Group #10604092 — subgroups

Subgroups — #10604092

Order 640.

184 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 66 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	#65729	C₂	1	2	yes	maximal
2	C₂×#3996	C₂	1	1	yes	maximal, characteristic
3	#65718	C₂	1	1	yes	maximal, characteristic
4	D₅×#108	C₂	1	1	yes	maximal, characteristic
5	#64902	C₂	1	2	yes	maximal
6	#4130	C₂²	1	1	yes	characteristic
7	D₅⋊M₁₆	C₂²	1	1	yes	characteristic
8	C₂×#700	C₂²	1	2	yes	—
9	D₅×M₁₆	C₂²	1	2	yes	—
10	C₂×C₈×D₅	C₂²	1	1	yes	characteristic
11	#3996	—	2	2	no	—
12	#3996	—	2	1	no	—
13	#3996	—	2	1	no	—
14	#3995	—	2	2	no	—
15	#3992	—	2	1	no	—
16	C₈×AGL(1,𝔽₅)	—	2	1	no	—
17	C₅×#108	C₄	1	1	yes	characteristic, complemented
18	#3954	C₄	1	2	yes	complemented
19	#3951	C₄	1	1	yes	characteristic, complemented
20	#1953	—	5	1	no	maximal, Sylow
21	C₂×C₄×D₅	C₂³	1	1	yes	characteristic
22	C₂²⋊AGL(1,𝔽₅)	Q₈	1	2	yes	—
23	#704	—	2	1	no	—
24	C₂×#170	—	2	2	no	—
25	C₄⋊AGL(1,𝔽₅)	—	2	1	no	—
26	C₄×AGL(1,𝔽₅)	—	2	1	no	—
27	#700	—	2	2	no	—
28	#700	—	2	2	no	—
29	#700	D₄	1	1	yes	characteristic
30	#700	D₄	1	1	yes	characteristic
31	D₅⋊C₈	—	2	1	no	—
32	C₅×M₁₆	C₂×C₄	1	2	yes	—
33	C₂×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	#681	C₂×C₄	1	2	yes	—
35	C₂×#168	C₂×C₄	1	1	yes	characteristic
36	#676	—	2	2	no	—
37	C₈×D₅	—	2	2	no	—
38	C₈×D₅	D₄	1	1	yes	characteristic
39	C₈×D₅	D₄	1	1	yes	characteristic
40	C₈×D₅	Q₈	1	1	yes	characteristic
41	C₈×D₅	Q₈	1	1	yes	characteristic
42	#430	—	5	2	no	—
43	C₂×#108	—	5	1	no	—
44	C₂×#108	—	5	1	no	—
45	#405	—	5	1	no	—
46	#344	—	5	2	no	—
47	C₂²×D₅	C₂×Q₈	1	1	yes	characteristic
48	C₂×AGL(1,𝔽₅)	—	2	2	no	—
49	C₂×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₅	#41	1	1	yes	characteristic
51	C₄×D₅	C₂×D₄	1	1	yes	characteristic
52	C₄×D₅	C₂×D₄	1	1	yes	characteristic
53	C₄×D₅	C₂×Q₈	1	1	yes	characteristic
54	C₄×D₅	#41	1	1	yes	characteristic
55	#170	—	2	1	no	—
56	#170	—	2	1	no	—
57	#170	—	4	2	no	—
58	C₅×C₈	—	2	2	no	—
59	C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
60	C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
61	#168	—	2	2	no	—
62	#168	C₄⋊C₄	1	1	yes	characteristic
63	#168	C₄⋊C₄	1	1	yes	characteristic
64	C₂×M₁₆	—	5	1	no	—
65	C₂×M₁₆	—	5	2	no	—
66	C₂×M₁₆	—	5	2	no	—
67	C₂²×C₈	—	5	1	no	—
68	#117	—	5	1	no	—
69	#108	—	5	2	no	—
70	#108	—	5	1	no	—
71	#108	—	10	1	no	—
72	#108	—	10	1	no	—
73	#108	—	10	2	no	—
74	#108	—	10	2	no	—
75	#108	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
76	#97	—	10	1	no	—
77	C₄×C₈	—	10	1	no	—
78	C₂²×C₅	C₄×Q₈	1	1	yes	characteristic
79	C₂×D₅	—	4	1	no	—
80	C₂×D₅	C₂²⋊Q₈	1	1	yes	characteristic
81	C₂×D₅	C₄⋊Q₈	1	1	yes	characteristic
82	AGL(1,𝔽₅)	—	4	2	no	—
83	C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₄×C₅	C₄×D₄	1	1	yes	characteristic
85	Dic₅	#121	1	1	yes	characteristic
86	Dic₅	#125	1	1	yes	characteristic
87	C₂²×C₄	—	5	1	no	—
88	M₁₆	—	5	1	no	—
89	M₁₆	—	5	1	no	—
90	M₁₆	—	5	2	no	—
91	M₁₆	—	10	2	no	—
92	M₁₆	—	10	1	no	—
93	M₁₆	—	10	2	no	—
94	M₁₆	—	10	2	no	—
95	M₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
96	C₂×C₈	—	5	1	no	—
97	C₂×C₈	—	5	1	no	—
98	C₂×C₈	—	5	1	no	—
99	C₂×C₈	—	5	1	no	—
100	C₂×C₈	—	5	1	no	—
101	C₂×C₈	—	10	2	no	—
102	C₂×C₈	—	10	1	no	—
103	C₂×C₈	—	10	2	no	—
104	C₂×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
105	C₄⋊C₄	—	10	1	no	—
106	C₂²⋊C₄	—	5	2	no	—
107	C₄²	—	10	1	no	—
108	C₂×C₅	—	2	1	no	—
109	C₂×C₅	#389	1	1	yes	characteristic
110	D₅	—	2	1	no	—
111	D₅	—	2	1	no	—
112	C₂³	—	5	1	no	—
113	C₂×C₄	—	5	1	no	—
114	C₂×C₄	—	5	1	no	—
115	C₂×C₄	—	5	1	no	—
116	C₂×C₄	—	5	1	no	—
117	C₂×C₄	—	5	1	no	—
118	C₂×C₄	—	10	2	no	—
119	C₂×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	5	1	no	—
122	C₈	—	5	1	no	—
123	C₈	—	10	1	no	—
124	C₈	—	10	1	no	—
125	C₈	—	10	2	no	—
126	C₈	—	20	2	no	—
127	C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
128	C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
129	C₅	#1953	1	1	yes	characteristic, complemented, Sylow
130	C₂²	—	5	1	no	—
131	C₂²	—	5	1	no	—
132	C₂²	—	20	1	no	—
133	C₂²	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
134	C₄	—	5	1	no	—
135	C₄	—	5	1	no	—
136	C₄	—	20	2	no	—
137	C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
138	C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	10	1	no	—
141	C₂	—	10	1	no	—
142	C₂	#65712	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#65729

