Group #13123178 — subgroups

Subgroups — #13123178

Order 1664.

924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88192	C₂	1	6	yes	maximal, complemented
3	C₂×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718882	C₂	1	6	yes	maximal, complemented
5	#11718596	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₃	C₂²	1	6	yes	—
7	C₂²×C₂²⋊D₁₃	C₂²	1	4	yes	—
8	C₂²×C₄×D₁₃	C₂²	1	3	yes	—
9	C₁₃×#127	—	2	4	no	—
10	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊Dic₁₃	C₂²	1	12	yes	—
12	#88192	—	2	24	no	—
13	C₂×C₄⋊Dic₁₃	C₂²	1	6	yes	—
14	C₂×C₄×Dic₁₃	C₂²	1	3	yes	—
15	C₄²×D₁₃	—	2	4	no	—
16	C₂⁴×C₁₃	C₂³	1	4	yes	—
17	C₂³×D₁₃	C₂³	1	1	yes	characteristic
18	C₂×D₄×C₁₃	—	2	12	no	—
19	C₂×D₄×C₁₃	—	2	12	no	—
20	C₂×D₄×C₁₃	—	2	12	no	—
21	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
22	C₂×C₂²⋊D₁₃	—	2	24	no	—
23	C₂²×Dic₁₃	C₂³	1	4	yes	—
24	C₂²×Dic₁₃	C₂³	1	3	yes	—
25	C₂×C₄×D₁₃	—	2	6	no	—
26	C₂×C₄×D₁₃	—	2	6	no	—
27	C₂×C₄×D₁₃	D₄	1	12	yes	complemented
28	C₄²×C₁₃	D₄	1	4	yes	complemented
29	C₂²⋊Dic₁₃	—	2	24	no	—
30	C₄⋊Dic₁₃	—	2	12	no	—
31	C₄×Dic₁₃	—	2	6	no	—
32	#2856	—	13	1	no	maximal, Sylow
33	C₂³×C₁₃	—	2	12	no	—
34	C₂³×C₁₃	—	2	4	no	—
35	C₂³×C₁₃	—	2	12	no	—
36	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
37	C₂²×D₁₃	—	2	1	no	—
38	C₂²×D₁₃	—	2	3	no	—
39	C₂²×D₁₃	C₂×D₄	1	6	yes	—
40	D₄×C₁₃	—	4	24	no	—
41	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
42	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
43	C₂×C₄×C₁₃	#41	1	6	yes	—
44	C₂²⋊D₁₃	—	4	16	no	—
45	C₂×Dic₁₃	—	2	3	no	—
46	C₂×Dic₁₃	—	2	3	no	—
47	C₂×Dic₁₃	—	2	12	no	—
48	C₂×Dic₁₃	C₂×D₄	1	6	yes	—
49	C₄×D₁₃	—	2	24	no	—
50	C₂×#127	D₁₃	1	1	yes	characteristic, complemented
51	C₂×#121	—	13	6	no	—
52	C₂²×C₄²	—	13	1	no	—
53	#390	—	13	6	no	—
54	#384	—	13	1	no	—
55	C₂²×C₁₃	—	4	12	no	—
56	C₂²×C₁₃	—	4	4	no	—
57	C₂²×C₁₃	—	4	12	no	—
58	C₂²×C₁₃	C₂²×D₄	1	3	yes	—
59	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
60	C₂²×C₁₃	C₂×#41	1	3	yes	—
