Group #13560202 — subgroups

Subgroups — D₇×#2840 #13560202

Order 1792.

3550 conjugacy classes of subgroups; only the 579 normal subgroups are shown.

normal subgroups only

This group has 3550 conjugacy classes of subgroups — too many to draw. Only its 579 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×#2840	C₂	1	1	yes	maximal, characteristic, complemented
2	#11734333	C₂	1	12	yes	maximal, complemented
3	D₇×#393	C₂	1	12	yes	maximal, complemented
4	D₇×#379	C₂	1	3	yes	maximal
5	#11731289	C₂	1	3	yes	maximal, complemented
6	C₂⁵×D₇	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×D₇	C₂²	1	4	yes	complemented
8	C₂×D₇×C₂²⋊C₄	C₂²	1	18	yes	—
9	C₇×#393	C₂²	1	12	yes	complemented
10	C₇×#379	C₂²	1	3	yes	—
11	#90019	C₂²	1	3	yes	—
12	#90003	C₂²	1	12	yes	complemented
13	#89727	C₂²	1	18	yes	—
14	#89719	C₂²	1	36	yes	complemented
15	#89445	C₂²	1	36	yes	complemented
16	D₇×#95	C₂²	1	12	yes	complemented
17	C₂×#2840	—	7	1	no	maximal, Sylow
18	C₂⁵×C₇	C₂³	1	1	yes	characteristic
19	C₂⁴×D₇	—	4	12	no	—
20	C₂⁴×D₇	Q₈	1	8	yes	complemented
21	C₂⁴×D₇	C₂³	1	3	yes	—
22	C₂⁴×D₇	C₂³	1	3	yes	—
23	C₂²×C₇×Q₈	C₂³	1	4	yes	complemented
24	C₂×Q₈×D₇	—	4	24	no	—
25	C₂²×C₄×D₇	C₂³	1	12	yes	—
26	C₂²×Dic₁₄	C₂³	1	12	yes	complemented
27	C₂×C₇×C₂²⋊C₄	C₂³	1	18	yes	—
28	C₂×C₂²⋊Dic₇	C₂³	1	18	yes	—
29	C₂×(C₂×D₇)⋊C₄	C₂³	1	36	yes	—
30	D₇×C₂²⋊C₄	—	4	36	no	—
31	C₇×#95	C₂³	1	12	yes	complemented
32	C₇⋊#95	C₂³	1	36	yes	complemented
33	#2840	—	7	12	no	—
34	#2840	—	7	3	no	—
35	#2840	D₇	1	1	yes	characteristic, direct product factor
36	C₂×#393	—	7	12	no	—
37	C₂×#379	—	7	3	no	—
38	C₂⁴×C₇	—	4	6	no	—
39	C₂⁴×C₇	C₂×Q₈	1	4	yes	complemented
40	C₂⁴×C₇	C₂⁴	1	3	yes	—
41	C₂³×D₇	—	2	36	no	—
42	C₂³×D₇	—	4	48	no	—
