Group #13380351 — subgroups

Subgroups — #13380351

Order 1792.

912 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 912 conjugacy classes of subgroups — too many to draw. Only its 255 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	C₄×C₇×#127	C₂	1	1	yes	maximal, characteristic
2	#11729365	C₂	1	12	yes	maximal, complemented
3	#11727147	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#127	C₄	1	1	yes	characteristic
5	C₂×C₄×C₇×D₄	C₂²	1	6	yes	—
6	C₂²×#1136	C₂²	1	4	yes	—
7	C₇×#390	C₄	1	6	yes	—
8	C₇×#384	C₄	1	1	yes	characteristic
9	C₄³×C₇	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊(C₇⋊C₈)	C₂²	1	12	yes	—
11	#89319	C₂²	1	12	yes	—
12	#23357	—	7	1	no	maximal, Sylow
13	C₂²×C₇×D₄	C₂×C₄	1	6	yes	—
14	C₂³×C₄×C₇	C₂³	1	4	yes	—
15	C₇×#127	—	2	4	no	—
16	C₄×C₇×D₄	—	4	6	no	—
17	C₄×C₇×D₄	—	4	6	no	—
18	C₂×C₇×C₄⋊C₄	C₂×C₄	1	6	yes	—
19	C₂×C₇×C₂²⋊C₄	C₂×C₄	1	12	yes	—
20	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂³	1	3	yes	—
23	C₂×#1136	—	4	12	no	—
24	C₂²×C₇⋊C₈	C₂³	1	8	yes	—
25	C₂²⋊(C₇⋊C₈)	—	4	12	no	—
26	C₄×#127	D₇	1	1	yes	characteristic
27	#1870	—	7	12	no	—
28	#1839	—	7	2	no	—
29	C₂⁴×C₇	C₂²×C₄	1	4	yes	—
30	C₂×C₇×D₄	—	2	12	no	—
31	C₂×C₇×D₄	—	4	6	no	—
32	C₂×C₇×D₄	—	4	6	no	—
33	C₂²×C₄×C₇	—	2	12	no	—
34	C₂²×C₄×C₇	—	2	12	no	—
35	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
36	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
37	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
38	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
39	C₇×C₄⋊C₄	—	4	6	no	—
40	C₇×C₂²⋊C₄	—	4	12	no	—
41	C₄²×C₇	—	2	6	no	—
42	C₄²×C₇	—	4	3	no	—
43	C₄²×C₇	M₁₆	1	4	yes	complemented
44	#1136	—	4	16	no	—
45	C₂×C₇⋊C₈	—	2	24	no	—
46	C₂²×M₁₆	—	7	4	no	—
47	C₂×#127	Dic₇	1	1	yes	characteristic
48	C₂×C₄×D₄	C₂×D₇	1	6	yes	—
49	C₂×C₂²⋊C₈	—	7	12	no	—
50	#390	Dic₇	1	6	yes	—
51	#384	Dic₇	1	1	yes	characteristic
52	C₄³	C₂×D₇	1	1	yes	characteristic
53	#336	—	7	12	no	—
54	C₂³×C₇	—	2	12	no	—
55	C₂³×C₇	—	2	4	no	—
56	C₂³×C₇	—	2	12	no	—
57	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
58	C₇×D₄	—	4	24	no	—
59	C₂×C₄×C₇	—	2	3	no	—
60	C₂×C₄×C₇	—	2	3	no	—
61	C₂×C₄×C₇	—	2	3	no	—
62	C₂×C₄×C₇	—	2	12	no	—
63	C₂×C₄×C₇	—	2	6	no	—
64	C₂×C₄×C₇	—	4	4	no	—