61	C₂×D₁₃	—	2	6	no	—
62	C₂×D₁₃	—	2	6	no	—
63	C₂×D₁₃	#127	1	4	yes	complemented
64	C₄×C₁₃	#121	1	12	yes	complemented
65	Dic₁₃	—	2	6	no	—
66	Dic₁₃	—	4	4	no	—
67	C₂²×D₄	—	13	4	no	—
68	C₂²×D₄	C₂×D₁₃	1	6	yes	—
69	C₂³×C₄	—	13	3	no	—
70	#127	—	2	4	no	—
71	#121	—	26	24	no	—
72	C₂×C₄⋊C₄	—	13	6	no	—
73	C₂×C₂²⋊C₄	—	13	12	no	—
74	C₂×C₄²	—	13	3	no	—
75	C₂×C₄²	—	26	4	no	—
76	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
77	C₂×C₁₃	—	8	4	no	—
78	C₂×C₁₃	C₂×#121	1	3	yes	—
79	C₂×C₁₃	C₂×#127	1	1	yes	characteristic
80	C₂×C₁₃	#532	1	3	yes	—
81	D₁₃	—	2	4	no	—
82	C₂⁴	—	13	1	no	—
83	C₂⁴	C₂²×D₁₃	1	4	yes	—
84	C₂×D₄	—	2	12	no	—
85	C₂×D₄	—	2	12	no	—
86	C₂×D₄	—	2	12	no	—
87	C₂×D₄	—	26	24	no	—
88	C₂²×C₄	—	13	3	no	—
89	C₂²×C₄	—	13	12	no	—
90	C₂²×C₄	—	13	4	no	—
91	C₂²×C₄	—	26	6	no	—
92	C₂²×C₄	—	26	6	no	—
93	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
94	C₄⋊C₄	—	26	12	no	—
95	C₂²⋊C₄	—	26	24	no	—
96	C₄²	—	26	6	no	—
97	C₄²	C₂²⋊D₁₃	1	4	yes	complemented
98	C₁₃	#2856	1	1	yes	characteristic, complemented, Sylow
99	C₂³	—	2	12	no	—
100	C₂³	—	2	12	no	—
101	C₂³	—	2	4	no	—
102	C₂³	—	13	6	no	—
103	C₂³	—	26	3	no	—
104	C₂³	—	26	1	no	—
105	C₂³	C₂³×D₁₃	1	1	yes	characteristic
106	D₄	—	4	24	no	—
107	D₄	—	52	16	no	—
108	C₂×C₄	—	13	6	no	—
109	C₂×C₄	—	26	3	no	—
110	C₂×C₄	—	26	3	no	—
111	C₂×C₄	—	26	24	no	—
112	C₂×C₄	—	26	12	no	—
113	C₂×C₄	D₄×D₁₃	1	6	yes	—
114	C₂×C₄	#6129	1	6	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
116	C₂²	—	4	12	no	—
117	C₂²	—	4	4	no	—
118	C₂²	—	4	12	no	—
119	C₂²	—	13	4	no	—
120	C₂²	—	26	6	no	—
121	C₂²	—	26	6	no	—
122	C₂²	C₂×D₄×D₁₃	1	3	yes	—
123	C₂²	C₂×#6129	1	3	yes	—
124	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
125	C₄	—	26	6	no	—
126	C₄	—	52	4	no	—
127	C₄	#88192	1	12	yes	complemented
128	C₂	—	8	4	no	—
129	C₂	—	26	4	no	—
130	C₂	D₁₃×#127	1	1	yes	characteristic
131	C₂	#11719421	1	3	yes	—
132	C₂	C₂×#88192	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₃×#127

