Group #47983991 — subgroups

Subgroups — #47983991

Order 35456.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#22579877	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579802	C₂	1	4	yes	maximal, complemented
3	#31799868	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#97×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₁₀₈	—	2	2	no	—
6	C₂²×D₁₁₀₈	C₂²	1	2	yes	—
7	#22579877	—	2	2	no	—
8	#22579877	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
10	#22579802	—	2	8	no	—
11	#22579787	—	2	2	no	—
12	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
14	#97×C₂₇₇	—	2	2	no	—
15	C₂³×D₂₇₇	—	2	2	no	—
16	C₂×D₁₁₀₈	—	2	8	no	—
17	C₂×D₁₁₀₈	—	2	4	no	—
18	C₂×D₁₁₀₈	—	2	2	no	—
19	C₂×D₁₁₀₈	—	2	4	no	—
20	C₂×D₁₁₀₈	—	4	8	no	—
21	C₂×D₁₁₀₈	C₂×C₄	1	4	yes	—
22	C₂²×Dic₂₇₇	—	2	2	no	—
23	C₄⋊Dic₂₇₇	—	2	4	no	—
24	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
25	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₂₇₇	—	2	2	no	—
27	C₂×C₈×C₂₇₇	—	2	2	no	—
28	C₂×C₈×C₂₇₇	D₄	1	4	yes	complemented
29	C₄²×C₂₇₇	C₂×C₄	1	2	yes	—
30	C₄²×C₂₇₇	D₄	1	2	yes	—
31	C₂²×D₂₇₇	—	2	4	no	—
32	C₂²×D₂₇₇	—	4	2	no	—
33	C₂²×D₂₇₇	—	4	2	no	—
34	C₂²×D₂₇₇	—	4	8	no	—
35	D₁₁₀₈	—	2	8	no	—
36	D₁₁₀₈	—	4	16	no	—
37	D₁₁₀₈	—	4	4	no	—
38	C₂×Dic₂₇₇	—	4	2	no	—
39	C₂×Dic₂₇₇	—	4	4	no	—
40	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₇₇	—	2	4	no	—
42	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₇₇	#41	1	2	yes	—
47	C₈×C₂₇₇	—	2	4	no	—
48	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₂₇₇	#124	1	1	yes	characteristic
50	C₂²×C₂₇₇	#127	1	1	yes	characteristic
51	C₂²×C₂₇₇	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×D₂₇₇	—	4	4	no	—
53	C₂×D₂₇₇	—	4	8	no	—
54	C₂×D₂₇₇	—	8	4	no	—
55	C₂×D₂₇₇	—	8	4	no	—
56	C₄×C₂₇₇	—	2	4	no	—
57	C₄×C₂₇₇	C₄×D₄	1	2	yes	—
58	C₄×C₂₇₇	#121	1	2	yes	—
59	Dic₂₇₇	—	8	2	no	—
60	C₂×C₂₇₇	#390	1	1	yes	characteristic
61	C₂×C₂₇₇	#418	1	2	yes	—
62	C₂×C₂₇₇	#490	1	2	yes	—
63	C₂×C₂₇₇	#496	1	2	yes	—
64	D₂₇₇	—	8	4	no	—
65	C₂₇₇	#1961	1	1	yes	characteristic, complemented, Sylow
66	#1961	—	277	1	no	maximal, Sylow
67	C₂×#127	—	277	1	no	—
68	C₂×#102	—	277	4	no	—
69	C₂×#97	D₂₇₇	1	1	yes	characteristic, complemented
70	#382	—	277	1	no	—
71	C₂²×D₄	—	277	2	no	—
72	C₂²×D₄	—	554	2	no	—
73	C₂²×C₈	C₂×D₂₇₇	1	2	yes	—
74	#127	—	277	4	no	—
75	#127	—	554	2	no	—
76	C₂×C₄⋊C₄	—	277	2	no	—
77	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
78	#102	—	554	8	no	—
79	#97	—	2	2	no	—
80	#95	—	554	2	no	—
81	C₂⁴	—	554	2	no	—
82	C₂×D₄	—	277	4	no	—
83	C₂×D₄	—	554	8	no	—
84	C₂×D₄	—	554	2	no	—
85	C₂×D₄	—	554	4	no	—
86	C₂×D₄	—	554	4	no	—
87	C₂×D₄	—	1108	8	no	—
88	C₂²×C₄	—	554	2	no	—
89	C₂²×C₄	D₁₁₀₈	1	2	yes	—
90	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₁₁₀₈	1	4	yes	complemented
94	C₄⋊C₄	—	554	4	no	—
95	C₄²	C₄×D₂₇₇	1	2	yes	—
96	C₄²	C₂²⋊D₂₇₇	1	2	yes	—
97	C₂³	—	554	4	no	—
98	C₂³	—	1108	2	no	—
99	C₂³	—	1108	2	no	—
100	C₂³	—	1108	8	no	—
101	C₂³	C₂×D₁₁₀₈	1	1	yes	characteristic
102	D₄	—	554	8	no	—
103	D₄	—	1108	16	no	—
104	D₄	—	1108	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	1108	2	no	—
107	C₂×C₄	—	1108	4	no	—
108	C₂×C₄	#17984540	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
111	C₂×C₄	C₂×D₁₁₀₈	1	2	yes	—
112	C₂×C₄	#17984563	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₂²	—	1108	4	no	—
115	C₂²	—	1108	8	no	—
116	C₂²	—	2216	4	no	—
117	C₂²	—	2216	4	no	—
118	C₂²	C₂×#17984540	1	1	yes	characteristic
119	C₂²	#22579868	1	1	yes	characteristic
120	C₂²	#22579877	1	1	yes	characteristic
121	C₂²	#22579908	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	2216	2	no	—
124	C₄	C₄×D₁₁₀₈	1	2	yes	—
125	C₄	#22579854	1	2	yes	—
126	C₂	—	2216	4	no	—
127	C₂	#31799904	1	1	yes	characteristic
128	C₂	#31800008	1	2	yes	—
129	C₂	#31800367	1	2	yes	—
130	C₂	#31800391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579877

