Group #47984325 — subgroups

Subgroups — #47984325

Order 35456.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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	#31800010	C₂	1	2	yes	maximal
2	#31799907	C₂	1	1	yes	maximal, characteristic
3	#31799771	C₂	1	1	yes	maximal, characteristic
4	#31799761	C₂	1	2	yes	maximal
5	#544×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₅₄	C₂²	1	1	yes	characteristic
7	C₂×#17984547	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
9	#22579803	—	2	2	no	—
10	#22579803	—	2	2	no	—
11	#22579787	—	2	2	no	—
12	C₄⋊Q₈×C₂₇₇	—	2	2	no	—
13	#125×C₂₇₇	—	2	2	no	—
14	C₂²⋊Q₈×C₂₇₇	—	2	2	no	—
15	#117×C₂₇₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₅₅₄	—	2	2	no	—
18	C₂×Dic₅₅₄	—	2	1	no	—
19	C₂×Dic₅₅₄	—	2	2	no	—
20	C₂×Dic₅₅₄	Q₈	1	2	yes	complemented
21	C₂²×Dic₂₇₇	—	2	1	no	—
22	C₂²×Dic₂₇₇	—	2	1	no	—
23	#17984547	—	2	2	no	—
24	#17984547	—	2	2	no	—
25	C₂×#14555660	—	2	2	no	—
26	C₄×Dic₂₇₇	—	2	2	no	—
27	C₂×Q₈×C₂₇₇	—	2	2	no	—
28	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
29	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₂₇₇	—	2	2	no	—
31	C₄⋊C₄×C₂₇₇	—	2	4	no	—
32	C₄⋊C₄×C₂₇₇	—	2	2	no	—
33	C₄⋊C₄×C₂₇₇	D₄	1	2	yes	—
34	C₄⋊C₄×C₂₇₇	D₄	1	2	yes	—
35	C₂²⋊C₄×C₂₇₇	—	2	2	no	—
36	C₄²×C₂₇₇	—	2	2	no	—
37	Dic₅₅₄	—	2	4	no	—
38	Dic₅₅₄	—	4	2	no	—
39	C₂×Dic₂₇₇	—	2	2	no	—
40	C₂×Dic₂₇₇	—	2	2	no	—
41	C₂×Dic₂₇₇	—	4	1	no	—
42	C₂×Dic₂₇₇	—	4	1	no	—
43	C₂×Dic₂₇₇	—	4	1	no	—
44	C₂×Dic₂₇₇	—	4	1	no	—
45	#14555660	—	4	2	no	—
46	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₂₇₇	—	8	2	no	—
48	C₂×C₄×C₂₇₇	—	2	2	no	—
49	C₂×C₄×C₂₇₇	—	2	2	no	—
50	C₂×C₄×C₂₇₇	—	2	2	no	—
51	C₂×C₄×C₂₇₇	—	4	2	no	—
52	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₂₇₇	#41	1	2	yes	—
55	C₂×C₄×C₂₇₇	#41	1	1	yes	characteristic
56	C₂²×C₂₇₇	—	2	1	no	—
57	C₂²×C₂₇₇	—	2	1	no	—
58	C₂²×C₂₇₇	#120	1	1	yes	characteristic
59	C₂²×C₂₇₇	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₂₇₇	#124	1	1	yes	characteristic
61	C₄×C₂₇₇	—	4	2	no	—
62	C₄×C₂₇₇	—	4	2	no	—
63	C₄×C₂₇₇	—	4	2	no	—
64	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₂₇₇	#124	1	2	yes	—
66	Dic₂₇₇	—	4	2	no	—
67	Dic₂₇₇	—	4	2	no	—
68	C₂×C₂₇₇	—	2	1	no	—
69	C₂×C₂₇₇	—	2	1	no	—
70	C₂×C₂₇₇	#393	1	1	yes	characteristic
71	C₂×C₂₇₇	#455	1	1	yes	characteristic
72	C₂×C₂₇₇	#456	1	1	yes	characteristic
73	C₂₇₇	#2028	1	1	yes	characteristic, complemented, Sylow
74	#2028	—	277	1	no	maximal, Sylow
75	#544	D₂₇₇	1	1	yes	characteristic
76	#419	—	277	2	no	—
77	#391	—	277	1	no	—
78	#343	—	277	1	no	—
79	#339	—	277	2	no	—
80	C₂²×Q₈	—	277	1	no	—
81	C₂×M₁₆	—	277	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	#125	—	2	2	no	—
84	C₂²⋊Q₈	—	2	2	no	—
85	#117	C₂×D₂₇₇	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
87	C₂×C₄²	—	277	1	no	—
88	#103	—	554	2	no	—
89	#103	—	554	2	no	—
90	#95	—	554	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	277	2	no	—
93	C₂×Q₈	—	554	1	no	—
94	C₂×Q₈	—	554	2	no	—
95	C₂×Q₈	—	554	2	no	—
96	C₂²×C₄	—	554	1	no	—
97	C₂²×C₄	—	554	1	no	—
98	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
100	M₁₆	—	554	2	no	—
101	M₁₆	—	554	2	no	—
102	C₂×C₈	—	554	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂²⋊D₂₇₇	1	2	yes	—
107	C₄⋊C₄	C₂²⋊D₂₇₇	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	554	2	no	—
111	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
112	Q₈	—	8	2	no	—
113	Q₈	—	554	4	no	—
114	Q₈	—	1108	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	554	2	no	—
120	C₂×C₄	—	554	2	no	—
121	C₂×C₄	—	1108	1	no	—
122	C₂×C₄	—	1108	1	no	—
123	C₂×C₄	—	1108	1	no	—
124	C₂×C₄	—	1108	1	no	—
125	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	2	yes	—
126	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
127	C₂×C₄	#17984564	1	2	yes	—
128	C₂×C₄	#17984564	1	1	yes	characteristic
129	C₈	—	1108	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#22579849	1	1	yes	characteristic
133	C₂²	#22579856	1	1	yes	characteristic
134	C₂²	#22579869	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	1108	2	no	—
139	C₄	—	1108	2	no	—
140	C₄	#22579856	1	2	yes	—
141	C₄	#22579869	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#31799913	1	1	yes	characteristic
145	C₂	#31800180	1	1	yes	characteristic
146	C₂	#31800185	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800010

