Group #47983216 — subgroups

Subgroups — #47983216

Order 35456.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#22579888	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579815	C₂	1	2	yes	maximal, complemented
3	#31799984	C₂	1	2	yes	maximal
4	#31799760	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17984563	C₂²	1	1	yes	characteristic
7	#22579888	—	2	4	no	—
8	C₂×#17984547	C₂²	1	2	yes	—
9	C₂×#17984546	C₄	1	2	yes	—
10	C₂²×#14555660	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₂₇₇	C₄	1	2	yes	—
12	#22579815	—	2	4	no	—
13	#22579798	—	2	2	no	—
14	#22579796	—	2	2	no	—
15	C₂×M₁₆×C₂₇₇	C₂²	1	2	yes	—
16	C₂×M₁₆×C₂₇₇	C₂²	1	1	yes	characteristic
17	#17984563	—	2	2	no	—
18	#17984563	—	2	2	no	—
19	C₂×Dic₅₅₄	D₄	1	1	yes	characteristic
20	C₂×D₁₁₀₈	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₂₇₇	Q₈	1	2	yes	—
22	C₂×C₄×D₂₇₇	C₂×C₄	1	2	yes	—
23	#17984547	—	2	2	no	—
24	#17984547	—	2	2	no	—
25	#17984546	—	2	4	no	—
26	C₂×#14555660	—	2	2	no	—
27	C₂×#14555660	C₂×C₄	1	1	yes	characteristic
28	C₂×#14555660	C₂×C₄	1	1	yes	characteristic
29	C₂×#14555660	D₄	1	4	yes	—
30	C₈×D₂₇₇	—	2	4	no	—
31	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₂₇₇	—	2	2	no	—
33	M₁₆×C₂₇₇	—	2	2	no	—
34	M₁₆×C₂₇₇	—	2	2	no	—
35	C₂×C₈×C₂₇₇	—	2	2	no	—
36	C₂×C₈×C₂₇₇	C₂×C₄	1	2	yes	—
37	C₂²×D₂₇₇	C₄⋊C₄	1	2	yes	—
38	Dic₅₅₄	—	2	2	no	—
39	D₁₁₀₈	—	2	2	no	—
40	C₂²⋊D₂₇₇	—	2	4	no	—
41	C₂×Dic₂₇₇	C₄⋊C₄	1	2	yes	—
42	C₄×D₂₇₇	—	2	2	no	—
43	C₄×D₂₇₇	—	2	2	no	—
44	#14555660	—	4	2	no	—
45	#14555660	C₂²⋊C₄	1	2	yes	—
46	#14555660	C₂²⋊C₄	1	2	yes	—
47	C₂³×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₇₇	#41	1	2	yes	—
52	C₈×C₂₇₇	—	2	2	no	—
53	C₈×C₂₇₇	—	4	2	no	—
54	C₂²×C₂₇₇	—	4	1	no	—
55	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₂₇₇	—	2	2	no	—
58	C₂×D₂₇₇	—	4	2	no	—
59	C₄×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₂₇₇	#117	1	1	yes	characteristic
61	C₄×C₂₇₇	#121	1	2	yes	—
62	Dic₂₇₇	—	2	2	no	—
63	C₂×C₂₇₇	—	2	1	no	—
64	C₂×C₂₇₇	—	2	2	no	—
65	C₂×C₂₇₇	#380	1	1	yes	characteristic
66	D₂₇₇	—	4	2	no	—
67	C₂₇₇	#1813	1	1	yes	characteristic, complemented, Sylow
68	#1813	—	277	1	no	maximal, Sylow
69	C₂×#131	—	277	1	no	—
70	C₂×#108	—	277	2	no	—
71	#413	—	277	2	no	—
72	#338	—	277	1	no	—
73	#338	D₂₇₇	1	1	yes	characteristic, complemented
74	C₂×#41	—	277	1	no	—
75	#131	—	554	4	no	—
76	C₂×M₁₆	—	277	2	no	—
77	C₂×M₁₆	—	277	2	no	—
78	C₂×M₁₆	C₂×D₂₇₇	1	2	yes	—
79	C₂×M₁₆	C₂×D₂₇₇	1	1	yes	characteristic
80	C₂²×C₈	—	277	1	no	—
81	C₂²×C₈	—	277	2	no	—
82	#108	—	554	4	no	—
83	#101	—	554	2	no	—
84	#100	—	554	2	no	—
85	#41	—	554	2	no	—
86	#41	—	554	2	no	—
87	C₂×Q₈	—	277	1	no	—
88	C₂×D₄	—	277	2	no	—
89	C₂×D₄	—	277	1	no	—
90	C₂²×C₄	—	277	2	no	—
91	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	554	4	no	—
96	M₁₆	—	554	2	no	—
97	M₁₆	—	554	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	277	1	no	—
100	C₂×C₈	—	277	1	no	—
101	C₂×C₈	—	277	4	no	—
102	C₂×C₈	—	554	4	no	—
103	C₂×C₈	—	554	2	no	—
104	C₂×C₈	C₄×D₂₇₇	1	2	yes	—
105	C₂³	—	277	2	no	—
106	C₂³	Q₈×D₂₇₇	1	1	yes	characteristic
107	Q₈	—	554	2	no	—
108	D₄	—	554	4	no	—
109	D₄	—	554	2	no	—
110	C₂×C₄	—	277	2	no	—
111	C₂×C₄	—	554	2	no	—
112	C₂×C₄	—	554	2	no	—
113	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
114	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
115	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
116	C₂×C₄	#17984563	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	277	2	no	—
120	C₈	—	277	2	no	—
121	C₈	—	1108	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	554	2	no	—
124	C₂²	—	1108	2	no	—
125	C₂²	C₄⋊C₄×D₂₇₇	1	1	yes	characteristic
126	C₂²	#22579858	1	2	yes	—
127	C₄	—	554	2	no	—
128	C₄	C₂²⋊C₄×D₂₇₇	1	1	yes	characteristic
129	C₄	#22579837	1	1	yes	characteristic
130	C₄	#22579853	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	1108	2	no	—
134	C₂	#31799861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579888

