Group #13268506 — subgroups

Subgroups — #13268506

Order 1792.

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

normal subgroups only

This group has 398 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	C₇×#2246	C₂	1	1	yes	maximal, characteristic, complemented
2	#11729044	C₂	1	2	yes	maximal, complemented
3	#11729030	C₂	1	2	yes	maximal, complemented
4	#11726814	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₆×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₇×D₁₆	—	2	2	no	—
7	C₇×#493	C₂²	1	2	yes	—
8	C₂×#6400	—	2	4	no	—
9	#89845	—	2	4	no	—
10	C₂×D₁₁₂	—	2	2	no	—
11	#89467	C₂²	1	2	yes	—
12	C₄×C₇×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₈×C₇⋊C₈	C₂²	1	1	yes	characteristic
15	#12515	—	7	1	no	maximal, Sylow
16	C₂×C₇×D₈	—	2	2	no	—
17	C₂×C₇×D₈	—	2	2	no	—
18	C₇×#127	—	2	2	no	—
19	C₂×#1141	—	4	8	no	—
20	C₂×D₅₆	—	2	2	no	—
21	C₂×D₅₆	—	2	2	no	—
22	#6437	—	2	2	no	—
23	C₇×D₁₆	—	4	4	no	—
24	C₂×C₇×C₁₆	D₄	1	2	yes	—
25	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
26	#6400	—	4	8	no	—
27	C₂×C₇⋊C₁₆	D₄	1	2	yes	—
28	C₄×C₇⋊C₈	D₄	1	2	yes	—
29	D₁₁₂	—	4	4	no	—
30	#2246	—	7	2	no	—
31	#2246	—	7	1	no	—
32	#2246	D₇	1	1	yes	characteristic, complemented
33	#1712	—	7	2	no	—
34	C₈×C₁₆	—	7	1	no	—
35	C₂×C₇×D₄	—	4	2	no	—
36	C₂×C₇×D₄	—	4	4	no	—
37	C₂×D₂₈	—	4	2	no	—
38	C₂×D₂₈	—	4	4	no	—
39	C₇×D₈	—	4	4	no	—
40	C₇×D₈	—	4	4	no	—
41	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
42	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
44	#1141	—	8	16	no	—
45	C₂×C₇⋊C₈	D₈	1	4	yes	—
46	D₅₆	—	4	4	no	—
47	D₅₆	—	4	4	no	—
48	C₇×C₁₆	D₈	1	4	yes	complemented
49	C₇⋊C₁₆	D₈	1	4	yes	complemented
50	C₂×D₁₆	—	2	2	no	—
51	C₂×D₁₆	—	14	4	no	—
52	C₂×D₁₆	—	14	2	no	—
53	#493	—	7	2	no	—
54	#493	—	14	4	no	—
55	#493	C₂×D₇	1	2	yes	—
56	C₄×C₁₆	—	7	1	no	—
57	C₄×C₁₆	C₂×D₇	1	1	yes	characteristic
58	C₈²	—	7	1	no	—
59	C₂³×C₇	—	8	2	no	—
60	C₂²×D₇	—	8	2	no	—
61	C₇×D₄	—	8	2	no	—
62	C₇×D₄	—	8	2	no	—
63	C₇×D₄	—	8	8	no	—
64	C₂×C₄×C₇	#127	1	1	yes	characteristic