C₂

yes

maximal

C₂×#3996

C₂

yes

maximal, characteristic

#65718

C₂

yes

maximal, characteristic

D₅×#108

C₂

yes

maximal, characteristic

#64902

C₂

yes

maximal

#4130

C₂²

yes

characteristic

D₅⋊M₁₆

C₂²

yes

characteristic

C₂×#700

C₂²

yes

—

D₅×M₁₆

C₂²

yes

—

C₂×C₈×D₅

C₂²

yes

—

—

—

—

—

—

C₄

yes

characteristic, complemented

#3954

C₄

yes

complemented

#3951

C₄

yes

characteristic, complemented

#1953

—

maximal, Sylow

C₂×C₄×D₅

C₂³

yes

characteristic

C₂²⋊AGL(1,𝔽₅)

Q₈

yes

—

—

—

—

—

—

—

D₄

yes

characteristic

#700

D₄

yes

characteristic

D₅⋊C₈

—

C₅×M₁₆

C₂×C₄

yes

—

C₂×C₅×C₈

C₂×C₄

yes

characteristic

#681

C₂×C₄

yes

—

C₂×#168

C₂×C₄

yes

characteristic

#676

—

C₈×D₅

—

C₈×D₅

D₄

yes

characteristic

C₈×D₅

D₄

yes

characteristic

C₈×D₅

Q₈

yes

characteristic

C₈×D₅

Q₈

yes

—

—

—

—

—

C₂²×D₅

C₂×Q₈

yes

characteristic

C₂×AGL(1,𝔽₅)

—

C₂×C₄×C₅

C₂²×C₄

yes

characteristic

C₂×Dic₅

#41

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×D₄

yes

characteristic

C₄×D₅

C₂×Q₈

yes

C₄×D₅

yes

—

—

—

C₅×C₈

—

C₅×C₈

C₄⋊C₄

yes

characteristic

C₅×C₈

C₄⋊C₄

yes

characteristic

#168

—

#168

C₄⋊C₄

yes

characteristic

#168

C₄⋊C₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

—

—

—

—

—

—

—

yes

characteristic, complemented

#97

—

C₄×C₈

—

C₂²×C₅

C₄×Q₈

yes

characteristic

C₂×D₅

—

C₂×D₅

C₂²⋊Q₈

yes

characteristic

C₂×D₅

C₄⋊Q₈

yes

characteristic

AGL(1,𝔽₅)

—

C₄×C₅

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅

C₄×D₄

yes

Dic₅

yes

Dic₅

yes

C₂²×C₄

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

C₂×AGL(1,𝔽₅)

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

C₂×AGL(1,𝔽₅)

yes

characteristic

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂×C₅

—

109

C₂×C₅

#389

yes

characteristic

110

D₅

—

111

D₅

—

112

C₂³

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×AGL(1,𝔽₅)

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

128

C₈

C₄⋊AGL(1,𝔽₅)

yes

characteristic

129

C₅

#1953

yes

characteristic, complemented, Sylow

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

Q₈×AGL(1,𝔽₅)

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

C₂×C₄⋊AGL(1,𝔽₅)

yes

characteristic

138

C₄

D₄×AGL(1,𝔽₅)

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

142

C₂

#65712

yes

characteristic

	subgroup	G/H
center	C₂	#65712
commutator	C₂×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₅×#108	C₄
socle	C₂×C₅	#389
prime radical	C₂²×D₅	C₂×Q₈
cosocle	C₂×C₄×D₅	C₂³
hypercenter	#108	AGL(1,𝔽₅)
hypocenter	C₅	#1953
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#65712

commutator

C₂×C₄×C₅

C₂²×C₄

Frattini

C₂×C₄

C₂²×AGL(1,𝔽₅)

C₄

C₂×C₅

C₂²×D₅

C₂×Q₈

cosocle

C₂×C₄×D₅

C₂³

C₅

G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. C₅×#108	C₄	semidirect
18. #3954	C₄	semidirect
19. #3951	C₄	semidirect
75. #108	AGL(1,𝔽₅)	semidirect
129. C₅	#1953	semidirect

H ≅ G/N

type

17. C₅×#108

C₄

semidirect

18. #3954

C₄

semidirect

19. #3951

C₄

semidirect

75. #108

AGL(1,𝔽₅)

semidirect

129. C₅

#1953

semidirect

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

subgroup	index
#65729	2
C₂×#3996	2
#65718	2
D₅×#108	2
#64902	2
#1953	5

subgroup

index

Subgroups — #10604092

Conjugacy classes of subgroups

Complements

Maximal subgroups