Group #10593834 — subgroups

Subgroups — #10593834

Order 576.

322 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 52 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×#3832	C₂	1	1	yes	maximal, characteristic
2	C₂×#3788	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#3787	C₂	1	1	yes	maximal, characteristic, complemented
4	#63778	C₂	1	4	yes	maximal, complemented
5	C₂×C₃²×Q₈	C₄	1	1	yes	characteristic
6	#3832	—	2	2	no	—
7	#3832	—	2	2	no	—
8	C₂×C₄⋊(C₃⋊D₃)	C₄	1	2	yes	—
9	C₂×C₄⋊(C₃⋊D₃)	C₄	1	1	yes	characteristic
10	C₂×C₄×C₃⋊D₃	C₂²	1	2	yes	—
11	C₂×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
12	C₂×#622	C₂²	1	2	yes	—
13	C₂×#622	C₂²	1	2	yes	—
14	#3788	—	2	2	no	—
15	#3787	—	2	2	no	—
16	C₂×#239	—	6	2	no	—
17	C₂²×C₃⋊D₃	C₂×C₄	1	2	yes	—
18	C₂²×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
19	C₃²×Q₈	—	2	2	no	—
20	C₂×C₃²×C₄	C₂×C₄	1	2	yes	—
21	C₂×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
23	C₄⋊(C₃⋊D₃)	—	2	4	no	—
24	C₄⋊(C₃⋊D₃)	—	2	2	no	—
25	C₄×C₃⋊D₃	—	2	2	no	—
26	C₄×C₃⋊D₃	—	2	2	no	—
27	C₄×C₃⋊D₃	D₄	1	2	yes	—
28	C₄×C₃⋊D₃	D₄	1	2	yes	—
29	#622	—	2	2	no	—
30	#622	—	2	2	no	—
31	#408	—	9	1	no	maximal, Sylow
32	C₂×C₃×Q₈	—	2	2	no	—
33	#239	—	6	8	no	—
34	#239	—	6	8	no	—
35	C₂×D₁₂	—	6	4	no	—
36	C₂×D₁₂	—	6	2	no	—
37	C₂×C₄×D₃	—	6	2	no	—
38	C₂×C₄×D₃	—	6	4	no	—
39	C₂²×C₃²	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₃⋊D₃	—	2	2	no	—
41	C₂×C₃⋊D₃	—	2	2	no	—
42	C₂×C₃⋊D₃	—	2	4	no	—
43	C₂×C₃⋊D₃	C₂²⋊C₄	1	2	yes	—
44	C₃²×C₄	—	2	2	no	—
45	C₃²×C₄	C₂²⋊C₄	1	2	yes	—
46	C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
47	C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
48	C₂×#41	—	9	1	no	—
49	C₂×M₁₆	—	9	1	no	—
50	C₂²×C₈	—	9	1	no	—
51	C₂²⋊C₈	—	9	4	no	—
52	C₂²×D₃	—	6	2	no	—
53	C₂²×D₃	—	6	4	no	—
54	C₃×Q₈	—	2	8	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×Dic₃	—	6	2	no	—
58	D₁₂	—	6	16	no	—
59	D₁₂	—	6	8	no	—
60	C₄×D₃	—	6	4	no	—
61	C₄×D₃	—	6	4	no	—
62	C₄×D₃	—	6	8	no	—
63	C₄×D₃	—	6	8	no	—
64	C₂×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃²	#131	1	2	yes	—
66	C₃⋊D₃	—	2	2	no	—
67	C₃⋊D₃	—	4	2	no	—
68	#41	—	18	2	no	—
69	#41	—	18	2	no	—
70	C₂×Q₈	#157	1	1	yes	characteristic
71	C₂×D₄	—	9	2	no	—
72	C₂×D₄	—	9	1	no	—
73	C₂²×C₄	—	9	1	no	—
74	C₂²×C₄	—	9	2	no	—
75	M₁₆	—	18	2	no	—
76	C₂×C₈	—	9	2	no	—
77	C₂×C₈	—	9	2	no	—
78	C₂×C₈	—	18	2	no	—
79	C₂²×C₃	—	2	2	no	—
80	C₂×D₃	—	6	4	no	—
81	C₂×D₃	—	6	8	no	—
82	C₂×D₃	—	12	4	no	—
83	C₂×D₃	—	12	8	no	—
84	C₃×C₄	—	2	8	no	—
85	C₃×C₄	—	2	4	no	—
86	Dic₃	—	6	2	no	—
87	Dic₃	—	6	2	no	—
88	C₃²	#408	1	1	yes	characteristic, complemented, Sylow
89	C₂³	—	9	1	no	—
90	C₂³	—	9	2	no	—
91	Q₈	—	2	2	no	—
92	D₄	—	18	4	no	—
93	D₄	—	18	2	no	—
94	C₂×C₄	—	9	1	no	—
95	C₂×C₄	—	9	2	no	—
96	C₂×C₄	—	9	2	no	—
97	C₂×C₄	—	18	2	no	—
98	C₂×C₄	—	18	2	no	—
99	C₂×C₄	C₂×#157	1	2	yes	—
100	C₂×C₄	C₂×#157	1	1	yes	characteristic
101	C₈	—	18	2	no	—
102	C₈	—	18	2	no	—
103	C₂×C₃	—	2	4	no	—
104	C₂×C₃	—	2	2	no	—
105	D₃	—	12	4	no	—
106	D₃	—	12	8	no	—
107	C₂²	—	9	2	no	—
108	C₂²	—	18	2	no	—
109	C₂²	—	18	4	no	—
110	C₂²	—	18	2	no	—
111	C₂²	C₂²×#157	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	9	1	no	—
114	C₄	—	9	1	no	—
115	C₄	C₂²⋊#157	1	2	yes	—
116	C₃	—	2	2	no	—
117	C₂	—	18	2	no	—
118	C₂	—	36	2	no	—
119	C₂	#64289	1	2	yes	—
120	C₂	C₂×C₂²⋊#157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#3832

