Group #13265359 — subgroups

Subgroups — #13265359

Order 1792.

220 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 99 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	D₇×#364	C₂	1	2	yes	maximal
2	#11726825	C₂	1	1	yes	maximal, characteristic
3	C₇×#1369	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725373	C₂	1	2	yes	maximal, complemented
5	#11725320	C₂	1	1	yes	maximal, characteristic, complemented
6	D₇×M₃₂	—	2	2	no	—
7	C₂×#6372	C₄	1	1	yes	characteristic
8	C₂×#6372	C₄	1	1	yes	characteristic
9	C₄×C₈×D₇	C₂²	1	1	yes	characteristic
10	C₇×#364	C₂²	1	2	yes	—
11	C₇×#346	C₂²	1	1	yes	characteristic
12	#89306	—	2	2	no	—
13	C₁₆×Dic₇	C₄	1	1	yes	characteristic
14	C₁₆×Dic₇	C₄	1	1	yes	characteristic
15	#89258	C₂²	1	2	yes	—
16	#89256	C₂²	1	1	yes	characteristic
17	#12002	—	7	1	no	maximal, Sylow
18	C₂×C₈×D₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×D₇	D₄	1	1	yes	characteristic
20	C₄²×D₇	Q₈	1	1	yes	characteristic
21	C₇×M₃₂	—	2	2	no	—
22	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
25	C₈×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₈×Dic₇	D₄	1	1	yes	characteristic
27	#6386	—	2	2	no	—
28	C₂×C₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
30	C₄×C₇⋊C₈	Q₈	1	1	yes	characteristic
31	#6372	—	2	2	no	—
32	#6372	—	4	2	no	—
33	#6372	C₈	1	4	yes	—
34	D₇×C₁₆	—	4	2	no	—
35	C₂×#364	—	7	2	no	—
36	C₄×M₃₂	—	7	1	no	—
37	#1369	—	7	2	no	—
38	#1369	—	7	1	no	—
39	#1369	D₇	1	1	yes	characteristic, complemented
40	C₂×C₄×D₇	—	2	1	no	—
41	C₂×C₄×D₇	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₇	C₂×Q₈	1	1	yes	characteristic
45	C₄×Dic₇	—	2	1	no	—
46	C₄×Dic₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₇	C₂×C₈	1	2	yes	—
50	C₈×D₇	M₁₆	1	2	yes	—
51	C₈×D₇	SD₁₆	1	2	yes	—
52	C₈×D₇	SD₁₆	1	2	yes	—
53	C₇×C₁₆	—	2	1	no	—
54	C₇×C₁₆	—	4	2	no	—
55	C₇×C₁₆	C₂×C₈	1	2	yes	—
56	C₇⋊C₁₆	—	2	1	no	—
57	C₇⋊C₁₆	—	4	2	no	—
58	C₇⋊C₁₆	C₂×C₈	1	2	yes	—
59	C₂×M₃₂	—	7	1	no	—
60	C₂×M₃₂	—	7	1	no	—
61	C₂×M₃₂	—	14	2	no	—
62	C₂×C₄×C₈	—	7	1	no	—
63	#364	—	7	2	no	—
64	#364	—	14	2	no	—
65	#364	C₂×D₇	1	2	yes	—
66	#346	—	7	1	no	—
67	#346	C₂×D₇	1	1	yes	characteristic
68	C₄×C₁₆	—	7	1	no	—
69	C₄×C₁₆	—	7	1	no	—
70	C₂²×D₇	#108	1	1	yes	characteristic
71	C₂×C₄×C₇	—	2	1	no	—
72	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₂×Dic₇	—	2	1	no	—
74	C₂×Dic₇	#108	1	1	yes	characteristic
