Group #47983255 — subgroups

Subgroups — #47983255

Order 35456.

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

normal subgroups only

This group has 344 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	C₂²×C₄⋊Dic₂₇₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#14555660	C₂	1	1	yes	maximal, characteristic
3	#31799764	C₂	1	4	yes	maximal, complemented
4	#380×C₂₇₇	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₇₇	—	2	1	no	—
6	C₂²×#14555660	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₇₇	—	2	1	no	—
8	C₂×C₄⋊Dic₂₇₇	—	2	2	no	—
9	C₂×C₄⋊Dic₂₇₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₂₇₇	C₂²	1	2	yes	—
11	C₂²⋊#14555660	—	2	2	no	—
12	C₂³×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	2	yes	—
14	C₂×C₂²⋊C₄×C₂₇₇	—	2	1	no	—
15	#95×C₂₇₇	—	2	1	no	—
16	C₂²×Dic₂₇₇	—	2	4	no	—
17	C₂²×Dic₂₇₇	—	2	2	no	—
18	C₂²×Dic₂₇₇	—	4	4	no	—
19	C₂×#14555660	—	2	2	no	—
20	C₂×#14555660	—	2	4	no	—
21	C₄⋊Dic₂₇₇	—	2	4	no	—
22	C₄⋊Dic₂₇₇	—	2	2	no	—
23	C₄⋊Dic₂₇₇	C₂×C₄	1	4	yes	—
24	C₂⁴×C₂₇₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₇₇	—	2	4	no	—
26	C₂²×C₄×C₂₇₇	—	2	2	no	—
27	C₂²×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₇₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
30	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₂₇₇	—	2	4	no	—
32	C₂²⋊C₄×C₂₇₇	—	4	2	no	—
33	C₂×Dic₂₇₇	—	2	4	no	—
34	C₂×Dic₂₇₇	—	4	8	no	—
35	C₂×Dic₂₇₇	—	4	4	no	—
36	#14555660	—	4	2	no	—
37	C₂³×C₂₇₇	—	2	2	no	—
38	C₂³×C₂₇₇	—	2	2	no	—
39	C₂³×C₂₇₇	C₂²⋊C₄	1	2	yes	—
40	C₂³×C₂₇₇	SD₁₆	1	2	yes	—
41	C₂³×C₂₇₇	Q₁₆	1	2	yes	—
42	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₇₇	—	2	2	no	—
44	C₂×C₄×C₂₇₇	—	2	4	no	—
45	C₂×C₄×C₂₇₇	—	2	4	no	—
46	C₂×C₄×C₂₇₇	—	4	2	no	—
47	C₂×C₄×C₂₇₇	—	4	2	no	—
48	C₂×C₄×C₂₇₇	—	4	2	no	—
49	C₂×C₄×C₂₇₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₂₇₇	#41	1	4	yes	—
53	C₂²×C₂₇₇	—	2	2	no	—
54	C₂²×C₂₇₇	—	2	2	no	—
55	C₂²×C₂₇₇	—	2	4	no	—
56	C₂²×C₂₇₇	—	2	4	no	—
57	C₂²×C₂₇₇	#103	1	4	yes	—
58	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₇₇	#123	1	2	yes	—
60	C₂²×C₂₇₇	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₂₇₇	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₂₇₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₂₇₇	—	2	2	no	—
64	C₄×C₂₇₇	—	8	2	no	—
65	C₄×C₂₇₇	#117	1	2	yes	—
66	C₄×C₂₇₇	#123	1	2	yes	—
67	Dic₂₇₇	—	4	4	no	—
68	C₂×C₂₇₇	—	2	4	no	—
