	1	yes	characteristic
9	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
10	C₁₃×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	#88201	—	2	8	no	—
14	#88198	—	2	8	no	—
15	C₂×#6103	C₂²	1	4	yes	—
16	C₂×#6103	C₂²	1	8	yes	—
17	C₂×#6101	C₂²	1	8	yes	—
18	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
19	#88129	—	2	4	no	—
20	#88071	C₂²	1	4	yes	—
21	C₂³×D₁₃	C₂³	1	2	yes	—
22	C₂×Q₈×C₁₃	—	2	4	no	—
23	C₂×Q₈×C₁₃	—	2	4	no	—
24	C₂×Q₈×C₁₃	—	2	4	no	—
25	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×Dic₁₃	C₂³	1	2	yes	—
29	C₂²×Dic₁₃	C₂³	1	4	yes	—
30	C₂×D₅₂	—	2	2	no	—
31	C₂×D₅₂	—	2	4	no	—
32	C₂×Dic₂₆	—	2	2	no	—
33	C₂×Dic₂₆	—	2	4	no	—
34	C₁₃×C₄⋊C₄	—	2	8	no	—
35	C₄²×C₁₃	D₄	1	4	yes	—
36	#6103	—	2	8	no	—
37	#6103	—	2	16	no	—
38	#6101	—	2	16	no	—
39	C₄×Dic₁₃	D₄	1	8	yes	—
40	#2658	—	13	1	no	maximal, Sylow
41	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
42	C₂²×D₁₃	—	2	4	no	—
43	C₂²×D₁₃	—	2	2	no	—
44	C₂²×D₁₃	—	2	8	no	—
45	Q₈×C₁₃	—	4	8	no	—
46	C₂×C₄×C₁₃	—	2	4	no	—
47	C₂×C₄×C₁₃	—	2	4	no	—
48	C₂×C₄×C₁₃	—	2	4	no	—
49	C₂×C₄×C₁₃	—	2	4	no	—
50	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
51	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
52	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
53	C₂×Dic₁₃	—	2	8	no	—
54	C₂×Dic₁₃	—	2	4	no	—
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	C₂×D₄	1	8	yes	—
57	D₅₂	—	4	4	no	—
58	Dic₂₆	—	4	4	no	—
59	C₂×C₄⋊Q₈	—	13	2	no	—
60	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
61	C₂×#124	—	13	2	no	—
62	C₂×#124	—	13	1	no	—
63	#399	—	13	8	no	—
64	#376	—	13	1	no	—
65	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
66	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₁₃	2₋¹⁺⁴	1	4	yes	—
68	C₂×D₁₃	—	4	2	no	—
69	C₂×D₁₃	—	4	8	no	—
70	C₂×D₁₃	—	4	4	no	—
71	C₄×C₁₃	—	2	4	no	—
72	C₄×C₁₃	—	4	4	no	—
73	C₄×C₁₃	#127	1	4	yes	—
74	Dic₁₃	—	2	8	no	—
75	Dic₁₃	—	4	2	no	—
76	C₂²×Q₈	—	13	1	no	—
77	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
78	C₂²×D₄	—	13	1	no	—
79	C₄⋊Q₈	—	2	4	no	—
80	C₄⋊Q₈	—	26	8	no	—
81	#124	—	26	8	no	—
82	#124	—	26	4	no	—
83	C₂×C₄⋊C₄	—	13	8	no	—
84	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
85	C₂×C₂²⋊C₄	—	13	8	no	—
86	C₂×C₂²⋊C₄	—	13	4	no	—
87	C₂×C₄²	—	13	2	no	—
88	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
89	#95	—	13	4	no	—
90	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
91	C₂×C₁₃	#536	1	4	yes	—
92	C₂×C₁₃	#536	1	2	yes	—
93	D₁₃	—	8	2	no	—
94	C₂⁴	—	13	2	no	—
95	C₂×Q₈	—	2	4	no	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	26	4	no	—
99	C₂×Q₈	—	26	2	no	—
100	C₂×D₄	—	26	4	no	—
101	C₂×D₄	—	26	2	no	—
102	C₂²×C₄	—	13	4	no	—
103	C₂²×C₄	—	13	2	no	—
104	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
105	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
107	C₄⋊C₄	—	2	8	no	—
108	C₄⋊C₄	—	26	16	no	—
109	C₂²⋊C₄	—	26	16	no	—
110	C₂²⋊C₄	—	26	8	no	—
111	C₄²	—	13	8	no	—
112	C₄²	C₂²⋊D₁₃	1	4	yes	—
113	C₁₃	#2658	1	1	yes	characteristic, complemented, Sylow
114	C₂³	—	26	4	no	—
115	C₂³	—	26	8	no	—
116	C₂³	—	26	2	no	—
117	C₂³	C₂³×D₁₃	1	1	yes	characteristic
118	Q₈	—	4	8	no	—
119	Q₈	—	52	4	no	—
120	D₄	—	52	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	13	8	no	—
126	C₂×C₄	—	26	8	no	—
127	C₂×C₄	—	26	4	no	—
128	C₂×C₄	—	26	2	no	—
129	C₂×C₄	D₄×D₁₃	1	4	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
132	C₂²	—	52	8	no	—
133	C₂²	—	52	2	no	—
134	C₂²	—	52	4	no	—
135	C₂²	C₂×D₄×D₁₃	1	2	yes	—
136	C₂²	#88254	1	4	yes	—
137	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	4	4	no	—
140	C₄	—	26	8	no	—
141	C₄	—	52	2	no	—
142	C₄	#88194	1	4	yes	—
143	C₂	—	104	2	no	—
144	C₂	#11719429	1	4	yes	—
145	C₂	C₂×#88194	1	1	yes	characteristic
146	C₂	#11719521	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×#88201

