Group #13190099 — subgroups

Subgroups — C₇×#12883 #13190099

Order 1792.

342 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 174 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×#2150	C₂	1	2	yes	maximal
2	C₄²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1376	C₂	1	4	yes	maximal
4	C₂×C₇×M₃₂	C₂²	1	4	yes	—
5	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
6	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₄³×C₇	C₄	1	1	yes	characteristic
10	C₇×C₄⋊C₁₆	—	2	4	no	—
11	#12883	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
14	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
15	C₂×C₄²×C₇	—	2	2	no	—
16	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
18	C₇×M₃₂	—	2	8	no	—
19	C₂×C₇×C₁₆	—	2	4	no	—
20	C₄×C₇×C₈	—	2	4	no	—
21	C₄×C₇×C₈	—	2	4	no	—
22	C₄×C₇×C₈	C₈	1	2	yes	—
23	C₄×C₇×C₈	C₈	1	2	yes	—
24	C₄×C₇×C₈	D₄	1	2	yes	—
25	C₄×C₇×C₈	Q₈	1	2	yes	—
26	#2150	C₂×C₇	1	2	yes	—
27	C₄²×C₈	C₂×C₇	1	1	yes	characteristic
28	#1376	C₂×C₇	1	4	yes	—
29	C₂²×C₄×C₇	—	2	2	no	—
30	C₂²×C₄×C₇	M₁₆	1	2	yes	—
31	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₇×C₈	—	2	2	no	—
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	—	2	2	no	—
35	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
37	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
38	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₇×C₈	#41	1	4	yes	—
40	C₄²×C₇	—	2	8	no	—
41	C₄²×C₇	—	2	2	no	—
42	C₄²×C₇	C₄⋊C₄	1	2	yes	—
43	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₇	C₂×C₈	1	1	yes	characteristic
45	C₄²×C₇	M₁₆	1	4	yes	—
46	C₇×C₁₆	—	4	4	no	—
47	C₂×M₃₂	C₂²×C₇	1	4	yes	—
48	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
49	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
50	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
51	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
52	C₄³	C₄×C₇	1	1	yes	characteristic
53	C₄⋊C₁₆	—	2	4	no	—
54	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
55	C₂×C₄×C₇	—	2	8	no	—
56	C₂×C₄×C₇	—	2	4	no	—
57	C₂×C₄×C₇	—	2	4	no	—
58	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
59	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
60	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₇	#117	1	2	yes	—
62	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
64	C₇×C₈	—	2	2	no	—
65	C₇×C₈	—	2	4	no	—
66	C₇×C₈	#124	1	2	yes	—
67	C₇×C₈	#125	1	2	yes	—
68	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
69	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
70	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
71	C₂×C₄²	—	2	2	no	—
72	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
73	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
74	M₃₂	—	2	8	no	—
75	C₂×C₁₆	—	2	4	no	—
76	C₄×C₈	—	2	4	no	—
77	C₄×C₈	—	2	4	no	—
78	C₄×C₈	C₇×C₈	1	2	yes	—
79	C₄×C₈	C₇×C₈	1	2	yes	—
80	C₄×C₈	C₇×D₄	1	2	yes	—
81	C₄×C₈	C₇×Q₈	1	2	yes	—
82	C₂²×C₇	—	2	2	no	—
83	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₇	#432	1	2	yes	—
85	C₄×C₇	—	2	8	no	—
86	C₄×C₇	—	2	2	no	—
87	C₄×C₇	#364	1	4	yes	—
88	C₄×C₇	#382	1	1	yes	characteristic
89	C₄×C₇	#423	1	1	yes	characteristic
90	C₄×C₇	#431	1	2	yes	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	C₇×M₁₆	1	2	yes	—
93	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
98	C₂×C₈	C₂×C₇×C₈	1	4	yes	—
99	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
100	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
101	C₂×C₈	C₇×#41	1	4	yes	—
102	C₄²	—	2	8	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₇×C₄⋊C₄	1	2	yes	—
105	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
106	C₄²	C₂×C₇×C₈	1	1	yes	characteristic
107	C₄²	C₇×M₁₆	1	4	yes	—
108	C₁₆	—	4	4	no	—
109	C₂×C₇	—	2	2	no	—
110	C₂×C₇	#1831	1	1	yes	characteristic
111	C₂×C₇	C₂×#364	1	2	yes	—
112	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	2	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
117	C₂×C₄	C₇×C₄⋊C₈	1	2	yes	—
118	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄	C₇×#117	1	2	yes	—
120	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
121	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
122	C₈	—	2	2	no	—
123	C₈	—	2	4	no	—
124	C₈	C₇×#124	1	2	yes	—
125	C₈	C₇×#125	1	2	yes	—
126	C₇	#12883	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²	—	2	2	no	—
128	C₂²	C₂×C₇×C₄⋊C₈	1	1	yes	characteristic
129	C₂²	C₇×#432	1	2	yes	—
130	C₄	—	2	8	no	—
131	C₄	—	2	2	no	—
132	C₄	C₇×#364	1	4	yes	—
133	C₄	C₇×#382	1	1	yes	characteristic
134	C₄	C₇×#423	1	1	yes	characteristic
135	C₄	C₇×#431	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	C₇×#1831	1	1	yes	characteristic
138	C₂	C₂×C₇×#364	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2150