65	C₂×C₄×C₇	—	4	12	no	—
66	C₂×C₄×C₇	C₂×M₁₆	1	6	yes	—
67	C₂×C₄×C₇	H(5,𝔽₂)	1	6	yes	—
68	C₇⋊C₈	—	4	8	no	—
69	C₂²×D₄	C₂×Dic₇	1	6	yes	—
70	C₂³×C₄	C₂²×D₇	1	4	yes	—
71	C₂×M₁₆	—	28	12	no	—
72	C₂²×C₈	—	7	8	no	—
73	#127	—	2	4	no	—
74	C₄×D₄	—	4	6	no	—
75	C₄×D₄	—	4	6	no	—
76	C₂×C₄⋊C₄	C₂×Dic₇	1	6	yes	—
77	C₂×C₂²⋊C₄	C₂×Dic₇	1	12	yes	—
78	C₂×C₄²	C₂×Dic₇	1	3	yes	—
79	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
80	C₂×C₄²	C₂²×D₇	1	3	yes	—
81	C₂²⋊C₈	—	28	12	no	—
82	C₂²×C₇	—	4	4	no	—
83	C₂²×C₇	—	4	12	no	—
84	C₂²×C₇	—	4	12	no	—
85	C₂²×C₇	#518	1	3	yes	—
86	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
87	C₂²×C₇	#568	1	3	yes	—
88	C₄×C₇	—	2	6	no	—
89	C₄×C₇	—	4	4	no	—
90	C₄×C₇	—	4	3	no	—
91	C₄×C₇	#560	1	4	yes	—
92	C₂⁴	C₂²×Dic₇	1	4	yes	—
93	C₂×D₄	—	2	12	no	—
94	C₂×D₄	—	4	6	no	—
95	C₂×D₄	—	4	6	no	—
96	C₂²×C₄	—	2	12	no	—
97	C₂²×C₄	—	2	12	no	—
98	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
99	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
100	C₂²×C₄	C₂²×Dic₇	1	3	yes	—
101	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
102	M₁₆	—	28	16	no	—
103	C₂×C₈	—	14	24	no	—
104	C₄⋊C₄	—	4	6	no	—
105	C₂²⋊C₄	—	4	12	no	—
106	C₄²	—	2	6	no	—
107	C₄²	—	4	3	no	—
108	C₄²	#1136	1	4	yes	complemented
109	C₂×C₇	—	8	4	no	—
110	C₂×C₇	#2380	1	1	yes	characteristic
111	C₂×C₇	#2975	1	3	yes	—
112	C₂×C₇	#2980	1	3	yes	—
113	C₂³	—	2	4	no	—
114	C₂³	—	2	12	no	—
115	C₂³	—	2	12	no	—
116	C₂³	C₂³×Dic₇	1	1	yes	characteristic
117	D₄	—	4	24	no	—
118	C₂×C₄	—	2	3	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	2	3	no	—
121	C₂×C₄	—	2	12	no	—
122	C₂×C₄	—	2	6	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	—	4	12	no	—
125	C₂×C₄	C₂×#1136	1	6	yes	—
126	C₂×C₄	#6548	1	6	yes	—
127	C₈	—	28	8	no	—
128	C₇	#23357	1	1	yes	characteristic, complemented, Sylow
129	C₂²	—	4	12	no	—
130	C₂²	—	4	12	no	—
131	C₂²	—	4	4	no	—
132	C₂²	C₂²×#1136	1	1	yes	characteristic
133	C₂²	#90489	1	3	yes	—
134	C₂²	#90510	1	3	yes	—
135	C₄	—	2	6	no	—
136	C₄	—	4	3	no	—
137	C₄	—	4	4	no	—
138	C₄	#90411	1	4	yes	—
139	C₂	—	8	4	no	—
140	C₂	#11739749	1	3	yes	—
141	C₂	#11740619	1	1	yes	characteristic
142	C₂	#11740724	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₇×#127

