Group #11719220 — subgroups

Subgroups — #11719220

Order 832.

218 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 95 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⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#88184	C₂	1	1	yes	maximal, characteristic
3	#88183	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
5	#88178	C₂	1	1	yes	maximal, characteristic, complemented
6	#88152	C₂	1	2	yes	maximal, complemented
7	#88150	C₂	1	2	yes	maximal, complemented
8	#88148	C₂	1	2	yes	maximal, complemented
9	#88143	C₂	1	2	yes	maximal, complemented
10	C₂²×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₃	C₂²	1	2	yes	—
14	C₂×D₅₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₃	C₂²	1	2	yes	—
16	C₂×Dic₂₆	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂²⋊Dic₁₃	C₂²	1	2	yes	—
20	#6103	C₂²	1	2	yes	—
21	#6103	C₂²	1	2	yes	—
22	#6103	C₂²	1	2	yes	—
23	#6103	C₂²	1	2	yes	—
24	C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
26	#6101	C₂²	1	2	yes	—
27	#6101	C₂²	1	2	yes	—
28	#6101	C₂²	1	2	yes	—
29	#6101	C₂²	1	2	yes	—
30	C₄×Dic₁₃	C₂²	1	2	yes	—
31	C₂³×C₁₃	C₂³	1	1	yes	characteristic
32	C₂²×D₁₃	C₂³	1	2	yes	—
33	C₂×C₄×C₁₃	—	2	2	no	—
34	C₂×C₄×C₁₃	—	4	1	no	—
35	C₂×C₄×C₁₃	—	4	1	no	—
36	C₂×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂×C₄×C₁₃	C₂³	1	1	yes	characteristic
40	C₂²⋊D₁₃	—	2	4	no	—
41	C₂×Dic₁₃	C₂³	1	2	yes	—
42	C₂×Dic₁₃	C₂³	1	2	yes	—
43	C₂×Dic₁₃	C₂³	1	2	yes	—
44	D₅₂	—	4	1	no	—
45	C₄×D₁₃	—	4	2	no	—
46	Dic₂₆	—	4	1	no	—
47	#539	—	13	1	no	maximal, Sylow
48	C₂²×C₁₃	—	2	1	no	—
49	C₂²×C₁₃	—	2	1	no	—
50	C₂²×C₁₃	#41	1	2	yes	—
51	C₂²×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₁₃	—	2	2	no	—
53	C₂×D₁₃	—	2	2	no	—
54	C₂×D₁₃	—	2	2	no	—
55	C₄×C₁₃	—	2	2	no	—
56	C₄×C₁₃	—	2	1	no	—
57	C₄×C₁₃	—	2	2	no	—
58	C₄×C₁₃	—	2	1	no	—
59	Dic₁₃	—	2	2	no	—
60	Dic₁₃	—	2	2	no	—
61	Dic₁₃	—	2	2	no	—
62	#126	—	13	2	no	—
63	#125	—	13	2	no	—
64	#123	—	13	2	no	—
65	#123	—	13	2	no	—
66	C₂²⋊Q₈	—	13	1	no	—
67	C₂²⋊Q₈	—	13	2	no	—
68	#121	—	13	1	no	—
69	C₄×D₄	—	13	2	no	—
70	C₂×C₄⋊C₄	D₁₃	1	1	yes	characteristic, complemented
71	C₂×C₁₃	—	2	2	no	—
72	C₂×C₁₃	C₂×#41	1	1	yes	characteristic
73	C₂×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
74	C₂×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
75	D₁₃	—	4	2	no	—
76	C₂×Q₈	—	13	1	no	—
77	C₂×D₄	—	13	2	no	—
78	C₂×D₄	—	13	1	no	—
79	C₂²×C₄	—	13	2	no	—
80	C₂²×C₄	C₂×D₁₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₁₃	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₁₃	1	1	yes	characteristic
83	C₄⋊C₄	—	13	2	no	—
84	C₄⋊C₄	—	13	1	no	—
85	C₄⋊C₄	—	13	2	no	—
86	C₄⋊C₄	—	13	2	no	—
87	C₄⋊C₄	—	13	1	no	—
88	C₄⋊C₄	—	13	2	no	—
89	C₄⋊C₄	C₂×D₁₃	1	2	yes	—
90	C₄⋊C₄	C₂×D₁₃	1	2	yes	—
91	C₂²⋊C₄	—	13	2	no	—
92	C₂²⋊C₄	—	13	2	no	—
93	C₂²⋊C₄	—	13	2	no	—
94	C₂²⋊C₄	—	13	2	no	—
95	C₂²⋊C₄	—	13	2	no	—
96	C₄²	—	13	2	no	—
97	C₁₃	#539	1	1	yes	characteristic, complemented, Sylow
98	C₂³	—	13	2	no	—
99	C₂³	C₂²×D₁₃	1	1	yes	characteristic
100	Q₈	—	52	1	no	—
101	D₄	—	26	4	no	—
102	D₄	—	52	1	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	13	2	no	—
107	C₂×C₄	—	13	2	no	—
108	C₂×C₄	—	13	2	no	—
109	C₂×C₄	—	52	2	no	—
110	C₂×C₄	C₂²×D₁₃	1	2	yes	—
111	C₂×C₄	C₂²×D₁₃	1	2	yes	—
112	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₃	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	26	2	no	—
117	C₂²	—	26	2	no	—
118	C₂²	—	26	2	no	—
119	C₂²	#6127	1	2	yes	—
120	C₂²	C₂³×D₁₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	—	2	1	no	—
125	C₄	—	26	2	no	—
126	C₄	—	26	2	no	—
127	C₄	—	26	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	52	2	no	—
130	C₂	C₂×#6127	1	1	yes	characteristic
131	C₂	#88251	1	1	yes	characteristic
132	C₂	#88254	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×C₄⋊C₄