43	C₂³×D₇	—	4	18	no	—
44	C₂³×D₇	—	4	64	no	—
45	C₂³×D₇	C₂×Q₈	1	12	yes	—
46	C₂³×D₇	C₂×Q₈	1	12	yes	complemented
47	C₂³×D₇	C₂⁴	1	3	yes	—
48	C₂³×D₇	C₂⁴	1	1	yes	characteristic
49	C₂×C₇×Q₈	—	4	12	no	—
50	C₂²×C₄×C₇	C₂⁴	1	12	yes	—
51	C₂²×Dic₇	C₂⁴	1	12	yes	—
52	Q₈×D₇	—	4	64	no	—
53	C₂×C₄×D₇	—	2	72	no	—
54	C₂×Dic₁₄	—	4	36	no	—
55	C₇×C₂²⋊C₄	—	4	18	no	—
56	C₂²⋊Dic₇	—	4	18	no	—
57	(C₂×D₇)⋊C₄	—	4	36	no	—
58	C₂⁶	—	7	1	no	—
59	C₂³×Q₈	—	7	4	no	—
60	C₂²×C₂²⋊C₄	—	7	18	no	—
61	#393	—	7	12	no	—
62	#393	—	7	36	no	—
63	#393	—	7	36	no	—
64	#393	C₂×D₇	1	12	yes	complemented
65	#379	—	7	3	no	—
66	#379	—	7	18	no	—
67	#379	C₂×D₇	1	3	yes	—
68	C₂×#95	—	7	12	no	—
69	C₂³×C₇	—	2	18	no	—
70	C₂³×C₇	—	4	12	no	—
71	C₂³×C₇	—	4	16	no	—
72	C₂³×C₇	C₂²×Q₈	1	6	yes	—
73	C₂³×C₇	C₂⁵	1	1	yes	characteristic
74	C₂²×D₇	—	2	18	no	—
75	C₂²×D₇	—	2	72	no	—
76	C₂²×D₇	—	4	36	no	—
77	C₂²×D₇	—	4	48	no	—
78	C₂²×D₇	—	4	128	no	—
79	C₂²×D₇	—	4	48	no	—
80	C₂²×D₇	C₂²×Q₈	1	6	yes	—
81	C₂²×D₇	C₂²×Q₈	1	2	yes	—
82	C₂²×D₇	H(5,𝔽₂)	1	12	yes	—
83	C₇×Q₈	—	4	16	no	—
84	C₂×C₄×C₇	—	2	36	no	—
85	C₂×Dic₇	—	2	36	no	—
86	C₄×D₇	—	4	48	no	—
87	Dic₁₄	—	4	48	no	—
88	C₂⁵	—	7	3	no	—
89	C₂⁵	—	7	8	no	—
90	C₂⁵	—	7	3	no	—
91	C₂⁵	—	28	12	no	—
92	C₂⁵	C₂²×D₇	1	1	yes	characteristic
93	C₂²×Q₈	—	7	12	no	—
94	C₂²×Q₈	—	28	24	no	—
95	C₂²×Q₈	C₂²×D₇	1	4	yes	complemented
96	C₂³×C₄	—	7	12	no	—
97	C₂×C₂²⋊C₄	—	7	18	no	—
98	C₂×C₂²⋊C₄	—	7	36	no	—
99	C₂×C₂²⋊C₄	—	28	36	no	—
100	C₂×C₂²⋊C₄	C₂²×D₇	1	18	yes	—
101	#95	—	7	36	no	—
102	#95	C₂²×D₇	1	12	yes	complemented
103	C₂²×C₇	—	2	18	no	—
104	C₂²×C₇	—	4	16	no	—
105	C₂²×C₇	—	4	12	no	—
106	C₂²×C₇	C₂³×Q₈	1	1	yes	characteristic
107	C₂²×C₇	C₂×H(5,𝔽₂)	1	6	yes	—
108	C₂×D₇	—	2	18	no	—
109	C₂×D₇	—	4	48	no	—
