Group #13120956 — subgroups

Subgroups — #13120956

Order 1664.

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

normal subgroups only

This group has 448 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
2	C₂×#88124	C₂	1	1	yes	maximal, characteristic
3	#11718890	C₂	1	4	yes	maximal
4	#11718641	C₂	1	4	yes	maximal
5	#11718638	C₂	1	2	yes	maximal
6	#11718637	C₂	1	2	yes	maximal
7	#11718593	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
9	C₁₃×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×#6101	C₂²	1	2	yes	—
16	C₂×#6101	C₂²	1	2	yes	—
17	C₂×#6101	C₂²	1	8	yes	—
18	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
19	#88124	—	2	4	no	—
20	#88071	C₂²	1	2	yes	—
21	#88071	C₂²	1	2	yes	—
22	#88071	C₂²	1	8	yes	—
23	C₂×Q₈×C₁₃	—	2	4	no	—
24	C₂×Q₈×C₁₃	—	4	4	no	—
25	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂²×Dic₁₃	C₂³	1	2	yes	—
30	C₂²×Dic₁₃	C₂³	1	2	yes	—
31	C₂²×Dic₁₃	C₂³	1	4	yes	—
32	C₁₃×C₄⋊C₄	—	2	4	no	—
33	C₁₃×C₄⋊C₄	—	2	4	no	—
34	C₄²×C₁₃	D₄	1	4	yes	—
35	C₄⋊Dic₁₃	—	2	2	no	—
36	C₄⋊Dic₁₃	—	2	2	no	—
37	#6101	—	2	4	no	—
38	#6101	—	2	4	no	—
39	#6101	—	4	8	no	—
40	C₄×Dic₁₃	Q₈	1	8	yes	complemented
41	#2660	—	13	1	no	maximal, Sylow
42	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
43	Q₈×C₁₃	—	4	8	no	—
44	C₂×C₄×C₁₃	—	2	2	no	—
45	C₂×C₄×C₁₃	—	2	2	no	—
46	C₂×C₄×C₁₃	—	2	4	no	—
47	C₂×C₄×C₁₃	—	2	8	no	—
48	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
52	C₂×Dic₁₃	—	2	4	no	—
53	C₂×Dic₁₃	—	2	4	no	—
54	C₂×Dic₁₃	—	2	2	no	—
55	C₂×Dic₁₃	—	2	2	no	—
56	C₂×Dic₁₃	—	2	4	no	—
57	C₂×Dic₁₃	—	2	4	no	—
58	C₂×Dic₁₃	C₂×Q₈	1	8	yes	—
59	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic
60	C₂×#125	—	13	1	no	—
61	#398	—	13	4	no	—
62	#395	—	13	4	no	—
63	#384	—	13	2	no	—
64	#382	—	13	2	no	—
65	#376	—	13	1	no	—
66	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
67	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
68	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
69	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
70	C₄×C₁₃	—	2	2	no	—
71	C₄×C₁₃	—	2	2	no	—
72	C₄×C₁₃	—	4	4	no	—
73	C₄×C₁₃	C₄⋊Q₈	1	4	yes	—
74	Dic₁₃	—	2	8	no	—
75	Dic₁₃	—	4	2	no	—
76	Dic₁₃	—	4	2	no	—
77	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
78	C₄⋊Q₈	—	2	4	no	—
79	#125	—	26	4	no	—
80	C₂×C₄⋊C₄	—	13	2	no	—
81	C₂×C₄⋊C₄	—	13	2	no	—
82	C₂×C₄⋊C₄	—	13	1	no	—
83	C₂×C₄⋊C₄	—	13	1	no	—
84	C₂×C₄⋊C₄	—	13	8	no	—
85	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
87	C₂×C₄²	—	13	2	no	—
88	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
89	#95	—	13	2	no	—
90	#95	—	13	2	no	—
91	#95	—	13	8	no	—
92	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
93	C₂×C₁₃	#535	1	1	yes	characteristic
94	C₂×C₁₃	#536	1	1	yes	characteristic
95	C₂×C₁₃	#544	1	4	yes	—
96	C₂×Q₈	—	2	4	no	—
97	C₂×Q₈	—	4	4	no	—
98	C₂²×C₄	—	13	4	no	—
99	C₂²×C₄	—	13	2	no	—
100	C₂²×C₄	—	13	2	no	—
101	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
102	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	26	4	no	—
108	C₄⋊C₄	—	26	4	no	—
109	C₄⋊C₄	—	26	2	no	—
110	C₄⋊C₄	—	26	2	no	—
111	C₄⋊C₄	—	52	8	no	—
112	C₄²	—	13	8	no	—
113	C₄²	C₂²⋊D₁₃	1	4	yes	—
114	C₁₃	#2660	1	1	yes	characteristic, complemented, Sylow
115	C₂³	C₂³×D₁₃	1	1	yes	characteristic
116	Q₈	—	4	8	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	8	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	13	8	no	—
122	C₂×C₄	—	26	4	no	—
123	C₂×C₄	—	26	4	no	—
124	C₂×C₄	—	26	4	no	—
125	C₂×C₄	—	26	4	no	—
126	C₂×C₄	—	26	2	no	—
127	C₂×C₄	—	26	2	no	—
128	C₂×C₄	Q₈×D₁₃	1	4	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
132	C₂²	#88251	1	2	yes	—
133	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
134	C₂²	#88254	1	2	yes	—
135	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	26	8	no	—
140	C₄	—	52	2	no	—
141	C₄	—	52	2	no	—
142	C₄	#88198	1	4	yes	—
143	C₂	#11719434	1	4	yes	—
144	C₂	#11719510	1	1	yes	characteristic
145	C₂	C₂×#88198	1	1	yes	characteristic
146	C₂	#11719521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#88124

