Group #12003796 — subgroups

Subgroups — #12003796

Order 1280.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2844	C₂	1	1	yes	maximal, characteristic
2	#10600343	C₂	1	3	yes	maximal
3	#10598121	C₂	1	3	yes	maximal
4	C₅×#400	—	2	3	no	—
5	C₅×#395	—	2	3	no	—
6	C₅×#382	C₂²	1	3	yes	—
7	C₂×#3940	—	2	3	no	—
8	C₂×#3939	—	2	3	no	—
9	#65224	C₂²	1	1	yes	characteristic
10	C₂×#3935	C₂²	1	3	yes	—
11	#10583	—	5	1	no	maximal, Sylow
12	C₂²×C₅×Q₈	—	2	1	no	—
13	C₂×C₅×C₄⋊C₄	—	2	3	no	—
14	C₂×C₅×C₄⋊C₄	D₄	1	3	yes	—
15	C₂×C₅×C₄⋊C₄	D₄	1	3	yes	—
16	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅	—	2	3	no	—
18	C₂²×#168	—	2	3	no	—
19	C₅×#95	—	2	3	no	—
20	C₅×#95	—	2	1	no	—
21	C₅×#95	—	2	6	no	—
22	#3940	—	4	6	no	—
23	#3939	—	4	6	no	—
24	#3935	—	2	3	no	—
25	#3935	Q₈	1	6	yes	complemented
26	#2844	D₅	1	1	yes	characteristic
27	#1847	—	5	3	no	—
28	#1295	—	5	3	no	—
29	C₂×C₅×Q₈	—	8	3	no	—
30	C₂²×C₄×C₅	—	2	3	no	—
31	C₂²×C₄×C₅	—	2	3	no	—
32	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
33	C₂²×Dic₅	—	4	1	no	—
34	C₅×C₄⋊C₄	—	4	3	no	—
35	C₅×C₄⋊C₄	—	4	3	no	—
36	C₅×C₄⋊C₄	—	8	3	no	—
37	C₄²×C₅	C₂×Q₈	1	3	yes	—
38	C₄²×C₅	#41	1	1	yes	characteristic
39	C₄⋊Dic₅	—	4	3	no	—
40	C₄⋊Dic₅	—	4	3	no	—
41	C₂×#168	—	2	6	no	—
42	C₂×#168	—	2	6	no	—
43	C₂×#168	—	2	6	no	—
44	C₂×#107	—	10	3	no	—
45	C₂×#106	—	10	3	no	—
46	C₂×#97	—	5	3	no	—
47	#400	—	2	3	no	—
48	#395	—	2	3	no	—
49	#384	—	5	1	no	—
50	#382	C₂×D₅	1	3	yes	—
51	C₂³×C₅	#120	1	1	yes	characteristic
52	C₅×Q₈	—	8	4	no	—
53	C₂×C₄×C₅	—	2	3	no	—
54	C₂×C₄×C₅	—	2	3	no	—
55	C₂×C₄×C₅	—	4	3	no	—
56	C₂×C₄×C₅	—	4	3	no	—
57	C₂×C₄×C₅	—	4	6	no	—
58	C₂×C₄×C₅	—	4	6	no	—
59	C₂×C₄×C₅	C₂²⋊Q₈	1	3	yes	—
60	C₂×C₄×C₅	C₄⋊Q₈	1	3	yes	—
61	C₂×Dic₅	—	8	3	no	—
62	#168	—	2	12	no	—
63	C₂²×Q₈	—	2	1	no	—
64	C₂²×C₈	—	10	3	no	—
65	C₂×C₄⋊C₄	—	2	3	no	—
66	C₂×C₄⋊C₄	—	10	3	no	—
67	C₂×C₄⋊C₄	C₂²⋊D₅	1	3	yes	—
68	C₂×C₄⋊C₄	C₂²⋊D₅	1	3	yes	—
69	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
70	#107	—	20	6	no	—
71	#106	—	20	6	no	—
72	#97	—	5	6	no	—
73	#97	—	10	3	no	—
74	#95	—	2	3	no	—
75	#95	—	2	1	no	—
76	#95	—	2	6	no	—
77	C₂²×C₅	#395	1	1	yes	characteristic
78	C₂²×C₅	#453	1	3	yes	—
79	C₂²×C₅	#456	1	3	yes	—
80	C₄×C₅	—	2	3	no	—
81	C₄×C₅	—	2	3	no	—
82	C₄×C₅	—	8	3	no	—
83	C₄×C₅	—	8	3	no	—
84	Dic₅	—	16	1	no	—
85	C₂×Q₈	—	8	3	no	—
86	C₂²×C₄	—	2	3	no	—
87	C₂²×C₄	—	2	3	no	—
88	C₂²×C₄	—	20	1	no	—
89	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
90	C₂×C₈	—	10	6	no	—
91	C₂×C₈	—	10	6	no	—
92	C₂×C₈	—	10	6	no	—
93	C₄⋊C₄	—	4	3	no	—
94	C₄⋊C₄	—	4	3	no	—
95	C₄⋊C₄	—	8	3	no	—
96	C₄⋊C₄	—	20	3	no	—
97	C₄⋊C₄	—	20	3	no	—
98	C₄²	Q₈×D₅	1	3	yes	—
99	C₄²	#713	1	1	yes	characteristic
100	C₂×C₅	#1655	1	1	yes	characteristic
101	C₂×C₅	#1660	1	3	yes	—
102	C₂×C₅	#2060	1	3	yes	—
103	C₂³	#4099	1	1	yes	characteristic
104	Q₈	—	8	4	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	—	4	6	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	40	3	no	—
112	C₂×C₄	#4090	1	3	yes	—
113	C₂×C₄	#4092	1	3	yes	—
114	C₈	—	10	12	no	—
115	C₅	#10583	1	1	yes	characteristic, complemented, Sylow
116	C₂²	#65518	1	1	yes	characteristic
117	C₂²	#65530	1	3	yes	—
118	C₂²	#65535	1	3	yes	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	—	8	3	no	—
122	C₄	—	8	3	no	—
123	C₄	—	80	1	no	—
124	C₂	#10602693	1	3	yes	—
125	C₂	#10602741	1	1	yes	characteristic
126	C₂	#10602749	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2844

