Group #13190101 — subgroups

Subgroups — C₇×#12885 #13190101

Order 1792.

234 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2153	C₂	1	1	yes	maximal, characteristic
2	C₇×#2151	C₂	1	1	yes	maximal, characteristic
3	C₇×#1747	C₂	1	1	yes	maximal, characteristic
4	C₇×#1379	C₂	1	2	yes	maximal
5	C₇×#1378	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₇×M₃₂	C₂²	1	2	yes	—
8	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
9	C₇×#405	C₂²	1	2	yes	—
10	C₇×#405	C₂²	1	1	yes	characteristic
11	C₇×#364	—	2	2	no	—
12	C₇×C₄⋊C₁₆	—	2	2	no	—
13	C₇×#338	C₄	1	1	yes	characteristic
14	C₇×H(3,ℤ/4)	C₄	1	1	yes	characteristic
15	C₇×H(3,ℤ/4)	C₄	1	2	yes	—
16	#12885	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₇×M₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
20	C₇×#117	C₂×C₄	1	2	yes	—
21	C₇×#117	C₂×C₄	1	1	yes	characteristic
22	C₇×M₃₂	—	2	2	no	—
23	C₇×M₃₂	—	2	2	no	—
24	C₇×M₃₂	—	2	2	no	—
25	C₂×C₇×C₁₆	—	2	1	no	—
26	C₂×C₇×C₁₆	—	2	1	no	—
27	C₂×C₇×C₁₆	—	2	2	no	—
28	C₂×C₇×C₁₆	—	2	2	no	—
29	C₇×#97	—	2	2	no	—
30	C₇×#97	Q₈	1	2	yes	—
31	C₄×C₇×C₈	—	2	2	no	—
32	C₄×C₇×C₈	D₄	1	2	yes	—
33	#2153	C₂×C₇	1	1	yes	characteristic
34	#2151	C₂×C₇	1	1	yes	characteristic
35	#1747	C₂×C₇	1	1	yes	characteristic
36	#1379	C₂×C₇	1	2	yes	—
37	#1378	C₂×C₇	1	2	yes	—
38	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
39	C₇×M₁₆	—	2	2	no	—
40	C₇×M₁₆	—	2	2	no	—
41	C₇×M₁₆	—	2	2	no	—
42	C₂×C₇×C₈	—	2	2	no	—
43	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
44	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
45	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₇×C₈	#41	1	2	yes	—
47	C₂×C₇×C₈	#41	1	2	yes	—
48	C₇×C₄⋊C₄	—	2	1	no	—
49	C₇×C₄⋊C₄	—	2	2	no	—
50	C₇×C₂²⋊C₄	—	2	1	no	—
51	C₇×C₂²⋊C₄	—	2	2	no	—
52	C₄²×C₇	—	2	2	no	—
53	C₄²×C₇	C₄⋊C₄	1	2	yes	—
54	C₇×C₁₆	—	4	1	no	—
55	C₇×C₁₆	—	4	1	no	—
56	C₇×C₁₆	—	4	2	no	—
57	C₂×M₃₂	C₂²×C₇	1	1	yes	characteristic
58	C₂×M₃₂	C₂²×C₇	1	2	yes	—
59	C₂²×C₁₆	C₂²×C₇	1	1	yes	characteristic
60	#405	C₂²×C₇	1	2	yes	—
61	#405	C₂²×C₇	1	1	yes	characteristic
62	#364	—	2	2	no	—
63	C₄⋊C₁₆	—	2	2	no	—
64	#338	C₄×C₇	1	1	yes	characteristic
65	H(3,ℤ/4)	C₄×C₇	1	1	yes	characteristic
66	H(3,ℤ/4)	C₄×C₇	1	2	yes	—
67	C₂³×C₇	#131	1	1	yes	characteristic
68	C₂×C₄×C₇	—	2	2	no	—
69	C₂×C₄×C₇	—	4	2	no	—