C₂

yes

maximal, characteristic

#11718890

C₂

yes

maximal

#11718641

C₂

yes

maximal

#11718638

C₂

yes

maximal

#11718637

C₂

yes

maximal

#11718593

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃

C₂²

yes

characteristic

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×#6101

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88124

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

#88071

C₂²

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₁₃×C₄⋊C₄

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

C₄⋊Dic₁₃

—

C₄⋊Dic₁₃

—

#6101

—

#6101

—

#6101

—

C₄×Dic₁₃

Q₈

yes

complemented

#2660

—

maximal, Sylow

C₂³×C₁₃

C₂⁴

yes

characteristic

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×C₄×C₁₃

C₂×Q₈

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×Q₈

yes

—

C₂×C₄⋊Q₈

D₁₃

yes

—

—

—

—

—

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂²×Q₈

yes

—

C₂²×C₁₃

H(5,𝔽₂)

yes

—

C₂²×C₁₃

2₋¹⁺⁴

yes

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

—

C₄×C₁₃

C₄⋊Q₈

yes

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

—

—

—

C₂×C₁₃

C₂×C₄⋊Q₈

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

C₂×C₁₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²×D₁₃

yes

—

102

C₂²×C₄

C₂²×D₁₃

yes

characteristic

103

C₂²×C₄

C₂²×D₁₃

yes

characteristic

104

C₂²×C₄

C₂²×D₁₃

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂²⋊D₁₃

yes

—

114

C₁₃

#2660

yes

characteristic, complemented, Sylow

115

C₂³

C₂³×D₁₃

yes

characteristic

116

Q₈

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

Q₈×D₁₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

132

C₂²

#88251

yes

—

133

C₂²

C₂×Q₈×D₁₃

yes

—

134

C₂²

#88254

yes

—

135

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

#88198

yes

—

143

C₂

#11719434

yes

—

144

C₂

#11719510

yes

characteristic

145

C₂

C₂×#88198

yes

characteristic

146

C₂

#11719521

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, prime radical, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2660
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, prime radical, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
40. C₄×Dic₁₃	Q₈	semidirect
114. C₁₃	#2660	semidirect

H ≅ G/N

type

40. C₄×Dic₁₃

Q₈

semidirect

114. C₁₃

#2660

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×#88124	2
#11718890	2
#11718641	2
#11718638	2
#11718637	2
#11718593	2
#2660	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

Subgroups — #13120956

Conjugacy classes of subgroups

Complements

Maximal subgroups