Group #47983328 — subgroups

Subgroups — #47983328

Order 35456.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#22579813	C₂	1	2	yes	maximal
2	C₂×#22579812	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
4	#31799867	C₂	1	2	yes	maximal
5	C₂×#106×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555660	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
9	#22579813	—	2	4	no	—
10	#22579812	—	2	2	no	—
11	C₈×Dic₂₇₇	C₄	1	4	yes	complemented
12	#22579787	—	2	4	no	—
13	C₂²×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
15	#106×C₂₇₇	—	2	2	no	—
16	C₂²×Dic₂₇₇	—	2	2	no	—
17	C₂²×Dic₂₇₇	D₄	1	2	yes	—
18	C₂×#14555660	C₂×C₄	1	2	yes	—
19	C₂×#14555660	D₄	1	2	yes	—
20	C₂×#14555660	Q₈	1	2	yes	—
21	C₄⋊Dic₂₇₇	—	2	2	no	—
22	C₄⋊Dic₂₇₇	—	2	2	no	—
23	C₄×Dic₂₇₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₇₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₇₇	—	2	2	no	—
26	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₂₇₇	D₄	1	2	yes	—
29	C₂×C₈×C₂₇₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₂₇₇	—	2	2	no	—
31	C₄⋊C₄×C₂₇₇	—	2	2	no	—
32	C₂×Dic₂₇₇	—	2	4	no	—
33	C₂×Dic₂₇₇	—	4	2	no	—
34	C₂×Dic₂₇₇	—	4	4	no	—
35	C₂×Dic₂₇₇	C₄⋊C₄	1	4	yes	—
36	#14555660	C₄⋊C₄	1	4	yes	—
37	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₇₇	—	4	2	no	—
39	C₂×C₄×C₂₇₇	—	4	4	no	—
40	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₂₇₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₇₇	#127	1	1	yes	characteristic
49	C₂²×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₇₇	#135	1	4	yes	—
51	C₄×C₂₇₇	—	8	2	no	—
52	C₄×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
55	C₄×C₂₇₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₂₇₇	—	2	4	no	—
57	Dic₂₇₇	—	8	2	no	—
58	C₂×C₂₇₇	#384	1	1	yes	characteristic
59	C₂×C₂₇₇	#427	1	2	yes	—
60	C₂×C₂₇₇	#495	1	2	yes	—
61	C₂×C₂₇₇	#499	1	2	yes	—
62	C₂₇₇	#1846	1	1	yes	characteristic, complemented, Sylow
63	#1846	—	277	1	no	maximal, Sylow
64	C₂×#107	—	277	2	no	—
65	C₂×#106	—	277	1	no	—
66	C₂×#106	D₂₇₇	1	1	yes	characteristic
67	C₂×C₄×C₈	—	277	1	no	—
68	#382	—	277	2	no	—
69	C₂²×C₈	—	277	1	no	—
70	C₂²×C₈	C₂×D₂₇₇	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	277	2	no	—
72	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
73	C₂×C₄²	—	277	1	no	—
74	#107	—	554	4	no	—
75	#106	—	2	2	no	—
76	#106	—	554	2	no	—
77	C₄×C₈	—	277	4	no	—
78	#95	—	554	4	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	277	2	no	—
81	C₂²×C₄	—	554	2	no	—
82	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
83	C₂×C₈	—	277	2	no	—
84	C₂×C₈	—	277	2	no	—
85	C₂×C₈	—	277	2	no	—
86	C₂×C₈	C₄×D₂₇₇	1	2	yes	—
87	C₂×C₈	C₂²⋊D₂₇₇	1	2	yes	—
88	C₂×C₈	Dic₅₅₄	1	2	yes	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	554	2	no	—
92	C₄⋊C₄	—	554	2	no	—
93	C₄²	—	277	2	no	—
94	C₄²	—	277	2	no	—
95	C₂³	D₄×D₂₇₇	1	1	yes	characteristic
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	277	4	no	—
99	C₂×C₄	—	554	4	no	—
100	C₂×C₄	—	1108	2	no	—
101	C₂×C₄	—	1108	4	no	—
102	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
103	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
108	C₈	—	277	4	no	—
109	C₈	#17984542	1	4	yes	—
110	C₂²	C₄⋊C₄×D₂₇₇	1	1	yes	characteristic
111	C₂²	#22579876	1	1	yes	characteristic
112	C₂²	#22579879	1	1	yes	characteristic
113	C₂²	#22579901	1	4	yes	—
114	C₄	—	8	2	no	—
115	C₄	—	554	4	no	—
116	C₄	—	2216	2	no	—
117	C₄	C₄⋊C₄×D₂₇₇	1	1	yes	characteristic
118	C₄	C₂×#17984542	1	1	yes	characteristic
119	C₄	#22579878	1	1	yes	characteristic
120	C₄	#22579879	1	1	yes	characteristic
121	C₂	#31799871	1	1	yes	characteristic
122	C₂	#31800037	1	2	yes	—
123	C₂	#31800383	1	2	yes	—
124	C₂	#31800402	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579813

