Group #47982248 — subgroups

Subgroups — #47982248

Order 35456.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#31800718	C₂	1	1	yes	maximal, characteristic, complemented
2	#31800194	C₂	1	2	yes	maximal, complemented
3	#31799751	C₂	1	1	yes	maximal, characteristic, complemented
4	#31799731	C₂	1	2	yes	maximal, complemented
5	#492×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17984548	—	2	2	no	—
7	#22579869	—	2	2	no	—
8	#22579855	—	2	2	no	—
9	C₄×D₁₁₀₈	C₂²	1	2	yes	—
10	#22579838	—	2	2	no	—
11	C₄⋊#14555660	C₂²	1	2	yes	—
12	#22579801	—	2	2	no	—
13	C₂×SD₁₆×C₂₇₇	—	2	2	no	—
14	C₄⋊Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
15	#127×C₂₇₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×D₁₁₀₈	—	2	2	no	—
18	C₂×D₁₁₀₈	D₄	1	2	yes	—
19	C₂×C₄×D₂₇₇	—	2	2	no	—
20	#17984548	—	4	4	no	—
21	C₂×#14555660	—	2	2	no	—
22	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
23	#17984540	—	2	4	no	—
24	#17984540	—	2	2	no	—
25	C₄×Dic₂₇₇	—	2	2	no	—
26	C₂×Q₈×C₂₇₇	D₄	1	2	yes	—
27	C₂×D₄×C₂₇₇	—	2	1	no	—
28	C₂×D₄×C₂₇₇	—	2	2	no	—
29	SD₁₆×C₂₇₇	—	4	4	no	—
30	C₂×C₈×C₂₇₇	—	2	1	no	—
31	C₄⋊C₄×C₂₇₇	—	2	2	no	—
32	C₄²×C₂₇₇	C₂³	1	1	yes	characteristic
33	C₂²×D₂₇₇	—	2	2	no	—
34	D₁₁₀₈	—	2	2	no	—
35	D₁₁₀₈	—	2	2	no	—
36	D₁₁₀₈	—	8	2	no	—
37	C₂×Dic₂₇₇	—	2	2	no	—
38	C₄×D₂₇₇	—	4	2	no	—
39	C₄×D₂₇₇	—	4	2	no	—
40	#14555660	—	4	2	no	—
41	C₂³×C₂₇₇	—	4	1	no	—
42	Q₈×C₂₇₇	—	4	2	no	—
43	D₄×C₂₇₇	—	4	2	no	—
44	D₄×C₂₇₇	—	4	2	no	—
45	D₄×C₂₇₇	—	4	2	no	—
46	C₂×C₄×C₂₇₇	—	2	2	no	—
47	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₂₇₇	—	2	2	no	—
50	C₂²×C₂₇₇	—	8	2	no	—
51	C₂²×C₂₇₇	—	8	1	no	—
52	C₂²×C₂₇₇	#120	1	1	yes	characteristic
53	C₂×D₂₇₇	—	4	2	no	—
54	C₂×D₂₇₇	—	4	2	no	—
55	C₂×D₂₇₇	—	4	2	no	—
56	C₄×C₂₇₇	—	2	1	no	—
57	C₄×C₂₇₇	—	4	2	no	—
58	C₄×C₂₇₇	#135	1	2	yes	—
59	C₄×C₂₇₇	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₂₇₇	—	4	2	no	—
61	C₂×C₂₇₇	—	16	1	no	—
62	C₂×C₂₇₇	#449	1	2	yes	—
63	C₂×C₂₇₇	#453	1	1	yes	characteristic
64	D₂₇₇	—	8	2	no	—
65	C₂₇₇	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	277	1	no	maximal, Sylow
67	#555	—	277	1	no	—
68	#492	D₂₇₇	1	1	yes	characteristic, complemented
69	#459	—	277	2	no	—
70	#333	—	277	1	no	—
71	D₄⋊C₈	—	277	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	554	2	no	—
74	C₄⋊Q₈	C₂×D₂₇₇	1	1	yes	characteristic
75	#127	C₂×D₂₇₇	1	1	yes	characteristic
76	#124	—	554	2	no	—
77	#121	—	554	2	no	—
78	C₄×D₄	—	277	2	no	—
79	#117	—	554	2	no	—
80	C₄⋊C₈	—	277	2	no	—
81	#102	—	554	2	no	—
82	C₄×C₈	C₂×D₂₇₇	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₂₇₇	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	277	2	no	—
87	C₂×D₄	—	554	2	no	—
88	C₂²×C₄	—	554	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	1108	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	554	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	277	2	no	—
95	C₂²⋊C₄	—	554	4	no	—
96	C₂²⋊C₄	—	554	2	no	—
97	C₄²	—	554	2	no	—
98	C₄²	C₂²×D₂₇₇	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	554	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	554	2	no	—
106	D₄	—	554	2	no	—
107	D₄	—	2216	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	554	2	no	—
110	C₂×C₄	—	1108	2	no	—
111	C₂×C₄	—	1108	2	no	—
112	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	1108	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	1108	2	no	—
119	C₂²	—	1108	2	no	—
120	C₂²	—	1108	2	no	—
121	C₂²	#22579848	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	1108	2	no	—
125	C₄	#22579901	1	2	yes	—
126	C₄	#22579907	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	2216	2	no	—
129	C₂	#31800146	1	2	yes	—
130	C₂	#31800169	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800718

