Group #28310049 — subgroups

Subgroups — #28310049

Order 13568.

236 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 86 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	#20291099	C₂	1	2	yes	maximal, complemented
2	#20289737	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#11719910	C₂	1	2	yes	maximal, complemented
4	#20289628	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×#88322	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	—	2	2	no	—
7	C₄⋊#11719910	C₂²	1	2	yes	—
8	C₂²⋊#11719910	—	2	2	no	—
9	C₄×#11719910	C₂²	1	1	yes	characteristic
10	C₄×#11719910	C₂²	1	1	yes	characteristic
11	D₄×#88322	C₂²	1	2	yes	—
12	#16810376	C₄	1	1	yes	characteristic
13	D₄⋊#88322	C₄	1	2	yes	—
14	C₈×#88322	C₂²	1	1	yes	characteristic
15	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
16	#13133291	—	4	4	no	—
17	C₂×#11719910	—	2	2	no	—
18	C₂×#11719910	—	2	1	no	—
19	C₂×#11719910	D₄	1	2	yes	—
20	C₂²×#88322	—	2	2	no	—
21	C₄⋊#88322	C₂×C₄	1	2	yes	—
22	C₂²⋊#88322	—	2	2	no	—
23	C₄×#88322	C₂×C₄	1	1	yes	characteristic
24	C₄×#88322	C₂³	1	1	yes	characteristic
25	C₂×D₈×C₅₃	C₈	1	1	yes	characteristic
26	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
27	#107×C₅₃	C₈	1	1	yes	characteristic
28	#102×C₅₃	C₈	1	2	yes	—
29	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	D₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×#88322	—	2	2	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	—	4	2	no	—
41	C₂×#88322	—	4	2	no	—
42	C₂×#88322	C₂×D₄	1	1	yes	characteristic
43	C₂×#88322	#41	1	1	yes	characteristic
44	#11719910	—	2	2	no	—
45	#11719910	—	4	1	no	—
46	#11719910	—	4	2	no	—
47	C₂³×C₅₃	—	2	2	no	—
48	D₄×C₅₃	—	2	2	no	—
49	D₄×C₅₃	—	2	2	no	—
50	C₂×C₄×C₅₃	—	2	2	no	—
51	C₂×C₄×C₅₃	—	4	2	no	—
52	C₂×C₄×C₅₃	—	4	2	no	—
53	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
56	C₈×C₅₃	—	2	1	no	—
57	C₈×C₅₃	M₃₂	1	2	yes	complemented
58	#88322	—	2	1	no	—
59	#88322	—	4	1	no	—
60	#88322	—	4	2	no	—
61	#88322	AGL(1,ℤ/8)	1	2	yes	—
62	#12064	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	4	2	no	—
64	C₂²×C₅₃	—	4	2	no	—
65	C₂²×C₅₃	—	4	2	no	—
66	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	4	2	no	—
69	C₄×C₅₃	#442	1	1	yes	characteristic
70	C₄×C₅₃	#444	1	1	yes	characteristic
71	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
72	C₄×C₅₃	#504	1	1	yes	characteristic
73	#2169	—	53	2	no	—
74	C₈×D₈	—	53	1	no	—
75	#1367	—	53	1	no	—
76	D₄⋊C₁₆	—	53	2	no	—
77	#1312	—	53	1	no	—
78	C₂×C₅₃	—	8	2	no	—
79	C₂×C₅₃	#1580	1	1	yes	characteristic
80	C₂×C₅₃	#2169	1	1	yes	characteristic
81	C₂×C₅₃	#2171	1	1	yes	characteristic
82	C₂×M₃₂	—	106	2	no	—
83	C₄×D₈	#6157	1	1	yes	characteristic
84	C₈×D₄	—	53	2	no	—
85	C₄⋊C₁₆	—	53	2	no	—
86	C₂²⋊C₁₆	—	106	2	no	—
87	C₄×C₁₆	—	53	1	no	—
88	C₄×C₁₆	—	53	1	no	—
89	#335	—	53	1	no	—
90	D₄⋊C₈	—	53	2	no	—
91	C₈²	—	53	1	no	—
92	C₅₃	#12064	1	1	yes	characteristic, complemented, Sylow
93	C₂×D₈	#88324	1	1	yes	characteristic
94	C₂²×C₈	—	106	2	no	—
95	C₄×D₄	C₂×#6157	1	2	yes	—
96	M₃₂	—	212	4	no	—
97	C₂×C₁₆	—	53	2	no	—
98	C₂×C₁₆	—	106	1	no	—
99	C₂×C₁₆	—	106	2	no	—
100	#107	#88324	1	1	yes	characteristic
101	C₄⋊C₈	—	53	2	no	—
102	#102	#88324	1	2	yes	—
103	C₂²⋊C₈	—	106	2	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	—	53	1	no	—
106	C₄×C₈	C₂×#6157	1	1	yes	characteristic
107	C₂×D₄	C₂×#88324	1	2	yes	—
108	C₂²×C₄	—	2	2	no	—
109	D₈	—	2	2	no	—
110	C₂×C₈	—	53	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	106	2	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	212	2	no	—
115	C₂×C₈	—	212	2	no	—
116	C₂×C₈	C₂×#88324	1	1	yes	characteristic
117	C₂×C₈	C₂×#88324	1	1	yes	characteristic
118	C₄⋊C₄	C₂×#88324	1	2	yes	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	C₂²×#6157	1	1	yes	characteristic
121	C₁₆	—	106	2	no	—
122	C₁₆	—	212	2	no	—
123	C₁₆	—	212	1	no	—
124	C₂³	—	2	2	no	—
125	D₄	—	2	2	no	—
126	D₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	D₄×#6157	1	1	yes	characteristic
131	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
132	C₂×C₄	#13133304	1	1	yes	characteristic
133	C₈	—	2	1	no	—
134	C₈	—	53	2	no	—
135	C₈	—	106	1	no	—
136	C₈	—	212	1	no	—
137	C₈	—	212	2	no	—
138	C₈	#13133291	1	2	yes	complemented
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	—	4	2	no	—
142	C₂²	D₄×#88324	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	#16811664	1	1	yes	characteristic
146	C₄	#16811670	1	1	yes	characteristic
147	C₄	C₂×#13133291	1	1	yes	characteristic
148	C₄	#16811681	1	1	yes	characteristic
149	C₂	—	8	2	no	—
150	C₂	#20290311	1	1	yes	characteristic
151	C₂	#20291099	1	1	yes	characteristic
152	C₂	#20291105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291099