110	C₂×D₇	—	4	12	no	—
111	C₂×D₇	—	4	48	no	—
112	C₂×D₇	#543	1	12	yes	—
113	C₂×D₇	H(3,𝔽₄)	1	16	yes	—
114	C₄×C₇	—	4	12	no	—
115	Dic₇	—	4	12	no	—
116	C₂⁴	—	4	6	no	—
117	C₂⁴	—	7	1	no	—
118	C₂⁴	—	7	12	no	—
119	C₂⁴	—	7	3	no	—
120	C₂⁴	—	7	12	no	—
121	C₂⁴	—	14	36	no	—
122	C₂⁴	—	28	18	no	—
123	C₂⁴	—	28	48	no	—
124	C₂⁴	—	28	64	no	—
125	C₂⁴	Q₈×D₇	1	4	yes	complemented
126	C₂⁴	C₂³×D₇	1	3	yes	—
127	C₂×Q₈	—	4	12	no	—
128	C₂×Q₈	—	28	36	no	—
129	C₂×Q₈	—	28	64	no	—
130	C₂²×C₄	—	7	12	no	—
131	C₂²×C₄	—	14	72	no	—
132	C₂²×C₄	C₂³×D₇	1	12	yes	—
133	C₂²⋊C₄	—	4	18	no	—
134	C₂²⋊C₄	—	28	36	no	—
135	C₂²⋊C₄	—	28	18	no	—
136	C₂×C₇	—	4	6	no	—
137	C₂×C₇	C₂×#543	1	3	yes	—
138	C₂×C₇	C₂×H(3,𝔽₄)	1	4	yes	—
139	D₇	—	4	6	no	—
140	D₇	#2840	1	8	yes	direct product factor
141	C₂³	—	2	18	no	—
142	C₂³	—	4	12	no	—
143	C₂³	—	4	16	no	—
144	C₂³	—	7	2	no	—
145	C₂³	—	7	12	no	—
146	C₂³	—	7	6	no	—
147	C₂³	—	14	72	no	—
148	C₂³	—	14	18	no	—
149	C₂³	—	28	36	no	—
150	C₂³	—	28	48	no	—
151	C₂³	—	28	128	no	—
152	C₂³	—	28	48	no	—
153	C₂³	C₂×Q₈×D₇	1	6	yes	—
154	C₂³	C₂⁴×D₇	1	1	yes	characteristic
155	Q₈	—	4	16	no	—
156	Q₈	—	28	48	no	—
157	C₂×C₄	—	2	36	no	—
158	C₂×C₄	—	14	36	no	—
159	C₂×C₄	—	28	48	no	—
160	C₇	C₂×#2840	1	1	yes	characteristic, complemented, Sylow
161	C₂²	—	2	18	no	—
162	C₂²	—	4	16	no	—
163	C₂²	—	4	12	no	—
164	C₂²	—	7	16	no	—
165	C₂²	—	7	12	no	—
166	C₂²	—	14	18	no	—
167	C₂²	—	28	48	no	—
168	C₂²	—	28	48	no	—
169	C₂²	—	28	12	no	—
170	C₂²	C₂²×Q₈×D₇	1	1	yes	characteristic
171	C₂²	D₇×H(5,𝔽₂)	1	6	yes	—
172	C₄	—	4	12	no	—
173	C₄	—	28	12	no	—
174	C₂	—	4	6	no	—
175	C₂	—	7	8	no	—
176	C₂	—	28	6	no	—
177	C₂	D₇×#543	1	3	yes	—
178	C₂	D₇×H(3,𝔽₄)	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇×#2840

