Group #13057434 — subgroups

Subgroups — #90391⋊C₃ #13057434

Order 1344.

440 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 162 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	ΓL(1,𝔽₈)×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#6459⋊C₃	C₂	1	4	yes	maximal, complemented
3	#6458⋊C₃	C₂	1	4	yes	maximal, complemented
4	#6457⋊C₃	C₂	1	4	yes	maximal, complemented
5	#6438⋊C₃	C₂	1	2	yes	maximal, complemented
6	#90391	C₃	1	1	yes	maximal, characteristic, complemented, Hall
7	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	2	yes	—
8	C₂×C₄⋊AGL(1,𝔽₇)	C₂²	1	4	yes	—
9	C₂×C₄×AGL(1,𝔽₇)	C₂²	1	4	yes	—
10	C₂×ΓL(1,𝔽₈)⋊Q₈	C₂²	1	2	yes	—
11	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	2	yes	—
12	C₄⋊C₄×ΓL(1,𝔽₈)	C₂²	1	4	yes	—
13	C₄²×ΓL(1,𝔽₈)	C₂²	1	1	yes	characteristic
14	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
15	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
16	(C₂×AGL(1,𝔽₇))⋊C₄	C₂²	1	4	yes	—
17	(ΓL(1,𝔽₈)⋊C₄)⋊C₄	C₂²	1	4	yes	—
18	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
19	#6459	C₂×C₃	1	4	yes	complemented
20	#6458	C₂×C₃	1	4	yes	complemented
21	#6457	C₂×C₃	1	4	yes	complemented
22	#6438	C₂×C₃	1	2	yes	complemented
23	C₃×#562	—	7	1	no	maximal, Hall
24	C₂²×AGL(1,𝔽₇)	C₂³	1	4	yes	—
25	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	2	yes	—
26	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	1	yes	characteristic
27	C₂×C₄×ΓL(1,𝔽₈)	C₂³	1	4	yes	—
28	C₂×ΓL(1,𝔽₈)⋊C₄	C₂³	1	4	yes	—
29	C₄⋊AGL(1,𝔽₇)	—	4	2	no	—
30	C₄⋊AGL(1,𝔽₇)	—	4	4	no	—
31	C₄×AGL(1,𝔽₇)	—	4	4	no	—
32	ΓL(1,𝔽₈)⋊Q₈	—	4	2	no	—
33	C₂×D₂₈	C₂²×C₃	1	2	yes	—
34	C₂×D₂₈	C₂²×C₃	1	4	yes	—
35	C₂×C₄×D₇	C₂²×C₃	1	4	yes	—
36	C₂×Dic₁₄	C₂²×C₃	1	2	yes	—
37	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
38	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
39	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
40	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
41	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
42	(C₂×D₇)⋊C₄	C₂²×C₃	1	4	yes	—
43	Dic₇⋊C₄	C₂²×C₃	1	4	yes	—
44	C₃×#125	—	7	1	no	—
45	C₃×#124	—	7	2	no	—
46	C₃×#123	—	7	4	no	—
47	C₃×C₂²⋊Q₈	—	7	4	no	—
48	C₃×#121	—	7	4	no	—
49	C₂²×ΓL(1,𝔽₈)	C₂⁴	1	1	yes	characteristic
50	C₂×AGL(1,𝔽₇)	—	2	4	no	—
51	C₂×AGL(1,𝔽₇)	—	2	4	no	—
52	C₂×AGL(1,𝔽₇)	—	2	4	no	—
53	C₄×ΓL(1,𝔽₈)	—	2	2	no	—
54	C₄×ΓL(1,𝔽₈)	—	2	4	no	—
55	C₄×ΓL(1,𝔽₈)	—	2	1	no	—
56	ΓL(1,𝔽₈)⋊C₄	—	2	4	no	—
57	#562	—	7	1	no	Sylow
58	C₂²×D₇	C₂³×C₃	1	4	yes	—
59	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
60	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
