Group #11730565 — subgroups

Subgroups — C₇×#2096 #11730565

Order 896.

254 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×C₇×#106	C₂	1	2	yes	maximal
2	C₂×C₇×#103	C₂	1	2	yes	maximal
3	C₇×#391	C₂	1	1	yes	maximal, characteristic
4	C₇×#384	C₂	1	1	yes	maximal, characteristic
5	C₇×#336	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₇×Q₈	C₂²	1	1	yes	characteristic
7	C₂²×C₇×C₈	C₂²	1	2	yes	—
8	C₂×C₇×C₄⋊C₄	—	2	2	no	—
9	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
12	C₇×#106	—	2	4	no	—
13	C₇×#103	—	2	4	no	—
14	C₇×#95	—	2	2	no	—
15	#2096	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂×C₇×Q₈	—	2	1	no	—
17	C₂×C₇×Q₈	—	2	2	no	—
18	C₂×C₇×Q₈	—	2	2	no	—
19	C₂×C₇×Q₈	Q₈	1	2	yes	—
20	C₂²×C₄×C₇	—	2	1	no	—
21	C₂²×C₄×C₇	—	2	1	no	—
22	C₂²×C₄×C₇	—	2	2	no	—
23	C₂²×C₄×C₇	D₄	1	2	yes	—
24	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂×C₇×C₈	—	2	2	no	—
26	C₂×C₇×C₈	—	2	2	no	—
27	C₂×C₇×C₈	D₄	1	4	yes	—
28	C₇×C₄⋊C₄	—	2	2	no	—
29	C₇×C₄⋊C₄	—	2	2	no	—
30	C₇×C₄⋊C₄	—	2	2	no	—
31	C₇×C₄⋊C₄	—	4	4	no	—
32	C₄²×C₇	—	2	2	no	—
33	C₂×#106	C₂×C₇	1	2	yes	—
34	C₂×#103	C₂×C₇	1	2	yes	—
35	#391	C₂×C₇	1	1	yes	characteristic
36	#384	C₂×C₇	1	1	yes	characteristic
37	#336	C₂×C₇	1	1	yes	characteristic
38	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
39	C₇×Q₈	—	2	4	no	—
40	C₇×Q₈	—	4	2	no	—
41	C₂×C₄×C₇	—	2	2	no	—
42	C₂×C₄×C₇	—	2	2	no	—
43	C₂×C₄×C₇	—	2	2	no	—
44	C₂×C₄×C₇	—	4	1	no	—
45	C₂×C₄×C₇	—	4	1	no	—
46	C₂×C₄×C₇	—	4	2	no	—
47	C₂×C₄×C₇	—	4	1	no	—
48	C₂×C₄×C₇	—	4	4	no	—
49	C₂×C₄×C₇	—	4	2	no	—
50	C₂×C₄×C₇	SD₁₆	1	4	yes	—
51	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₇	#41	1	2	yes	—
54	C₂×C₄×C₇	#41	1	1	yes	characteristic
55	C₇×C₈	—	2	4	no	—
56	C₂²×Q₈	C₂²×C₇	1	1	yes	characteristic
57	C₂²×C₈	C₂²×C₇	1	2	yes	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	C₂²×C₇	1	2	yes	—
60	C₂×C₄⋊C₄	C₂²×C₇	1	1	yes	characteristic
61	C₂×C₄²	C₂²×C₇	1	1	yes	characteristic
62	#106	—	2	4	no	—
63	#103	—	2	4	no	—
64	#95	—	2	2	no	—
65	C₂²×C₇	C₂²⋊Q₈	1	1	yes	characteristic
66	C₂²×C₇	#123	1	1	yes	characteristic
67	C₂²×C₇	#127	1	1	yes	characteristic
68	C₂²×C₇	C₂×SD₁₆	1	2	yes	—
69	C₂²×C₇	#137	1	2	yes	—
70	C₄×C₇	—	2	4	no	—
71	C₄×C₇	—	4	2	no	—
72	C₄×C₇	—	8	2	no	—
73	C₄×C₇	—	8	1	no	—
74	C₄×C₇	C₂²⋊Q₈	1	2	yes	—
75	C₄×C₇	#123	1	2	yes	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	C₇×Q₈	1	2	yes	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	C₇×D₄	1	2	yes	—
84	C₂²×C₄	C₂³×C₇	1	1	yes	characteristic
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	C₇×D₄	1	4	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	4	4	no	—
92	C₄²	—	2	2	no	—
93	C₂×C₇	#399	1	1	yes	characteristic
94	C₂×C₇	#475	1	2	yes	—
95	C₂×C₇	#483	1	2	yes	—
96	C₂×C₇	#492	1	1	yes	characteristic
97	C₂×C₇	#497	1	1	yes	characteristic
98	C₂³	C₂×C₇×D₄	1	1	yes	characteristic
99	Q₈	—	2	4	no	—
100	Q₈	—	4	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₇×SD₁₆	1	4	yes	—
111	C₂×C₄	C₂×C₇×D₄	1	2	yes	—
112	C₂×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
113	C₂×C₄	C₇×#41	1	2	yes	—
114	C₂×C₄	C₇×#41	1	1	yes	characteristic
115	C₈	—	2	4	no	—
116	C₇	#2096	1	1	yes	characteristic, direct product factor, Sylow
117	C₂²	C₇×C₂²⋊Q₈	1	1	yes	characteristic
118	C₂²	C₇×#123	1	1	yes	characteristic
119	C₂²	C₇×#127	1	1	yes	characteristic
120	C₂²	C₂×C₇×SD₁₆	1	2	yes	—
121	C₂²	C₇×#137	1	2	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	4	2	no	—
124	C₄	—	8	2	no	—
125	C₄	—	8	1	no	—
126	C₄	C₇×C₂²⋊Q₈	1	2	yes	—
127	C₄	C₇×#123	1	2	yes	—
128	C₂	C₇×#399	1	1	yes	characteristic
129	C₂	C₇×#475	1	2	yes	—
130	C₂	C₇×#483	1	2	yes	—
131	C₂	C₇×#492	1	1	yes	characteristic
132	C₂	C₇×#497	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₇×#106

