Group #11857017 — subgroups

Subgroups — #11857017

Order 1152.

454 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 119 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	#10593736	C₂	1	2	yes	maximal
2	C₄²×#157	C₂	1	1	yes	maximal, characteristic
3	#10593554	C₂	1	4	yes	maximal
4	#10593550	C₂	1	8	yes	maximal
5	C₂×C₄⋊#157	C₂²	1	4	yes	—
6	C₂×C₄×#157	C₂²	1	4	yes	—
7	C₂×C₄×#157	C₂²	1	2	yes	—
8	C₂×#3777	C₂²	1	8	yes	—
9	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	#63781	C₂²	1	8	yes	—
11	#63747	C₂²	1	8	yes	—
12	C₂²×#157	C₂³	1	8	yes	—
13	C₂²×#157	C₂³	1	4	yes	—
14	C₂×C₄×C₃⋊D₃	C₂³	1	2	yes	—
15	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
16	C₄⋊#157	—	2	8	no	—
17	C₄×#157	—	2	8	no	—
18	C₄×#157	—	2	4	no	—
19	#3777	—	2	16	no	—
20	C₃²×C₄²	Q₈	1	1	yes	characteristic
21	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
22	C₄×C₃⋊Dic₃	Q₈	1	2	yes	—
23	#2864	—	9	1	no	maximal, Sylow
24	C₄²×D₃	—	12	1	no	—
25	C₂²×C₃⋊D₃	C₂⁴	1	1	yes	characteristic
26	C₂×#157	—	2	4	no	—
27	C₂×#157	—	2	4	no	—
28	C₂×#157	—	2	16	no	—
29	C₂×#157	—	2	8	no	—
30	C₂×#157	#41	1	8	yes	—
31	C₂×C₃²×C₄	C₂×Q₈	1	2	yes	—
32	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₃⋊Dic₃	C₂×Q₈	1	2	yes	—
34	C₂×C₃⋊Dic₃	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₃⋊D₃	—	2	4	no	—
36	C₄×C₃⋊D₃	#41	1	4	yes	—
37	#389	—	9	4	no	—
38	#387	—	9	8	no	—
39	#382	—	9	2	no	—
40	C₄³	—	9	1	no	—
41	C₂×C₄×D₃	—	12	1	no	—
42	C₂×C₄×D₃	—	12	2	no	—
43	C₃×C₄²	—	4	1	no	—
44	C₄×Dic₃	—	12	2	no	—
45	C₄×Dic₃	—	12	1	no	—
46	C₂²×C₃²	C₂²×Q₈	1	1	yes	characteristic
47	C₂×C₃⋊D₃	C₂×#41	1	2	yes	—
48	C₂×C₃⋊D₃	C₂×#41	1	4	yes	—
49	#157	—	2	8	no	—
50	#157	—	4	8	no	—
51	C₃²×C₄	—	2	2	no	—
52	C₃²×C₄	#125	1	2	yes	—
53	C₃⋊Dic₃	—	2	2	no	—
54	C₃⋊Dic₃	#125	1	2	yes	—
55	C₂×C₄⋊C₄	—	9	4	no	—
56	C₂×C₄⋊C₄	—	9	8	no	—
57	C₂×C₄²	—	9	1	no	—
58	C₂×C₄²	—	9	4	no	—
59	C₂×C₄²	—	9	2	no	—
60	#95	—	9	8	no	—
61	#95	—	9	8	no	—
62	C₂²×D₃	—	12	1	no	—
63	C₂×C₃×C₄	—	4	2	no	—
64	C₂×C₃×C₄	—	4	1	no	—
65	C₂×Dic₃	—	12	1	no	—
66	C₂×Dic₃	—	12	2	no	—
67	C₄×D₃	—	12	4	no	—
68	C₄×D₃	—	12	8	no	—
69	C₂×C₃²	C₂×#125	1	1	yes	characteristic
70	C₂×C₃²	#533	1	2	yes	—
71	C₃⋊D₃	#529	1	4	yes	—
72	C₂²×C₄	—	9	1	no	—
73	C₂²×C₄	—	9	4	no	—
74	C₂²×C₄	—	9	2	no	—
75	C₂²×C₄	—	9	8	no	—
76	C₄⋊C₄	—	18	16	no	—
77	C₄⋊C₄	—	18	8	no	—
78	C₄²	—	9	2	no	—
79	C₄²	—	9	1	no	—
80	C₄²	—	18	4	no	—
81	C₄²	—	18	8	no	—
82	C₄²	PSU(3,𝔽₂)	1	1	yes	characteristic
83	C₂²×C₃	—	4	1	no	—
84	C₂×D₃	—	12	2	no	—
85	C₂×D₃	—	12	4	no	—
86	C₃×C₄	—	4	4	no	—
87	C₃×C₄	—	4	2	no	—
88	Dic₃	—	12	2	no	—
89	Dic₃	—	12	4	no	—
90	C₃²	#2864	1	1	yes	characteristic, complemented, Sylow
91	C₂³	—	9	1	no	—
92	C₂×C₄	—	9	1	no	—
93	C₂×C₄	—	9	4	no	—
94	C₂×C₄	—	9	8	no	—
95	C₂×C₄	—	9	2	no	—
96	C₂×C₄	—	18	4	no	—
97	C₂×C₄	—	18	4	no	—
98	C₂×C₄	—	18	4	no	—
99	C₂×C₄	—	18	8	no	—
100	C₂×C₄	—	18	16	no	—
101	C₂×C₄	C₂×PSU(3,𝔽₂)	1	2	yes	—
102	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
103	C₂×C₃	—	4	1	no	—
104	C₂×C₃	—	4	2	no	—
105	D₃	—	12	4	no	—
106	C₂²	—	9	2	no	—
107	C₂²	—	9	4	no	—
108	C₂²	C₂²×PSU(3,𝔽₂)	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	9	2	no	—
111	C₄	—	18	8	no	—
112	C₄	—	18	2	no	—
113	C₄	—	36	8	no	—
114	C₄	#64243	1	2	yes	—
115	C₃	—	4	1	no	—
116	C₂	—	9	4	no	—
117	C₂	C₂×#64243	1	1	yes	characteristic
118	C₂	#10596542	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10593736

