Group #13123194 — subgroups

Subgroups — #13123194

Order 1664.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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	4	yes	maximal, complemented
3	C₂×#88128	C₂	1	1	yes	maximal, characteristic
4	#11718649	C₂	1	8	yes	maximal, complemented
5	C₄²×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₂²×D₅₂	C₂²	1	2	yes	—
8	C₂²×D₅₂	C₂²	1	4	yes	—
9	C₁₃×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
12	#88201	—	2	16	no	—
13	C₂×#6103	C₂²	1	16	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	4	yes	—
15	C₂×C₄×Dic₁₃	C₂²	1	2	yes	—
16	#88128	—	2	4	no	—
17	C₂³×D₁₃	C₂³	1	4	yes	—
18	C₂×Q₈×C₁₃	—	2	8	no	—
19	C₂×Q₈×C₁₃	—	2	4	no	—
20	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
23	C₂²×Dic₁₃	C₂³	1	4	yes	—
24	C₂×D₅₂	—	2	8	no	—
25	C₂×D₅₂	—	2	4	no	—
26	C₂×D₅₂	—	2	8	no	—
27	C₂×D₅₂	—	2	16	no	—
28	C₁₃×C₄⋊C₄	—	2	8	no	—
29	C₄²×C₁₃	D₄	1	4	yes	—
30	#6103	—	2	32	no	—
31	C₄×Dic₁₃	—	2	8	no	—
32	C₄×Dic₁₃	D₄	1	8	yes	complemented
33	#2858	—	13	1	no	maximal, Sylow
34	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
35	C₂²×D₁₃	—	2	8	no	—
36	C₂²×D₁₃	—	2	4	no	—
37	C₂²×D₁₃	—	2	16	no	—
38	Q₈×C₁₃	—	4	8	no	—
39	C₂×C₄×C₁₃	—	2	8	no	—
40	C₂×C₄×C₁₃	—	2	4	no	—
41	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₃	#41	1	8	yes	—
45	C₂×Dic₁₃	—	2	8	no	—
46	C₂×Dic₁₃	C₂×D₄	1	8	yes	—
47	D₅₂	—	4	16	no	—
48	D₅₂	—	4	8	no	—
49	C₂×C₄⋊Q₈	D₁₃	1	1	yes	characteristic, complemented
50	C₂×#127	—	13	1	no	—
51	C₂×#124	—	13	4	no	—
52	#390	—	13	8	no	—
53	C₄³	—	13	1	no	—
54	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
55	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
56	C₂²×C₁₃	C₂×#41	1	4	yes	—
57	C₂×D₁₃	—	4	8	no	—
58	C₂×D₁₃	—	4	4	no	—
59	C₂×D₁₃	—	4	16	no	—
60	C₄×C₁₃	—	4	4	no	—
61	C₄×C₁₃	#124	1	8	yes	complemented
62	C₄×C₁₃	#127	1	4	yes	—
63	Dic₁₃	—	2	8	no	—
64	C₂²×Q₈	C₂×D₁₃	1	2	yes	—
65	C₂²×D₄	—	13	2	no	—
66	C₂²×D₄	—	13	4	no	—
67	C₄⋊Q₈	—	2	4	no	—
68	#127	—	26	4	no	—
69	#124	—	26	16	no	—
70	C₂×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
71	C₂×C₂²⋊C₄	—	13	16	no	—
72	C₂×C₄²	—	13	2	no	—
73	C₂×C₄²	—	13	4	no	—
74	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
75	C₂×C₁₃	C₂×#124	1	2	yes	—
76	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
77	C₂×C₁₃	#532	1	4	yes	—
78	D₁₃	—	8	4	no	—
79	C₂⁴	—	13	4	no	—
80	C₂×Q₈	—	2	8	no	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×D₄	—	26	8	no	—
83	C₂×D₄	—	26	16	no	—
84	C₂×D₄	—	26	4	no	—
85	C₂×D₄	—	26	8	no	—
86	C₂²×C₄	—	13	4	no	—
87	C₂²×C₄	C₂²×D₁₃	1	4	yes	—
88	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
90	C₄⋊C₄	—	2	8	no	—
91	C₂²⋊C₄	—	26	32	no	—
92	C₄²	—	13	8	no	—
93	C₄²	—	26	8	no	—
94	C₄²	C₂²⋊D₁₃	1	4	yes	—
95	C₁₃	#2858	1	1	yes	characteristic, complemented, Sylow
96	C₂³	—	26	8	no	—
97	C₂³	—	26	16	no	—
98	C₂³	—	26	4	no	—
99	C₂³	C₂³×D₁₃	1	1	yes	characteristic
100	Q₈	—	4	8	no	—
101	D₄	—	52	8	no	—
102	D₄	—	52	16	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	13	8	no	—
106	C₂×C₄	—	26	8	no	—
107	C₂×C₄	D₄×D₁₃	1	4	yes	—
108	C₂×C₄	#6131	1	8	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₃	1	4	yes	—
111	C₂²	—	52	16	no	—
112	C₂²	—	52	4	no	—
113	C₂²	—	52	8	no	—
114	C₂²	C₂×D₄×D₁₃	1	2	yes	—
115	C₂²	C₂×#6131	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	26	8	no	—
119	C₄	#88194	1	4	yes	—
120	C₄	#88201	1	8	yes	complemented
121	C₂	—	104	4	no	—
122	C₂	#11719430	1	4	yes	—
123	C₂	C₂×#88194	1	1	yes	characteristic
124	C₂	C₂×#88201	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, complemented

