Group #13185355 — subgroups

Subgroups — C₇×#8139 #13185355

Order 1792.

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

normal subgroups only

This group has 330 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×#1726	C₂	1	1	yes	maximal, characteristic
2	C₇×#1452	C₂	1	2	yes	maximal
3	C₇×#1275	C₂	1	2	yes	maximal
4	C₇×#1274	C₂	1	2	yes	maximal
5	C₇×#431	C₂²	1	4	yes	—
6	C₂×C₇×#97	C₄	1	2	yes	—
7	C₂×C₇×#97	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₇×#376	C₂²	1	1	yes	characteristic
11	C₇×#322	C₄	1	8	yes	complemented
12	#8139	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
18	C₇×C₄⋊C₈	—	2	4	no	—
19	C₇×C₂²⋊C₈	—	2	4	no	—
20	C₇×#97	—	2	4	no	—
21	C₇×#97	—	2	2	no	—
22	C₄×C₇×C₈	—	2	4	no	—
23	C₄×C₇×C₈	C₂×C₄	1	8	yes	—
24	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
25	C₇×#95	—	2	2	no	—
26	#1726	C₂×C₇	1	1	yes	characteristic
27	#1452	C₂×C₇	1	2	yes	—
28	#1275	C₂×C₇	1	2	yes	—
29	#1274	C₂×C₇	1	2	yes	—
30	C₂²×C₄×C₇	—	2	2	no	—
31	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₇	#41	1	2	yes	—
33	C₂×C₇×C₈	—	2	2	no	—
34	C₂×C₇×C₈	—	2	2	no	—
35	C₂×C₇×C₈	—	2	2	no	—
36	C₂×C₇×C₈	—	2	2	no	—
37	C₂×C₇×C₈	—	2	8	no	—
38	C₂×C₇×C₈	C₄²	1	2	yes	—
39	C₂×C₇×C₈	C₄²	1	2	yes	—
40	C₂×C₇×C₈	M₁₆	1	4	yes	—
41	C₄²×C₇	—	2	2	no	—
42	C₄²×C₇	M₁₆	1	4	yes	—
43	C₄²×C₇	C₂²×C₄	1	2	yes	—
44	C₄²×C₇	#41	1	2	yes	—
45	#431	C₂²×C₇	1	4	yes	—
46	C₂×#97	C₄×C₇	1	2	yes	—
47	C₂×#97	C₂²×C₇	1	1	yes	characteristic
48	C₂×C₄×C₈	C₄×C₇	1	2	yes	—
49	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
50	#376	C₂²×C₇	1	1	yes	characteristic
51	#322	C₄×C₇	1	8	yes	complemented
52	C₂³×C₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₇	—	2	4	no	—
54	C₂×C₄×C₇	—	4	2	no	—
55	C₂×C₄×C₇	—	4	4	no	—
56	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
57	C₂×C₄×C₇	#117	1	4	yes	—
58	C₂×C₄×C₇	#117	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
60	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
61	C₇×C₈	—	2	2	no	—
62	C₇×C₈	—	2	2	no	—
63	C₇×C₈	—	2	4	no	—
64	C₇×C₈	—	4	8	no	—
65	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
66	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
67	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
68	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
69	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
70	C₄⋊C₈	—	2	4	no	—
71	C₂²⋊C₈	—	2	4	no	—
72	#97	—	2	4	no	—
73	#97	—	2	2	no	—
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	C₂×C₄×C₇	1	8	yes	—
76	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
77	#95	—	2	2	no	—
78	C₂²×C₇	—	2	2	no	—
79	C₂²×C₇	#376	1	1	yes	characteristic
80	C₂²×C₇	#432	1	2	yes	—
81	C₄×C₇	—	2	4	no	—
82	C₄×C₇	—	4	4	no	—
83	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
84	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
85	C₄×C₇	#432	1	2	yes	—
86	C₂²×C₄	—	2	2	no	—
87	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
88	C₂²×C₄	C₇×#41	1	2	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	8	no	—
94	C₂×C₈	C₄²×C₇	1	2	yes	—
95	C₂×C₈	C₄²×C₇	1	2	yes	—
96	C₂×C₈	C₇×M₁₆	1	4	yes	—
97	C₄²	—	2	2	no	—
98	C₄²	C₇×M₁₆	1	4	yes	—
99	C₄²	C₂²×C₄×C₇	1	2	yes	—
100	C₄²	C₇×#41	1	2	yes	—
101	C₂×C₇	—	2	2	no	—
102	C₂×C₇	#1745	1	1	yes	characteristic
103	C₂×C₇	#1747	1	2	yes	—
104	C₂³	C₇×#117	1	1	yes	characteristic
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	4	no	—
108	C₂×C₄	C₂×C₄²×C₇	1	1	yes	characteristic
109	C₂×C₄	C₇×#117	1	4	yes	—
110	C₂×C₄	C₇×#117	1	1	yes	characteristic
111	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
112	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	—	4	8	no	—
117	C₇	#8139	1	1	yes	characteristic, direct product factor, Sylow
118	C₂²	—	2	2	no	—
119	C₂²	C₇×#376	1	1	yes	characteristic
120	C₂²	C₇×#432	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₄×C₇×M₁₆	1	1	yes	characteristic
124	C₄	C₄×C₇×M₁₆	1	1	yes	characteristic
125	C₄	C₇×#432	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	C₇×#1745	1	1	yes	characteristic
128	C₂	C₇×#1747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#1726