C₂

yes

maximal, characteristic, complemented

#11734333

C₂

yes

maximal, complemented

D₇×#393

C₂

yes

maximal, complemented

D₇×#379

C₂

yes

maximal

#11731289

C₂

yes

maximal, complemented

C₂⁵×D₇

C₂²

yes

characteristic

C₂²×Q₈×D₇

C₂²

yes

complemented

C₂×D₇×C₂²⋊C₄

C₂²

yes

—

C₇×#393

C₂²

yes

complemented

C₇×#379

C₂²

yes

—

#90019

C₂²

yes

—

#90003

C₂²

yes

complemented

#89727

C₂²

yes

—

#89719

C₂²

yes

complemented

#89445

C₂²

yes

complemented

D₇×#95

C₂²

yes

complemented

C₂×#2840

—

maximal, Sylow

C₂⁵×C₇

C₂³

yes

characteristic

C₂⁴×D₇

—

C₂⁴×D₇

Q₈

yes

complemented

C₂⁴×D₇

C₂³

yes

—

C₂⁴×D₇

C₂³

yes

—

C₂²×C₇×Q₈

C₂³

yes

complemented

C₂×Q₈×D₇

—

C₂²×C₄×D₇

C₂³

yes

—

C₂²×Dic₁₄

C₂³

yes

complemented

C₂×C₇×C₂²⋊C₄

C₂³

yes

—

C₂×C₂²⋊Dic₇

C₂³

yes

—

C₂×(C₂×D₇)⋊C₄

C₂³

yes

—

D₇×C₂²⋊C₄

—

C₇×#95

C₂³

yes

complemented

C₇⋊#95

C₂³

yes

—

—

D₇

yes

characteristic, direct product factor

C₂×#393

—

C₂×#379

—

C₂⁴×C₇

—

C₂⁴×C₇

C₂×Q₈

yes

complemented

C₂⁴×C₇

C₂⁴

yes

—

C₂³×D₇

—

C₂³×D₇

—

C₂³×D₇

—

C₂³×D₇

—

C₂³×D₇

C₂×Q₈

yes

—

C₂³×D₇

C₂×Q₈

yes

complemented

C₂³×D₇

C₂⁴

yes

—

C₂³×D₇

C₂⁴

yes

characteristic

C₂×C₇×Q₈

—

C₂²×C₄×C₇

C₂⁴

yes

—

C₂²×Dic₇

C₂⁴

yes

—

Q₈×D₇

—

C₂×C₄×D₇

—

C₂×Dic₁₄

—

C₇×C₂²⋊C₄

—

C₂²⋊Dic₇

—

(C₂×D₇)⋊C₄

—

C₂⁶

—

C₂³×Q₈

—

C₂²×C₂²⋊C₄

—

—

—

—

C₂×D₇

yes

—

—

C₂×D₇

yes

—

C₂×#95

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

—

C₂³×C₇

C₂²×Q₈

yes

—

C₂³×C₇

C₂⁵

yes

characteristic

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

C₂²×D₇

—

128

—

C₂²×D₇

—

C₂²×D₇

C₂²×Q₈

yes

—

C₂²×D₇

C₂²×Q₈

yes

—

C₂²×D₇

H(5,𝔽₂)

yes

—

C₇×Q₈

—

C₂×C₄×C₇

—

C₂×Dic₇

—

C₄×D₇

—

Dic₁₄

—

C₂⁵

—

C₂⁵

—

C₂⁵

—

C₂⁵

—

C₂⁵

C₂²×D₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂²×D₇

yes

complemented

C₂³×C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

100

C₂×C₂²⋊C₄

C₂²×D₇

yes

—

101

#95

—

102

#95

C₂²×D₇

yes

complemented

103

C₂²×C₇

—

104

C₂²×C₇

—

105

C₂²×C₇

—

106

C₂²×C₇

C₂³×Q₈

yes

characteristic

107

C₂²×C₇

C₂×H(5,𝔽₂)

yes

—

108

C₂×D₇

—

109

C₂×D₇

—

110

C₂×D₇

—

111

C₂×D₇

—

112

C₂×D₇

#543

yes

—

113

C₂×D₇

H(3,𝔽₄)

yes

—

114

C₄×C₇

—

115

Dic₇

—

116

C₂⁴

—

117

C₂⁴

—

118

C₂⁴

—

119

C₂⁴

—

120

C₂⁴

—

121

C₂⁴

—

122

C₂⁴

—

123

C₂⁴

—

124

C₂⁴

—

125

C₂⁴

Q₈×D₇

yes

complemented

126

C₂⁴

C₂³×D₇

yes

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂²×C₄

—

131

C₂²×C₄

—

132

C₂²×C₄

C₂³×D₇

yes

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₂²⋊C₄

—

136

C₂×C₇

—

137

C₂×C₇

C₂×#543

yes

—

138

C₂×C₇

C₂×H(3,𝔽₄)

yes

—

139

D₇

—

140

D₇

#2840

yes

direct product factor

141

C₂³

—

142

C₂³

—

143

C₂³

—

144

C₂³

—

145

C₂³

—

146

C₂³

—

147

C₂³

—

148

C₂³

—

149

C₂³

—

150

C₂³

—

151

C₂³

—

128

—

152

C₂³

—

153

C₂³

C₂×Q₈×D₇

yes

—

154

C₂³

C₂⁴×D₇

yes

characteristic

155

Q₈

—

156

Q₈

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₇

C₂×#2840

yes

characteristic, complemented, Sylow

161

C₂²

—

162

C₂²

—

163

C₂²

—

164

C₂²

—

165

C₂²

—

166

C₂²

—

167

C₂²

—

168

C₂²

—

169

C₂²

—

170

C₂²

C₂²×Q₈×D₇

yes

characteristic

171

C₂²

D₇×H(5,𝔽₂)

yes

—

172

C₄

—

173

C₄

—

174

C₂

—

175

C₂

—

176

C₂

—

177

C₂

D₇×#543

yes

—

178

C₂

D₇×H(3,𝔽₄)