C₂

yes

maximal

C₂×C₇×#103

C₂

yes

maximal

C₇×#391

C₂

yes

maximal, characteristic

C₇×#384

C₂

yes

maximal, characteristic

C₇×#336

C₂

yes

maximal, characteristic

C₂²×C₇×Q₈

C₂²

yes

characteristic

C₂²×C₇×C₈

C₂²

yes

—

C₂×C₇×C₄⋊C₄

—

C₂×C₇×C₄⋊C₄

C₂²

yes

—

C₂×C₇×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₄²×C₇

C₂²

yes

characteristic

C₇×#106

—

C₇×#103

—

C₇×#95

—

#2096

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

—

C₂×C₇×Q₈

Q₈

yes

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

—

C₂²×C₄×C₇

D₄

yes

—

C₂²×C₄×C₇

C₂³

yes

characteristic

C₂×C₇×C₈

—

C₂×C₇×C₈

—

C₂×C₇×C₈

D₄

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₄²×C₇

—

C₂×#106

C₂×C₇

yes

—

C₂×#103

C₂×C₇

yes

—

#391

C₂×C₇

yes

characteristic

#384

C₂×C₇

yes

characteristic

#336

C₂×C₇

yes

characteristic

C₂³×C₇

C₂×D₄

yes

characteristic

C₇×Q₈

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

SD₁₆

yes

—

C₂×C₄×C₇

C₂×D₄

yes

—

C₂×C₄×C₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₇

#41

yes

—

C₂×C₄×C₇

#41

yes

characteristic

C₇×C₈

—

C₂²×Q₈

C₂²×C₇

yes

characteristic

C₂²×C₈

C₂²×C₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₇

yes

—

C₂×C₄⋊C₄

C₂²×C₇

yes

characteristic

C₂×C₄²

C₂²×C₇

yes

—

—

—

C₂²×C₇

C₂²⋊Q₈

yes

C₂²×C₇

yes

C₂²×C₇

yes

C₂²×C₇

C₂×SD₁₆

yes

—

C₂²×C₇

#137

yes

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

C₂²⋊Q₈

yes

—

C₄×C₇

#123

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₇×Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₇×D₄

yes

—

C₂²×C₄

C₂³×C₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂×C₇

#399

yes

characteristic

C₂×C₇

#475

yes

—

C₂×C₇

#483

yes

—

C₂×C₇

yes

C₂×C₇

yes

C₂³

C₂×C₇×D₄

yes

characteristic

Q₈

—

100

Q₈

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₇×SD₁₆

yes

—

111

C₂×C₄

C₂×C₇×D₄

yes

—

112

C₂×C₄

C₂×C₇×Q₈

yes

characteristic

113

C₂×C₄

C₇×#41

yes

—

114

C₂×C₄

C₇×#41

yes

characteristic

115

C₈

—

116

C₇

#2096

yes

characteristic, direct product factor, Sylow

117

C₂²

C₇×C₂²⋊Q₈

yes

characteristic

118

C₂²

C₇×#123

yes

characteristic

119

C₂²

C₇×#127

yes

characteristic

120

C₂²

C₂×C₇×SD₁₆

yes

—

121

C₂²

C₇×#137

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₇×C₂²⋊Q₈

yes

—

127

C₄

C₇×#123

yes

—

128

C₂

C₇×#399

yes

characteristic

129

C₂

C₇×#475

yes

—

130

C₂

C₇×#483

yes

—

131

C₂

C₇×#492

yes

characteristic

132

C₂

C₇×#497

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₇

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₇ ◁ C₂×C₄²×C₇ ◁ G

N	H ≅ G/N	type
15. #2096	C₇	direct (N × H)
116. C₇	#2096	direct (N × H)

H ≅ G/N

type

15. #2096

C₇

direct (N × H)

116. C₇

#2096

direct (N × H)

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

subgroup	index
C₂×C₇×#106	2
C₂×C₇×#103	2
C₇×#391	2
C₇×#384	2
C₇×#336	2
#2096	7

subgroup

index

C₂×C₇×#106

C₂×C₇×#103

Subgroups — C₇×#2096 #11730565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#2096 #11730565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#2096 #11730565