69	C₂×C₂₇₇	#381	1	1	yes	characteristic
70	C₂×C₂₇₇	C₂×#103	1	1	yes	characteristic
71	C₂×C₂₇₇	#418	1	1	yes	characteristic
72	C₂×C₂₇₇	#481	1	2	yes	—
73	C₂×C₂₇₇	#484	1	2	yes	—
74	C₂₇₇	#1824	1	1	yes	characteristic, complemented, Sylow
75	#1824	—	277	1	no	maximal, Sylow
76	C₂²×C₄⋊C₄	—	277	1	no	—
77	C₂×C₂²⋊C₈	—	277	1	no	—
78	#380	D₂₇₇	1	1	yes	characteristic
79	#340	—	277	4	no	—
80	C₂³×C₄	—	554	1	no	—
81	C₂³×C₄	C₂×D₂₇₇	1	1	yes	characteristic
82	C₂²×C₈	—	277	2	no	—
83	C₂×C₄⋊C₄	—	277	4	no	—
84	C₂×C₄⋊C₄	—	277	2	no	—
85	C₂×C₄⋊C₄	—	554	1	no	—
86	C₂×C₄⋊C₄	—	554	2	no	—
87	C₂×C₄⋊C₄	C₂×D₂₇₇	1	2	yes	—
88	C₂×C₂²⋊C₄	—	2	1	no	—
89	C₂²⋊C₈	—	554	2	no	—
90	#95	—	2	1	no	—
91	C₂⁴	C₂²⋊D₂₇₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	554	4	no	—
95	C₂²×C₄	—	554	2	no	—
96	C₂²×C₄	—	1108	4	no	—
97	C₂²×C₄	C₄×D₂₇₇	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₂₇₇	1	1	yes	characteristic
99	C₂²×C₄	D₁₁₀₈	1	2	yes	—
100	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
101	C₂×C₈	—	554	2	no	—
102	C₂×C₈	—	554	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	277	4	no	—
105	C₄⋊C₄	—	554	4	no	—
106	C₄⋊C₄	—	554	2	no	—
107	C₂²⋊C₄	—	4	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#17984540	1	2	yes	—
111	C₂³	#17984551	1	2	yes	—
112	C₂³	#17984553	1	2	yes	—
113	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	554	4	no	—
121	C₂×C₄	—	1108	8	no	—
122	C₂×C₄	—	1108	4	no	—
123	C₂×C₄	#17984540	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₂₇₇	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₁₀₈	1	1	yes	characteristic
126	C₂×C₄	#17984562	1	4	yes	—
127	C₈	—	1108	2	no	—
128	C₂²	—	2	4	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	2	no	—
131	C₂²	—	2	4	no	—
132	C₂²	#22579803	1	4	yes	—
133	C₂²	C₂×#17984540	1	1	yes	characteristic
134	C₂²	#22579861	1	2	yes	—
135	C₂²	C₂×#17984551	1	1	yes	characteristic
136	C₂²	C₂×#17984553	1	1	yes	characteristic
137	C₂²	#22579905	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	8	2	no	—
140	C₄	—	1108	4	no	—
141	C₄	#22579836	1	2	yes	—
142	C₄	#22579864	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#31799863	1	1	yes	characteristic
145	C₂	C₂×#22579803	1	1	yes	characteristic
146	C₂	#31800006	1	1	yes	characteristic
147	C₂	#31800318	1	2	yes	—
148	C₂	#31800335	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₂₇₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊#14555660

