Group #13107217 — subgroups

Subgroups — #13107217

Order 1600.

888 conjugacy classes of subgroups; only the 239 normal subgroups are shown.

normal subgroups only

This group has 888 conjugacy classes of subgroups — too many to draw. Only its 239 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	C₅²×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717216	C₂	1	4	yes	maximal
3	#11717212	C₂	1	2	yes	maximal
4	#11717208	C₂	1	2	yes	maximal, complemented
5	#11717207	C₂	1	4	yes	maximal, complemented
6	C₄²×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic
7	#11717190	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
9	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
10	C₂×#6045	C₂²	1	4	yes	—
11	C₅²×C₄⋊C₄	C₂²	1	4	yes	—
12	C₅²×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄²×C₅²	C₂²	1	1	yes	characteristic
14	#87808	C₂²	1	4	yes	—
15	C₄⋊#1039	C₂²	1	2	yes	—
16	#87806	C₂²	1	4	yes	—
17	#87806	C₂²	1	4	yes	—
18	C₄×#1039	C₂²	1	2	yes	—
19	C₄×#1039	C₂²	1	1	yes	characteristic
20	C₄×#1039	C₂²	1	4	yes	—
21	#66022	—	5	6	no	maximal
22	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
23	C₂×C₄×C₅²	C₂³	1	4	yes	—
24	C₂×C₄×C₅²	C₂³	1	2	yes	—
25	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
26	C₂×#1039	C₂³	1	4	yes	—
27	C₂×#1039	C₂³	1	1	yes	characteristic
28	C₂×#1039	C₂³	1	2	yes	—
29	C₄×C₅⋊D₅	—	2	4	no	—
30	C₄×C₅⋊D₅	Q₈	1	4	yes	complemented
31	#6045	—	2	8	no	—
32	C₅×#125	D₅	1	6	yes	complemented
33	#4042	—	5	24	no	—
34	#4038	—	5	12	no	—
35	#4034	—	5	12	no	—
36	#4033	—	5	24	no	—
37	C₄²×D₅	—	5	6	no	—
38	#4016	—	5	6	no	—
39	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
40	C₂×C₅⋊D₅	—	2	2	no	—
41	C₂×C₅⋊D₅	C₂×Q₈	1	2	yes	—
42	C₄×C₅²	—	2	4	no	—
43	C₄×C₅²	—	2	2	no	—
44	C₄×C₅²	C₂×Q₈	1	2	yes	—
45	#1039	—	2	4	no	—
46	#1039	C₂×Q₈	1	2	yes	—
47	#1039	#41	1	4	yes	—
48	C₂×C₄×D₅	—	5	6	no	—
49	C₂×C₄×D₅	—	5	12	no	—
50	C₂×Dic₁₀	—	5	24	no	—
51	C₅×C₄⋊C₄	C₂×D₅	1	24	yes	—
52	C₅×C₄⋊C₄	C₂×D₅	1	12	yes	—
53	C₄²×C₅	C₂×D₅	1	6	yes	—
54	#685	—	5	24	no	—
55	C₄⋊Dic₅	—	5	12	no	—
56	#683	—	5	24	no	—
