Group #13061790 — subgroups

Subgroups — #13061790

Order 1344.

440 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 131 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	C₃×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#10621379	C₂	1	4	yes	maximal, complemented
3	#10621378	C₂	1	4	yes	maximal, complemented
4	#10621377	C₂	1	4	yes	maximal, complemented
5	#10621358	C₂	1	2	yes	maximal, complemented
6	#90391	—	3	1	no	maximal, Hall
7	C₂×D₈₄	C₂²	1	2	yes	—
8	C₂×D₈₄	C₂²	1	4	yes	—
9	C₂×C₄×D₂₁	C₂²	1	4	yes	—
10	C₂×Dic₄₂	C₂²	1	2	yes	—
11	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
14	#66631	C₂²	1	4	yes	—
15	#66631	C₂²	1	4	yes	—
16	#66631	C₂²	1	4	yes	—
17	#66629	C₂²	1	4	yes	—
18	C₇×#125	D₃	1	1	yes	characteristic, complemented
19	#6459	—	3	4	no	—
20	#6458	—	3	4	no	—
21	#6457	—	3	4	no	—
22	#6438	—	3	2	no	—
23	#5700	—	7	1	no	maximal, Hall
24	C₂²×D₂₁	C₂³	1	4	yes	—
25	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
26	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
28	C₂×Dic₂₁	C₂³	1	4	yes	—
29	D₈₄	—	4	2	no	—
30	D₈₄	—	4	4	no	—
31	C₄×D₂₁	—	4	4	no	—
32	Dic₄₂	—	4	2	no	—
33	C₂×D₂₈	—	3	2	no	—
34	C₂×D₂₈	—	3	4	no	—
35	C₂×C₄×D₇	—	3	4	no	—
36	C₂×Dic₁₄	—	3	2	no	—
37	C₇×C₄⋊C₄	C₂×D₃	1	2	yes	—
38	C₇×C₄⋊C₄	C₂×D₃	1	4	yes	—
39	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
40	(C₂×D₇)⋊C₄	—	3	4	no	—
41	(C₂×D₇)⋊C₄	—	3	4	no	—
42	(C₂×D₇)⋊C₄	—	3	4	no	—
43	Dic₇⋊C₄	—	3	4	no	—
44	C₃×#125	D₇	1	1	yes	characteristic, complemented
45	#896	—	7	4	no	—
46	#895	—	7	4	no	—
47	#894	—	7	4	no	—
48	#875	—	7	2	no	—
49	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
50	C₂×D₂₁	—	2	4	no	—
51	C₂×D₂₁	—	2	4	no	—
52	C₂×D₂₁	—	2	4	no	—
53	C₃×C₄×C₇	—	2	2	no	—
54	C₃×C₄×C₇	—	2	4	no	—
55	C₃×C₄×C₇	—	2	1	no	—
56	Dic₂₁	—	2	4	no	—
57	#562	—	21	1	no	Sylow
58	C₂²×D₇	—	3	4	no	—
59	C₂×C₄×C₇	C₂²×D₃	1	4	yes	—
60	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
62	C₂×Dic₇	—	3	4	no	—
63	D₂₈	—	12	2	no	—
64	D₂₈	—	12	4	no	—
65	C₄×D₇	—	12	4	no	—
66	Dic₁₄	—	12	2	no	—
67	C₂×D₁₂	—	7	2	no	—
68	C₂×D₁₂	—	7	4	no	—
69	C₂×C₄×D₃	—	7	4	no	—
70	C₂×Dic₆	—	7	2	no	—
71	C₃×C₄⋊C₄	C₂×D₇	1	2	yes	—
72	C₃×C₄⋊C₄	C₂×D₇	1	4	yes	—