70	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₇	#117	1	2	yes	—
72	C₂×C₄×C₇	#131	1	2	yes	—
73	C₂×C₄×C₇	#131	1	1	yes	characteristic
74	C₇×C₈	—	2	2	no	—
75	C₇×C₈	—	4	2	no	—
76	C₇×C₈	#124	1	1	yes	characteristic
77	C₇×C₈	#124	1	1	yes	characteristic
78	C₇×C₈	#125	1	2	yes	—
79	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
80	C₂×M₁₆	C₂×C₄×C₇	1	1	yes	characteristic
81	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
82	#117	C₂×C₄×C₇	1	2	yes	—
83	#117	C₂×C₄×C₇	1	1	yes	characteristic
84	M₃₂	—	2	2	no	—
85	M₃₂	—	2	2	no	—
86	M₃₂	—	2	2	no	—
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	—	2	2	no	—
90	C₂×C₁₆	—	2	2	no	—
91	#97	—	2	2	no	—
92	#97	C₇×Q₈	1	2	yes	—
93	C₄×C₈	—	2	2	no	—
94	C₄×C₈	C₇×D₄	1	2	yes	—
95	C₂²×C₇	—	4	1	no	—
96	C₂²×C₇	#424	1	1	yes	characteristic
97	C₂²×C₇	#433	1	2	yes	—
98	C₄×C₇	—	4	2	no	—
99	C₄×C₇	—	8	2	no	—
100	C₄×C₇	#382	1	1	yes	characteristic
101	C₄×C₇	#424	1	1	yes	characteristic
102	C₄×C₇	#433	1	2	yes	—
103	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
104	M₁₆	—	2	2	no	—
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	C₇×C₄⋊C₄	1	2	yes	—
109	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
110	C₂×C₈	C₂×C₇×Q₈	1	1	yes	characteristic
111	C₂×C₈	C₇×#41	1	2	yes	—
112	C₂×C₈	C₇×#41	1	2	yes	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	1	no	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	C₇×C₄⋊C₄	1	2	yes	—
119	C₁₆	—	4	1	no	—
120	C₁₆	—	4	1	no	—
121	C₁₆	—	4	2	no	—
122	C₂×C₇	—	2	1	no	—
123	C₂×C₇	—	2	2	no	—
124	C₂×C₇	#1832	1	1	yes	characteristic
125	C₂³	C₇×#131	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄	C₇×#117	1	2	yes	—
130	C₂×C₄	C₇×#131	1	2	yes	—
131	C₂×C₄	C₇×#131	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	4	2	no	—
134	C₈	C₇×#124	1	1	yes	characteristic
135	C₈	C₇×#124	1	1	yes	characteristic
136	C₈	C₇×#125	1	2	yes	—
137	C₇	#12885	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²	—	4	1	no	—
139	C₂²	C₇×#424	1	1	yes	characteristic
140	C₂²	C₇×#433	1	2	yes	—
141	C₄	—	4	2	no	—
142	C₄	—	8	2	no	—
143	C₄	C₇×#382	1	1	yes	characteristic
144	C₄	C₇×#424	1	1	yes	characteristic
145	C₄	C₇×#433	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	2	no	—
148	C₂	C₇×#1832	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2153

