Group #13101316 — subgroups

Subgroups — #13101316

Order 1600.

360 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 108 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	C₂×C₄×#1039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#681	C₂	1	2	yes	maximal
3	#11716613	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
5	C₂²×#1039	C₄	1	2	yes	—
6	C₄×#1039	C₄	1	1	yes	characteristic
7	C₄×#1039	C₄	1	1	yes	characteristic
8	C₄×#1039	C₂²	1	2	yes	—
9	C₅×#681	—	2	4	no	—
10	C₂×C₅×#168	C₂²	1	4	yes	—
11	#65407	—	5	2	no	maximal
12	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₅²	D₄	1	2	yes	—
15	C₂×C₄×C₅²	Q₈	1	2	yes	—
16	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
17	C₂×#1039	—	2	4	no	—
18	C₂×#1039	C₂×C₄	1	2	yes	—
19	C₂×#1039	C₂×C₄	1	2	yes	—
20	C₅×#168	—	2	4	no	—
21	C₂×C₅×M₁₆	—	5	2	no	—
22	C₂×C₄×Dic₅	—	5	2	no	—
23	C₂×C₄×Dic₅	—	10	2	no	—
24	C₂×#681	D₅	1	2	yes	—
25	#3946	—	5	8	no	—
26	C₂²×C₅²	—	2	2	no	—
27	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
28	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅²	C₄⋊C₄	1	2	yes	—
30	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
31	C₄×C₅²	C₂×Q₈	1	1	yes	characteristic
32	#1039	—	2	2	no	—
33	#1039	M₁₆	1	4	yes	complemented
34	C₂²×C₄×C₅	—	2	2	no	—
35	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
36	C₂²×Dic₅	—	5	4	no	—
37	C₂²×Dic₅	—	10	4	no	—
38	C₅×M₁₆	—	10	4	no	—
39	C₂×C₅×C₈	—	5	4	no	—
40	C₄×Dic₅	—	5	2	no	—
41	C₄×Dic₅	—	5	2	no	—
42	C₄×Dic₅	—	5	4	no	—
43	C₄×Dic₅	—	10	2	no	—
44	C₄×Dic₅	—	10	2	no	—
45	C₄×Dic₅	—	10	4	no	—
46	#681	—	2	4	no	—
47	C₂×#168	C₂×D₅	1	4	yes	—
48	#423	—	25	1	no	Sylow
49	C₂×C₅²	—	2	2	no	—
50	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅²	C₂×M₁₆	1	2	yes	—
52	C₂³×C₅	—	2	2	no	—
53	C₂³×C₅	C₄×D₅	1	2	yes	—
54	C₂×C₄×C₅	—	2	2	no	—
55	C₂×C₄×C₅	—	2	4	no	—
56	C₂×C₄×C₅	—	2	4	no	—
57	C₂×C₄×C₅	—	2	2	no	—
58	C₂×C₄×C₅	Dic₁₀	1	4	yes	—
59	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
60	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
61	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
62	C₂×Dic₅	—	5	4	no	—
63	C₂×Dic₅	—	5	4	no	—
64	C₂×Dic₅	—	10	4	no	—
65	C₂×Dic₅	—	10	4	no	—
66	C₂×Dic₅	—	10	8	no	—
67	C₂×Dic₅	—	10	16	no	—
68	C₅×C₈	—	10	4	no	—
69	#168	—	2	4	no	—
70	C₂×M₁₆	—	5	2	no	—
71	C₂×C₄²	—	25	1	no	—
72	C₄⋊C₈	—	25	4	no	—
73	C₅²	#423	1	1	yes	characteristic, complemented, Sylow
74	C₂²×C₅	—	2	8	no	—
75	C₂²×C₅	—	2	4	no	—
76	C₂²×C₅	—	2	2	no	—
77	C₂²×C₅	—	2	4	no	—
78	C₂²×C₅	#683	1	4	yes	—
79	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
80	C₄×C₅	—	2	2	no	—
81	C₄×C₅	—	2	2	no	—
82	C₄×C₅	—	2	4	no	—
83	C₄×C₅	#683	1	4	yes	—
84	C₄×C₅	C₂×Dic₁₀	1	2	yes	—
85	C₄×C₅	C₂×C₂²⋊D₅	1	2	yes	—
86	Dic₅	—	5	8	no	—
87	Dic₅	—	10	8	no	—
88	Dic₅	—	10	8	no	—
89	Dic₅	—	10	4	no	—
90	C₂²×C₄	—	25	2	no	—
91	C₂²×C₄	D₅²	1	1	yes	characteristic
92	M₁₆	—	10	4	no	—
93	C₂×C₈	—	5	4	no	—
94	C₄²	—	25	1	no	—
95	C₄²	—	25	1	no	—
96	C₄²	—	25	2	no	—
97	C₂×C₅	—	2	8	no	—
98	C₂×C₅	—	2	4	no	—
99	C₂×C₅	—	2	4	no	—
100	C₂×C₅	—	2	2	no	—
101	C₂×C₅	D₅×M₁₆	1	4	yes	—
102	C₂×C₅	C₂×#683	1	2	yes	—
103	C₂³	#6036	1	1	yes	characteristic
104	C₂×C₄	—	25	2	no	—
105	C₂×C₄	—	25	2	no	—
106	C₂×C₄	—	50	4	no	—
107	C₂×C₄	#6036	1	1	yes	characteristic
108	C₂×C₄	#6037	1	2	yes	—
109	C₂×C₄	#6039	1	2	yes	—
110	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
111	C₈	—	10	4	no	—
112	C₅	—	2	2	no	—
113	C₅	#65407	1	2	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#87781	1	2	yes	—
116	C₂²	C₂×#6036	1	1	yes	characteristic
117	C₄	—	25	4	no	—
118	C₄	—	50	2	no	—
119	C₄	#87781	1	2	yes	—
120	C₄	C₂×#6037	1	1	yes	characteristic
121	C₄	C₂×#6039	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#11716936	1	2	yes	—
124	C₂	C₂×#87781	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#1039