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂⁴×D₇
commutator, socle, cosocle	C₂³×C₇	C₂⁵
Fitting	C₇×#2840	C₂
prime radical	C₂⁵×D₇	C₂²
hypercenter	#2840	D₇
hypocenter	C₇	C₂×#2840
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ #2840

subgroup

G/H

center, Frattini

C₂³

C₂⁴×D₇

commutator, socle, cosocle

C₂³×C₇

C₂⁵

Fitting

C₇×#2840

C₂

prime radical

C₂⁵×D₇

C₂²

D₇

C₇

G ▷ C₂³×C₇ ▷ C₇

C₁ ◁ C₂³ ◁ #2840

N	H ≅ G/N	type
1. C₇×#2840	C₂	semidirect
2. #11734333	C₂	semidirect
3. D₇×#393	C₂	semidirect
5. #11731289	C₂	semidirect
7. C₂²×Q₈×D₇	C₂²	semidirect
9. C₇×#393	C₂²	semidirect
12. #90003	C₂²	semidirect
14. #89719	C₂²	semidirect
15. #89445	C₂²	semidirect
16. D₇×#95	C₂²	semidirect
20. C₂⁴×D₇	Q₈	semidirect
23. C₂²×C₇×Q₈	C₂³	semidirect
26. C₂²×Dic₁₄	C₂³	semidirect
31. C₇×#95	C₂³	semidirect
32. C₇⋊#95	C₂³	semidirect
35. #2840	D₇	direct (N × H)
39. C₂⁴×C₇	C₂×Q₈	semidirect
46. C₂³×D₇	C₂×Q₈	semidirect
64. #393	C₂×D₇	semidirect
95. C₂²×Q₈	C₂²×D₇	semidirect
102. #95	C₂²×D₇	semidirect
125. C₂⁴	Q₈×D₇	semidirect
140. D₇	#2840	direct (N × H)
160. C₇	C₂×#2840	semidirect

H ≅ G/N

type

1. C₇×#2840

C₂

semidirect

2. #11734333

C₂

semidirect

3. D₇×#393

C₂

semidirect

5. #11731289

C₂

semidirect

7. C₂²×Q₈×D₇

C₂²

semidirect

9. C₇×#393

C₂²

semidirect

12. #90003

C₂²

semidirect

14. #89719

C₂²

semidirect

15. #89445

C₂²

semidirect

16. D₇×#95

C₂²

semidirect

20. C₂⁴×D₇

Q₈

semidirect

23. C₂²×C₇×Q₈

C₂³

semidirect

26. C₂²×Dic₁₄

C₂³

semidirect

31. C₇×#95

C₂³

semidirect

32. C₇⋊#95

C₂³

semidirect

35. #2840

D₇

direct (N × H)

39. C₂⁴×C₇

C₂×Q₈

semidirect

46. C₂³×D₇

C₂×Q₈

semidirect

64. #393

C₂×D₇

semidirect

95. C₂²×Q₈

C₂²×D₇

semidirect

102. #95

C₂²×D₇

semidirect

125. C₂⁴

Q₈×D₇

semidirect

140. D₇

#2840

direct (N × H)

160. C₇

C₂×#2840

semidirect

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

subgroup	index
C₇×#2840	2
#11734333	2
D₇×#393	2
D₇×#379	2
#11731289	2
C₂×#2840	7

subgroup

index

Subgroups — D₇×#2840 #13560202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D7×#2840 #13560202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₇×#2840 #13560202