65	C₂×C₄×C₇	C₂×D₈	1	2	yes	—
66	D₂₈	—	8	2	no	—
67	D₂₈	—	8	2	no	—
68	D₂₈	—	8	8	no	—
69	C₇×C₈	C₂×D₈	1	2	yes	—
70	C₇×C₈	C₂×D₈	1	2	yes	—
71	C₇⋊C₈	D₁₆	1	8	yes	complemented
72	C₂×D₈	—	2	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	—	14	2	no	—
75	C₂×D₈	—	14	2	no	—
76	C₂×D₈	—	28	8	no	—
77	#127	—	2	2	no	—
78	#127	—	14	2	no	—
79	D₁₆	—	4	4	no	—
80	D₁₆	—	28	8	no	—
81	D₁₆	—	28	4	no	—
82	C₂×C₁₆	—	7	2	no	—
83	C₂×C₁₆	C₂²⋊D₇	1	2	yes	—
84	C₄×C₈	—	7	2	no	—
85	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
86	C₂²×C₇	—	16	2	no	—
87	C₂²×C₇	—	16	4	no	—
88	C₂²×C₇	#493	1	1	yes	characteristic
89	C₂×D₇	—	16	2	no	—
90	C₂×D₇	—	16	4	no	—
91	C₄×C₇	#493	1	1	yes	characteristic
92	C₄×C₇	#493	1	1	yes	characteristic
93	C₄×C₇	C₂×D₁₆	1	4	yes	—
94	C₂×D₄	—	4	4	no	—
95	C₂×D₄	—	4	2	no	—
96	C₂×D₄	—	28	4	no	—
97	C₂×D₄	—	28	2	no	—
98	D₈	—	4	4	no	—
99	D₈	—	4	4	no	—
100	D₈	—	28	4	no	—
101	D₈	—	28	4	no	—
102	D₈	—	56	16	no	—
103	C₂×C₈	—	7	4	no	—
104	C₂×C₈	D₄×D₇	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
106	C₄²	D₄×D₇	1	1	yes	characteristic
107	C₁₆	—	7	4	no	—
108	C₁₆	#1141	1	4	yes	complemented
109	C₂×C₇	—	32	2	no	—
110	C₂×C₇	#1712	1	1	yes	characteristic
111	C₂×C₇	#2246	1	2	yes	—
112	D₇	—	32	2	no	—
113	C₂³	—	8	2	no	—
114	C₂³	—	56	2	no	—
115	D₄	—	8	2	no	—
116	D₄	—	8	8	no	—
117	D₄	—	8	2	no	—
118	D₄	—	56	2	no	—
119	D₄	—	56	8	no	—
120	D₄	—	56	2	no	—
121	C₂×C₄	D₇×D₈	1	2	yes	—
122	C₂×C₄	#6503	1	1	yes	characteristic
123	C₈	—	7	8	no	—
124	C₈	D₇×D₈	1	2	yes	—
125	C₈	C₂×#1141	1	2	yes	—
126	C₇	#12515	1	1	yes	characteristic, complemented, Sylow
127	C₂²	—	16	4	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	112	4	no	—
130	C₂²	—	112	2	no	—
131	C₂²	#89923	1	1	yes	characteristic
132	C₄	D₇×D₁₆	1	4	yes	—
133	C₄	#89845	1	1	yes	characteristic
134	C₄	#89923	1	1	yes	characteristic
135	C₂	—	32	2	no	—
136	C₂	—	224	2	no	—
137	C₂	#11729030	1	1	yes	characteristic
138	C₂	#11729218	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2246