C₂

yes

maximal, characteristic

#31799764

C₂

yes

maximal, complemented

#380×C₂₇₇

C₂

yes

maximal, characteristic

C₂³×Dic₂₇₇

—

C₂²×#14555660

C₂²

yes

—

C₂×C₄⋊Dic₂₇₇

—

C₂×C₄⋊Dic₂₇₇

—

C₂×C₄⋊Dic₂₇₇

C₄

yes

complemented

C₂×C₄⋊Dic₂₇₇

C₂²

yes

—

C₂²⋊#14555660

—

C₂³×C₄×C₂₇₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₇₇

—

#95×C₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

C₂²×Dic₂₇₇

—

C₂×#14555660

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

C₂×C₄

yes

—

C₂⁴×C₂₇₇

D₄

yes

characteristic

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

D₄

yes

characteristic

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₂²⋊C₄×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

C₂²⋊C₄

yes

—

C₂³×C₂₇₇

SD₁₆

yes

—

C₂³×C₂₇₇

Q₁₆

yes

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

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

#41

yes

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

#103

yes

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

#123

yes

—

C₂²×C₂₇₇

C₂×SD₁₆

yes

characteristic

C₂²×C₂₇₇

C₂×Q₁₆

yes

characteristic

C₂²×C₂₇₇

AGL(1,ℤ/8)

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#117

yes

—

C₄×C₂₇₇

#123

yes

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

—

C₂×C₂₇₇

#484

yes

—

C₂₇₇

#1824

yes

characteristic, complemented, Sylow

#1824

—

277

maximal, Sylow

C₂²×C₄⋊C₄

—

277

—

C₂×C₂²⋊C₈

—

277

—

#380

D₂₇₇

yes

characteristic

#340

—

277

—

C₂³×C₄

—

554

—

C₂³×C₄

C₂×D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

277

—

C₂×C₄⋊C₄

—

554

—

C₂×C₄⋊C₄

—

554

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

554

—

#95

—

C₂⁴

C₂²⋊D₂₇₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

554

—

C₂²×C₄

—

554

—

C₂²×C₄

—

1108

—

C₂²×C₄

C₄×D₂₇₇

yes

—

C₂²×C₄

C₂²⋊D₂₇₇

yes

characteristic

C₂²×C₄

D₁₁₀₈

yes

—

100

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

101

C₂×C₈

—

554

—

102

C₂×C₈

—

554

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

277

—

105

C₄⋊C₄

—

554

—

106

C₄⋊C₄

—

554

—

107

C₂²⋊C₄

—

108

C₂³

—

109

C₂³

—

110

C₂³

#17984540

yes

—

111

C₂³

#17984551

yes

—

112

C₂³

#17984553

yes

—

113

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

554

—

121

C₂×C₄

—

1108

—

122

C₂×C₄

—

1108

—

123

C₂×C₄

#17984540

yes

—

124

C₂×C₄

C₂×C₄×D₂₇₇

yes

characteristic

125

C₂×C₄

C₂×D₁₁₀₈

yes

characteristic

126

C₂×C₄

#17984562

yes

—

127

C₈

—

1108

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#22579803

yes

—

133

C₂²

C₂×#17984540

yes

characteristic

134

C₂²

#22579861

yes

—

135

C₂²

C₂×#17984551

yes

characteristic

136

C₂²

C₂×#17984553

yes

characteristic

137

C₂²

#22579905

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

1108

—

141

C₄

#22579836

yes

—

142

C₄

#22579864

yes

—

143

C₂

—

144

C₂

#31799863

yes

characteristic

145

C₂

C₂×#22579803

yes

characteristic

146

C₂

#31800006

yes

characteristic

147

C₂

#31800318

yes

—

148

C₂

#31800335

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₇₇
commutator	C₂×C₄×C₂₇₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#380×C₂₇₇	C₂
socle	C₂³×C₂₇₇	C₂×D₄
prime radical	C₂⁴×C₂₇₇	D₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#380	D₂₇₇
hypocenter	C₂₇₇	#1824
LCS	G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₇₇

commutator

C₂×C₄×C₂₇₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#380×C₂₇₇

C₂

socle

C₂³×C₂₇₇

C₂×D₄

prime radical

C₂⁴×C₂₇₇

D₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
3. #31799764	C₂	semidirect
9. C₂×C₄⋊Dic₂₇₇	C₄	semidirect
74. C₂₇₇	#1824	semidirect

H ≅ G/N

type

3. #31799764

C₂

semidirect

9. C₂×C₄⋊Dic₂₇₇

C₄

semidirect

74. C₂₇₇

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₂₇₇	2
C₂×C₂²⋊#14555660	2
#31799764	2
#380×C₂₇₇	2
#1824	277

subgroup

index

C₂²×C₄⋊Dic₂₇₇

C₂×C₂²⋊#14555660

#31799764

#380×C₂₇₇

#1824

277

Subgroups — #47983255

Conjugacy classes of subgroups

Complements

Maximal subgroups