C₂

yes

maximal, characteristic, complemented

#88184

C₂

yes

maximal, characteristic

#88183

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₃

C₂

yes

maximal, complemented

#88178

C₂

yes

maximal, characteristic, complemented

#88152

C₂

yes

maximal, complemented

#88150

C₂

yes

maximal, complemented

#88148

C₂

yes

maximal, complemented

#88143

C₂

yes

maximal, complemented

C₂²×C₄×C₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₃

C₂²

yes

characteristic

C₂²×C₄×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊D₁₃

C₂²

yes

—

C₂×D₅₂

C₂²

yes

characteristic

C₂×C₄×D₁₃

C₂²

yes

—

C₂×Dic₂₆

C₂²

yes

characteristic

C₁₃×C₄⋊C₄

C₂²

yes

—

C₁₃×C₄⋊C₄

C₂²

yes

—

C₂²⋊Dic₁₃

C₂²

yes

—

#6103

C₂²

yes

—

#6103

C₂²

yes

—

#6103

C₂²

yes

—

#6103

C₂²

yes

—

C₄⋊Dic₁₃

C₂²

yes

characteristic

C₄⋊Dic₁₃

C₂²

yes

characteristic

#6101

C₂²

yes

—

#6101

C₂²

yes

—

#6101

C₂²

yes

—

#6101

C₂²

yes

—

C₄×Dic₁₃

C₂²

yes

—

C₂³×C₁₃

C₂³

yes

characteristic

C₂²×D₁₃

C₂³

yes

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

—

C₂×C₄×C₁₃

C₂³

yes

—

C₂×C₄×C₁₃

C₂³

yes

characteristic

C₂×C₄×C₁₃

C₂³

yes

—

C₂×C₄×C₁₃

C₂³

yes

characteristic

C₂²⋊D₁₃

—

C₂×Dic₁₃

C₂³

yes

—

C₂×Dic₁₃

C₂³

yes

—

C₂×Dic₁₃

C₂³

yes

—

D₅₂

—

C₄×D₁₃

—

Dic₂₆

—

#539

—

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

—

Dic₁₃

—

Dic₁₃

—

Dic₁₃

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

D₁₃

yes

characteristic, complemented

C₂×C₁₃

—

C₂×C₁₃

C₂×#41

yes

characteristic

C₂×C₁₃

H(5,𝔽₂)

yes

characteristic

C₂×C₁₃

2₋¹⁺⁴

yes

characteristic

D₁₃

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₁₃

yes

characteristic

C₂²×C₄

C₂×D₁₃

yes

characteristic

C₂²×C₄

C₂×D₁₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₃

yes

—

C₄⋊C₄

C₂×D₁₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₃

#539

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

C₂²×D₁₃

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²×D₁₃

yes

—

111

C₂×C₄

C₂²×D₁₃

yes

—

112

C₂×C₄

C₂²×D₁₃

yes

characteristic

113

C₂×C₄

C₂²×D₁₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#6127

yes

—

120

C₂²

C₂³×D₁₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₂

—

129

C₂

—

130

C₂

C₂×#6127

yes

characteristic

131

C₂

#88251

yes

characteristic

132

C₂

#88254

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁴
Fitting	C₂×C₁₃×C₄⋊C₄	C₂
prime radical	#88184	C₂
hypercenter	C₂×C₄⋊C₄	D₁₃
hypocenter	C₁₃	#539
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₃

commutator, socle, cosocle

C₂²×C₁₃

C₂⁴

Fitting

C₂×C₁₃×C₄⋊C₄

C₂

prime radical

#88184

C₂

hypercenter

C₂×C₄⋊C₄

D₁₃

C₁₃

G ▷ C₂²×C₁₃ ▷ C₁₃

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. C₂×C₁₃×C₄⋊C₄	C₂	semidirect
3. #88183	C₂	semidirect
4. C₄×C₂²⋊D₁₃	C₂	semidirect
5. #88178	C₂	semidirect
6. #88152	C₂	semidirect
7. #88150	C₂	semidirect
8. #88148	C₂	semidirect
9. #88143	C₂	semidirect
70. C₂×C₄⋊C₄	D₁₃	semidirect
97. C₁₃	#539	semidirect

H ≅ G/N

type

1. C₂×C₁₃×C₄⋊C₄

C₂

semidirect

3. #88183

C₂

semidirect

4. C₄×C₂²⋊D₁₃

C₂

semidirect

5. #88178

C₂

semidirect

6. #88152

C₂

semidirect

7. #88150

C₂

semidirect

8. #88148

C₂

semidirect

9. #88143

C₂

semidirect

70. C₂×C₄⋊C₄

D₁₃

semidirect

97. C₁₃

#539

semidirect

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

subgroup	index
C₂×C₁₃×C₄⋊C₄	2
#88184	2
#88183	2
C₄×C₂²⋊D₁₃	2
#88178	2
#88152	2
#88150	2
#88148	2
#88143	2
#539	13

subgroup

index

C₂×C₁₃×C₄⋊C₄

C₄×C₂²⋊D₁₃

Subgroups — #11719220

Conjugacy classes of subgroups

Complements

Maximal subgroups