C₂

yes

maximal

C₂×#22579812

C₂

yes

maximal, characteristic

C₂×C₈×Dic₂₇₇

C₂

yes

maximal, characteristic

#31799867

C₂

yes

maximal

C₂×#106×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579813

—

#22579812

—

C₈×Dic₂₇₇

C₄

yes

complemented

#22579787

—

C₂²×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

#106×C₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

D₄

yes

—

C₂×#14555660

C₂×C₄

yes

—

C₂×#14555660

D₄

yes

—

C₂×#14555660

Q₈

yes

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄×Dic₂₇₇

C₂×C₄

yes

—

C₄×Dic₂₇₇

Q₈

yes

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₂×C₈×C₂₇₇

D₄

yes

—

C₂×C₈×C₂₇₇

Q₈

yes

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₄⋊C₄

yes

—

#14555660

C₄⋊C₄

yes

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₈×C₂₇₇

C₄⋊C₄

yes

—

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₇

#127

yes

characteristic

C₂²×C₂₇₇

C₄⋊Q₈

yes

characteristic

C₂²×C₂₇₇

#135

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

C₄×C₂₇₇

C₄⋊Q₈

yes

characteristic

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

#384

yes

characteristic

C₂×C₂₇₇

#427

yes

—

C₂×C₂₇₇

#495

yes

—

C₂×C₂₇₇

#499

yes

—

C₂₇₇

#1846

yes

characteristic, complemented, Sylow

#1846

—

277

maximal, Sylow

C₂×#107

—

277

—

C₂×#106

—

277

—

C₂×#106

D₂₇₇

yes

characteristic

C₂×C₄×C₈

—

277

—

#382

—

277

—

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

characteristic

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

—

C₂×C₄²

—

277

—

#107

—

554

—

#106

—

#106

—

554

—

C₄×C₈

—

277

—

#95

—

554

—

C₂²×C₄

—

C₂²×C₄

—

277

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

—

277

—

C₂×C₈

C₄×D₂₇₇

yes

—

C₂×C₈

C₂²⋊D₂₇₇

yes

—

C₂×C₈

Dic₅₅₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

554

—

C₄⋊C₄

—

554

—

C₄²

—

277

—

C₄²

—

277

—

C₂³

D₄×D₂₇₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

277

—

C₂×C₄

—

554

—

100

C₂×C₄

—

1108

—

101

C₂×C₄

—

1108

—

102

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

103

C₂×C₄

D₄×D₂₇₇

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

105

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

106

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

107

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

108

C₈

—

277

—

109

C₈

#17984542

yes

—

110

C₂²

C₄⋊C₄×D₂₇₇

yes

characteristic

111

C₂²

#22579876

yes

characteristic

112

C₂²

#22579879

yes

characteristic

113

C₂²

#22579901

yes

—

114

C₄

—

115

C₄

—

554

—

116

C₄

—

2216

—

117

C₄

C₄⋊C₄×D₂₇₇

yes

characteristic

118

C₄

C₂×#17984542

yes

characteristic

119

C₄

#22579878

yes

characteristic

120

C₄

#22579879

yes

characteristic

121

C₂

#31799871

yes

characteristic

122

C₂

#31800037

yes

—

123

C₂

#31800383

yes

—

124

C₂

#31800402

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₇₇
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	C₂×#106×C₂₇₇	C₂
socle, prime radical	C₂³×C₂₇₇	C₂×D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×#106	D₂₇₇
hypocenter	C₂₇₇	#1846
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₂₇₇

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

C₂×#106×C₂₇₇

C₂

socle, prime radical

C₂³×C₂₇₇

C₂×D₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
11. C₈×Dic₂₇₇	C₄	semidirect
62. C₂₇₇	#1846	semidirect

H ≅ G/N

type

11. C₈×Dic₂₇₇

C₄

semidirect

62. C₂₇₇

#1846

semidirect

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

subgroup	index
C₂×#22579813	2
C₂×#22579812	2
C₂×C₈×Dic₂₇₇	2
#31799867	2
C₂×#106×C₂₇₇	2
#1846	277

subgroup

index

C₂×C₈×Dic₂₇₇

C₂×#106×C₂₇₇

277

Subgroups — #47983328

Conjugacy classes of subgroups

Complements

Maximal subgroups