73	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
74	(C₂×D₃)⋊C₄	—	7	4	no	—
75	(C₂×D₃)⋊C₄	—	7	4	no	—
76	(C₂×D₃)⋊C₄	—	7	4	no	—
77	Dic₃⋊C₄	—	7	4	no	—
78	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
79	C₂×C₃×C₇	2₋¹⁺⁴	1	1	yes	characteristic
80	D₂₁	—	4	4	no	—
81	#125	D₂₁	1	1	yes	characteristic, complemented
82	#124	—	21	2	no	—
83	#123	—	21	4	no	—
84	C₂²⋊Q₈	—	21	4	no	—
85	#121	—	21	4	no	—
86	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
87	C₂×D₇	—	6	4	no	—
88	C₂×D₇	—	6	4	no	—
89	C₂×D₇	—	6	4	no	—
90	C₄×C₇	—	2	1	no	—
91	C₄×C₇	—	2	2	no	—
92	C₄×C₇	—	2	4	no	—
93	Dic₇	—	6	4	no	—
94	C₂²×D₃	—	7	4	no	—
95	C₂×C₃×C₄	C₂²×D₇	1	4	yes	—
96	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
98	C₂×Dic₃	—	7	4	no	—
99	D₁₂	—	28	2	no	—
100	D₁₂	—	28	4	no	—
101	C₄×D₃	—	28	4	no	—
102	Dic₆	—	28	2	no	—
103	C₃×C₇	#562	1	1	yes	characteristic, complemented, Hall
104	C₂×Q₈	—	21	2	no	—
105	C₂×D₄	—	21	4	no	—
106	C₂×D₄	—	21	2	no	—
107	C₂²×C₄	—	21	4	no	—
108	C₄⋊C₄	—	21	4	no	—
109	C₄⋊C₄	C₂×D₂₁	1	4	yes	—
110	C₄⋊C₄	C₂×D₂₁	1	2	yes	—
111	C₂²⋊C₄	—	21	4	no	—
112	C₂²⋊C₄	—	21	4	no	—
113	C₂²⋊C₄	—	21	4	no	—
114	C₄²	C₂×D₂₁	1	1	yes	characteristic
115	C₂×C₇	#1007	1	1	yes	characteristic
116	C₂×C₇	#1009	1	2	yes	—
117	D₇	—	12	4	no	—
118	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
119	C₂×D₃	—	14	4	no	—
120	C₂×D₃	—	14	4	no	—
121	C₂×D₃	—	14	4	no	—
122	C₃×C₄	—	2	1	no	—
123	C₃×C₄	—	2	2	no	—
124	C₃×C₄	—	2	4	no	—
125	Dic₃	—	14	4	no	—
126	C₂³	—	21	4	no	—
127	Q₈	—	84	2	no	—
128	D₄	—	84	4	no	—
129	D₄	—	84	2	no	—
130	C₂×C₄	—	21	4	no	—
131	C₂×C₄	—	84	4	no	—
132	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₂₁	1	2	yes	—
134	C₂×C₄	C₂²×D₂₁	1	4	yes	—
135	C₇	#5700	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₃	#6551	1	1	yes	characteristic
137	C₂×C₃	#6553	1	2	yes	—
138	D₃	—	28	4	no	—
139	C₂²	—	42	4	no	—
140	C₂²	—	42	4	no	—
141	C₂²	—	42	4	no	—
142	C₂²	C₂³×D₂₁	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	4	no	—
145	C₄	—	2	2	no	—
146	C₄	—	42	4	no	—
147	C₃	#90391	1	1	yes	characteristic, complemented, Sylow
148	C₂	—	84	4	no	—
149	C₂	#10621660	1	1	yes	characteristic
150	C₂	#10621662	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₇×#125

