Group #47983150 — subgroups

Subgroups — #47983150

Order 35456.

312 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 101 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×#22579837	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579809	C₂	1	2	yes	maximal, complemented
3	C₂×#22579792	C₂	1	2	yes	maximal
4	#31799757	C₂	1	1	yes	maximal, characteristic, complemented
5	#336×C₂₇₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₂₇₇	C₂²	1	1	yes	characteristic
7	C₂²×#14555660	C₂²	1	2	yes	—
8	#22579837	—	2	4	no	—
9	C₂×#17984542	C₄	1	2	yes	—
10	C₂×#17984540	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
12	#22579809	—	2	4	no	—
13	#22579792	—	2	4	no	—
14	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
16	C₂³×D₂₇₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
18	C₂²×Dic₂₇₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×D₂₇₇	—	2	2	no	—
20	C₂×C₄×D₂₇₇	—	2	2	no	—
21	C₂×C₄×D₂₇₇	D₄	1	2	yes	—
22	C₂×C₄×D₂₇₇	Q₈	1	2	yes	—
23	C₂×#14555660	—	2	2	no	—
24	C₂×#14555660	—	2	2	no	—
25	C₂×#14555660	—	2	2	no	—
26	#17984542	—	2	4	no	—
27	#17984540	—	2	4	no	—
28	C₄×Dic₂₇₇	D₄	1	4	yes	—
29	C₂²×C₄×C₂₇₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₇₇	—	2	2	no	—
32	C₂×C₈×C₂₇₇	—	2	2	no	—
33	C₂×C₈×C₂₇₇	—	2	2	no	—
34	C₄²×C₂₇₇	—	2	2	no	—
35	C₂²×D₂₇₇	—	2	1	no	—
36	C₂²×D₂₇₇	—	2	1	no	—
37	C₂²×D₂₇₇	—	2	2	no	—
38	C₂²×D₂₇₇	—	2	2	no	—
39	C₂²×D₂₇₇	C₄⋊C₄	1	2	yes	—
40	C₂×Dic₂₇₇	—	2	1	no	—
41	C₂×Dic₂₇₇	—	2	1	no	—
42	C₂×Dic₂₇₇	—	2	2	no	—
43	C₂×Dic₂₇₇	—	2	2	no	—
44	C₂×Dic₂₇₇	C₂²⋊C₄	1	4	yes	—
45	C₂×Dic₂₇₇	C₄⋊C₄	1	2	yes	—
46	C₄×D₂₇₇	—	2	4	no	—
47	C₄×D₂₇₇	—	2	4	no	—
48	#14555660	—	4	2	no	—
49	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₇₇	—	2	2	no	—
51	C₂×C₄×C₂₇₇	—	2	2	no	—
52	C₂×C₄×C₂₇₇	—	2	2	no	—
53	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
56	C₂×C₄×C₂₇₇	#41	1	2	yes	—
57	C₈×C₂₇₇	—	4	2	no	—
58	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₇₇	#117	1	1	yes	characteristic
61	C₂²×C₂₇₇	#131	1	2	yes	—
62	C₂²×C₂₇₇	#131	1	2	yes	—
63	C₂×D₂₇₇	—	2	2	no	—
64	C₂×D₂₇₇	—	2	4	no	—
65	C₂×D₂₇₇	—	4	2	no	—
66	C₂×D₂₇₇	—	4	1	no	—
67	C₂×D₂₇₇	—	4	1	no	—
68	C₄×C₂₇₇	—	4	2	no	—
69	C₄×C₂₇₇	#121	1	2	yes	—
70	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
71	Dic₂₇₇	—	2	2	no	—
72	Dic₂₇₇	—	2	4	no	—
