Group #47983149 — subgroups

Subgroups — #47983149

Order 35456.

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

normal subgroups only

This group has 312 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×#22579839	C₂	1	1	yes	maximal, characteristic
2	C₂×#22579809	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#14555660	C₂	1	1	yes	maximal, characteristic
4	#31799756	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555660	C₂²	1	2	yes	—
7	#22579839	—	2	4	no	—
8	C₂×C₄⋊Dic₂₇₇	C₄	1	2	yes	—
9	C₂×C₂²⋊Dic₂₇₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₂₇₇	C₂²	1	1	yes	characteristic
12	#22579809	—	2	4	no	—
13	C₂²⋊#14555660	—	2	2	no	—
14	C₂³×C₄×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₂₇₇	C₂²	1	2	yes	—
16	C₂²⋊C₈×C₂₇₇	—	2	2	no	—
17	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
18	C₂²×Dic₂₇₇	C₂×C₄	1	2	yes	—
19	C₂×#14555660	—	2	2	no	—
20	C₂×#14555660	—	2	2	no	—
21	C₂×#14555660	—	2	2	no	—
22	C₄⋊Dic₂₇₇	—	2	4	no	—
23	C₂²⋊Dic₂₇₇	—	2	4	no	—
24	C₄×Dic₂₇₇	—	2	2	no	—
25	C₄×Dic₂₇₇	D₄	1	4	yes	—
26	C₂⁴×C₂₇₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₇₇	—	2	2	no	—
28	C₂²×C₄×C₂₇₇	—	2	2	no	—
29	C₂²×C₄×C₂₇₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
31	C₂²×C₄×C₂₇₇	Q₈	1	2	yes	—
32	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₂₇₇	—	2	2	no	—
34	C₂×C₈×C₂₇₇	—	2	2	no	—
35	C₂×C₈×C₂₇₇	—	2	2	no	—
36	C₂×Dic₂₇₇	—	2	2	no	—
37	C₂×Dic₂₇₇	—	2	2	no	—
38	C₂×Dic₂₇₇	—	2	2	no	—
39	C₂×Dic₂₇₇	—	2	2	no	—
40	C₂×Dic₂₇₇	—	2	2	no	—
41	C₂×Dic₂₇₇	C₂²⋊C₄	1	4	yes	—
42	#14555660	—	4	2	no	—
43	C₂³×C₂₇₇	—	2	1	no	—
44	C₂³×C₂₇₇	—	2	1	no	—
45	C₂³×C₂₇₇	—	2	2	no	—
46	C₂³×C₂₇₇	—	2	2	no	—
47	C₂³×C₂₇₇	C₄⋊C₄	1	2	yes	—
48	C₂³×C₂₇₇	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₂₇₇	—	2	1	no	—
50	C₂×C₄×C₂₇₇	—	2	1	no	—
51	C₂×C₄×C₂₇₇	—	2	4	no	—
52	C₂×C₄×C₂₇₇	—	2	4	no	—
53	C₂×C₄×C₂₇₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×C₂₇₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₂₇₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₇₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₂₇₇	#41	1	2	yes	—
58	C₈×C₂₇₇	—	4	2	no	—
59	C₂²×C₂₇₇	—	2	2	no	—
60	C₂²×C₂₇₇	—	2	2	no	—
61	C₂²×C₂₇₇	—	2	2	no	—
62	C₂²×C₂₇₇	—	4	2	no	—
63	C₂²×C₂₇₇	—	4	2	no	—
64	C₂²×C₂₇₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₂₇₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₂₇₇	#117	1	1	yes	characteristic
67	C₂²×C₂₇₇	#131	1	2	yes	—
68	C₂²×C₂₇₇	#131	1	2	yes	—