C₂

yes

maximal

#31799907

C₂

yes

maximal, characteristic

#31799771

C₂

yes

maximal, characteristic

#31799761

C₂

yes

maximal

#544×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×Dic₅₅₄

C₂²

yes

characteristic

C₂×#17984547

C₂²

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

—

—

—

C₄⋊Q₈×C₂₇₇

—

#125×C₂₇₇

—

C₂²⋊Q₈×C₂₇₇

—

#117×C₂₇₇

C₂²

yes

—

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

characteristic

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

—

C₂×Dic₅₅₄

Q₈

yes

complemented

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

#17984547

—

#17984547

—

C₂×#14555660

—

C₄×Dic₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

—

C₄⋊C₄×C₂₇₇

D₄

yes

—

C₄⋊C₄×C₂₇₇

D₄

yes

—

C₂²⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

—

Dic₅₅₄

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₂×C₄×C₂₇₇

#41

yes

characteristic

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#120

yes

characteristic

C₂²×C₂₇₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₇₇

#124

yes

characteristic

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

C₄×C₂₇₇

#124

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂₇₇

yes

characteristic, complemented, Sylow

#2028

—

277

maximal, Sylow

#544

D₂₇₇

yes

characteristic

#419

—

277

—

#391

—

277

—

#343

—

277

—

#339

—

277

—

C₂²×Q₈

—

277

—

C₂×M₁₆

—

277

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₂₇₇

yes

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

characteristic

C₂×C₄²

—

277

—

#103

—

554

—

#103

—

554

—

#95

—

554

—

C₂×Q₈

—

C₂×Q₈

—

277

—

C₂×Q₈

—

554

—

C₂×Q₈

—

554

—

C₂×Q₈

—

554

—

C₂²×C₄

—

554

—

C₂²×C₄

—

554

—

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

100

M₁₆

—

554

—

101

M₁₆

—

554

—

102

C₂×C₈

—

554

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²⋊D₂₇₇

yes

—

107

C₄⋊C₄

C₂²⋊D₂₇₇

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

554

—

111

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

112

Q₈

—

113

Q₈

—

554

—

114

Q₈

—

1108

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

554

—

120

C₂×C₄

—

554

—

121

C₂×C₄

—

1108

—

122

C₂×C₄

—

1108

—

123

C₂×C₄

—

1108

—

124

C₂×C₄

—

1108

—

125

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

—

126

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

127

C₂×C₄

#17984564

yes

—

128

C₂×C₄

#17984564

yes

characteristic

129

C₈

—

1108

—

130

C₂²

—

131

C₂²

—

132

C₂²

#22579849

yes

characteristic

133

C₂²

#22579856

yes

characteristic

134

C₂²

#22579869

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

1108

—

139

C₄

—

1108

—

140

C₄

#22579856

yes

—

141

C₄

#22579869

yes

—

142

C₂

—

143

C₂

—

144

C₂

#31799913

yes

characteristic

145

C₂

#31800180

yes

characteristic

146

C₂

#31800185

yes

characteristic

	subgroup	G/H
center	C₂²	#22579849
commutator, cosocle	C₂²×C₄×C₂₇₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#544×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#120
prime radical	C₂³×C₂₇₇	C₂×D₄
hypercenter	#544	D₂₇₇
hypocenter	C₂₇₇	#2028
LCS	G ▷ C₂²×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#22579849

commutator, cosocle

C₂²×C₄×C₂₇₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#544×C₂₇₇

C₂

socle

C₂²×C₂₇₇

#120

prime radical

C₂³×C₂₇₇

C₂×D₄

D₂₇₇

C₂₇₇

G ▷ C₂²×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
20. C₂×Dic₅₅₄	Q₈	semidirect
73. C₂₇₇	#2028	semidirect

H ≅ G/N

type

20. C₂×Dic₅₅₄

Q₈

semidirect

73. C₂₇₇

#2028

semidirect

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

subgroup	index
#31800010	2
#31799907	2
#31799771	2
#31799761	2
#544×C₂₇₇	2
#2028	277

subgroup

index

#544×C₂₇₇

277

Subgroups — #47984325

Conjugacy classes of subgroups

Complements

Maximal subgroups