Group #47984328 — subgroups

Subgroups — #47984328

Order 35456.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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	#31800633	C₂	1	1	yes	maximal, characteristic
2	#31800011	C₂	1	2	yes	maximal
3	#31799771	C₂	1	1	yes	maximal, characteristic
4	#31799762	C₂	1	2	yes	maximal
5	#391×C₂₇₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#17984547	C₂²	1	2	yes	—
7	#22579879	—	2	2	no	—
8	#22579871	—	2	2	no	—
9	#22579859	—	2	2	no	—
10	#22579839	C₂²	1	2	yes	—
11	#22579804	—	2	2	no	—
12	#22579804	—	2	2	no	—
13	C₂²×Q₈×C₂₇₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₂₇₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₂₇₇	C₂²	1	1	yes	characteristic
16	#95×C₂₇₇	—	2	2	no	—
17	C₂×Dic₅₅₄	—	2	2	no	—
18	#17984547	—	2	2	no	—
19	#17984547	—	2	2	no	—
20	C₂×#14555660	—	2	2	no	—
21	C₄⋊Dic₂₇₇	—	2	2	no	—
22	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
23	C₄⋊Dic₂₇₇	D₄	1	2	yes	—
24	#17984542	—	2	4	no	—
25	C₂²⋊Dic₂₇₇	—	2	2	no	—
26	C₄×Dic₂₇₇	—	2	2	no	—
27	C₂×Q₈×C₂₇₇	—	2	1	no	—
28	C₂×Q₈×C₂₇₇	—	2	2	no	—
29	C₂×Q₈×C₂₇₇	—	2	2	no	—
30	C₂×Q₈×C₂₇₇	Q₈	1	2	yes	complemented
31	C₂²×C₄×C₂₇₇	—	2	1	no	—
32	C₂²×C₄×C₂₇₇	—	2	1	no	—
33	C₂²×C₄×C₂₇₇	D₄	1	2	yes	—
34	C₂²×C₄×C₂₇₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₂₇₇	—	2	2	no	—
36	C₄²×C₂₇₇	—	2	2	no	—
37	Dic₅₅₄	—	8	2	no	—
38	C₂×Dic₂₇₇	—	2	2	no	—
39	C₂×Dic₂₇₇	—	2	2	no	—
40	#14555660	—	4	2	no	—
41	C₂³×C₂₇₇	C₂×D₄	1	1	yes	characteristic
42	Q₈×C₂₇₇	—	2	4	no	—
43	Q₈×C₂₇₇	—	4	2	no	—
44	C₂×C₄×C₂₇₇	—	2	2	no	—
45	C₂×C₄×C₂₇₇	—	2	2	no	—
46	C₂×C₄×C₂₇₇	—	2	2	no	—
47	C₂×C₄×C₂₇₇	—	4	1	no	—
48	C₂×C₄×C₂₇₇	—	4	1	no	—
49	C₂×C₄×C₂₇₇	—	4	1	no	—
50	C₂×C₄×C₂₇₇	—	4	1	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	—	277	1	no	—
76	#419	—	277	2	no	—
77	#391	D₂₇₇	1	1	yes	characteristic
78	#343	—	277	1	no	—
79	#339	—	277	2	no	—
80	C₂²×Q₈	C₂×D₂₇₇	1	1	yes	characteristic
81	C₂×M₁₆	—	277	2	no	—
82	C₄⋊Q₈	—	554	2	no	—
83	#125	—	554	2	no	—
84	C₂²⋊Q₈	—	554	2	no	—
85	#117	—	277	2	no	—
86	C₂×C₄⋊C₄	C₂×D₂₇₇	1	1	yes	characteristic
87	C₂×C₄²	C₂×D₂₇₇	1	1	yes	characteristic
88	#103	—	554	2	no	—
89	#103	—	554	2	no	—
90	#95	—	2	2	no	—
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	554	2	no	—
95	C₂×Q₈	Dic₅₅₄	1	2	yes	complemented
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	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	—	277	2	no	—
105	C₄⋊C₄	—	277	2	no	—
106	C₄⋊C₄	—	554	4	no	—
107	C₄⋊C₄	—	554	2	no	—
108	C₂²⋊C₄	—	554	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	554	2	no	—
111	C₂³	C₂×C₂²⋊D₂₇₇	1	1	yes	characteristic
112	Q₈	—	2	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	2216	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	554	2	no	—
124	C₂×C₄	—	554	2	no	—
125	C₂×C₄	D₄×D₂₇₇	1	2	yes	—
126	C₂×C₄	C₂×Dic₅₅₄	1	1	yes	characteristic
127	C₂×C₄	#17984562	1	2	yes	—
128	C₂×C₄	#17984563	1	1	yes	characteristic
129	C₈	—	1108	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#22579848	1	1	yes	characteristic
133	C₂²	#22579859	1	1	yes	characteristic
134	C₂²	#22579867	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	#22579857	1	2	yes	—
141	C₄	#22579866	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#31799914	1	1	yes	characteristic
145	C₂	#31800179	1	1	yes	characteristic
146	C₂	#31800184	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31800633