73	C₂×C₂₇₇	#380	1	1	yes	characteristic
74	C₂×C₂₇₇	#408	1	2	yes	—
75	C₂×C₂₇₇	#424	1	2	yes	—
76	C₂×C₂₇₇	#433	1	1	yes	characteristic
77	C₂×C₂₇₇	#433	1	1	yes	characteristic
78	D₂₇₇	—	4	2	no	—
79	C₂₇₇	#1798	1	1	yes	characteristic, complemented, Sylow
80	#1798	—	277	1	no	maximal, Sylow
81	C₂×#117	—	277	1	no	—
82	C₂×C₄⋊C₈	—	277	2	no	—
83	C₂×C₂²⋊C₈	—	277	2	no	—
84	#336	—	277	1	no	—
85	#336	D₂₇₇	1	1	yes	characteristic, complemented
86	C₂³×C₄	—	277	1	no	—
87	C₂²×C₈	—	277	2	no	—
88	C₂²×C₈	C₂×D₂₇₇	1	2	yes	—
89	#117	—	554	4	no	—
90	C₂×C₄⋊C₄	—	277	2	no	—
91	C₂×C₂²⋊C₄	—	277	2	no	—
92	C₂×C₄²	—	277	1	no	—
93	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
94	C₄⋊C₈	—	554	4	no	—
95	C₂²⋊C₈	—	554	4	no	—
96	C₂⁴	—	277	1	no	—
97	C₂²×C₄	—	277	2	no	—
98	C₂²×C₄	—	277	1	no	—
99	C₂²×C₄	—	277	2	no	—
100	C₂²×C₄	—	277	2	no	—
101	C₂²×C₄	—	554	2	no	—
102	C₂²×C₄	—	554	2	no	—
103	C₂²×C₄	C₄×D₂₇₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	554	2	no	—
109	C₂×C₈	—	554	2	no	—
110	C₂×C₈	—	554	2	no	—
111	C₄⋊C₄	—	554	4	no	—
112	C₂²⋊C₄	—	554	4	no	—
113	C₄²	—	2	2	no	—
114	C₄²	—	277	4	no	—
115	C₂³	—	277	2	no	—
116	C₂³	—	554	1	no	—
117	C₂³	—	554	1	no	—
118	C₂³	—	554	2	no	—
119	C₂³	—	554	2	no	—
120	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	277	4	no	—
125	C₂×C₄	—	277	2	no	—
126	C₂×C₄	—	554	1	no	—
127	C₂×C₄	—	554	1	no	—
128	C₂×C₄	—	554	2	no	—
129	C₂×C₄	—	554	2	no	—
130	C₂×C₄	—	554	4	no	—
131	C₂×C₄	—	554	4	no	—
132	C₂×C₄	D₄×D₂₇₇	1	1	yes	characteristic
133	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
134	C₂×C₄	Q₈×D₂₇₇	1	1	yes	characteristic
135	C₂×C₄	#17984563	1	2	yes	—
136	C₈	—	4	2	no	—
137	C₈	—	1108	2	no	—
138	C₂²	—	554	2	no	—
139	C₂²	—	554	4	no	—
140	C₂²	—	1108	1	no	—
141	C₂²	—	1108	2	no	—
142	C₂²	—	1108	1	no	—
143	C₂²	C₂²⋊C₄×D₂₇₇	1	1	yes	characteristic
144	C₂²	C₄⋊C₄×D₂₇₇	1	1	yes	characteristic
145	C₂²	#22579837	1	1	yes	characteristic
146	C₂²	#22579886	1	2	yes	—
147	C₂²	#22579888	1	2	yes	—
148	C₄	—	4	2	no	—
149	C₄	—	554	2	no	—
150	C₄	—	554	4	no	—
151	C₄	#22579853	1	2	yes	—
152	C₄	#22579858	1	2	yes	—
153	C₂	—	1108	2	no	—
154	C₂	#31799861	1	1	yes	characteristic
155	C₂	#31799967	1	2	yes	—
156	C₂	#31800027	1	2	yes	—
157	C₂	#31800061	1	1	yes	characteristic
158	C₂	#31800064	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579837