C₂

yes

maximal, characteristic

#11729365

C₂

yes

maximal, complemented

#11727147

C₂

yes

maximal, complemented

C₂×C₇×#127

C₄

yes

characteristic

C₂×C₄×C₇×D₄

C₂²

yes

—

C₂²×#1136

C₂²

yes

—

C₇×#390

C₄

yes

—

C₇×#384

C₄

yes

characteristic

C₄³×C₇

C₂²

yes

characteristic

C₂×C₂²⋊(C₇⋊C₈)

C₂²

yes

—

#89319

C₂²

yes

—

#23357

—

maximal, Sylow

C₂²×C₇×D₄

C₂×C₄

yes

—

C₂³×C₄×C₇

C₂³

yes

—

C₇×#127

—

C₄×C₇×D₄

—

C₄×C₇×D₄

—

C₂×C₇×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₇×C₂²⋊C₄

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₇

C₂³

yes

—

C₂×#1136

—

C₂²×C₇⋊C₈

C₂³

yes

—

C₂²⋊(C₇⋊C₈)

—

C₄×#127

D₇

yes

characteristic

#1870

—

#1839

—

C₂⁴×C₇

C₂²×C₄

yes

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂²×C₄

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

—

C₂²×C₄×C₇

C₂⁴

yes

characteristic

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

M₁₆

yes

complemented

#1136

—

C₂×C₇⋊C₈

—

C₂²×M₁₆

—

C₂×#127

Dic₇

yes

characteristic

C₂×C₄×D₄

C₂×D₇

yes

—

C₂×C₂²⋊C₈

—

#390

Dic₇

yes

—

#384

Dic₇

yes

characteristic

C₄³

C₂×D₇

yes

characteristic

#336

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂³×C₄

yes

characteristic

C₇×D₄

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₂×C₄×C₇

H(5,𝔽₂)

yes

—

C₇⋊C₈

—

C₂²×D₄

C₂×Dic₇

yes

—

C₂³×C₄

C₂²×D₇

yes

—

C₂×M₁₆

—

C₂²×C₈

—

#127

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

C₂×Dic₇

yes

—

C₂×C₂²⋊C₄

C₂×Dic₇

yes

—

C₂×C₄²

C₂×Dic₇

yes

—

C₂×C₄²

C₂×Dic₇

yes

characteristic

C₂×C₄²

C₂²×D₇

yes

—

C₂²⋊C₈

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#518

yes

—

C₂²×C₇

C₂²×M₁₆

yes

characteristic

C₂²×C₇

#568

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#560

yes

—

C₂⁴

C₂²×Dic₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₇

yes

—

C₂²×C₄

C₂²×Dic₇

yes

—

100

C₂²×C₄

C₂²×Dic₇

yes

—

101

C₂²×C₄

C₂³×D₇

yes

characteristic

102

M₁₆

—

103

C₂×C₈

—

104

C₄⋊C₄

—

105

C₂²⋊C₄

—

106

C₄²

—

107

C₄²

—

108

C₄²

#1136

yes

complemented

109

C₂×C₇

—

110

C₂×C₇

#2380

yes

characteristic

111

C₂×C₇

#2975

yes

—

112

C₂×C₇

#2980

yes

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

C₂³×Dic₇

yes

characteristic

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×#1136

yes

—

126

C₂×C₄

#6548

yes

—

127

C₈

—

128

C₇

#23357

yes

characteristic, complemented, Sylow

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂²×#1136

yes

characteristic

133

C₂²

#90489

yes

—

134

C₂²

#90510

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#90411

yes

—

139

C₂

—

140

C₂

#11739749

yes

—

141

C₂

#11740619

yes

characteristic

142

C₂

#11740724

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇
commutator, socle	C₂³×C₇	C₂³×C₄
Fitting	C₄×C₇×#127	C₂
prime radical	C₂×C₇×#127	C₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₄×#127	D₇
hypocenter	C₇	#23357
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#127

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₇

commutator, socle

C₂³×C₇

C₂³×C₄

Fitting

C₄×C₇×#127

C₂

prime radical

C₂×C₇×#127

C₄

cosocle

C₂²×C₄×C₇

C₂⁴

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂²×C₄ ◁ C₄×#127

N	H ≅ G/N	type
2. #11729365	C₂	semidirect
3. #11727147	C₂	semidirect
43. C₄²×C₇	M₁₆	semidirect
108. C₄²	#1136	semidirect
128. C₇	#23357	semidirect

H ≅ G/N

type

2. #11729365

C₂

semidirect

3. #11727147

C₂

semidirect

43. C₄²×C₇

M₁₆

semidirect

108. C₄²

#1136

semidirect

128. C₇

#23357

semidirect

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

subgroup	index
C₄×C₇×#127	2
#11729365	2
#11727147	2
#23357	7

subgroup

index

C₄×C₇×#127

#11729365

#11727147

#23357

Subgroups — #13380351

Conjugacy classes of subgroups

Complements

Maximal subgroups