C₂

yes

maximal, characteristic

#10600343

C₂

yes

maximal

#10598121

C₂

yes

—

—

C₂²

yes

—

C₂×#3940

—

C₂×#3939

—

#65224

C₂²

yes

characteristic

C₂×#3935

C₂²

yes

—

#10583

—

maximal, Sylow

C₂²×C₅×Q₈

—

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂²×#168

—

C₅×#95

—

C₅×#95

—

C₅×#95

—

—

—

—

Q₈

yes

complemented

#2844

D₅

yes

characteristic

#1847

—

#1295

—

C₂×C₅×Q₈

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×Q₈

yes

—

C₄²×C₅

#41

yes

characteristic

C₄⋊Dic₅

—

C₄⋊Dic₅

—

—

—

—

—

—

C₂×#97

—

—

—

—

C₂×D₅

yes

—

C₂³×C₅

#120

yes

characteristic

C₅×Q₈

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×Dic₅

—

#168

—

C₂²×Q₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²×D₅

yes

—

—

—

—

—

—

—

C₂²×C₅

#395

yes

characteristic

C₂²×C₅

#453

yes

—

C₂²×C₅

#456

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₅

yes

—

C₄²

#713

yes

characteristic

100

C₂×C₅

#1655

yes

characteristic

101

C₂×C₅

#1660

yes

—

102

C₂×C₅

#2060

yes

—

103

C₂³

#4099

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#4090

yes

—

113

C₂×C₄

#4092

yes

—

114

C₈

—

115

C₅

#10583

yes

characteristic, complemented, Sylow

116

C₂²

#65518

yes

characteristic

117

C₂²

#65530

yes

—

118

C₂²

#65535

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₂

#10602693

yes

—

125

C₂

#10602741

yes

characteristic

126

C₂

#10602749

yes

—

	subgroup	G/H
center	C₂³	#4099
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2844	C₂
socle, prime radical	C₂³×C₅	#120
hypercenter	#2844	D₅
hypocenter	C₅	#10583
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#4099

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
25. #3935	Q₈	semidirect
115. C₅	#10583	semidirect

H ≅ G/N

type

25. #3935

Q₈

semidirect

115. C₅

#10583

semidirect

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

subgroup	index
C₅×#2844	2
#10600343	2
#10598121	2
#10583	5

subgroup

index

Subgroups — #12003796

Conjugacy classes of subgroups

Complements

Maximal subgroups