C₂

yes

maximal, characteristic, complemented

#10621379

C₂

yes

maximal, complemented

#10621378

C₂

yes

maximal, complemented

#10621377

C₂

yes

maximal, complemented

#10621358

C₂

yes

maximal, complemented

#90391

—

maximal, Hall

C₂×D₈₄

C₂²

yes

—

C₂×D₈₄

C₂²

yes

—

C₂×C₄×D₂₁

C₂²

yes

—

C₂×Dic₄₂

C₂²

yes

—

C₃×C₇×C₄⋊C₄

C₂²

yes

—

C₃×C₇×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₇

C₂²

yes

characteristic

#66631

C₂²

yes

—

#66631

C₂²

yes

—

#66631

C₂²

yes

—

#66629

C₂²

yes

—

C₇×#125

D₃

yes

characteristic, complemented

—

—

—

—

—

C₂²×D₂₁

C₂³

yes

—

C₂×C₃×C₄×C₇

C₂³

yes

—

C₂×C₃×C₄×C₇

C₂³

yes

characteristic

C₂×C₃×C₄×C₇

C₂³

yes

—

C₂×Dic₂₁

C₂³

yes

—

D₈₄

—

D₈₄

—

C₄×D₂₁

—

Dic₄₂

—

C₂×D₂₈

—

C₂×D₂₈

—

C₂×C₄×D₇

—

C₂×Dic₁₄

—

C₇×C₄⋊C₄

C₂×D₃

yes

—

C₇×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₇

C₂×D₃

yes

characteristic

(C₂×D₇)⋊C₄

—

(C₂×D₇)⋊C₄

—

(C₂×D₇)⋊C₄

—

Dic₇⋊C₄

—

C₃×#125

D₇

yes

characteristic, complemented

—

—

—

—

C₂²×C₃×C₇

C₂⁴

yes

characteristic

C₂×D₂₁

—

C₂×D₂₁

—

C₂×D₂₁

—

C₃×C₄×C₇

—

C₃×C₄×C₇

—

C₃×C₄×C₇

—

Dic₂₁

—

#562

—

Sylow

C₂²×D₇

—

C₂×C₄×C₇

C₂²×D₃

yes

—

C₂×C₄×C₇

C₂²×D₃

yes

characteristic

C₂×C₄×C₇

C₂²×D₃

yes

—

C₂×Dic₇

—

D₂₈

—

D₂₈

—

C₄×D₇

—

Dic₁₄

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×D₇

yes

—

C₃×C₄⋊C₄

C₂×D₇

yes

—

C₃×C₄²

C₂×D₇

yes

characteristic

(C₂×D₃)⋊C₄

—

(C₂×D₃)⋊C₄

—

(C₂×D₃)⋊C₄

—

Dic₃⋊C₄

—

C₂×C₃×C₇

H(5,𝔽₂)

yes

—

C₂×C₃×C₇

2₋¹⁺⁴

yes

characteristic

D₂₁

—

#125

D₂₁

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

#121

—

C₂²×C₇

C₂³×D₃

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

C₂²×D₃

—

C₂×C₃×C₄

C₂²×D₇

yes

—

C₂×C₃×C₄

C₂²×D₇

yes

characteristic

C₂×C₃×C₄

C₂²×D₇

yes

—

C₂×Dic₃

—

D₁₂

—

100

D₁₂

—

101

C₄×D₃

—

102

Dic₆

—

103

C₃×C₇

#562

yes

characteristic, complemented, Hall

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×D₂₁

yes

—

110

C₄⋊C₄

C₂×D₂₁

yes

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

C₂×D₂₁

yes

characteristic

115

C₂×C₇

#1007

yes

characteristic

116

C₂×C₇

#1009

yes

—

117

D₇

—

118

C₂²×C₃

C₂³×D₇

yes

characteristic

119

C₂×D₃

—

120

C₂×D₃

—

121

C₂×D₃

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

—

125

Dic₃

—

126

C₂³

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²×D₂₁

yes

characteristic

133

C₂×C₄

C₂²×D₂₁

yes

—

134

C₂×C₄

C₂²×D₂₁

yes

—

135

C₇

#5700

yes

characteristic, complemented, Sylow

136

C₂×C₃

#6551

yes

characteristic

137

C₂×C₃

#6553

yes

—

138

D₃

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×D₂₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₃

#90391

yes

characteristic, complemented, Sylow

148

C₂

—

149

C₂

#10621660

yes

characteristic

150

C₂

#10621662

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁
commutator, socle, cosocle	C₂²×C₃×C₇	C₂⁴
Fitting	C₃×C₇×#125	C₂
hypercenter	#125	D₂₁
hypocenter	C₃×C₇	#562
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₁

commutator, socle, cosocle

C₂²×C₃×C₇

C₂⁴

Fitting

C₃×C₇×#125

C₂

D₂₁

C₃×C₇

G ▷ C₂²×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₃×C₇×#125	C₂	semidirect
2. #10621379	C₂	semidirect
3. #10621378	C₂	semidirect
4. #10621377	C₂	semidirect
5. #10621358	C₂	semidirect
18. C₇×#125	D₃	semidirect
44. C₃×#125	D₇	semidirect
81. #125	D₂₁	semidirect
103. C₃×C₇	#562	semidirect
135. C₇	#5700	semidirect
147. C₃	#90391	semidirect

H ≅ G/N

type

1. C₃×C₇×#125

C₂

semidirect

2. #10621379

C₂

semidirect

3. #10621378

C₂

semidirect

4. #10621377

C₂

semidirect

5. #10621358

C₂

semidirect

18. C₇×#125

D₃

semidirect

44. C₃×#125

D₇

semidirect

81. #125

D₂₁

semidirect

103. C₃×C₇

#562

semidirect

135. C₇

#5700

semidirect

147. C₃

#90391

semidirect

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

subgroup	index
C₃×C₇×#125	2
#10621379	2
#10621378	2
#10621377	2
#10621358	2
#90391	3
#5700	7

subgroup

index

C₃×C₇×#125

Subgroups — #13061790

Conjugacy classes of subgroups

Complements

Maximal subgroups