C₂

yes

maximal, characteristic

C₇×#1452

C₂

yes

maximal

C₇×#1275

C₂

yes

maximal

C₇×#1274

C₂

yes

maximal

C₇×#431

C₂²

yes

—

C₂×C₇×#97

C₄

yes

—

C₂×C₇×#97

C₂²

yes

characteristic

C₂×C₄×C₇×C₈

C₄

yes

—

C₂×C₄×C₇×C₈

C₂²

yes

characteristic

C₇×#376

C₂²

yes

characteristic

C₇×#322

C₄

yes

complemented

#8139

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂×C₄

yes

—

C₂×C₄²×C₇

C₂³

yes

characteristic

C₇×C₄⋊C₈

—

C₇×C₂²⋊C₈

—

C₇×#97

—

C₇×#97

—

C₄×C₇×C₈

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₄×C₇×C₈

C₂×C₄

yes

—

C₇×#95

—

#1726

C₂×C₇

yes

characteristic

#1452

C₂×C₇

yes

—

#1275

C₂×C₇

yes

—

#1274

C₂×C₇

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₇

#41

yes

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄²

yes

—

C₂×C₇×C₈

C₄²

yes

—

C₂×C₇×C₈

M₁₆

yes

—

C₄²×C₇

—

C₄²×C₇

M₁₆

yes

—

C₄²×C₇

C₂²×C₄

yes

—

C₄²×C₇

#41

yes

—

#431

C₂²×C₇

yes

—

C₂×#97

C₄×C₇

yes

—

C₂×#97

C₂²×C₇

yes

characteristic

C₂×C₄×C₈

C₄×C₇

yes

—

C₂×C₄×C₈

C₂²×C₇

yes

characteristic

#376

C₂²×C₇

yes

characteristic

#322

C₄×C₇

yes

complemented

C₂³×C₇

#117

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#117

yes

characteristic

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₂×C₄×C₇

C₂×M₁₆

yes

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₇×C₈

—

C₂²×C₈

C₂×C₄×C₇

yes

characteristic

C₂²×C₈

C₂×C₄×C₇

yes

characteristic

C₂²×C₈

C₂×C₄×C₇

yes

—

C₂×C₄²

C₂×C₄×C₇

yes

—

C₂×C₄²

C₂³×C₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×C₄×C₇

yes

—

C₄×C₈

C₂×C₄×C₇

yes

—

#95

—

C₂²×C₇

—

C₂²×C₇

#376

yes

characteristic

C₂²×C₇

#432

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₄×M₁₆

yes

characteristic

C₄×C₇

C₄×M₁₆

yes

characteristic

C₄×C₇

#432

yes

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₇

yes

characteristic

C₂²×C₄

C₇×#41

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²×C₇

yes

—

C₂×C₈

C₄²×C₇

yes

—

C₂×C₈

C₇×M₁₆

yes

—

C₄²

—

C₄²

C₇×M₁₆

yes

—

C₄²

C₂²×C₄×C₇

yes

—

100

C₄²

C₇×#41

yes

—

101

C₂×C₇

—

102

C₂×C₇

#1745

yes

characteristic

103

C₂×C₇

#1747

yes

—

104

C₂³

C₇×#117

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂×C₄²×C₇

yes

characteristic

109

C₂×C₄

C₇×#117

yes

—

110

C₂×C₄

C₇×#117

yes

characteristic

111

C₂×C₄

C₂×C₇×M₁₆

yes

—

112

C₂×C₄

C₂×C₇×M₁₆

yes

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₇

#8139

yes

characteristic, direct product factor, Sylow

118

C₂²

—

119

C₂²

C₇×#376

yes

characteristic

120

C₂²

C₇×#432

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₄×C₇×M₁₆

yes

characteristic

124

C₄

C₄×C₇×M₁₆

yes

characteristic

125

C₄

C₇×#432

yes

—

126

C₂

—

127

C₂

C₇×#1745

yes

characteristic

128

C₂

C₇×#1747

yes

—

	subgroup	G/H
center	C₂×C₄×C₇	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
socle	C₂²×C₇	#376
prime radical	C₂³×C₇	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₇

C₂×C₄²

commutator

C₂×C₄

C₂×C₄²×C₇

Frattini, cosocle

C₂×C₄²

C₂³×C₇

C₂²×C₇

C₂³×C₇

G ▷ C₂×C₄ ▷ C₁

C₁ ◁ C₂×C₄×C₇ ◁ G

N	H ≅ G/N	type
11. C₇×#322	C₄	semidirect
12. #8139	C₇	direct (N × H)
51. #322	C₄×C₇	semidirect
117. C₇	#8139	direct (N × H)

H ≅ G/N

type

11. C₇×#322

C₄

semidirect

12. #8139

C₇

direct (N × H)

51. #322

C₄×C₇

semidirect

117. C₇

#8139

direct (N × H)

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

subgroup	index
C₇×#1726	2
C₇×#1452	2
C₇×#1275	2
C₇×#1274	2
#8139	7

subgroup

index

Subgroups — C₇×#8139 #13185355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#8139 #13185355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#8139 #13185355