75	C₄×D₇	—	4	2	no	—
76	C₄×D₇	C₄⋊C₈	1	2	yes	—
77	C₄×D₇	#106	1	2	yes	—
78	C₇×C₈	C₂²×C₈	1	1	yes	characteristic
79	C₇×C₈	C₂×M₁₆	1	1	yes	characteristic
80	C₇×C₈	C₂×SD₁₆	1	2	yes	—
81	C₇⋊C₈	C₄⋊C₈	1	2	yes	—
82	C₇⋊C₈	#106	1	1	yes	characteristic
83	C₇⋊C₈	#106	1	1	yes	characteristic
84	C₂²×C₈	—	7	1	no	—
85	C₂²×C₈	—	7	1	no	—
86	C₂×C₄²	—	7	1	no	—
87	M₃₂	—	2	2	no	—
88	M₃₂	—	7	4	no	—
89	M₃₂	—	14	2	no	—
90	M₃₂	—	14	2	no	—
91	M₃₂	—	28	2	no	—
92	C₂×C₁₆	—	7	1	no	—
93	C₂×C₁₆	—	7	1	no	—
94	C₂×C₁₆	—	28	2	no	—
95	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
96	C₂×C₁₆	C₄×D₇	1	1	yes	characteristic
97	C₄×C₈	—	7	1	no	—
98	C₄×C₈	—	7	1	no	—
99	C₄×C₈	—	7	1	no	—
100	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
101	C₂²×C₇	C₂×#108	1	1	yes	characteristic
102	C₂×D₇	—	4	1	no	—
103	C₂×D₇	#334	1	2	yes	—
104	C₄×C₇	—	4	1	no	—
105	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
106	C₄×C₇	C₂×#106	1	1	yes	characteristic
107	Dic₇	—	2	2	no	—
108	Dic₇	#334	1	2	yes	—
109	C₂²×C₄	—	7	1	no	—
110	C₂²×C₄	—	14	1	no	—
111	C₂×C₈	—	7	1	no	—
112	C₂×C₈	—	7	1	no	—
113	C₂×C₈	—	7	2	no	—
114	C₂×C₈	—	7	2	no	—
115	C₂×C₈	—	7	2	no	—
116	C₂×C₈	—	7	2	no	—
117	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
118	C₂×C₈	D₄×D₇	1	1	yes	characteristic
119	C₄²	—	7	1	no	—
120	C₄²	—	14	1	no	—
121	C₄²	Q₈×D₇	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	—	4	2	no	—
124	C₁₆	—	7	2	no	—
125	C₁₆	—	14	1	no	—
126	C₁₆	—	28	2	no	—
127	C₁₆	C₈×D₇	1	2	yes	—
128	C₂×C₇	—	2	1	no	—
129	C₂×C₇	C₂×#334	1	1	yes	characteristic
130	D₇	—	2	2	no	—
131	C₂³	—	7	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	7	1	no	—
134	C₂×C₄	—	7	2	no	—
135	C₂×C₄	—	7	2	no	—
136	C₂×C₄	—	14	1	no	—
137	C₂×C₄	—	28	2	no	—
138	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
139	C₈	—	7	2	no	—
140	C₈	—	7	1	no	—
141	C₈	—	7	1	no	—
142	C₈	C₂×C₈×D₇	1	1	yes	characteristic
143	C₈	D₇×M₁₆	1	1	yes	characteristic
144	C₈	D₇×SD₁₆	1	2	yes	—
145	C₇	#12002	1	1	yes	characteristic, complemented, Sylow
146	C₂²	—	7	2	no	—
147	C₂²	—	28	1	no	—
148	C₂²	D₇×#108	1	1	yes	characteristic
149	C₄	—	4	1	no	—
150	C₄	—	7	2	no	—
151	C₄	—	14	2	no	—
152	C₄	D₇×C₄⋊C₈	1	1	yes	characteristic
153	C₄	D₇×#106	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	14	2	no	—
156	C₂	D₇×#334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₇×#364