C₂

yes

maximal, characteristic

C₂×#3788

C₂

yes

maximal, characteristic, complemented

C₂×#3787

C₂

yes

maximal, characteristic, complemented

#63778

C₂

yes

maximal, complemented

C₂×C₃²×Q₈

C₄

yes

characteristic

#3832

—

#3832

—

C₂×C₄⋊(C₃⋊D₃)

C₄

yes

—

C₂×C₄⋊(C₃⋊D₃)

C₄

yes

characteristic

C₂×C₄×C₃⋊D₃

C₂²

yes

—

C₂×C₄×C₃⋊D₃

C₂²

yes

characteristic

C₂×#622

C₂²

yes

—

C₂×#622

C₂²

yes

—

#3788

—

#3787

—

C₂×#239

—

C₂²×C₃⋊D₃

C₂×C₄

yes

—

C₂²×C₃⋊D₃

C₂×C₄

yes

characteristic

C₃²×Q₈

—

C₂×C₃²×C₄

C₂×C₄

yes

—

C₂×C₃²×C₄

C₂×C₄

yes

characteristic

C₂×C₃⋊Dic₃

C₂³

yes

characteristic

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

D₄

yes

—

C₄×C₃⋊D₃

D₄

yes

—

—

—

—

C₂×C₃×Q₈

—

#239

—

#239

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₂²×C₃²

C₂²×C₄

yes

characteristic

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

C₂²⋊C₄

yes

—

C₃²×C₄

—

C₃²×C₄

C₂²⋊C₄

yes

—

C₃⋊Dic₃

C₂×D₄

yes

characteristic

C₃⋊Dic₃

C₂×D₄

yes

characteristic

C₂×#41

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²⋊C₈

—

C₂²×D₃

—

C₂²×D₃

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₃²

#131

yes

—

C₃⋊D₃

—

C₃⋊D₃

—

#41

—

#41

—

C₂×Q₈

#157

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#408

yes

characteristic, complemented, Sylow

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#157

yes

—

100

C₂×C₄

C₂×#157

yes

characteristic

101

C₈

—

102

C₈

—

103

C₂×C₃

—

104

C₂×C₃

—

105

D₃

—

106

D₃

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²×#157

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

C₂²⋊#157

yes

—

116

C₃

—

117

C₂

—

118

C₂

—

119

C₂

#64289

yes

—

120

C₂

C₂×C₂²⋊#157

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#157
commutator, socle	C₂²×C₃²	C₂²×C₄
Fitting	C₂×C₃²×Q₈	C₄
prime radical	C₂×#3832	C₂
cosocle	C₂×C₃⋊Dic₃	C₂³
hypercenter	C₂×Q₈	#157
hypocenter	C₃²	#408
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂²×#157

commutator, socle

C₂²×C₃²

C₂²×C₄

Fitting

C₂×C₃²×Q₈

C₄

prime radical

C₂×#3832

C₂

cosocle

C₂×C₃⋊Dic₃

C₂³

C₂×Q₈

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂² ◁ C₂×Q₈

N	H ≅ G/N	type
2. C₂×#3788	C₂	semidirect
3. C₂×#3787	C₂	semidirect
4. #63778	C₂	semidirect
88. C₃²	#408	semidirect

H ≅ G/N

type

2. C₂×#3788

C₂

semidirect

3. C₂×#3787

C₂

semidirect

4. #63778

C₂

semidirect

88. C₃²

#408

semidirect

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

subgroup	index
C₂×#3832	2
C₂×#3788	2
C₂×#3787	2
#63778	2
#408	9

subgroup

index

Subgroups — #10593834

Conjugacy classes of subgroups

Complements

Maximal subgroups