57	#683	—	5	24	no	—
58	C₄×Dic₅	—	5	12	no	—
59	C₄×Dic₅	—	5	6	no	—
60	C₄×Dic₅	—	5	24	no	—
61	#533	—	25	1	no	Sylow
62	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₅²	C₂×#41	1	2	yes	—
64	C₅⋊D₅	—	2	2	no	—
65	C₂²×D₅	—	5	6	no	—
66	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
67	C₂×C₄×C₅	C₂²×D₅	1	24	yes	—
68	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
69	C₂×Dic₅	—	5	6	no	—
70	C₂×Dic₅	—	5	12	no	—
71	C₂×Dic₅	—	5	24	no	—
72	C₄×D₅	—	5	24	no	—
73	C₄×D₅	—	10	24	no	—
74	Dic₁₀	—	10	48	no	—
75	C₄⋊Q₈	—	25	2	no	—
76	#125	—	25	1	no	—
77	#125	C₅⋊D₅	1	1	yes	characteristic, complemented
78	C₂²⋊Q₈	—	25	4	no	—
79	C₄×Q₈	—	25	4	no	—
80	#117	—	25	2	no	—
81	C₂×C₄²	—	25	1	no	—
82	C₅²	#533	1	1	yes	characteristic, complemented, Sylow
83	C₂²×C₅	C₂³×D₅	1	6	yes	—
84	C₂×D₅	—	5	12	no	—
85	C₂×D₅	—	10	12	no	—
86	C₄×C₅	—	2	24	no	—
87	C₄×C₅	—	2	12	no	—
88	C₄×C₅	Q₈×D₅	1	12	yes	—
89	Dic₅	—	5	12	no	—
90	Dic₅	—	5	24	no	—
91	Dic₅	—	10	24	no	—
92	C₂×Q₈	—	25	4	no	—
93	C₂²×C₄	—	25	1	no	—
94	C₂²×C₄	—	25	2	no	—
95	C₄⋊C₄	—	25	2	no	—
96	C₄⋊C₄	—	25	4	no	—
97	C₄⋊C₄	—	25	4	no	—
98	C₄⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
99	C₄⋊C₄	C₂×C₅⋊D₅	1	4	yes	—
100	C₂²⋊C₄	—	25	4	no	—
101	C₄²	—	25	2	no	—
102	C₄²	—	25	1	no	—
103	C₄²	—	25	4	no	—
104	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
105	C₂×C₅	C₂×Q₈×D₅	1	6	yes	—
106	C₂×C₅	D₅×#41	1	12	yes	—
107	D₅	—	10	12	no	—
108	C₂³	—	25	1	no	—
109	Q₈	—	50	8	no	—
110	C₂×C₄	—	25	1	no	—
111	C₂×C₄	—	25	2	no	—
112	C₂×C₄	—	25	4	no	—
113	C₂×C₄	—	25	4	no	—
114	C₂×C₄	—	50	4	no	—
115	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
116	C₂×C₄	C₂²×C₅⋊D₅	1	4	yes	—
117	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
118	C₅	#66022	1	6	yes	complemented
119	C₂²	—	25	2	no	—
120	C₂²	—	50	2	no	—
121	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	25	2	no	—
125	C₄	—	25	4	no	—
126	C₄	—	50	4	no	—
127	C₄	Q₈×C₅⋊D₅	1	2	yes	—
128	C₂	—	50	2	no	—
129	C₂	C₂×Q₈×C₅⋊D₅	1	1	yes	characteristic
130	C₂	#41×C₅⋊D₅	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅²×#125