C₂

yes

maximal, characteristic

C₇×#2151

C₂

yes

maximal, characteristic

C₇×#1747

C₂

yes

maximal, characteristic

C₇×#1379

C₂

yes

maximal

C₇×#1378

C₂

yes

maximal

C₂×C₇×M₃₂

C₂²

yes

characteristic

C₂×C₇×M₃₂

C₂²

yes

—

C₂²×C₇×C₁₆

C₂²

yes

characteristic

C₇×#405

C₂²

yes

—

C₇×#405

C₂²

yes

characteristic

C₇×#364

—

C₇×C₄⋊C₁₆

—

C₇×#338

C₄

yes

characteristic

C₇×H(3,ℤ/4)

C₄

yes

characteristic

C₇×H(3,ℤ/4)

C₄

yes

—

#12885

C₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₇×M₁₆

C₂×C₄

yes

—

C₂×C₇×M₁₆

C₂×C₄

yes

characteristic

C₂²×C₇×C₈

C₂³

yes

characteristic

C₇×#117

C₂×C₄

yes

—

C₇×#117

C₂×C₄

yes

characteristic

C₇×M₃₂

—

C₇×M₃₂

—

C₇×M₃₂

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

—

C₂×C₇×C₁₆

—

C₇×#97

—

C₇×#97

Q₈

yes

—

C₄×C₇×C₈

—

C₄×C₇×C₈

D₄

yes

—

#2153

C₂×C₇

yes

characteristic

#2151

C₂×C₇

yes

characteristic

#1747

C₂×C₇

yes

characteristic

#1379

C₂×C₇

yes

—

#1378

C₂×C₇

yes

—

C₂²×C₄×C₇

C₂²×C₄

yes

characteristic

C₇×M₁₆

—

C₇×M₁₆

—

C₇×M₁₆

—

C₂×C₇×C₈

—

C₂×C₇×C₈

C₄⋊C₄

yes

—

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₂×C₇×C₈

C₂×Q₈

yes

characteristic

C₂×C₇×C₈

#41

yes

—

C₂×C₇×C₈

#41

yes

—

C₇×C₄⋊C₄

—

C₇×C₄⋊C₄

—

C₇×C₂²⋊C₄

—

C₇×C₂²⋊C₄

—

C₄²×C₇

—

C₄²×C₇

C₄⋊C₄

yes

—

C₇×C₁₆

—

C₇×C₁₆

—

C₇×C₁₆

—

C₂×M₃₂

C₂²×C₇

yes

characteristic

C₂×M₃₂

C₂²×C₇

yes

—

C₂²×C₁₆

C₂²×C₇

yes

characteristic

#405

C₂²×C₇

yes

—

#405

C₂²×C₇

yes

characteristic

#364

—

C₄⋊C₁₆

—

#338

C₄×C₇

yes

characteristic

H(3,ℤ/4)

C₄×C₇

yes

characteristic

H(3,ℤ/4)

C₄×C₇

yes

—

C₂³×C₇

#131

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇

#117

yes

—

C₂×C₄×C₇

#131

yes

—

C₂×C₄×C₇

#131

yes

characteristic

C₇×C₈

—

C₇×C₈

—

C₇×C₈

yes

C₇×C₈

yes

C₇×C₈

yes

—

C₂×M₁₆

C₂×C₄×C₇

yes

—

C₂×M₁₆

C₂×C₄×C₇

yes

characteristic

C₂²×C₈

C₂³×C₇

yes

characteristic

#117

C₂×C₄×C₇

yes

—

#117

C₂×C₄×C₇

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

C₇×Q₈

yes

—

C₄×C₈

—

C₄×C₈

C₇×D₄

yes

—

C₂²×C₇

—

C₂²×C₇

#424

yes

characteristic

C₂²×C₇

#433

yes

—

C₄×C₇

—

C₄×C₇

—

100

C₄×C₇

#382

yes

characteristic

101

C₄×C₇

#424

yes

characteristic

102

C₄×C₇

#433

yes

—

103

C₂²×C₄

C₂²×C₄×C₇

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

C₇×C₄⋊C₄

yes

—

109

C₂×C₈

C₂×C₇×D₄

yes

characteristic

110

C₂×C₈

C₂×C₇×Q₈

yes

characteristic

111

C₂×C₈

C₇×#41

yes

—

112

C₂×C₈

C₇×#41

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₇×C₄⋊C₄

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₂×C₇

—

123

C₂×C₇

—

124

C₂×C₇

#1832

yes

characteristic

125

C₂³

C₇×#131

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₇×C₄⋊C₄

yes

characteristic

129

C₂×C₄

C₇×#117

yes

—

130

C₂×C₄

C₇×#131

yes

—

131

C₂×C₄

C₇×#131

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

C₇×#124

yes

characteristic

135

C₈

C₇×#124

yes

characteristic

136

C₈

C₇×#125

yes

—

137

C₇

#12885

yes

characteristic, direct product factor, Sylow

138

C₂²

—

139

C₂²

C₇×#424

yes

characteristic

140

C₂²

C₇×#433

yes

—

141

C₄

—

142

C₄

—

143

C₄

C₇×#382

yes

characteristic

144

C₄

C₇×#424

yes

characteristic

145

C₄

C₇×#433

yes

—

146

C₂

—

147

C₂

—

148

C₂

C₇×#1832

yes

characteristic

	subgroup	G/H
center	C₇×C₈	#124
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂×C₇	#1832
prime radical	C₂³×C₇	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

subgroup

G/H

center

C₇×C₈

#124

commutator

C₂²×C₄

C₂²×C₄×C₇

Frattini, cosocle

C₂²×C₈

C₂³×C₇

C₂×C₇

C₂³×C₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₇×C₈ ◁ C₂²×C₇×C₈ ◁ G

N	H ≅ G/N	type
16. #12885	C₇	direct (N × H)
137. C₇	#12885	direct (N × H)

H ≅ G/N

type

16. #12885

C₇

direct (N × H)

137. C₇

#12885

direct (N × H)

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

subgroup	index
C₇×#2153	2
C₇×#2151	2
C₇×#1747	2
C₇×#1379	2
C₇×#1378	2
#12885	7

subgroup

index

Subgroups — C₇×#12885 #13190101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C7×#12885 #13190101

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12885 #13190101