Group #12014034 — subgroups

Subgroups — #12014034

Order 1280.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₅×#1709	C₂	1	1	yes	maximal, characteristic
2	#10602105	C₂	1	2	yes	maximal, complemented
3	#10602084	C₂	1	2	yes	maximal
4	#10598858	C₂	1	1	yes	maximal, characteristic, complemented
5	#10597928	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#494	C₂²	1	1	yes	characteristic
7	C₅×#493	C₂²	1	1	yes	characteristic
8	C₅×#486	—	2	2	no	—
9	C₂×#3961	—	2	4	no	—
10	C₂×#3959	—	2	4	no	—
11	#64988	C₂²	1	2	yes	—
12	#64971	—	2	2	no	—
13	C₅×C₈²	C₂²	1	1	yes	characteristic
14	C₄×#672	C₂²	1	2	yes	—
15	#12520	—	5	1	no	maximal, Sylow
16	C₂×C₅×Q₁₆	—	2	2	no	—
17	C₂×C₅×D₈	—	2	2	no	—
18	C₅×C₄⋊Q₈	—	2	1	no	—
19	C₅×#127	—	2	1	no	—
20	C₂×Dic₂₀	—	2	2	no	—
21	C₂×Dic₂₀	—	2	2	no	—
22	#4015	—	2	2	no	—
23	C₅×#102	—	4	4	no	—
24	C₄×C₅×C₈	D₄	1	2	yes	—
25	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
26	#3961	—	4	8	no	—
27	#3959	—	4	8	no	—
28	#3950	—	4	4	no	—
29	C₂×#672	D₄	1	4	yes	—
30	#2248	—	5	2	no	—
31	#2247	—	5	2	no	—
32	#1713	—	5	1	no	—
33	#1709	D₅	1	1	yes	characteristic
34	#1311	—	5	1	no	—
35	C₂×C₅×Q₈	—	4	1	no	—
36	C₂×C₅×D₄	—	4	1	no	—
37	C₂×C₅×D₄	—	4	2	no	—
38	C₂×Dic₁₀	—	4	2	no	—
39	C₅×Q₁₆	—	4	4	no	—
40	C₅×D₈	—	4	4	no	—
41	C₂×C₅×C₈	D₈	1	4	yes	—
42	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
43	C₅×C₄⋊C₄	—	4	2	no	—
44	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₅	—	4	4	no	—
46	Dic₂₀	—	4	4	no	—
47	Dic₂₀	—	4	4	no	—
48	#672	D₈	1	8	yes	—
49	C₂×Q₃₂	—	10	4	no	—
50	C₂×SD₃₂	—	10	4	no	—
51	#500	—	10	2	no	—
52	#494	—	5	2	no	—
53	#494	C₂×D₅	1	1	yes	characteristic
54	#493	C₂×D₅	1	1	yes	characteristic
55	#486	—	2	2	no	—
56	C₄×C₁₆	—	5	2	no	—
57	C₈²	C₂×D₅	1	1	yes	characteristic
58	C₂³×C₅	—	8	1	no	—
59	C₅×Q₈	—	8	2	no	—
60	C₅×D₄	—	8	2	no	—
61	C₅×D₄	—	8	4	no	—
62	C₂×C₄×C₅	—	8	1	no	—
63	C₂×C₄×C₅	#127	1	1	yes	characteristic
64	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
65	C₂×Dic₅	—	8	2	no	—
66	Dic₁₀	—	8	2	no	—
67	Dic₁₀	—	8	2	no	—
68	C₅×C₈	—	2	4	no	—
69	C₅×C₈	C₂×D₈	1	4	yes	—
70	C₂×Q₁₆	—	2	2	no	—
71	C₂×Q₁₆	—	10	2	no	—
72	C₂×Q₁₆	—	10	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₄⋊Q₈	—	2	1	no	—
75	C₄⋊Q₈	—	10	2	no	—
76	#127	—	2	1	no	—
77	Q₃₂	—	20	8	no	—
78	SD₃₂	—	20	8	no	—
79	C₂×C₁₆	—	5	4	no	—
80	#107	—	20	4	no	—
81	#102	—	4	4	no	—
82	C₄×C₈	C₂²⋊D₅	1	2	yes	—
83	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
84	C₂²×C₅	—	16	1	no	—
85	C₂²×C₅	—	16	2	no	—
86	C₂²×C₅	#493	1	1	yes	characteristic
87	C₄×C₅	—	16	1	no	—
88	C₄×C₅	#493	1	2	yes	—
89	C₄×C₅	#510	1	4	yes	—
90	Dic₅	—	16	2	no	—
91	C₂×Q₈	—	4	1	no	—
92	C₂×Q₈	—	20	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	1	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	20	4	no	—
97	Q₁₆	—	20	4	no	—
98	D₈	—	4	4	no	—
99	C₂×C₈	#686	1	4	yes	—
100	C₂×C₈	D₄×D₅	1	2	yes	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	20	4	no	—
103	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
104	C₁₆	—	5	8	no	—
105	C₂×C₅	—	32	1	no	—
106	C₂×C₅	#1712	1	1	yes	characteristic
107	C₂×C₅	#2251	1	2	yes	—
108	C₂³	—	8	1	no	—
109	Q₈	—	8	2	no	—
110	Q₈	—	40	2	no	—
111	Q₈	—	40	2	no	—
112	D₄	—	8	2	no	—
113	D₄	—	8	4	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	40	2	no	—
116	C₂×C₄	C₂×#686	1	2	yes	—
117	C₂×C₄	#4086	1	1	yes	characteristic
118	C₈	—	2	4	no	—
119	C₈	D₅×D₈	1	4	yes	—
120	C₅	#12520	1	1	yes	characteristic, complemented, Sylow
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#65362	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	80	2	no	—
126	C₄	#65440	1	2	yes	—
127	C₄	#65484	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#10602069	1	1	yes	characteristic
130	C₂	#10602115	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#1709