C₂

yes

maximal, characteristic, complemented

#11717216

C₂

yes

maximal

#11717212

C₂

yes

maximal

#11717208

C₂

yes

maximal, complemented

#11717207

C₂

yes

maximal, complemented

C₄²×C₅⋊D₅

C₂

yes

maximal, characteristic

#11717190

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₅⋊D₅

C₂²

yes

—

C₂×C₄×C₅⋊D₅

C₂²

yes

characteristic

C₂×#6045

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₅²×C₄⋊C₄

C₂²

yes

—

C₄²×C₅²

C₂²

yes

characteristic

#87808

C₂²

yes

—

C₄⋊#1039

C₂²

yes

—

#87806

C₂²

yes

—

#87806

C₂²

yes

—

C₄×#1039

C₂²

yes

—

C₄×#1039

C₂²

yes

characteristic

C₄×#1039

C₂²

yes

—

#66022

—

maximal

C₂²×C₅⋊D₅

C₂³

yes

characteristic

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₂×#1039

C₂³

yes

characteristic

C₂×#1039

C₂³

yes

—

C₄×C₅⋊D₅

—

C₄×C₅⋊D₅

Q₈

yes

complemented

#6045

—

C₅×#125

D₅

yes

—

—

—

—

C₄²×D₅

—

#4016

—

C₂²×C₅²

C₂⁴

yes

characteristic

C₂×C₅⋊D₅

—

C₂×C₅⋊D₅

C₂×Q₈

yes

—

C₄×C₅²

—

C₄×C₅²

—

C₄×C₅²

C₂×Q₈

yes

—

#1039

—

#1039

C₂×Q₈

yes

—

#1039

#41

yes

—

C₂×C₄×D₅

—

C₂×C₄×D₅

—

C₂×Dic₁₀

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₅×C₄⋊C₄

C₂×D₅

yes

—

C₄²×C₅

C₂×D₅

yes

—

#685

—

C₄⋊Dic₅

—

#683

—

#683

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#533

—

Sylow

C₂×C₅²

C₂²×Q₈

yes

characteristic

C₂×C₅²

C₂×#41

yes

—

C₅⋊D₅

—

C₂²×D₅

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₄×D₅

—

C₄×D₅

—

Dic₁₀

—

C₄⋊Q₈

—

#125

—

#125

C₅⋊D₅

yes

characteristic, complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₅²

#533

yes

characteristic, complemented, Sylow

C₂²×C₅

C₂³×D₅

yes

—

C₂×D₅

—

C₂×D₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

Q₈×D₅

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

C₄⋊C₄

C₂×C₅⋊D₅

yes

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×C₅⋊D₅

yes

characteristic

105

C₂×C₅

C₂×Q₈×D₅

yes

—

106

C₂×C₅

D₅×#41

yes

—

107

D₅

—

108

C₂³

—

109

Q₈

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂²×C₅⋊D₅

yes

—

116

C₂×C₄

C₂²×C₅⋊D₅

yes

—

117

C₂×C₄

C₂²×C₅⋊D₅

yes

characteristic

118

C₅

#66022

yes

complemented

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×C₅⋊D₅

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

Q₈×C₅⋊D₅

yes

—

128

C₂

—

129

C₂

C₂×Q₈×C₅⋊D₅

yes

characteristic

130

C₂

#41×C₅⋊D₅

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₅²×#125	C₂
prime radical	C₂²×C₅⋊D₅	C₂³
hypercenter	#125	C₅⋊D₅
hypocenter	C₅²	#533
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×C₅⋊D₅

commutator, socle, cosocle

C₂²×C₅²

C₂⁴

Fitting

C₅²×#125

C₂

prime radical

C₂²×C₅⋊D₅

C₂³

C₅⋊D₅

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. C₅²×#125	C₂	semidirect
4. #11717208	C₂	semidirect
5. #11717207	C₂	semidirect
7. #11717190	C₂	semidirect
30. C₄×C₅⋊D₅	Q₈	semidirect
32. C₅×#125	D₅	semidirect
77. #125	C₅⋊D₅	semidirect
82. C₅²	#533	semidirect
118. C₅	#66022	semidirect

H ≅ G/N

type

1. C₅²×#125

C₂

semidirect

4. #11717208

C₂

semidirect

5. #11717207

C₂

semidirect

7. #11717190

C₂

semidirect

30. C₄×C₅⋊D₅

Q₈

semidirect

32. C₅×#125

D₅

semidirect

77. #125

C₅⋊D₅

semidirect

82. C₅²

#533

semidirect

118. C₅

#66022

semidirect

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

subgroup	index
C₅²×#125	2
#11717216	2
#11717212	2
#11717208	2
#11717207	2
C₄²×C₅⋊D₅	2
#11717190	2
#66022	5

subgroup

index

C₅²×#125

C₄²×C₅⋊D₅

Subgroups — #13107217

Conjugacy classes of subgroups

Complements

Maximal subgroups