C₂

yes

maximal, characteristic

#31800011

C₂

yes

maximal

#31799771

C₂

yes

maximal, characteristic

#31799762

C₂

yes

maximal

#391×C₂₇₇

C₂

yes

maximal, characteristic

C₂×#17984547

C₂²

yes

—

—

—

—

C₂²

yes

—

#22579804

—

#22579804

—

C₂²×Q₈×C₂₇₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₇₇

C₂²

yes

characteristic

C₂×C₄²×C₂₇₇

C₂²

yes

characteristic

#95×C₂₇₇

—

C₂×Dic₅₅₄

—

#17984547

—

#17984547

—

C₂×#14555660

—

C₄⋊Dic₂₇₇

—

C₄⋊Dic₂₇₇

D₄

yes

—

C₄⋊Dic₂₇₇

D₄

yes

—

#17984542

—

C₂²⋊Dic₂₇₇

—

C₄×Dic₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

—

C₂×Q₈×C₂₇₇

Q₈

yes

complemented

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

—

C₂²×C₄×C₂₇₇

D₄

yes

—

C₂²×C₄×C₂₇₇

C₂³

yes

characteristic

C₄⋊C₄×C₂₇₇

—

C₄²×C₂₇₇

—

Dic₅₅₄

—

C₂×Dic₂₇₇

—

C₂×Dic₂₇₇

—

#14555660

—

C₂³×C₂₇₇

C₂×D₄

yes

characteristic

Q₈×C₂₇₇

—

Q₈×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×C₂₇₇

—

C₂×C₄×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

—

277

—

#419

—

277

—

#391

D₂₇₇

yes

characteristic

#343

—

277

—

#339

—

277

—

C₂²×Q₈

C₂×D₂₇₇

yes

characteristic

C₂×M₁₆

—

277

—

C₄⋊Q₈

—

554

—

#125

—

554

—

C₂²⋊Q₈

—

554

—

#117

—

277

—

C₂×C₄⋊C₄

C₂×D₂₇₇

yes

characteristic

C₂×C₄²

C₂×D₂₇₇

yes

characteristic

#103

—

554

—

#103

—

554

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

554

—

C₂×Q₈

Dic₅₅₄

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

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

—

277

—

105

C₄⋊C₄

—

277

—

106

C₄⋊C₄

—

554

—

107

C₄⋊C₄

—

554

—

108

C₂²⋊C₄

—

554

—

109

C₄²

—

110

C₄²

—

554

—

111

C₂³

C₂×C₂²⋊D₂₇₇

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

2216

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

554

—

124

C₂×C₄

—

554

—

125

C₂×C₄

D₄×D₂₇₇

yes

—

126

C₂×C₄

C₂×Dic₅₅₄

yes

characteristic

127

C₂×C₄

#17984562

yes

—

128

C₂×C₄

#17984563

yes

characteristic

129

C₈

—

1108

—

130

C₂²

—

131

C₂²

—

132

C₂²

#22579848

yes

characteristic

133

C₂²

#22579859

yes

characteristic

134

C₂²

#22579867

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

1108

—

139

C₄

—

1108

—

140

C₄

#22579857

yes

—

141

C₄

#22579866

yes

—

142

C₂

—

143

C₂

—

144

C₂

#31799914

yes

characteristic

145

C₂

#31800179

yes

characteristic

146

C₂

#31800184

yes

characteristic

	subgroup	G/H
center	C₂²	#22579848
commutator, cosocle	C₂²×C₄×C₂₇₇	C₂³
Frattini	C₂²×C₄	C₂²×D₂₇₇
Fitting	#391×C₂₇₇	C₂
socle	C₂²×C₂₇₇	#120
prime radical	C₂³×C₂₇₇	C₂×D₄
hypercenter	#391	D₂₇₇
hypocenter	C₂₇₇	#2028
LCS	G ▷ C₂²×C₄×C₂₇₇ ▷ C₂×C₂₇₇ ▷ C₂₇₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #391

subgroup

G/H

center

C₂²

#22579848

commutator, cosocle

C₂²×C₄×C₂₇₇

C₂³

Frattini

C₂²×C₄

C₂²×D₂₇₇

Fitting

#391×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 ◁ #391

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

H ≅ G/N

type

30. C₂×Q₈×C₂₇₇

Q₈

semidirect

73. C₂₇₇

#2028

semidirect

95. C₂×Q₈

Dic₅₅₄

semidirect

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

subgroup	index
#31800633	2
#31800011	2
#31799771	2
#31799762	2
#391×C₂₇₇	2
#2028	277

subgroup

index

#391×C₂₇₇

277

Subgroups — #47984328

Conjugacy classes of subgroups

Complements

Maximal subgroups