C₂

yes

maximal, characteristic

#10602105

C₂

yes

maximal, complemented

#10602084

C₂

yes

maximal

#10598858

C₂

yes

maximal, characteristic, complemented

#10597928

C₂

yes

maximal, characteristic, complemented

C₅×#494

C₂²

yes

characteristic

C₅×#493

C₂²

yes

—

—

—

C₂²

yes

—

#64971

—

C₅×C₈²

C₂²

yes

characteristic

C₄×#672

C₂²

yes

—

#12520

—

maximal, Sylow

C₂×C₅×Q₁₆

—

C₂×C₅×D₈

—

C₅×C₄⋊Q₈

—

C₅×#127

—

C₂×Dic₂₀

—

C₂×Dic₂₀

—

#4015

—

C₅×#102

—

C₄×C₅×C₈

D₄

yes

—

C₄×C₅×C₈

C₂³

yes

—

—

—

D₄

yes

—

—

—

—

D₅

yes

characteristic

#1311

—

C₂×C₅×Q₈

—

C₂×C₅×D₄

—

C₂×C₅×D₄

—

C₂×Dic₁₀

—

C₅×Q₁₆

—

C₅×D₈

—

C₂×C₅×C₈

D₈

yes

—

C₂×C₅×C₈

C₂×D₄

yes

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×D₄

yes

characteristic

C₄⋊Dic₅

—

Dic₂₀

—

Dic₂₀

—

#672

D₈

yes

—

C₂×Q₃₂

—

C₂×SD₃₂

—

#500

—

#494

—

#494

C₂×D₅

yes

characteristic

#493

C₂×D₅

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₅

yes

characteristic

C₂³×C₅

—

C₅×Q₈

—

C₅×D₄

—

C₅×D₄

—

C₂×C₄×C₅

—

C₂×C₄×C₅

#127

yes

characteristic

C₂×C₄×C₅

C₂×D₈

yes

—

C₂×Dic₅

—

Dic₁₀

—

Dic₁₀

—

C₅×C₈

—

C₅×C₈

C₂×D₈

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

Q₃₂

—

SD₃₂

—

C₂×C₁₆

—

#107

—

#102

—

C₄×C₈

C₂²⋊D₅

yes

—

C₄×C₈

C₂²×D₅

yes

characteristic

C₂²×C₅

—

C₂²×C₅

—

C₂²×C₅

#493

yes

characteristic

C₄×C₅

—

C₄×C₅

#493

yes

—

C₄×C₅

#510

yes

—

Dic₅

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

C₂×C₈

#686

yes

—

100

C₂×C₈

D₄×D₅

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

C₂×C₂²⋊D₅

yes

characteristic

104

C₁₆

—

105

C₂×C₅

—

106

C₂×C₅

#1712

yes

characteristic

107

C₂×C₅

#2251

yes

—

108

C₂³

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×#686

yes

—

117

C₂×C₄

#4086

yes

characteristic

118

C₈

—

119

C₈

D₅×D₈

yes

—

120

C₅

#12520

yes

characteristic, complemented, Sylow

121

C₂²

—

122

C₂²

—

123

C₂²

#65362

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

#65440

yes

—

127

C₄

#65484

yes

—

128

C₂

—

129

C₂

#10602069

yes

characteristic

130

C₂

#10602115

yes

—

	subgroup	G/H
center	C₂²	#65362
commutator, cosocle	C₄×C₅×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#1709	C₂
socle	C₂²×C₅	#493
prime radical	C₅×#493	C₂²
hypercenter	#1709	D₅
hypocenter	C₅	#12520
LCS	G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#65362

commutator, cosocle

C₄×C₅×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₅

C₂

C₂²×C₅

C₂²

D₅

C₅

G ▷ C₄×C₅×C₈ ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #10602105	C₂	semidirect
4. #10598858	C₂	semidirect
5. #10597928	C₂	semidirect
120. C₅	#12520	semidirect

H ≅ G/N

type

2. #10602105

C₂

semidirect

4. #10598858

C₂

semidirect

5. #10597928

C₂

semidirect

120. C₅

#12520

semidirect

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

subgroup	index
C₅×#1709	2
#10602105	2
#10602084	2
#10598858	2
#10597928	2
#12520	5

subgroup

index

Subgroups — #12014034

Conjugacy classes of subgroups

Complements

Maximal subgroups