Group #13187427 — subgroups

Subgroups — C₇×#10211 #13187427

Order 1792.

406 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3237	C₂	1	3	yes	maximal
2	C₇×#1515	C₂	1	3	yes	maximal
3	C₇×#1308	C₂	1	1	yes	maximal, characteristic
4	C₇×#533	C₂²	1	3	yes	—
5	C₇×#501	—	2	3	no	—
6	C₇×#491	—	2	3	no	—
7	C₇×#485	—	2	6	no	—
8	C₇×#477	—	2	3	no	—
9	C₇×#475	—	2	3	no	—
10	C₇×#432	C₂²	1	3	yes	—
11	C₇×#383	C₂²	1	1	yes	characteristic
12	C₇×#328	—	2	6	no	—
13	#10211	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	C₇×C₄⋊Q₈	D₄	1	3	yes	—
15	C₇×C₄⋊Q₈	D₄	1	3	yes	—
16	C₇×#125	—	2	3	no	—
17	C₇×C₂²⋊Q₈	—	2	3	no	—
18	C₇×C₂²⋊Q₈	—	2	3	no	—
19	C₄×C₇×Q₈	—	2	3	no	—
20	C₄×C₇×Q₈	—	2	3	no	—
21	C₇×#117	—	2	3	no	—
22	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
23	C₇×#107	—	2	6	no	—
24	C₇×#106	—	2	6	no	—
25	C₇×C₄⋊C₈	—	2	3	no	—
26	C₇×#103	—	2	6	no	—
27	C₇×#103	—	2	6	no	—
28	C₇×C₂²⋊C₈	D₄	1	6	yes	—
29	C₇×#97	—	2	3	no	—
30	C₇×#95	—	2	3	no	—
31	#3237	C₂×C₇	1	3	yes	—
32	#1515	C₂×C₇	1	3	yes	—
33	#1308	C₂×C₇	1	1	yes	characteristic
34	C₂×C₇×Q₈	—	2	3	no	—
35	C₂×C₇×Q₈	—	2	3	no	—
36	C₂²×C₄×C₇	—	4	1	no	—
37	C₂²×C₄×C₇	C₂×D₄	1	3	yes	—
38	C₂×C₇×C₈	—	2	6	no	—
39	C₇×C₄⋊C₄	—	2	3	no	—
40	C₇×C₄⋊C₄	—	2	3	no	—
41	C₇×C₄⋊C₄	—	2	3	no	—
42	C₇×C₄⋊C₄	—	2	3	no	—
43	C₇×C₄⋊C₄	—	4	3	no	—
44	C₇×C₄⋊C₄	—	4	3	no	—
45	C₇×C₂²⋊C₄	—	4	3	no	—
46	C₄²×C₇	—	4	3	no	—
47	C₄²×C₇	C₂×D₄	1	3	yes	—
48	C₄²×C₇	#41	1	1	yes	characteristic
49	#533	C₂²×C₇	1	3	yes	—
50	#501	—	2	3	no	—
51	#491	—	2	3	no	—
52	#485	—	2	6	no	—
53	#477	—	2	3	no	—
54	#475	—	2	3	no	—
55	#432	C₂²×C₇	1	3	yes	—
56	#383	C₂²×C₇	1	1	yes	characteristic
57	#328	—	2	6	no	—
58	C₂³×C₇	#127	1	1	yes	characteristic
59	C₇×Q₈	—	4	6	no	—
60	C₇×Q₈	—	8	3	no	—
61	C₂×C₄×C₇	—	4	3	no	—
62	C₂×C₄×C₇	—	4	3	no	—
63	C₂×C₄×C₇	—	4	3	no	—
64	C₂×C₄×C₇	—	8	3	no	—
65	C₂×C₄×C₇	#120	1	3	yes	—
66	C₂×C₄×C₇	#121	1	3	yes	—
67	C₇×C₈	—	4	6	no	—
68	C₄⋊Q₈	C₇×D₄	1	3	yes	—
69	C₄⋊Q₈	C₇×D₄	1	3	yes	—
70	#125	—	2	3	no	—
71	C₂²⋊Q₈	—	2	3	no	—
72	C₂²⋊Q₈	—	2	3	no	—
73	C₄×Q₈	—	2	3	no	—
74	C₄×Q₈	—	2	3	no	—
75	#117	—	2	3	no	—
76	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
77	#107	—	2	6	no	—
78	#106	—	2	6	no	—
79	C₄⋊C₈	—	2	3	no	—
80	#103	—	2	6	no	—
81	#103	—	2	6	no	—
82	C₂²⋊C₈	C₇×D₄	1	6	yes	—
83	#97	—	2	3	no	—
84	#95	—	2	3	no	—
85	C₂²×C₇	—	2	3	no	—
86	C₂²×C₇	#392	1	1	yes	characteristic
87	C₄×C₇	—	2	3	no	—
88	C₄×C₇	—	2	3	no	—
89	C₄×C₇	—	8	3	no	—
90	C₄×C₇	—	8	3	no	—
91	C₄×C₇	—	16	1	no	—
92	C₂×Q₈	—	2	3	no	—
93	C₂×Q₈	—	2	3	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	C₂×C₇×D₄	1	3	yes	—
96	C₂×C₈	—	2	6	no	—
97	C₄⋊C₄	—	2	3	no	—
98	C₄⋊C₄	—	2	3	no	—
99	C₄⋊C₄	—	2	3	no	—
100	C₄⋊C₄	—	2	3	no	—
101	C₄⋊C₄	—	4	3	no	—
102	C₄⋊C₄	—	4	3	no	—
103	C₂²⋊C₄	—	4	3	no	—
104	C₄²	—	4	3	no	—
105	C₄²	C₂×C₇×D₄	1	3	yes	—
106	C₄²	C₇×#41	1	1	yes	characteristic
107	C₂×C₇	—	4	1	no	—
108	C₂×C₇	#2009	1	3	yes	—
109	C₂³	C₇×#127	1	1	yes	characteristic
110	Q₈	—	4	6	no	—
111	Q₈	—	8	3	no	—
112	C₂×C₄	—	4	3	no	—
113	C₂×C₄	—	4	3	no	—
114	C₂×C₄	—	4	3	no	—
115	C₂×C₄	—	8	3	no	—
116	C₂×C₄	C₇×#120	1	3	yes	—
117	C₂×C₄	C₇×#121	1	3	yes	—
118	C₈	—	4	6	no	—
119	C₇	#10211	1	1	yes	characteristic, direct product factor, Sylow
120	C₂²	—	2	3	no	—
121	C₂²	C₇×#392	1	1	yes	characteristic
122	C₄	—	2	3	no	—
123	C₄	—	2	3	no	—
124	C₄	—	8	3	no	—
125	C₄	—	8	3	no	—
126	C₄	—	16	1	no	—
127	C₂	—	4	1	no	—
128	C₂	C₇×#2009	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#3237