C₂

yes

maximal, characteristic, complemented

#31800194

C₂

yes

maximal, complemented

#31799751

C₂

yes

maximal, characteristic, complemented

#31799731

C₂

yes

maximal, complemented

#492×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂×#17984548

—

#22579869

—

#22579855

—

C₄×D₁₁₀₈

C₂²

yes

—

#22579838

—

C₄⋊#14555660

C₂²

yes

—

#22579801

—

C₂×SD₁₆×C₂₇₇

—

C₄⋊Q₈×C₂₇₇

C₂²

yes

characteristic

#127×C₂₇₇

C₂²

yes

characteristic

C₄×C₈×C₂₇₇

C₂²

yes

characteristic

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

D₄

yes

—

C₂×C₄×D₂₇₇

—

#17984548

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

D₄

yes

—

#17984540

—

#17984540

—

C₄×Dic₂₇₇

—

C₂×Q₈×C₂₇₇

D₄

yes

—

C₂×D₄×C₂₇₇

—

C₂×D₄×C₂₇₇

—

SD₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

C₂³

yes

characteristic

C₂²×D₂₇₇

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

C₂×Dic₂₇₇

—

C₄×D₂₇₇

—

C₄×D₂₇₇

—

#14555660

—

C₂³×C₂₇₇

—

Q₈×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

D₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#135

yes

—

C₄×C₂₇₇

AGL(1,ℤ/8)

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#449

yes

—

C₂×C₂₇₇

#453

yes

characteristic

D₂₇₇

—

C₂₇₇

#1624

yes

characteristic, complemented, Sylow

#1624

—

277

maximal, Sylow

#555

—

277

—

#492

D₂₇₇

yes

characteristic, complemented

#459

—

277

—

#333

—

277

—

D₄⋊C₈

—

277

—

C₂×SD₁₆

—

C₂×D₈

—

554

—

C₄⋊Q₈

C₂×D₂₇₇

yes

characteristic

#127

C₂×D₂₇₇

yes

characteristic

#124

—

554

—

#121

—

554

—

C₄×D₄

—

277

—

#117

—

554

—

C₄⋊C₈

—

277

—

#102

—

554

—

C₄×C₈

C₂×D₂₇₇

yes

characteristic

C₂×Q₈

C₂²⋊D₂₇₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

277

—

C₂×D₄

—

554

—

C₂²×C₄

—

554

—

SD₁₆

—

D₈

—

1108

—

C₂×C₈

—

C₂×C₈

—

554

—

C₄⋊C₄

—

C₄⋊C₄

—

277

—

C₂²⋊C₄

—

554

—

C₂²⋊C₄

—

554

—

C₄²

—

554

—

C₄²

C₂²×D₂₇₇

yes

characteristic

C₂³

—

100

C₂³

—

554

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

554

—

106

D₄

—

554

—

107

D₄

—

2216

—

108

C₂×C₄

—

109

C₂×C₄

—

554

—

110

C₂×C₄

—

1108

—

111

C₂×C₄

—

1108

—

112

C₂×C₄

D₄×D₂₇₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

114

C₈

—

115

C₈

—

1108

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

1108

—

119

C₂²

—

1108

—

120

C₂²

—

1108

—

121

C₂²

#22579848

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

1108

—

125

C₄

#22579901

yes

—

126

C₄

#22579907

yes

—

127

C₂

—

128

C₂

—

2216

—

129

C₂

#31800146

yes

—

130

C₂

#31800169

yes

characteristic

	subgroup	G/H
center	C₂²	#22579848
commutator, cosocle	C₄²×C₂₇₇	C₂³
Frattini	C₄²	C₂²×D₂₇₇
Fitting	#492×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#120
hypercenter	#492	D₂₇₇
hypocenter	C₂₇₇	#1624
LCS	G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#22579848

commutator, cosocle

C₄²×C₂₇₇

C₂³

Frattini

C₄²

C₂²×D₂₇₇

Fitting

#492×C₂₇₇

C₂

C₂²×C₂₇₇

D₂₇₇

C₂₇₇

G ▷ C₄²×C₂₇₇ ▷ C₂²×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #31800718	C₂	semidirect
2. #31800194	C₂	semidirect
3. #31799751	C₂	semidirect
4. #31799731	C₂	semidirect
5. #492×C₂₇₇	C₂	semidirect
65. C₂₇₇	#1624	semidirect
68. #492	D₂₇₇	semidirect

H ≅ G/N

type

1. #31800718

C₂

semidirect

2. #31800194

C₂

semidirect

3. #31799751

C₂

semidirect

4. #31799731

C₂

semidirect

5. #492×C₂₇₇

C₂

semidirect

65. C₂₇₇

#1624

semidirect

68. #492

D₂₇₇

semidirect

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

subgroup	index
#31800718	2
#31800194	2
#31799751	2
#31799731	2
#492×C₂₇₇	2
#1624	277

subgroup

index

#492×C₂₇₇

277

Subgroups — #47982248

Conjugacy classes of subgroups

Complements

Maximal subgroups