C₂×#88128

C₂

yes

maximal, characteristic

#11718649

C₂

yes

maximal, complemented

C₄²×Dic₁₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₃

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

—

C₂²×D₅₂

C₂²

yes

—

C₁₃×C₄⋊Q₈

—

C₂×C₁₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

#88201

—

C₂×#6103

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

#88128

—

C₂³×D₁₃

C₂³

yes

—

C₂×Q₈×C₁₃

—

C₂×Q₈×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×C₄×C₁₃

C₂³

yes

characteristic

C₂²×C₄×C₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₂×D₅₂

—

C₁₃×C₄⋊C₄

—

C₄²×C₁₃

D₄

yes

—

#6103

—

C₄×Dic₁₃

—

C₄×Dic₁₃

D₄

yes

complemented

#2858

—

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×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

D₅₂

—

D₅₂

—

C₂×C₄⋊Q₈

D₁₃

yes

characteristic, complemented

C₂×#127

—

C₂×#124

—

#390

—

C₄³

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₄×C₁₃

—

C₄×C₁₃

#124

yes

complemented

C₄×C₁₃

#127

yes

—

Dic₁₃

—

C₂²×Q₈

C₂×D₁₃

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₄⋊Q₈

—

#127

—

#124

—

C₂×C₄⋊C₄

C₂×D₁₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₁₃

C₂×#124

yes

—

C₂×C₁₃

C₂×#127

yes

characteristic

C₂×C₁₃

#532

yes

—

D₁₃

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₂²×C₄

C₂²×D₁₃

yes

characteristic

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₁₃

yes

—

C₁₃

#2858

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₃

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

D₄×D₁₃

yes

—

108

C₂×C₄

#6131

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₁₃

yes

—

115

C₂²

C₂×#6131

yes

—

116

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#88194

yes

—

120

C₄

#88201

yes

complemented

121

C₂

—

104

—

122

C₂

#11719430

yes

—

123

C₂

C₂×#88194

yes

characteristic

124

C₂

C₂×#88201

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊Q₈	C₂
prime radical	C₂×#88128	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₃
hypocenter	C₁₃	#2858
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

C₂×#88128

C₂

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
2. C₂×#88201	C₂	semidirect
4. #11718649	C₂	semidirect
5. C₄²×Dic₁₃	C₂	semidirect
32. C₄×Dic₁₃	D₄	semidirect
49. C₂×C₄⋊Q₈	D₁₃	semidirect
61. C₄×C₁₃	#124	semidirect
95. C₁₃	#2858	semidirect
120. C₄	#88201	semidirect

H ≅ G/N

type

1. C₂×C₁₃×C₄⋊Q₈

C₂

semidirect

2. C₂×#88201

C₂

semidirect

4. #11718649

C₂

semidirect

5. C₄²×Dic₁₃

C₂

semidirect

32. C₄×Dic₁₃

D₄

semidirect

49. C₂×C₄⋊Q₈

D₁₃

semidirect

61. C₄×C₁₃

#124

semidirect

95. C₁₃

#2858

semidirect

120. C₄

#88201

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×#88128	2
#11718649	2
C₄²×Dic₁₃	2
#2858	13

subgroup

index

C₂×C₁₃×C₄⋊Q₈

C₄²×Dic₁₃

Subgroups — #13123194

Conjugacy classes of subgroups

Complements

Maximal subgroups