Group #13186012 — subgroups

Subgroups — C₇×#8796 #13186012

Order 1792.

228 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 118 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×#1568	C₂	1	2	yes	maximal
2	C₇×#1450	C₂	1	1	yes	maximal, characteristic
3	C₇×#1287	C₂	1	4	yes	maximal
4	C₇×#432	C₂²	1	4	yes	—
5	C₇×#431	C₂²	1	2	yes	—
6	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇×#335	—	2	2	no	—
8	C₇×#334	—	2	2	no	—
9	C₇×#322	C₄	1	2	yes	—
10	C₇×C₈²	C₄	1	2	yes	—
11	#8796	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
13	C₂²×C₇×C₈	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
15	C₇×C₄⋊C₈	—	2	4	no	—
16	C₇×C₄⋊C₈	—	2	2	no	—
17	C₇×C₂²⋊C₈	—	2	2	no	—
18	C₇×C₂²⋊C₈	—	2	4	no	—
19	C₇×#97	—	2	4	no	—
20	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
21	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₇×C₈	D₄	1	1	yes	characteristic
24	C₄×C₇×C₈	D₄	1	1	yes	characteristic
25	#1568	C₂×C₇	1	2	yes	—
26	#1450	C₂×C₇	1	1	yes	characteristic
27	#1287	C₂×C₇	1	4	yes	—
28	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
29	C₂²×C₄×C₇	#41	1	2	yes	—
30	C₂×C₇×C₈	—	2	2	no	—
31	C₂×C₇×C₈	—	2	2	no	—
32	C₂×C₇×C₈	—	2	4	no	—
33	C₂×C₇×C₈	—	4	4	no	—
34	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₇×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₇	#41	1	2	yes	—
41	#432	C₂²×C₇	1	4	yes	—
42	#431	C₂²×C₇	1	2	yes	—
43	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
44	#335	—	2	2	no	—
45	#334	—	2	2	no	—
46	#322	C₄×C₇	1	2	yes	—
47	C₈²	C₄×C₇	1	2	yes	—
48	C₂³×C₇	#125	1	1	yes	characteristic
49	C₂×C₄×C₇	—	2	1	no	—
50	C₂×C₄×C₇	—	2	1	no	—
51	C₂×C₄×C₇	—	4	2	no	—
52	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₇	#117	1	2	yes	—
54	C₂×C₄×C₇	#124	1	2	yes	—
55	C₂×C₄×C₇	#125	1	1	yes	characteristic
56	C₇×C₈	—	2	1	no	—
57	C₇×C₈	—	2	1	no	—
58	C₇×C₈	—	4	4	no	—
59	C₇×C₈	—	8	4	no	—
60	C₇×C₈	#108	1	2	yes	—
61	C₇×C₈	#108	1	2	yes	—
62	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
63	C₂²×C₈	C₇×Q₈	1	1	yes	characteristic
64	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
65	C₄⋊C₈	—	2	4	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₂²⋊C₈	—	2	4	no	—
69	#97	—	2	4	no	—
70	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
71	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
72	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
73	C₄×C₈	C₇×D₄	1	1	yes	characteristic
74	C₄×C₈	C₇×D₄	1	1	yes	characteristic
75	C₂²×C₇	—	2	2	no	—
76	C₂²×C₇	—	2	1	no	—
77	C₂²×C₇	#382	1	1	yes	characteristic
78	C₄×C₇	—	2	2	no	—
79	C₄×C₇	—	2	2	no	—
80	C₄×C₇	—	2	1	no	—
81	C₄×C₇	C₂×#108	1	1	yes	characteristic
82	C₄×C₇	C₂×#108	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
84	C₂²×C₄	C₇×#41	1	2	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	4	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₈	C₇×C₄⋊C₄	1	1	yes	characteristic
93	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
94	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
95	C₄²	C₇×#41	1	2	yes	—
96	C₂×C₇	—	4	1	no	—
97	C₂×C₇	#1833	1	2	yes	—
98	C₂×C₇	#1837	1	1	yes	characteristic
99	C₂³	C₇×#125	1	1	yes	characteristic
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₇×#117	1	2	yes	—
105	C₂×C₄	C₇×#124	1	2	yes	—
106	C₂×C₄	C₇×#125	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	4	4	no	—
110	C₈	—	8	4	no	—
111	C₈	C₇×#108	1	2	yes	—
112	C₈	C₇×#108	1	2	yes	—
113	C₇	#8796	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	C₇×#382	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂×C₇×#108	1	1	yes	characteristic
121	C₄	C₂×C₇×#108	1	1	yes	characteristic
122	C₂	—	4	1	no	—
123	C₂	C₇×#1833	1	2	yes	—
124	C₂	C₇×#1837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1568