61	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
62	C₂×Dic₇	C₂³×C₃	1	4	yes	—
63	D₂₈	—	4	2	no	—
64	D₂₈	—	4	4	no	—
65	C₄×D₇	—	4	4	no	—
66	Dic₁₄	—	4	2	no	—
67	C₂×C₃×Q₈	—	7	2	no	—
68	C₂×C₃×D₄	—	7	4	no	—
69	C₂×C₃×D₄	—	7	2	no	—
70	C₂²×C₃×C₄	—	7	4	no	—
71	C₃×C₄⋊C₄	—	7	4	no	—
72	C₃×C₄⋊C₄	—	7	4	no	—
73	C₃×C₄⋊C₄	—	7	2	no	—
74	C₃×C₂²⋊C₄	—	7	4	no	—
75	C₃×C₂²⋊C₄	—	7	4	no	—
76	C₃×C₂²⋊C₄	—	7	4	no	—
77	C₃×C₄²	—	7	1	no	—
78	C₂×ΓL(1,𝔽₈)	H(5,𝔽₂)	1	2	yes	—
79	C₂×ΓL(1,𝔽₈)	2₋¹⁺⁴	1	1	yes	characteristic
80	AGL(1,𝔽₇)	—	4	4	no	—
81	#125	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
82	#124	—	7	2	no	—
83	#123	—	7	4	no	—
84	C₂²⋊Q₈	—	7	4	no	—
85	#121	—	7	4	no	—
86	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
87	C₂×D₇	—	2	4	no	—
88	C₂×D₇	—	2	4	no	—
89	C₂×D₇	—	2	4	no	—
90	C₄×C₇	—	2	2	no	—
91	C₄×C₇	—	2	4	no	—
92	C₄×C₇	—	2	1	no	—
93	Dic₇	—	2	4	no	—
94	C₂³×C₃	—	7	4	no	—
95	C₃×Q₈	—	28	2	no	—
96	C₃×D₄	—	28	4	no	—
97	C₃×D₄	—	28	2	no	—
98	C₂×C₃×C₄	—	7	4	no	—
99	C₂×C₃×C₄	—	7	1	no	—
100	C₂×C₃×C₄	—	7	2	no	—
101	C₂×C₃×C₄	—	7	4	no	—
102	C₂×C₃×C₄	—	28	4	no	—
103	ΓL(1,𝔽₈)	#562	1	1	yes	characteristic, complemented, Hall
104	C₂×Q₈	—	7	2	no	—
105	C₂×D₄	—	7	4	no	—
106	C₂×D₄	—	7	2	no	—
107	C₂²×C₄	—	7	4	no	—
108	C₄⋊C₄	—	7	4	no	—
109	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	4	yes	—
110	C₄⋊C₄	C₂×AGL(1,𝔽₇)	1	2	yes	—
111	C₂²⋊C₄	—	7	4	no	—
112	C₂²⋊C₄	—	7	4	no	—
113	C₂²⋊C₄	—	7	4	no	—
114	C₄²	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic
115	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
116	C₂×C₇	C₃×2₋¹⁺⁴	1	1	yes	characteristic
117	D₇	—	4	4	no	—
118	C₂²×C₃	—	7	1	no	—
119	C₂²×C₃	—	14	4	no	—
120	C₂²×C₃	—	14	4	no	—
121	C₂²×C₃	—	14	4	no	—
122	C₃×C₄	—	14	4	no	—
123	C₃×C₄	—	14	2	no	—
124	C₃×C₄	—	14	1	no	—
125	C₃×C₄	—	14	4	no	—
126	C₂³	—	7	4	no	—
127	Q₈	—	28	2	no	—
128	D₄	—	28	4	no	—
129	D₄	—	28	2	no	—
130	C₂×C₄	—	7	4	no	—
131	C₂×C₄	—	28	4	no	—
132	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	1	yes	characteristic
133	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	2	yes	—
134	C₂×C₄	C₂²×AGL(1,𝔽₇)	1	4	yes	—
135	C₇	C₃×#562	1	1	yes	characteristic, complemented, Sylow
136	C₂×C₃	—	7	1	no	—
137	C₂×C₃	—	7	2	no	—
138	C₂×C₃	—	28	4	no	—
139	C₂²	—	14	4	no	—
140	C₂²	—	14	4	no	—
141	C₂²	—	14	4	no	—
142	C₂²	C₂³×AGL(1,𝔽₇)	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	4	no	—
145	C₄	—	2	2	no	—
146	C₄	—	14	4	no	—
147	C₃	—	7	1	no	Sylow
148	C₂	—	28	4	no	—
149	C₂	#10621516	1	1	yes	characteristic
150	C₂	#10621518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