C₂

yes

maximal

C₄²×#157

C₂

yes

maximal, characteristic

#10593554

C₂

yes

maximal

#10593550

C₂

yes

maximal

C₂×C₄⋊#157

C₂²

yes

—

C₂×C₄×#157

C₂²

yes

—

C₂×C₄×#157

C₂²

yes

—

C₂×#3777

C₂²

yes

—

C₄²×C₃⋊D₃

C₂²

yes

characteristic

#63781

C₂²

yes

—

#63747

C₂²

yes

—

C₂²×#157

C₂³

yes

—

C₂²×#157

C₂³

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

—

—

—

—

C₃²×C₄²

Q₈

yes

characteristic

C₄×C₃⋊Dic₃

Q₈

yes

characteristic

C₄×C₃⋊Dic₃

Q₈

yes

—

#2864

—

maximal, Sylow

C₄²×D₃

—

C₂²×C₃⋊D₃

C₂⁴

yes

—

—

—

—

yes

—

C₂×C₃²×C₄

C₂×Q₈

yes

—

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃⋊Dic₃

C₂×Q₈

yes

—

C₂×C₃⋊Dic₃

C₂×Q₈

yes

characteristic

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

#41

yes

—

#389

—

#387

—

#382

—

C₄³

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂²×Q₈

yes

characteristic

C₂×C₃⋊D₃

C₂×#41

yes

—

C₂×C₃⋊D₃

C₂×#41

yes

—

#157

—

#157

—

C₃²×C₄

—

C₃²×C₄

#125

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

#125

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

#95

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

C₂×#125

yes

characteristic

C₂×C₃²

#533

yes

—

C₃⋊D₃

#529

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

PSU(3,𝔽₂)

yes

characteristic

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#2864

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

—

102

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

103

C₂×C₃

—

104

C₂×C₃

—

105

D₃

—

106

C₂²

—

107

C₂²

—

108

C₂²

C₂²×PSU(3,𝔽₂)

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

#64243

yes

—

115

C₃

—

116

C₂

—

117

C₂

C₂×#64243

yes

characteristic

118

C₂

#10596542

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂²×PSU(3,𝔽₂)
commutator, prime radical, cosocle	C₂²×C₃⋊D₃	C₂⁴
Fitting	C₃²×C₄²	Q₈
socle	C₂²×C₃²	C₂²×Q₈
hypercenter	C₄²	PSU(3,𝔽₂)
hypocenter	C₃²	#2864
LCS	G ▷ C₂²×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄²

subgroup

G/H

center, Frattini

C₂²

C₂²×PSU(3,𝔽₂)

commutator, prime radical, cosocle

C₂²×C₃⋊D₃

C₂⁴

Fitting

C₃²×C₄²

Q₈

socle

C₂²×C₃²

C₂²×Q₈

C₄²

C₃²

G ▷ C₂²×C₃⋊D₃ ▷ C₃²

C₁ ◁ C₂² ◁ C₄²

N	H ≅ G/N	type
90. C₃²	#2864	semidirect

H ≅ G/N

type

90. C₃²

#2864

semidirect

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

subgroup	index
#10593736	2
C₄²×#157	2
#10593554	2
#10593550	2
#2864	9

subgroup

index

C₄²×#157

Subgroups — #11857017

Conjugacy classes of subgroups

Complements

Maximal subgroups