C₂

yes

maximal

C₇×#1450

C₂

yes

maximal, characteristic

C₇×#1287

C₂

yes

maximal

C₇×#432

C₂²

yes

—

C₇×#431

C₂²

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

—

—

C₄

yes

—

C₇×C₈²

C₄

yes

—

#8796

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×C₈

Q₈

yes

characteristic

C₂²×C₇×C₈

Q₈

yes

characteristic

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₇×C₂²⋊C₈

—

C₇×#97

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂×C₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

C₄×C₇×C₈

D₄

yes

characteristic

#1568

C₂×C₇

yes

—

#1450

C₂×C₇

yes

characteristic

#1287

C₂×C₇

yes

—

C₂²×C₄×C₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

characteristic

C₂×C₇×C₈

C₄⋊C₄

yes

characteristic

C₂×C₇×C₈

C₄⋊C₄

yes

characteristic

C₂×C₇×C₈

C₄⋊C₄

yes

characteristic

C₄²×C₇

C₂²×C₄

yes

characteristic

C₄²×C₇

C₂×D₄

yes

characteristic

C₄²×C₇

#41

yes

—

#432

C₂²×C₇

yes

—

#431

C₂²×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

—

—

C₄×C₇

yes

—

C₈²

C₄×C₇

yes

—

C₂³×C₇

#125

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#124

yes

—

C₂×C₄×C₇

#125

yes

characteristic

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

#108

yes

—

C₇×C₈

#108

yes

—

C₂²×C₈

C₇×Q₈

yes

characteristic

C₂²×C₈

C₇×Q₈

yes

characteristic

C₂×C₄²

C₂³×C₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₇

yes

—

C₄×C₈

C₂×C₄×C₇

yes

characteristic

C₄×C₈

C₂×C₄×C₇

yes

characteristic

C₄×C₈

C₇×D₄

yes

characteristic

C₄×C₈

C₇×D₄

yes

characteristic

C₂²×C₇

—

C₂²×C₇

—

C₂²×C₇

#382

yes

characteristic

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

yes

C₄×C₇

yes

C₂²×C₄

C₂×C₇×Q₈

yes

characteristic

C₂²×C₄

C₇×#41

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇×C₄⋊C₄

yes

characteristic

C₂×C₈

C₇×C₄⋊C₄

yes

characteristic

C₂×C₈

C₇×C₄⋊C₄

yes

characteristic

C₂×C₈

C₇×C₄⋊C₄

yes

characteristic

C₄²

C₂²×C₄×C₇

yes

characteristic

C₄²

C₂×C₇×D₄

yes

characteristic

C₄²

C₇×#41

yes

—

C₂×C₇

—

C₂×C₇

#1833

yes

—

C₂×C₇

yes

C₂³

yes

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂×C₇×C₄⋊C₄

yes

characteristic

104

C₂×C₄

C₇×#117

yes

—

105

C₂×C₄

C₇×#124

yes

—

106

C₂×C₄

C₇×#125

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

C₇×#108

yes

—

112

C₈

C₇×#108

yes

—

113

C₇

#8796

yes

characteristic, direct product factor, Sylow

114

C₂²

—

115

C₂²

—

116

C₂²

C₇×#382

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×C₇×#108

yes

characteristic

121

C₄

C₂×C₇×#108

yes

characteristic

122

C₂

—

123

C₂

C₇×#1833

yes

—

124

C₂

C₇×#1837

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₇	#382
commutator	C₄²	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
prime radical	C₂³×C₇	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₇

#382

commutator

C₄²

C₂²×C₄×C₇

Frattini, cosocle

C₂×C₄²

C₂³×C₇

C₂³×C₇

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
11. #8796	C₇	direct (N × H)
113. C₇	#8796	direct (N × H)

H ≅ G/N

type

11. #8796

C₇

direct (N × H)

113. C₇

#8796

direct (N × H)

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

subgroup	index
C₇×#1568	2
C₇×#1450	2
C₇×#1287	2
#8796	7

subgroup

index

Subgroups — C₇×#8796 #13186012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#8796 #13186012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8796 #13186012