C₂

yes

maximal, characteristic

C₂×#22579815

C₂

yes

maximal, complemented

#31799984

C₂

yes

maximal

#31799760

C₂

yes

maximal, characteristic, complemented

#338×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂×#17984563

C₂²

yes

characteristic

#22579888

—

C₂×#17984547

C₂²

yes

—

C₂×#17984546

C₄

yes

—

C₂²×#14555660

C₂²

yes

characteristic

C₂×C₈×D₂₇₇

C₄

yes

—

#22579815

—

#22579798

—

#22579796

—

C₂×M₁₆×C₂₇₇

C₂²

yes

—

C₂×M₁₆×C₂₇₇

C₂²

yes

characteristic

#17984563

—

#17984563

—

C₂×Dic₅₅₄

D₄

yes

characteristic

C₂×D₁₁₀₈

D₄

yes

characteristic

C₂×C₂²⋊D₂₇₇

Q₈

yes

—

C₂×C₄×D₂₇₇

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#14555660

C₂×C₄

yes

characteristic

C₂×#14555660

D₄

yes

—

C₈×D₂₇₇

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

—

M₁₆×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

C₂×C₄

yes

—

C₂²×D₂₇₇

C₄⋊C₄

yes

—

Dic₅₅₄

—

D₁₁₀₈

—

C₂²⋊D₂₇₇

—

C₂×Dic₂₇₇

C₄⋊C₄

yes

—

C₄×D₂₇₇

—

C₄×D₂₇₇

—

#14555660

—

#14555660

C₂²⋊C₄

yes

—

#14555660

C₂²⋊C₄

yes

—

C₂³×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₂²⋊Q₈

yes

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

C₂×C₂²⋊C₄

yes

C₄×C₂₇₇

yes

C₄×C₂₇₇

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#380

yes

characteristic

D₂₇₇

—

C₂₇₇

#1813

yes

characteristic, complemented, Sylow

#1813

—

277

maximal, Sylow

C₂×#131

—

277

—

C₂×#108

—

277

—

#413

—

277

—

#338

—

277

—

#338

D₂₇₇

yes

characteristic, complemented

C₂×#41

—

277

—

#131

—

554

—

C₂×M₁₆

—

277

—

C₂×M₁₆

—

277

—

C₂×M₁₆

C₂×D₂₇₇

yes

—

C₂×M₁₆

C₂×D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

C₂²×C₈

—

277

—

#108

—

554

—

#101

—

554

—

#100

—

554

—

#41

—

554

—

#41

—

554

—

C₂×Q₈

—

277

—

C₂×D₄

—

277

—

C₂×D₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

554

—

M₁₆

—

554

—

M₁₆

—

554

—

C₂×C₈

—

C₂×C₈

—

277

—

100

C₂×C₈

—

277

—

101

C₂×C₈

—

277

—

102

C₂×C₈

—

554

—

103

C₂×C₈

—

554

—

104

C₂×C₈

C₄×D₂₇₇

yes

—

105

C₂³

—

277

—

106

C₂³

Q₈×D₂₇₇

yes

characteristic

107

Q₈

—

554

—

108

D₄

—

554

—

109

D₄

—

554

—

110

C₂×C₄

—

277

—

111

C₂×C₄

—

554

—

112

C₂×C₄

—

554

—

113

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

114

C₂×C₄

D₄×D₂₇₇

yes

—

115

C₂×C₄

D₄×D₂₇₇

yes

characteristic

116

C₂×C₄

#17984563

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

277

—

120

C₈

—

277

—

121

C₈

—

1108

—

122

C₂²

—

123

C₂²

—

554

—

124

C₂²

—

1108

—

125

C₂²

C₄⋊C₄×D₂₇₇

yes

characteristic

126

C₂²

#22579858

yes

—

127

C₄

—

554

—

128

C₄

C₂²⋊C₄×D₂₇₇

yes

characteristic

129

C₄

#22579837

yes

characteristic

130

C₄

#22579853

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

1108

—

134

C₂

#31799861

yes

characteristic

	subgroup	G/H
center	C₄	#22579837
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#338×C₂₇₇	C₂
socle	C₂×C₂₇₇	#380
prime radical	C₂×#17984563	C₂²
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#338	D₂₇₇
hypocenter	C₂₇₇	#1813
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#22579837

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#338×C₂₇₇

C₂

C₂×C₂₇₇

C₂²

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#22579815	C₂	semidirect
4. #31799760	C₂	semidirect
5. #338×C₂₇₇	C₂	semidirect
67. C₂₇₇	#1813	semidirect
73. #338	D₂₇₇	semidirect

H ≅ G/N

type

2. C₂×#22579815

C₂

semidirect

4. #31799760

C₂

semidirect

5. #338×C₂₇₇

C₂

semidirect

67. C₂₇₇

#1813

semidirect

73. #338

D₂₇₇

semidirect

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

subgroup	index
C₂×#22579888	2
C₂×#22579815	2
#31799984	2
#31799760	2
#338×C₂₇₇	2
#1813	277

subgroup

index

#338×C₂₇₇

277

Subgroups — #47983216

Conjugacy classes of subgroups

Complements

Maximal subgroups