C₂

yes

maximal, characteristic

C₂×C₅×#681

C₂

yes

maximal

#11716613

C₂

yes

maximal, complemented

C₂²×C₄×C₅²

C₂²

yes

characteristic

C₂²×#1039

C₄

yes

—

C₄×#1039

C₄

yes

characteristic

C₄×#1039

C₄

yes

characteristic

C₄×#1039

C₂²

yes

—

C₅×#681

—

C₂×C₅×#168

C₂²

yes

—

#65407

—

maximal

C₂³×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

C₂×C₄

yes

characteristic

C₂×C₄×C₅²

D₄

yes

—

C₂×C₄×C₅²

Q₈

yes

—

C₂×C₄×C₅²

C₂³

yes

characteristic

C₂×#1039

—

C₂×#1039

C₂×C₄

yes

—

C₂×#1039

C₂×C₄

yes

—

C₅×#168

—

C₂×C₅×M₁₆

—

C₂×C₄×Dic₅

—

C₂×C₄×Dic₅

—

C₂×#681

D₅

yes

—

#3946

—

C₂²×C₅²

—

C₂²×C₅²

C₄⋊C₄

yes

—

C₂²×C₅²

C₂²×C₄

yes

characteristic

C₄×C₅²

C₄⋊C₄

yes

—

C₄×C₅²

C₂×D₄

yes

characteristic

C₄×C₅²

C₂×Q₈

yes

characteristic

#1039

—

#1039

M₁₆

yes

complemented

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₅

yes

—

C₂²×Dic₅

—

C₂²×Dic₅

—

C₅×M₁₆

—

C₂×C₅×C₈

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

C₄×Dic₅

—

#681

—

C₂×#168

C₂×D₅

yes

—

#423

—

Sylow

C₂×C₅²

—

C₂×C₅²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₅²

C₂×M₁₆

yes

—

C₂³×C₅

—

C₂³×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

Dic₁₀

yes

—

C₂×C₄×C₅

C₄×D₅

yes

—

C₂×C₄×C₅

C₂²⋊D₅

yes

—

C₂×C₄×C₅

C₂²×D₅

yes

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₂×Dic₅

—

C₅×C₈

—

#168

—

C₂×M₁₆

—

C₂×C₄²

—

C₄⋊C₈

—

C₅²

#423

yes

characteristic, complemented, Sylow

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#683

yes

—

C₂²×C₅

C₂×C₄×D₅

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

#683

yes

—

C₄×C₅

C₂×Dic₁₀

yes

—

C₄×C₅

C₂×C₂²⋊D₅

yes

—

Dic₅

—

Dic₅

—

Dic₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

D₅²

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₂×C₅

—

C₂×C₅

—

C₂×C₅

—

100

C₂×C₅

—

101

C₂×C₅

D₅×M₁₆

yes

—

102

C₂×C₅

C₂×#683

yes

—

103

C₂³

#6036

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#6036

yes

characteristic

108

C₂×C₄

#6037

yes

—

109

C₂×C₄

#6039

yes

—

110

C₂×C₄

C₂×D₅²

yes

characteristic

111

C₈

—

112

C₅

—

113

C₅

#65407

yes

complemented

114

C₂²

—

115

C₂²

#87781

yes

—

116

C₂²

C₂×#6036

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#87781

yes

—

120

C₄

C₂×#6037

yes

characteristic

121

C₄

C₂×#6039

yes

characteristic

122

C₂

—

123

C₂

#11716936

yes

—

124

C₂

C₂×#87781

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₅²
commutator, socle	C₂²×C₅²	C₂²×C₄
Fitting	C₂²×C₄×C₅²	C₂²
prime radical	C₂³×C₅²	C₂×C₄
cosocle	C₂×C₄×C₅²	C₂³
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	#423
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂×D₅²

commutator, socle

C₂²×C₅²

C₂²×C₄

Fitting

C₂²×C₄×C₅²

C₂²

prime radical

C₂³×C₅²

C₂×C₄

cosocle

C₂×C₄×C₅²

C₂³

hypercenter

C₂²×C₄

D₅²

C₅²

G ▷ C₂²×C₅² ▷ C₅²

C₁ ◁ C₂×C₄ ◁ C₂²×C₄

N	H ≅ G/N	type
3. #11716613	C₂	semidirect
33. #1039	M₁₆	semidirect
73. C₅²	#423	semidirect
113. C₅	#65407	semidirect

H ≅ G/N

type

3. #11716613

C₂

semidirect

33. #1039

M₁₆

semidirect

73. C₅²

#423

semidirect

113. C₅

#65407

semidirect

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

subgroup	index
C₂×C₄×#1039	2
C₂×C₅×#681	2
#11716613	2
#65407	5

subgroup

index

C₂×C₄×#1039

C₂×C₅×#681

#11716613

#65407

Subgroups — #13101316

Conjugacy classes of subgroups

Complements

Maximal subgroups