C₂

yes

maximal, characteristic

C₂×#22579802

C₂

yes

maximal, complemented

#31799868

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₁₀₈

—

C₂²×D₁₁₀₈

C₂²

yes

—

#22579877

—

#22579877

C₄

yes

complemented

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

#22579802

—

#22579787

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

#97×C₂₇₇

—

C₂³×D₂₇₇

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

—

C₂×D₁₁₀₈

C₂×C₄

yes

—

C₂²×Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

D₄

yes

complemented

C₄²×C₂₇₇

C₂×C₄

yes

—

C₄²×C₂₇₇

D₄

yes

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

D₁₁₀₈

—

D₁₁₀₈

—

D₁₁₀₈

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂²⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₄×D₄

yes

—

C₄×C₂₇₇

#121

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

#390

yes

characteristic

C₂×C₂₇₇

#418

yes

—

C₂×C₂₇₇

#490

yes

—

C₂×C₂₇₇

#496

yes

—

D₂₇₇

—

C₂₇₇

#1961

yes

characteristic, complemented, Sylow

#1961

—

277

maximal, Sylow

C₂×#127

—

277

—

C₂×#102

—

277

—

C₂×#97

D₂₇₇

yes

characteristic, complemented

#382

—

277

—

C₂²×D₄

—

277

—

C₂²×D₄

—

554

—

C₂²×C₈

C₂×D₂₇₇

yes

—

#127

—

277

—

#127

—

554

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#102

—

554

—

#97

—

#95

—

554

—

C₂⁴

—

554

—

C₂×D₄

—

277

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

554

—

C₂×D₄

—

1108

—

C₂²×C₄

—

554

—

C₂²×C₄

D₁₁₀₈

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₁₀₈

yes

complemented

C₄⋊C₄

—

554

—

C₄²

C₄×D₂₇₇

yes

—

C₄²

C₂²⋊D₂₇₇

yes

—

C₂³

—

554

—

C₂³

—

1108

—

C₂³

—

1108

—

100

C₂³

—

1108

—

101

C₂³

C₂×D₁₁₀₈

yes

characteristic

102

D₄

—

554

—

103

D₄

—

1108

—

104

D₄

—

1108

—

105

C₂×C₄

—

106

C₂×C₄

—

1108

—

107

C₂×C₄

—

1108

—

108

C₂×C₄

#17984540

yes

—

109

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

111

C₂×C₄

C₂×D₁₁₀₈

yes

—

112

C₂×C₄

#17984563

yes

—

113

C₈

—

114

C₂²

—

1108

—

115

C₂²

—

1108

—

116

C₂²

—

2216

—

117

C₂²

—

2216

—

118

C₂²

C₂×#17984540

yes

characteristic

119

C₂²

#22579868

yes

characteristic

120

C₂²

#22579877

yes

characteristic

121

C₂²

#22579908

yes

—

122

C₄

—

123

C₄

—

2216

—

124

C₄

C₄×D₁₁₀₈

yes

—

125

C₄

#22579854

yes

—

126

C₂

—

2216

—

127

C₂

#31799904

yes

characteristic

128

C₂

#31800008

yes

—

129

C₂

#31800367

yes

—

130

C₂

#31800391

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₀₈
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	C₂×#97×C₂₇₇	C₂
socle	C₂³×C₂₇₇	C₂×D₄
prime radical	C₂×#22579877	C₂
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×#97	D₂₇₇
hypocenter	C₂₇₇	#1961
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₁₀₈

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

C₂×#97×C₂₇₇

C₂

socle

C₂³×C₂₇₇

C₂×D₄

prime radical

C₂×#22579877

C₂

cosocle

C₂²×C₄×C₂₇₇

C₂³

hypercenter

C₂×#97

D₂₇₇

hypocenter

C₂₇₇

#1961

LCS

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#22579802	C₂	semidirect
3. #31799868	C₂	semidirect
4. C₂×#97×C₂₇₇	C₂	semidirect
8. #22579877	C₄	semidirect
28. C₂×C₈×C₂₇₇	D₄	semidirect
65. C₂₇₇	#1961	semidirect
69. C₂×#97	D₂₇₇	semidirect
93. C₂×C₈	D₁₁₀₈	semidirect

H ≅ G/N

type

2. C₂×#22579802

C₂

semidirect

3. #31799868

C₂

semidirect

4. C₂×#97×C₂₇₇

C₂

semidirect

8. #22579877

C₄

semidirect

28. C₂×C₈×C₂₇₇

D₄

semidirect

65. C₂₇₇

#1961

semidirect

69. C₂×#97

D₂₇₇

semidirect

93. C₂×C₈

D₁₁₀₈

semidirect

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

subgroup	index
C₂×#22579877	2
C₂×#22579802	2
#31799868	2
C₂×#97×C₂₇₇	2
#1961	277

subgroup

index

C₂×#97×C₂₇₇

277

Subgroups — #47983991

Conjugacy classes of subgroups

Complements

Maximal subgroups