C₂

yes

maximal, characteristic

C₂×#22579809

C₂

yes

maximal, complemented

C₂×#22579792

C₂

yes

maximal

#31799757

C₂

yes

maximal, characteristic, complemented

#336×C₂₇₇

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₂₇₇

C₂²

yes

characteristic

C₂²×#14555660

C₂²

yes

—

#22579837

—

C₂×#17984542

C₄

yes

—

C₂×#17984540

C₄

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579809

—

#22579792

—

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

C₂³×D₂₇₇

C₂×C₄

yes

characteristic

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂²×Dic₂₇₇

C₂×C₄

yes

characteristic

C₂×C₄×D₂₇₇

—

C₂×C₄×D₂₇₇

—

C₂×C₄×D₂₇₇

D₄

yes

—

C₂×C₄×D₂₇₇

Q₈

yes

—

—

—

—

—

—

C₄×Dic₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂×C₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₄²×C₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

—

C₂²×D₂₇₇

C₄⋊C₄

yes

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₂²⋊C₄

yes

—

C₂×Dic₂₇₇

C₄⋊C₄

yes

—

C₄×D₂₇₇

—

C₄×D₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

—

C₂²×C₂₇₇

#131

yes

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₂×D₂₇₇

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#121

yes

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

#380

yes

characteristic

C₂×C₂₇₇

#408

yes

—

C₂×C₂₇₇

#424

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

D₂₇₇

—

C₂₇₇

#1798

yes

characteristic, complemented, Sylow

#1798

—

277

maximal, Sylow

C₂×#117

—

277

—

C₂×C₄⋊C₈

—

277

—

C₂×C₂²⋊C₈

—

277

—

#336

—

277

—

#336

D₂₇₇

yes

characteristic, complemented

C₂³×C₄

—

277

—

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

—

#117

—

554

—

C₂×C₄⋊C₄

—

277

—

C₂×C₂²⋊C₄

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

C₄⋊C₈

—

554

—

C₂²⋊C₈

—

554

—

C₂⁴

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

C₂²×C₄

—

277

—

100

C₂²×C₄

—

277

—

101

C₂²×C₄

—

554

—

102

C₂²×C₄

—

554

—

103

C₂²×C₄

C₄×D₂₇₇

yes

—

104

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

554

—

109

C₂×C₈

—

554

—

110

C₂×C₈

—

554

—

111

C₄⋊C₄

—

554

—

112

C₂²⋊C₄

—

554

—

113

C₄²

—

114

C₄²

—

277

—

115

C₂³

—

277

—

116

C₂³

—

554

—

117

C₂³

—

554

—

118

C₂³

—

554

—

119

C₂³

—

554

—

120

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

277

—

125

C₂×C₄

—

277

—

126

C₂×C₄

—

554

—

127

C₂×C₄

—

554

—

128

C₂×C₄

—

554

—

129

C₂×C₄

—

554

—

130

C₂×C₄

—

554

—

131

C₂×C₄

—

554

—

132

C₂×C₄

D₄×D₂₇₇

yes

characteristic

133

C₂×C₄

D₄×D₂₇₇

yes

—

134

C₂×C₄

Q₈×D₂₇₇

yes

characteristic

135

C₂×C₄

#17984563

yes

—

136

C₈

—

137

C₈

—

1108

—

138

C₂²

—

554

—

139

C₂²

—

554

—

140

C₂²

—

1108

—

141

C₂²

—

1108

—

142

C₂²

—

1108

—

143

C₂²

C₂²⋊C₄×D₂₇₇

yes

characteristic

144

C₂²

C₄⋊C₄×D₂₇₇

yes

characteristic

145

C₂²

#22579837

yes

characteristic

146

C₂²

#22579886

yes

—

147

C₂²

#22579888

yes

—

148

C₄

—

149

C₄

—

554

—

150

C₄

—

554

—

151

C₄

#22579853

yes

—

152

C₄

#22579858

yes

—

153

C₂

—

1108

—

154

C₂

#31799861

yes

characteristic

155

C₂

#31799967

yes

—

156

C₂

#31800027

yes

—

157

C₂

#31800061

yes

characteristic

158

C₂

#31800064

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₇₇
commutator, socle	C₂³×C₂₇₇	C₂²×C₄
Fitting	#336×C₂₇₇	C₂
prime radical	C₂³×D₂₇₇	C₂×C₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	#336	D₂₇₇
hypocenter	C₂₇₇	#1798
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₇₇

commutator, socle

C₂³×C₂₇₇

C₂²×C₄

Fitting

#336×C₂₇₇

C₂

prime radical

C₂³×D₂₇₇

C₂×C₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

D₂₇₇

C₂₇₇

G ▷ C₂³×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
2. C₂×#22579809	C₂	semidirect
4. #31799757	C₂	semidirect
5. #336×C₂₇₇	C₂	semidirect
79. C₂₇₇	#1798	semidirect
85. #336	D₂₇₇	semidirect

H ≅ G/N

type

2. C₂×#22579809

C₂

semidirect

4. #31799757

C₂

semidirect

5. #336×C₂₇₇

C₂

semidirect

79. C₂₇₇

#1798

semidirect

85. #336

D₂₇₇

semidirect

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

subgroup	index
C₂×#22579837	2
C₂×#22579809	2
C₂×#22579792	2
#31799757	2
#336×C₂₇₇	2
#1798	277

subgroup

index

#336×C₂₇₇

277

Subgroups — #47983150

Conjugacy classes of subgroups

Complements

Maximal subgroups