C₂

yes

maximal

C₇×#1515

C₂

yes

maximal

C₇×#1308

C₂

yes

maximal, characteristic

C₇×#533

C₂²

yes

—

—

—

—

—

—

C₂²

yes

—

C₇×#383

C₂²

yes

characteristic

C₇×#328

—

#10211

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₇×C₄⋊Q₈

D₄

yes

—

C₇×C₄⋊Q₈

D₄

yes

—

C₇×#125

—

C₇×C₂²⋊Q₈

—

C₇×C₂²⋊Q₈

—

C₄×C₇×Q₈

—

C₄×C₇×Q₈

—

C₇×#117

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×#107

—

C₇×#106

—

C₇×C₄⋊C₈

—

C₇×#103

—

C₇×#103

—

C₇×C₂²⋊C₈

D₄

yes

—

C₇×#97

—

C₇×#95

—

#3237

C₂×C₇

yes

—

#1515

C₂×C₇

yes

—

#1308

C₂×C₇

yes

characteristic

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂×D₄

yes

—

C₂×C₇×C₈

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₂×D₄

yes

—

C₄²×C₇

#41

yes

characteristic

#533

C₂²×C₇

yes

—

—

—

—

—

—

C₂²×C₇

yes

—

#383

C₂²×C₇

yes

characteristic

#328

—

C₂³×C₇

#127

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

#120

yes

—

C₂×C₄×C₇

#121

yes

—

C₇×C₈

—

C₄⋊Q₈

C₇×D₄

yes

—

C₄⋊Q₈

C₇×D₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂³×C₇

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

C₇×D₄

yes

—

#97

—

#95

—

C₂²×C₇

—

C₂²×C₇

#392

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₇×D₄

yes

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₂²⋊C₄

—

104

C₄²

—

105

C₄²

C₂×C₇×D₄

yes

—

106

C₄²

C₇×#41

yes

characteristic

107

C₂×C₇

—

108

C₂×C₇

#2009

yes

—

109

C₂³

C₇×#127

yes

characteristic

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₇×#120

yes

—

117

C₂×C₄

C₇×#121

yes

—

118

C₈

—

119

C₇

#10211

yes

characteristic, direct product factor, Sylow

120

C₂²

—

121

C₂²

C₇×#392

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₂

—

128

C₂

C₇×#2009

yes

—

	subgroup	G/H
center, socle	C₂²×C₇	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₇

prime radical

C₂³×C₇

#127

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
13. #10211	C₇	direct (N × H)
119. C₇	#10211	direct (N × H)

H ≅ G/N

type

13. #10211

C₇

direct (N × H)

119. C₇

#10211

direct (N × H)

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

subgroup	index
C₇×#3237	2
C₇×#1515	2
C₇×#1308	2
#10211	7

subgroup

index

Subgroups — C₇×#10211 #13187427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#10211 #13187427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#10211 #13187427