C₂

yes

maximal

C₄²×C₇×C₈

C₂

yes

maximal, characteristic

C₇×#1376

C₂

yes

maximal

C₂×C₇×M₃₂

C₂²

yes

—

C₂×C₄×C₇×C₈

C₄

yes

—

C₂×C₄×C₇×C₈

C₄

yes

characteristic

C₂×C₄×C₇×C₈

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₄³×C₇

C₄

yes

characteristic

C₇×C₄⋊C₁₆

—

#12883

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂²×C₇×C₈

C₂³

yes

characteristic

C₂×C₄²×C₇

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

characteristic

C₇×M₃₂

—

C₂×C₇×C₁₆

—

C₄×C₇×C₈

—

C₄×C₇×C₈

—

C₄×C₇×C₈

C₈

yes

—

C₄×C₇×C₈

C₈

yes

—

C₄×C₇×C₈

D₄

yes

—

C₄×C₇×C₈

Q₈

yes

—

#2150

C₂×C₇

yes

—

C₄²×C₈

C₂×C₇

yes

characteristic

#1376

C₂×C₇

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

M₁₆

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₂×C₈

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×Q₈

yes

characteristic

C₂×C₇×C₈

#41

yes

—

C₄²×C₇

—

C₄²×C₇

—

C₄²×C₇

C₄⋊C₄

yes

—

C₄²×C₇

C₂×C₈

yes

characteristic

C₄²×C₇

C₂×C₈

yes

characteristic

C₄²×C₇

M₁₆

yes

—

C₇×C₁₆

—

C₂×M₃₂

C₂²×C₇

yes

—

C₂×C₄×C₈

C₄×C₇

yes

—

C₂×C₄×C₈

C₄×C₇

yes

characteristic

C₂×C₄×C₈

C₂²×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

characteristic

C₄³

C₄×C₇

yes

characteristic

C₄⋊C₁₆

—

C₂³×C₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₄⋊C₈

yes

—

C₂×C₄×C₇

C₄⋊C₈

yes

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

#124

yes

—

C₇×C₈

#125

yes

—

C₂²×C₈

C₂×C₄×C₇

yes

characteristic

C₂²×C₈

C₂×C₄×C₇

yes

—

C₂²×C₈

C₂³×C₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₇

yes

—

C₂×C₄²

C₂×C₄×C₇

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₇×C₈

yes

—

C₄×C₈

C₇×C₈

yes

—

C₄×C₈

C₇×D₄

yes

—

C₄×C₈

C₇×Q₈

yes

—

C₂²×C₇

—

C₂²×C₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₇

#432

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

#364

yes

—

C₄×C₇

yes

C₄×C₇

yes

C₄×C₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₇×M₁₆

yes

—

C₂²×C₄

C₂²×C₄×C₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇×C₄⋊C₄

yes

—

C₂×C₈

C₂×C₇×C₈

yes

—

C₂×C₈

C₂×C₇×D₄

yes

characteristic

100

C₂×C₈

C₂×C₇×Q₈

yes

characteristic

101

C₂×C₈

C₇×#41

yes

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₇×C₄⋊C₄

yes

—

105

C₄²

C₂×C₇×C₈

yes

characteristic

106

C₄²

C₂×C₇×C₈

yes

characteristic

107

C₄²

C₇×M₁₆

yes

—

108

C₁₆

—

109

C₂×C₇

—

110

C₂×C₇

#1831

yes

characteristic

111

C₂×C₇

C₂×#364

yes

—

112

C₂³

C₂×C₇×M₁₆

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₇×C₄⋊C₈

yes

—

117

C₂×C₄

C₇×C₄⋊C₈

yes

—

118

C₂×C₄

C₂×C₇×C₄⋊C₄

yes

characteristic

119

C₂×C₄

C₇×#117

yes

—

120

C₂×C₄

C₂²×C₇×C₈

yes

characteristic

121

C₂×C₄

C₂×C₇×M₁₆

yes

—

122

C₈

—

123

C₈

—

124

C₈

C₇×#124

yes

—

125

C₈

C₇×#125

yes

—

126

C₇

#12883

yes

characteristic, direct product factor, Sylow

127

C₂²

—

128

C₂²

C₂×C₇×C₄⋊C₈

yes

characteristic

129

C₂²

C₇×#432

yes

—

130

C₄

—

131

C₄

—

132

C₄

C₇×#364

yes

—

133

C₄

C₇×#382

yes

characteristic

134

C₄

C₇×#423

yes

characteristic

135

C₄

C₇×#431

yes

—

136

C₂

—

137

C₂

C₇×#1831

yes

characteristic

138

C₂

C₂×C₇×#364

yes

—

	subgroup	G/H
center	C₂×C₇×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

subgroup

G/H

center

C₂×C₇×C₈

C₂×D₄

commutator

C₂×C₄

C₂²×C₇×C₈

Frattini, cosocle

C₂²×C₈

C₂³×C₇

socle

C₂²×C₇

C₂×C₄⋊C₈

prime radical

C₂³×C₇

C₂×M₁₆

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₇×C₈ ◁ C₂×C₄×C₇×C₈ ◁ G

N	H ≅ G/N	type
11. #12883	C₇	direct (N × H)
126. C₇	#12883	direct (N × H)

H ≅ G/N

type

11. #12883

C₇

direct (N × H)

126. C₇

#12883

direct (N × H)

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

subgroup	index
C₇×#2150	2
C₄²×C₇×C₈	2
C₇×#1376	2
#12883	7

subgroup

index

C₇×#2150

C₄²×C₇×C₈

C₇×#1376

#12883

Subgroups — C₇×#12883 #13190099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#12883 #13190099

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12883 #13190099