Group #46835440 — subgroups

Subgroups — #46835440

Order 34688.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#31774627	C₂	1	1	yes	maximal, characteristic, complemented
2	#31774094	C₂	1	2	yes	maximal
3	#31773646	C₂	1	1	yes	maximal, characteristic, complemented
4	#31773629	C₂	1	2	yes	maximal, complemented
5	#492×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16864642	—	2	2	no	—
7	#22576159	—	2	2	no	—
8	#22576158	—	2	2	no	—
9	C₄×Dic₅₄₂	C₂²	1	2	yes	—
10	#22576125	—	2	2	no	—
11	C₄⋊(C₂₇₁⋊C₈)	C₂²	1	2	yes	—
12	#22576081	—	2	2	no	—
13	C₂×SD₁₆×C₂₇₁	—	2	2	no	—
14	C₄⋊Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
15	#127×C₂₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₄₂	—	2	2	no	—
18	C₂×Dic₅₄₂	D₄	1	2	yes	—
19	#16864642	—	4	4	no	—
20	C₂×C₂₇₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₂₇₁	D₄	1	2	yes	—
22	Dic₂₇₁⋊C₄	—	2	4	no	—
23	Dic₂₇₁⋊C₄	—	2	2	no	—
24	C₄×Dic₂₇₁	—	2	2	no	—
25	C₄×Dic₂₇₁	—	2	2	no	—
26	C₂×Q₈×C₂₇₁	D₄	1	2	yes	—
27	C₂×D₄×C₂₇₁	—	2	2	no	—
28	C₂×D₄×C₂₇₁	—	2	1	no	—
29	SD₁₆×C₂₇₁	—	4	4	no	—
30	C₂×C₈×C₂₇₁	—	2	1	no	—
31	C₄⋊C₄×C₂₇₁	—	2	2	no	—
32	C₄²×C₂₇₁	C₂³	1	1	yes	characteristic
33	Dic₅₄₂	—	2	2	no	—
34	Dic₅₄₂	—	8	2	no	—
35	Dic₅₄₂	SD₁₆	1	4	yes	complemented
36	C₂×Dic₂₇₁	—	2	2	no	—
37	C₂×Dic₂₇₁	—	2	2	no	—
38	C₂₇₁⋊C₈	—	4	2	no	—
39	C₂³×C₂₇₁	—	4	1	no	—
40	Q₈×C₂₇₁	—	4	2	no	—
41	D₄×C₂₇₁	—	4	2	no	—
42	D₄×C₂₇₁	—	4	2	no	—
43	D₄×C₂₇₁	—	4	2	no	—
44	C₂×C₄×C₂₇₁	—	2	2	no	—
45	C₂×C₄×C₂₇₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
47	C₈×C₂₇₁	—	2	2	no	—
48	C₂²×C₂₇₁	—	8	2	no	—
49	C₂²×C₂₇₁	—	8	1	no	—
50	C₂²×C₂₇₁	#120	1	1	yes	characteristic
51	C₄×C₂₇₁	—	2	1	no	—
52	C₄×C₂₇₁	—	4	2	no	—
53	C₄×C₂₇₁	C₂×SD₁₆	1	2	yes	—
54	C₄×C₂₇₁	#137	1	2	yes	—
55	Dic₂₇₁	—	2	4	no	—
56	Dic₂₇₁	—	4	2	no	—
57	C₂×C₂₇₁	—	16	1	no	—
58	C₂×C₂₇₁	#448	1	2	yes	—
59	C₂×C₂₇₁	#453	1	1	yes	characteristic
60	C₂₇₁	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	271	1	no	maximal, Sylow
62	Q₈²	—	271	1	no	—
63	#492	D₂₇₁	1	1	yes	characteristic
64	#460	—	271	2	no	—
65	#333	—	271	1	no	—
66	Q₈⋊C₈	—	271	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	542	2	no	—
69	C₄⋊Q₈	—	542	2	no	—
70	C₄⋊Q₈	—	542	2	no	—
71	C₄⋊Q₈	C₂×D₂₇₁	1	1	yes	characteristic
72	#127	C₂×D₂₇₁	1	1	yes	characteristic
73	C₄×Q₈	—	271	2	no	—
74	C₄×Q₈	—	542	2	no	—
75	C₄⋊C₈	—	271	2	no	—
76	#102	—	542	2	no	—
77	C₄×C₈	C₂×D₂₇₁	1	1	yes	characteristic
78	C₂×Q₈	—	271	2	no	—
79	C₂×Q₈	—	542	2	no	—
80	C₂×Q₈	C₂²⋊D₂₇₁	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	1084	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	542	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	271	2	no	—
89	C₄⋊C₄	—	542	4	no	—
90	C₄⋊C₄	—	542	2	no	—
91	C₄²	—	542	2	no	—
92	C₄²	—	542	2	no	—
93	C₄²	C₂²×D₂₇₁	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	271	4	no	—
97	Q₈	—	542	2	no	—
98	Q₈	—	2168	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	542	2	no	—
104	C₂×C₄	—	542	2	no	—
105	C₂×C₄	D₄×D₂₇₁	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₂₇₁	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	1084	2	no	—
109	C₂²	—	8	2	no	—
110	C₂²	—	8	1	no	—
111	C₂²	#22576128	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	542	4	no	—
115	C₄	—	1084	2	no	—
116	C₄	SD₁₆×D₂₇₁	1	2	yes	—
117	C₄	#22576189	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#31774034	1	2	yes	—
120	C₂	#31774064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31774627