C₂

yes

maximal

C₂×#88198

C₂

yes

maximal, complemented

C₂×#88129

C₂

yes

maximal, characteristic

#11718653

C₂

yes

maximal, complemented

#11718593

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

characteristic

C₂²×Dic₂₆

C₂²

yes

characteristic

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

—

—

C₂²

yes

—

C₂×#6103

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88129

—

#88071

C₂²

yes

—

C₂³×D₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×Dic₂₆

—

C₂×Dic₂₆

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

#6103

—

#6103

—

#6101

—

C₄×Dic₁₃

D₄

yes

—

#2658

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

—

Q₈×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

D₅₂

—

Dic₂₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₁₃

yes

characteristic, complemented

—

—

—

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#127

yes

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂²×D₄

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#124

—

#124

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

#95

—

C₂×C₁₃

C₂×#127

yes

characteristic

C₂×C₁₃

#536

yes

—

C₂×C₁₃

#536

yes

—

D₁₃

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²×D₁₃

yes

—

105

C₂²×C₄

C₂²×D₁₃

yes

—

106

C₂²×C₄

C₂²×D₁₃

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

C₂²⋊D₁₃

yes

—

113

C₁₃

#2658

yes

characteristic, complemented, Sylow

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂³×D₁₃

yes

characteristic

118

Q₈

—

119

Q₈

—

120

D₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

D₄×D₁₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂×D₄×D₁₃

yes

—

136

C₂²

#88254

yes

—

137

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#88194

yes

—

143

C₂

—

104

—

144

C₂

#11719429

yes

—

145

C₂

C₂×#88194

yes

characteristic

146

C₂

#11719521

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2658
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

prime radical

C₂²×D₅₂

C₂²

hypercenter

C₂×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
3. C₂×#88198	C₂	semidirect
5. #11718653	C₂	semidirect
6. #11718593	C₂	semidirect
60. C₂×C₄⋊Q₈	D₁₃	semidirect
113. C₁₃	#2658	semidirect

H ≅ G/N

type

1. C₂×C₁₃×C₄⋊Q₈

C₂

semidirect

3. C₂×#88198

C₂

semidirect

5. #11718653

C₂

semidirect

6. #11718593

C₂

semidirect

60. C₂×C₄⋊Q₈

D₁₃

semidirect

113. C₁₃

#2658

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊Q₈	2
C₂×#88201	2
C₂×#88198	2
C₂×#88129	2
#11718653	2
#11718593	2
#2658	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

Subgroups — #13120948

Conjugacy classes of subgroups

Complements

Maximal subgroups