69	C₄×C₂₇₇	—	2	2	no	—
70	C₄×C₂₇₇	#121	1	2	yes	—
71	C₄×C₂₇₇	C₂²⋊Q₈	1	2	yes	—
72	Dic₂₇₇	—	2	4	no	—
73	Dic₂₇₇	—	4	2	no	—
74	C₂×C₂₇₇	—	4	2	no	—
75	C₂×C₂₇₇	#380	1	1	yes	characteristic
76	C₂×C₂₇₇	#408	1	2	yes	—
77	C₂×C₂₇₇	#424	1	1	yes	characteristic
78	C₂×C₂₇₇	#424	1	1	yes	characteristic
79	C₂×C₂₇₇	#433	1	2	yes	—
80	C₂₇₇	#1798	1	1	yes	characteristic, complemented, Sylow
81	#1798	—	277	1	no	maximal, Sylow
82	C₂×#117	—	277	1	no	—
83	C₂×C₄⋊C₈	—	277	2	no	—
84	C₂×C₂²⋊C₈	—	277	1	no	—
85	C₂×C₂²⋊C₈	D₂₇₇	1	1	yes	characteristic
86	#336	—	277	2	no	—
87	C₂³×C₄	C₂×D₂₇₇	1	1	yes	characteristic
88	C₂²×C₈	—	277	2	no	—
89	C₂²×C₈	C₂×D₂₇₇	1	2	yes	—
90	#117	—	554	4	no	—
91	C₂×C₄⋊C₄	—	277	2	no	—
92	C₂×C₂²⋊C₄	—	277	2	no	—
93	C₂×C₄²	—	277	1	no	—
94	C₂×C₄²	—	277	1	no	—
95	C₄⋊C₈	—	554	4	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	554	2	no	—
98	C₂⁴	C₄×D₂₇₇	1	1	yes	characteristic
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	277	2	no	—
102	C₂²×C₄	—	277	2	no	—
103	C₂²×C₄	C₄×D₂₇₇	1	1	yes	characteristic
104	C₂²×C₄	C₂²⋊D₂₇₇	1	2	yes	—
105	C₂²×C₄	Dic₅₅₄	1	2	yes	—
106	C₂²×C₄	C₂²×D₂₇₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	554	2	no	—
111	C₂×C₈	—	554	2	no	—
112	C₂×C₈	—	554	2	no	—
113	C₄⋊C₄	—	554	4	no	—
114	C₂²⋊C₄	—	554	4	no	—
115	C₄²	—	277	4	no	—
116	C₄²	—	554	2	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	#17984542	1	2	yes	—
122	C₂³	C₂×C₄×D₂₇₇	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	277	4	no	—
128	C₂×C₄	—	554	2	no	—
129	C₂×C₄	—	554	2	no	—
130	C₂×C₄	—	554	2	no	—
131	C₂×C₄	—	554	2	no	—
132	C₂×C₄	—	554	2	no	—
133	C₂×C₄	#17984542	1	2	yes	—
134	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
135	C₂×C₄	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
136	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
137	C₂×C₄	#17984562	1	2	yes	—
138	C₈	—	4	2	no	—
139	C₈	—	1108	2	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂²⋊C₄×D₂₇₇	1	1	yes	characteristic
146	C₂²	C₂×#17984542	1	1	yes	characteristic
147	C₂²	#22579836	1	1	yes	characteristic
148	C₂²	#22579886	1	2	yes	—
149	C₂²	#22579888	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	554	4	no	—
152	C₄	—	1108	2	no	—
153	C₄	#22579852	1	2	yes	—
154	C₄	#22579857	1	2	yes	—
155	C₂	—	4	2	no	—
156	C₂	#31799860	1	1	yes	characteristic
157	C₂	#31799967	1	2	yes	—
158	C₂	#31800028	1	1	yes	characteristic
159	C₂	#31800030	1	1	yes	characteristic
160	C₂	#31800060	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22579839