C₂

yes

maximal

#11726825

C₂

yes

maximal, characteristic

C₇×#1369

C₂

yes

maximal, characteristic, complemented

#11725373

C₂

yes

maximal, complemented

#11725320

C₂

yes

maximal, characteristic, complemented

D₇×M₃₂

—

C₂×#6372

C₄

yes

characteristic

C₂×#6372

C₄

yes

characteristic

C₄×C₈×D₇

C₂²

yes

characteristic

C₇×#364

C₂²

yes

—

C₇×#346

C₂²

yes

characteristic

#89306

—

C₁₆×Dic₇

C₄

yes

characteristic

C₁₆×Dic₇

C₄

yes

characteristic

#89258

C₂²

yes

—

#89256

C₂²

yes

characteristic

#12002

—

maximal, Sylow

C₂×C₈×D₇

C₂×C₄

yes

characteristic

C₂×C₈×D₇

D₄

yes

characteristic

C₄²×D₇

Q₈

yes

characteristic

C₇×M₃₂

—

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₂×C₇×C₁₆

C₂×C₄

yes

characteristic

C₄×C₇×C₈

C₂³

yes

characteristic

C₈×Dic₇

C₂×C₄

yes

characteristic

C₈×Dic₇

D₄

yes

characteristic

#6386

—

C₂×C₇⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₇⋊C₁₆

C₂×C₄

yes

characteristic

C₄×C₇⋊C₈

Q₈

yes

—

—

C₈

yes

—

D₇×C₁₆

—

C₂×#364

—

C₄×M₃₂

—

#1369

—

#1369

—

#1369

D₇

yes

characteristic, complemented

C₂×C₄×D₇

—

C₂×C₄×D₇

C₄⋊C₄

yes

characteristic

C₂×C₇×C₈

C₂²×C₄

yes

characteristic

C₂×C₇×C₈

C₂×D₄

yes

characteristic

C₄²×C₇

C₂×Q₈

yes

characteristic

C₄×Dic₇

—

C₄×Dic₇

C₄⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₄⋊C₄

yes

characteristic

C₂×C₇⋊C₈

C₄⋊C₄

yes

characteristic

C₈×D₇

C₂×C₈

yes

—

C₈×D₇

M₁₆

yes

—

C₈×D₇

SD₁₆

yes

—

C₈×D₇

SD₁₆

yes

—

C₇×C₁₆

—

C₇×C₁₆

—

C₇×C₁₆

C₂×C₈

yes

—

C₇⋊C₁₆

—

C₇⋊C₁₆

—

C₇⋊C₁₆

C₂×C₈

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×C₄×C₈

—

#364

—

#364

—

#364

C₂×D₇

yes

—

#346

—

#346

C₂×D₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₂²×D₇

#108

yes

characteristic

C₂×C₄×C₇

—

C₂×C₄×C₇

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇

—

C₂×Dic₇

#108

yes

characteristic

C₄×D₇

—

C₄×D₇

C₄⋊C₈

yes

—

C₄×D₇

#106

yes

—

C₇×C₈

C₂²×C₈

yes

characteristic

C₇×C₈

C₂×M₁₆

yes

characteristic

C₇×C₈

C₂×SD₁₆

yes

—

C₇⋊C₈

C₄⋊C₈

yes

—

C₇⋊C₈

yes

C₇⋊C₈

yes

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇

yes

characteristic

C₂×C₁₆

C₄×D₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

C₂²×D₇

yes

characteristic

101

C₂²×C₇

C₂×#108

yes

characteristic

102

C₂×D₇

—

103

C₂×D₇

#334

yes

—

104

C₄×C₇

—

105

C₄×C₇

C₂×C₄⋊C₈

yes

characteristic

106

C₄×C₇

C₂×#106

yes

characteristic

107

Dic₇

—

108

Dic₇

#334

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

C₂×C₄×D₇

yes

characteristic

118

C₂×C₈

D₄×D₇

yes

characteristic

119

C₄²

—

120

C₄²

—

121

C₄²

Q₈×D₇

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₁₆

—

127

C₁₆

C₈×D₇

yes

—

128

C₂×C₇

—

129

C₂×C₇

C₂×#334

yes

characteristic

130

D₇

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

D₇×C₄⋊C₄

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

C₂×C₈×D₇

yes

characteristic

143

C₈

D₇×M₁₆

yes

characteristic

144

C₈

D₇×SD₁₆

yes

—

145

C₇

#12002

yes

characteristic, complemented, Sylow

146

C₂²

—

147

C₂²

—

148

C₂²

D₇×#108

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

D₇×C₄⋊C₈

yes

characteristic

153

C₄

D₇×#106

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

D₇×#334

yes

characteristic

	subgroup	G/H
center	C₄	D₇×#106
commutator	C₇×C₈	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₇×#1369	C₂
socle	C₂×C₇	C₂×#334
prime radical	C₂²×D₇	#108
cosocle	C₄×C₇×C₈	C₂³
hypercenter	#1369	D₇
hypocenter	C₇	#12002
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

subgroup

G/H

center

C₄

D₇×#106

commutator

C₇×C₈

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₇

C₂

C₂×C₇

C₂²×D₇

C₄×C₇×C₈

C₂³

D₇

C₇

G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1369

N	H ≅ G/N	type
3. C₇×#1369	C₂	semidirect
4. #11725373	C₂	semidirect
5. #11725320	C₂	semidirect
39. #1369	D₇	semidirect
145. C₇	#12002	semidirect

H ≅ G/N

type

3. C₇×#1369

C₂

semidirect

4. #11725373

C₂

semidirect

5. #11725320

C₂

semidirect

39. #1369

D₇

semidirect

145. C₇

#12002

semidirect

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

subgroup	index
D₇×#364	2
#11726825	2
C₇×#1369	2
#11725373	2
#11725320	2
#12002	7

subgroup

index

Subgroups — #13265359

Conjugacy classes of subgroups

Complements

Maximal subgroups