ΓL(1,𝔽₈)×#125

C₂

yes

maximal, characteristic, complemented

#6459⋊C₃

C₂

yes

maximal, complemented

#6458⋊C₃

C₂

yes

maximal, complemented

#6457⋊C₃

C₂

yes

maximal, complemented

#6438⋊C₃

C₂

yes

maximal, complemented

#90391

C₃

yes

maximal, characteristic, complemented, Hall

C₂×C₄⋊AGL(1,𝔽₇)

C₂²

yes

—

C₂×C₄⋊AGL(1,𝔽₇)

C₂²

yes

—

C₂×C₄×AGL(1,𝔽₇)

C₂²

yes

—

C₂×ΓL(1,𝔽₈)⋊Q₈

C₂²

yes

—

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄⋊C₄×ΓL(1,𝔽₈)

C₂²

yes

—

C₄²×ΓL(1,𝔽₈)

C₂²

yes

characteristic

(C₂×AGL(1,𝔽₇))⋊C₄

C₂²

yes

—

(C₂×AGL(1,𝔽₇))⋊C₄

C₂²

yes

—

(C₂×AGL(1,𝔽₇))⋊C₄

C₂²

yes

—

(ΓL(1,𝔽₈)⋊C₄)⋊C₄

C₂²

yes

—

C₇×#125

C₂×C₃

yes

characteristic, complemented

#6459

C₂×C₃

yes

complemented

#6458

C₂×C₃

yes

complemented

#6457

C₂×C₃

yes

complemented

#6438

C₂×C₃

yes

complemented

C₃×#562

—

maximal, Hall

C₂²×AGL(1,𝔽₇)

C₂³

yes

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

—

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

characteristic

C₂×C₄×ΓL(1,𝔽₈)

C₂³

yes

—

C₂×ΓL(1,𝔽₈)⋊C₄

C₂³

yes

—

—

—

—

—

C₂×D₂₈

C₂²×C₃

yes

—

C₂×D₂₈

C₂²×C₃

yes

—

C₂×C₄×D₇

C₂²×C₃

yes

—

C₂×Dic₁₄

C₂²×C₃

yes

—

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₇×C₄⋊C₄

C₂²×C₃

yes

—

C₄²×C₇

C₂²×C₃

yes

characteristic

(C₂×D₇)⋊C₄

C₂²×C₃

yes

—

(C₂×D₇)⋊C₄

C₂²×C₃

yes

—

(C₂×D₇)⋊C₄

C₂²×C₃

yes

—

Dic₇⋊C₄

C₂²×C₃

yes

—

C₃×#125

—

C₃×#124

—

C₃×#123

—

C₃×C₂²⋊Q₈

—

C₃×#121

—

C₂²×ΓL(1,𝔽₈)