C₂

yes

maximal, characteristic

C₂×#22579809

C₂

yes

maximal, complemented

C₂×C₂²⋊#14555660

C₂

yes

maximal, characteristic

#31799756

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₂₇₇

C₂

yes

maximal, characteristic

C₂²×#14555660

C₂²

yes

—

#22579839

—

C₂×C₄⋊Dic₂₇₇

C₄

yes

—

C₂×C₂²⋊Dic₂₇₇

C₄

yes

—

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

C₂×C₄×Dic₂₇₇

C₂²

yes

characteristic

#22579809

—

C₂²⋊#14555660

—

C₂³×C₄×C₂₇₇

C₂²

yes

characteristic

C₂²×C₈×C₂₇₇

C₂²

yes

—

C₂²⋊C₈×C₂₇₇

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂²×Dic₂₇₇

C₂×C₄

yes

—

C₂×#14555660

—

C₂×#14555660

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₄×Dic₂₇₇

D₄

yes

—

C₂⁴×C₂₇₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

Q₈

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×C₈×C₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

C₂²⋊C₄

yes

—

#14555660

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

—

C₂³×C₂₇₇

C₄⋊C₄

yes

—

C₂³×C₂₇₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

C₄⋊C₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

—

C₂×C₄×C₂₇₇

C₂×D₄

yes

characteristic

C₂×C₄×C₂₇₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₇₇

#41

yes

—

C₈×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

—

C₂²×C₂₇₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₇₇

C₂×C₄⋊C₄

yes

C₂²×C₂₇₇

yes

C₂²×C₂₇₇

yes

—

C₂²×C₂₇₇

#131

yes

—

C₄×C₂₇₇

—

C₄×C₂₇₇

#121

yes

—

C₄×C₂₇₇

C₂²⋊Q₈

yes

—

Dic₂₇₇

—

Dic₂₇₇

—

C₂×C₂₇₇

—

C₂×C₂₇₇

#380

yes

characteristic

C₂×C₂₇₇

#408

yes

—

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

C₂×C₂₇₇

yes

—

C₂₇₇

#1798

yes

characteristic, complemented, Sylow

#1798

—

277

maximal, Sylow

C₂×#117

—

277

—

C₂×C₄⋊C₈

—

277

—

C₂×C₂²⋊C₈

—

277

—

C₂×C₂²⋊C₈

D₂₇₇

yes

characteristic

#336

—

277

—

C₂³×C₄

C₂×D₂₇₇

yes

characteristic

C₂²×C₈

—

277

—

C₂²×C₈

C₂×D₂₇₇

yes

—

#117

—

554

—

C₂×C₄⋊C₄

—

277

—

C₂×C₂²⋊C₄

—

277

—

C₂×C₄²

—

277

—

C₂×C₄²

—

277

—

C₄⋊C₈

—

554

—

C₂²⋊C₈

—

C₂²⋊C₈

—

554

—

C₂⁴

C₄×D₂₇₇

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

277

—

102

C₂²×C₄

—

277

—

103

C₂²×C₄

C₄×D₂₇₇

yes

characteristic

104

C₂²×C₄

C₂²⋊D₂₇₇

yes

—

105

C₂²×C₄

Dic₅₅₄

yes

—

106

C₂²×C₄

C₂²×D₂₇₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

554

—

111

C₂×C₈

—

554

—

112

C₂×C₈

—

554

—

113

C₄⋊C₄

—

554

—

114

C₂²⋊C₄

—

554

—

115

C₄²

—

277

—

116

C₄²

—

554

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

#17984542

yes

—

122

C₂³

C₂×C₄×D₂₇₇

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

277

—

128

C₂×C₄

—

554

—

129

C₂×C₄

—

554

—

130

C₂×C₄

—

554

—

131

C₂×C₄

—

554

—

132

C₂×C₄

—

554

—

133

C₂×C₄

#17984542

yes

—

134

C₂×C₄

D₄×D₂₇₇

yes

—

135

C₂×C₄

C₂×C₂²⋊D₂₇₇

yes

characteristic

136

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

137

C₂×C₄

#17984562

yes

—

138

C₈

—

139

C₈

—

1108

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂²⋊C₄×D₂₇₇

yes

characteristic

146

C₂²

C₂×#17984542

yes

characteristic

147

C₂²

#22579836

yes

characteristic

148

C₂²

#22579886

yes

—

149

C₂²

#22579888

yes

—

150

C₄

—

151

C₄

—

554

—

152

C₄

—

1108

—

153

C₄

#22579852

yes

—

154

C₄

#22579857

yes

—

155

C₂

—

156

C₂

#31799860

yes

characteristic

157

C₂

#31799967

yes

—

158

C₂

#31800028

yes

characteristic

159

C₂

#31800030

yes

characteristic

160

C₂

#31800060

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₇₇
commutator, socle	C₂³×C₂₇₇	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₂₇₇	C₂
prime radical	C₂⁴×C₂₇₇	C₂×C₄
cosocle	C₂²×C₄×C₂₇₇	C₂³
hypercenter	C₂×C₂²⋊C₈	D₂₇₇
hypocenter	C₂₇₇	#1798
LCS	G ▷ C₂³×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₇₇

commutator, socle

C₂³×C₂₇₇

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₂₇₇

C₂

prime radical

C₂⁴×C₂₇₇

C₂×C₄

cosocle

C₂²×C₄×C₂₇₇

C₂³

hypercenter

C₂×C₂²⋊C₈

D₂₇₇

C₂₇₇

G ▷ C₂³×C₂₇₇ ▷ C₂₇₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×#22579809	C₂	semidirect
4. #31799756	C₂	semidirect
80. C₂₇₇	#1798	semidirect

H ≅ G/N

type

2. C₂×#22579809

C₂

semidirect

4. #31799756

C₂

semidirect

80. C₂₇₇

#1798

semidirect

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

subgroup	index
C₂×#22579839	2
C₂×#22579809	2
C₂×C₂²⋊#14555660	2
#31799756	2
C₂×C₂²⋊C₈×C₂₇₇	2
#1798	277

subgroup

index

C₂×C₂²⋊#14555660

C₂×C₂²⋊C₈×C₂₇₇

277

Subgroups — #47983149

Conjugacy classes of subgroups

Complements

Maximal subgroups