C₂

yes

maximal, characteristic, complemented

#31774094

C₂

yes

maximal

#31773646

C₂

yes

maximal, characteristic, complemented

#31773629

C₂

yes

maximal, complemented

#492×C₂₇₁

C₂

yes

maximal, characteristic

C₂×#16864642

—

#22576159

—

#22576158

—

C₄×Dic₅₄₂

C₂²

yes

—

#22576125

—

C₄⋊(C₂₇₁⋊C₈)

C₂²

yes

—

#22576081

—

C₂×SD₁₆×C₂₇₁

—

C₄⋊Q₈×C₂₇₁

C₂²

yes

characteristic

#127×C₂₇₁

C₂²

yes

characteristic

C₄×C₈×C₂₇₁

C₂²

yes

characteristic

C₂×Dic₅₄₂

—

C₂×Dic₅₄₂

D₄

yes

—

#16864642

—

C₂×C₂₇₁⋊C₈

—

C₄⋊Dic₂₇₁

D₄

yes

—

Dic₂₇₁⋊C₄

—

Dic₂₇₁⋊C₄

—

C₄×Dic₂₇₁

—

C₄×Dic₂₇₁

—

C₂×Q₈×C₂₇₁

D₄

yes

—

C₂×D₄×C₂₇₁

—

C₂×D₄×C₂₇₁

—

SD₁₆×C₂₇₁

—

C₂×C₈×C₂₇₁

—

C₄⋊C₄×C₂₇₁

—

C₄²×C₂₇₁

C₂³

yes

characteristic

Dic₅₄₂

—

Dic₅₄₂

—

Dic₅₄₂

SD₁₆

yes

complemented

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

—

C₂₇₁⋊C₈

—

C₂³×C₂₇₁

—

Q₈×C₂₇₁

—

D₄×C₂₇₁

—

D₄×C₂₇₁

—

D₄×C₂₇₁

—

C₂×C₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

—

C₂×C₄×C₂₇₁

C₂×D₄

yes

characteristic

C₈×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

#120

yes

characteristic

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂×SD₁₆

yes

—

C₄×C₂₇₁

#137

yes

—

Dic₂₇₁

—

Dic₂₇₁

—

C₂×C₂₇₁

—

C₂×C₂₇₁

#448

yes

—

C₂×C₂₇₁

#453

yes

characteristic

C₂₇₁

#1623

yes

characteristic, complemented, Sylow

#1623

—

271

maximal, Sylow

Q₈²

—

271

—

#492

D₂₇₁

yes

characteristic

#460

—

271

—

#333

—

271

—

Q₈⋊C₈

—

271

—

C₂×SD₁₆

—

C₂×SD₁₆

—

542

—

C₄⋊Q₈

—

542

—

C₄⋊Q₈

—

542

—

C₄⋊Q₈

C₂×D₂₇₁

yes

characteristic

#127

C₂×D₂₇₁

yes

characteristic

C₄×Q₈

—

271

—

C₄×Q₈

—

542

—

C₄⋊C₈

—

271

—

#102

—

542

—

C₄×C₈

C₂×D₂₇₁

yes

characteristic

C₂×Q₈

—

271

—

C₂×Q₈

—

542

—

C₂×Q₈

C₂²⋊D₂₇₁

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

1084

—

C₂×C₈

—

C₂×C₈

—

542

—

C₄⋊C₄

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

542

—

C₄⋊C₄

—

542

—

C₄²

—

542

—

C₄²

—

542

—

C₄²

C₂²×D₂₇₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

271

—

Q₈

—

542

—

Q₈

—

2168

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

542

—

104

C₂×C₄

—

542

—

105

C₂×C₄

D₄×D₂₇₁

yes

—

106

C₂×C₄

C₂×C₂²⋊D₂₇₁

yes

characteristic

107

C₈

—

108

C₈

—

1084

—

109

C₂²

—

110

C₂²

—

111

C₂²

#22576128

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

542

—

115

C₄

—

1084

—

116

C₄

SD₁₆×D₂₇₁

yes

—

117

C₄

#22576189

yes

—

118

C₂

—

119

C₂

#31774034

yes

—

120

C₂

#31774064

yes

characteristic

	subgroup	G/H
center	C₂²	#22576128
commutator, cosocle	C₄²×C₂₇₁	C₂³
Frattini	C₄²	C₂²×D₂₇₁
Fitting	#492×C₂₇₁	C₂
socle	C₂²×C₂₇₁	#120
prime radical	#127×C₂₇₁	C₂²
hypercenter	#492	D₂₇₁
hypocenter	C₂₇₁	#1623
LCS	G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22576128

commutator, cosocle

C₄²×C₂₇₁

C₂³

Frattini

C₄²

C₂²×D₂₇₁

Fitting

#492×C₂₇₁

C₂

socle

C₂²×C₂₇₁

#120

prime radical

#127×C₂₇₁

C₂²

D₂₇₁

C₂₇₁

G ▷ C₄²×C₂₇₁ ▷ C₂²×C₂₇₁ ▷ C₂₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #31774627	C₂	semidirect
3. #31773646	C₂	semidirect
4. #31773629	C₂	semidirect
35. Dic₅₄₂	SD₁₆	semidirect
60. C₂₇₁	#1623	semidirect

H ≅ G/N

type

1. #31774627

C₂

semidirect

3. #31773646

C₂

semidirect

4. #31773629

C₂

semidirect

35. Dic₅₄₂

SD₁₆

semidirect

60. C₂₇₁

#1623

semidirect

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

subgroup	index
#31774627	2
#31774094	2
#31773646	2
#31773629	2
#492×C₂₇₁	2
#1623	271

subgroup

index

#492×C₂₇₁

271

Subgroups — #46835440

Conjugacy classes of subgroups

Complements

Maximal subgroups