C₂

yes

maximal, characteristic, complemented

#11729044

C₂

yes

maximal, complemented

#11729030

C₂

yes

maximal, complemented

#11726814

C₂

yes

maximal, characteristic, complemented

C₁₆×C₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂×C₇×D₁₆

—

C₇×#493

C₂²

yes

—

C₂×#6400

—

#89845

—

C₂×D₁₁₂

—

#89467

C₂²

yes

—

C₄×C₇×C₁₆

C₂²

yes

characteristic

C₄×C₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₇⋊C₈

C₂²

yes

characteristic

#12515

—

maximal, Sylow

C₂×C₇×D₈

—

C₂×C₇×D₈

—

C₇×#127

—

C₂×#1141

—

C₂×D₅₆

—

C₂×D₅₆

—

#6437

—

C₇×D₁₆

—

C₂×C₇×C₁₆

D₄

yes

—

C₄×C₇×C₈

C₂³

yes

characteristic

#6400

—

C₂×C₇⋊C₁₆

D₄

yes

—

C₄×C₇⋊C₈

D₄

yes

—

D₁₁₂

—

#2246

—

#2246

—

#2246

D₇

yes

characteristic, complemented

#1712

—

C₈×C₁₆

—

C₂×C₇×D₄

—

C₂×C₇×D₄

—

C₂×D₂₈

—

C₂×D₂₈

—

C₇×D₈

—

C₇×D₈

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

#1141

—

C₂×C₇⋊C₈

D₈

yes

—

D₅₆

—

D₅₆

—

C₇×C₁₆

D₈

yes

complemented

C₇⋊C₁₆

D₈

yes

complemented

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

#493

—

#493

—

#493

C₂×D₇

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇

yes

characteristic

C₈²

—

C₂³×C₇

—

C₂²×D₇

—

C₇×D₄

—

C₇×D₄

—

C₇×D₄

—

C₂×C₄×C₇

#127

yes

characteristic

C₂×C₄×C₇

C₂×D₈

yes

—

D₂₈

—

D₂₈

—

D₂₈

—

C₇×C₈

C₂×D₈

yes

—

C₇×C₈

C₂×D₈

yes

—

C₇⋊C₈

D₁₆

yes

complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

—

D₁₆

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇

yes

—

C₄×C₈

—

C₄×C₈

C₂²×D₇

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#493

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

C₂×D₁₆

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

100

D₈

—

101

D₈

—

102

D₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₇

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₇

yes

characteristic

106

C₄²

D₄×D₇

yes

characteristic

107

C₁₆

—

108

C₁₆

#1141

yes

complemented

109

C₂×C₇

—

110

C₂×C₇

#1712

yes

characteristic

111

C₂×C₇

#2246

yes

—

112

D₇

—

113

C₂³

—

114

C₂³

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

C₂×C₄

D₇×D₈

yes

—

122

C₂×C₄

#6503

yes

characteristic

123

C₈

—

124

C₈

D₇×D₈

yes

—

125

C₈

C₂×#1141

yes

—

126

C₇

#12515

yes

characteristic, complemented, Sylow

127

C₂²

—

128

C₂²

—

129

C₂²

—

112

—

130

C₂²

—

112

—

131

C₂²

#89923

yes

characteristic

132

C₄

D₇×D₁₆

yes

—

133

C₄

#89845

yes

characteristic

134

C₄

#89923

yes

characteristic

135

C₂

—

136

C₂

—

224

—

137

C₂

#11729030

yes

characteristic

138

C₂

#11729218

yes

—

	subgroup	G/H
center	C₂²	#89923
commutator, cosocle	C₄×C₇×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#2246	C₂
socle	C₂²×C₇	#493
hypercenter	#2246	D₇
hypocenter	C₇	#12515
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#89923

commutator, cosocle

C₄×C₇×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂²×C₇

D₇

C₇

G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. C₇×#2246	C₂	semidirect
2. #11729044	C₂	semidirect
3. #11729030	C₂	semidirect
4. #11726814	C₂	semidirect
5. C₁₆×C₇⋊C₈	C₂	semidirect
32. #2246	D₇	semidirect
48. C₇×C₁₆	D₈	semidirect
49. C₇⋊C₁₆	D₈	semidirect
71. C₇⋊C₈	D₁₆	semidirect
108. C₁₆	#1141	semidirect
126. C₇	#12515	semidirect

H ≅ G/N

type

1. C₇×#2246

C₂

semidirect

2. #11729044

C₂

semidirect

3. #11729030

C₂

semidirect

4. #11726814

C₂

semidirect

5. C₁₆×C₇⋊C₈

C₂

semidirect

32. #2246

D₇

semidirect

48. C₇×C₁₆

D₈

semidirect

49. C₇⋊C₁₆

D₈

semidirect

71. C₇⋊C₈

D₁₆

semidirect

108. C₁₆

#1141

semidirect

126. C₇

#12515

semidirect

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

subgroup	index
C₇×#2246	2
#11729044	2
#11729030	2
#11726814	2
C₁₆×C₇⋊C₈	2
#12515	7

subgroup

index

C₁₆×C₇⋊C₈

Subgroups — #13268506

Conjugacy classes of subgroups

Complements

Maximal subgroups