C₂

yes

maximal, characteristic, complemented

C₂×#88192

C₂

yes

maximal, complemented

C₂×C₄²×D₁₃

C₂

yes

maximal, characteristic, complemented

#11718882

C₂

yes

maximal, complemented

#11718596

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₃

C₂²

yes

—

C₂²×C₂²⋊D₁₃

C₂²

yes

—

C₂²×C₄×D₁₃

C₂²

yes

—

C₁₃×#127

—

C₂×C₄²×C₁₃

C₂²

yes

characteristic

C₂×C₂²⋊Dic₁₃

C₂²

yes

—

#88192

—

C₂×C₄⋊Dic₁₃

C₂²

yes

—

C₂×C₄×Dic₁₃

C₂²

yes

—

C₄²×D₁₃

—

C₂⁴×C₁₃

C₂³

yes

—

C₂³×D₁₃

C₂³

yes

characteristic

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂×D₄×C₁₃

—

C₂²×C₄×C₁₃

C₂³

yes

—

C₂×C₂²⋊D₁₃

—

C₂²×Dic₁₃

C₂³

yes

—

C₂²×Dic₁₃

C₂³

yes

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

—

C₂×C₄×D₁₃

D₄

yes

complemented

C₄²×C₁₃

D₄

yes

complemented

C₂²⋊Dic₁₃

—

C₄⋊Dic₁₃

—

C₄×Dic₁₃

—

#2856

—

maximal, Sylow

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

—

C₂³×C₁₃

C₂⁴

yes

characteristic

C₂²×D₁₃

—

C₂²×D₁₃

—

C₂²×D₁₃

C₂×D₄

yes

—

D₄×C₁₃

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

C₂×D₄

yes

—

C₂×C₄×C₁₃

#41

yes

—

C₂²⋊D₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

—

C₂×Dic₁₃

C₂×D₄

yes

—

C₄×D₁₃

—

C₂×#127

D₁₃

yes

characteristic, complemented

C₂×#121

—

C₂²×C₄²

—

#390

—

#384

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

—

C₂²×C₁₃

C₂²×D₄

yes

—

C₂²×C₁₃

C₂²×D₄

yes

characteristic

C₂²×C₁₃

C₂×#41

yes

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

yes

C₄×C₁₃

yes

Dic₁₃

—

Dic₁₃

—

C₂²×D₄

—

C₂²×D₄

C₂×D₁₃

yes

—

C₂³×C₄

—

#127

—

#121

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₃

yes

characteristic

C₂×C₁₃

—

C₂×C₁₃

C₂×#121

yes

—

C₂×C₁₃

C₂×#127

yes

characteristic

C₂×C₁₃

#532

yes

—

D₁₃

—

C₂⁴

—

C₂⁴

C₂²×D₁₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²⋊D₁₃

yes

complemented

C₁₃

#2856

yes

characteristic, complemented, Sylow

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂³×D₁₃

yes

characteristic

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

D₄×D₁₃

yes

—

114

C₂×C₄

#6129

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×D₄×D₁₃

yes

—

123

C₂²

C₂×#6129

yes

—

124

C₂²

C₂²×C₂²⋊D₁₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

#88192

yes

complemented

128

C₂

—

129

C₂

—

130

C₂

D₁₃×#127

yes

characteristic

131

C₂

#11719421

yes

—

132

C₂

C₂×#88192

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃
commutator, socle, cosocle	C₂³×C₁₃	C₂⁴
Fitting	C₂×C₁₃×#127	C₂
hypercenter	C₂×#127	D₁₃
hypocenter	C₁₃	#2856
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃

commutator, socle, cosocle

C₂³×C₁₃

C₂⁴

Fitting

C₂×C₁₃×#127

C₂

D₁₃

C₁₃

G ▷ C₂³×C₁₃ ▷ C₁₃

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×C₁₃×#127	C₂	semidirect
2. C₂×#88192	C₂	semidirect
3. C₂×C₄²×D₁₃	C₂	semidirect
4. #11718882	C₂	semidirect
5. #11718596	C₂	semidirect
27. C₂×C₄×D₁₃	D₄	semidirect
28. C₄²×C₁₃	D₄	semidirect
50. C₂×#127	D₁₃	semidirect
63. C₂×D₁₃	#127	semidirect
64. C₄×C₁₃	#121	semidirect
97. C₄²	C₂²⋊D₁₃	semidirect
98. C₁₃	#2856	semidirect
127. C₄	#88192	semidirect

H ≅ G/N

type

1. C₂×C₁₃×#127

C₂

semidirect

2. C₂×#88192

C₂

semidirect

3. C₂×C₄²×D₁₃

C₂

semidirect

4. #11718882

C₂

semidirect

5. #11718596

C₂

semidirect

27. C₂×C₄×D₁₃

D₄

semidirect

28. C₄²×C₁₃

D₄

semidirect

50. C₂×#127

D₁₃

semidirect

63. C₂×D₁₃

#127

semidirect

64. C₄×C₁₃

#121

semidirect

97. C₄²

C₂²⋊D₁₃

semidirect

98. C₁₃

#2856

semidirect

127. C₄

#88192

semidirect

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

subgroup	index
C₂×C₁₃×#127	2
C₂×#88192	2
C₂×C₄²×D₁₃	2
#11718882	2
#11718596	2
#2856	13

subgroup

index

C₂×C₁₃×#127

C₂×C₄²×D₁₃

Subgroups — #13123178

Conjugacy classes of subgroups

Complements

Maximal subgroups