C₂⁴

yes

—

—

—

—

—

—

—

—

C₂²×D₇

C₂³×C₃

yes

—

C₂×C₄×C₇

C₂³×C₃

yes

—

C₂×C₄×C₇

C₂³×C₃

yes

characteristic

C₂×C₄×C₇

C₂³×C₃

yes

—

C₂×Dic₇

C₂³×C₃

yes

—

D₂₈

—

D₂₈

—

C₄×D₇

—

Dic₁₄

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂²×C₃×C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₂×ΓL(1,𝔽₈)

H(5,𝔽₂)

yes

—

C₂×ΓL(1,𝔽₈)

2₋¹⁺⁴

yes

—

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

#121

—

C₂²×C₇

C₂⁴×C₃

yes

characteristic

C₂×D₇

—

C₂×D₇

—

C₂×D₇

—

C₄×C₇

—

C₄×C₇

—

C₄×C₇

—

Dic₇

—

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

—

103

ΓL(1,𝔽₈)

#562

yes

characteristic, complemented, Hall

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂²×C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×AGL(1,𝔽₇)

yes

—

110

C₄⋊C₄

C₂×AGL(1,𝔽₇)

yes

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

114

C₄²

C₂×AGL(1,𝔽₇)

yes

characteristic

115

C₂×C₇

C₃×H(5,𝔽₂)

yes

—

116

C₂×C₇

C₃×2₋¹⁺⁴

yes

characteristic

117

D₇

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

—

122

C₃×C₄

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₂³

—

127

Q₈

—

128

D₄

—

129

D₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

characteristic

133

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

—

134

C₂×C₄

C₂²×AGL(1,𝔽₇)

yes

—

135

C₇

C₃×#562

yes

characteristic, complemented, Sylow

136

C₂×C₃

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂³×AGL(1,𝔽₇)

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₃

—

Sylow

148

C₂

—

149

C₂

#10621516

yes

characteristic

150

C₂

#10621518

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁴×C₃
Fitting	C₇×#125	C₂×C₃
hypercenter	#125	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#562
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×AGL(1,𝔽₇)

commutator, socle, cosocle

C₂²×C₇

C₂⁴×C₃

C₂×C₃

C₇

G ▷ C₂²×C₇ ▷ C₇

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×#125	C₂	semidirect
2. #6459⋊C₃	C₂	semidirect
3. #6458⋊C₃	C₂	semidirect
4. #6457⋊C₃	C₂	semidirect
5. #6438⋊C₃	C₂	semidirect
6. #90391	C₃	semidirect
18. C₇×#125	C₂×C₃	semidirect
19. #6459	C₂×C₃	semidirect
20. #6458	C₂×C₃	semidirect
21. #6457	C₂×C₃	semidirect
22. #6438	C₂×C₃	semidirect
81. #125	AGL(1,𝔽₇)	semidirect
103. ΓL(1,𝔽₈)	#562	semidirect
135. C₇	C₃×#562	semidirect

H ≅ G/N

type

1. ΓL(1,𝔽₈)×#125

C₂

semidirect

2. #6459⋊C₃

C₂

semidirect

3. #6458⋊C₃

C₂

semidirect

4. #6457⋊C₃

C₂

semidirect

5. #6438⋊C₃

C₂

semidirect

6. #90391

C₃

semidirect

18. C₇×#125

C₂×C₃

semidirect

19. #6459

C₂×C₃

semidirect

20. #6458

C₂×C₃

semidirect

21. #6457

C₂×C₃

semidirect

22. #6438

C₂×C₃

semidirect

81. #125

AGL(1,𝔽₇)

semidirect

103. ΓL(1,𝔽₈)

#562

semidirect

135. C₇

C₃×#562

semidirect

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

subgroup	index
ΓL(1,𝔽₈)×#125	2
#6459⋊C₃	2
#6458⋊C₃	2
#6457⋊C₃	2
#6438⋊C₃	2
#90391	3
C₃×#562	7

subgroup

index

Subgroups — #90391⋊C₃ #13057434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90391⋊C3 #13057434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #90391⋊C₃ #13057434