C₂

yes

maximal, complemented

#20289737

C₂

yes

maximal, characteristic, complemented

D₄⋊#11719910

C₂

yes

maximal, complemented

#20289628

C₂

yes

maximal, characteristic, complemented

D₈×#88322

C₂

yes

maximal, characteristic

C₂×#13133291

—

C₄⋊#11719910

C₂²

yes

—

C₂²⋊#11719910

—

C₄×#11719910

C₂²

yes

characteristic

C₄×#11719910

C₂²

yes

characteristic

D₄×#88322

C₂²

yes

—

#16810376

C₄

yes

characteristic

D₄⋊#88322

C₄

yes

—

C₈×#88322

C₂²

yes

characteristic

C₄×D₈×C₅₃

C₄

yes

—

—

—

D₄

yes

—

C₂²×#88322

—

C₄⋊#88322

C₂×C₄

yes

—

C₂²⋊#88322

—

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂³

yes

characteristic

C₂×D₈×C₅₃

C₈

yes

characteristic

C₄×D₄×C₅₃

C₂×C₄

yes

—

#107×C₅₃

C₈

yes

characteristic

#102×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₅₃

C₂×C₈

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

—

—

—

—

C₂×D₄

yes

yes

—

—

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

M₃₂

yes

—

—

—

yes

—

#12064

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×M₃₂

yes

C₄×C₅₃

yes

—

C₈×D₈

—

#1367

—

D₄⋊C₁₆

—

#1312

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×M₃₂

—

106

—

C₄×D₈

#6157

yes

characteristic

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

106

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

—

C₅₃

#12064

yes

characteristic, complemented, Sylow

C₂×D₈

#88324

yes

characteristic

C₂²×C₈

—

106

—

C₄×D₄

C₂×#6157

yes

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

100

#107

#88324

yes

characteristic

101

C₄⋊C₈

—

102

#102

#88324

yes

—

103

C₂²⋊C₈

—

106

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×#6157

yes

characteristic

107

C₂×D₄

C₂×#88324

yes

—

108

C₂²×C₄

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

106

—

113

C₂×C₈

—

106

—

114

C₂×C₈

—

212

—

115

C₂×C₈

—

212

—

116

C₂×C₈

C₂×#88324

yes

characteristic

117

C₂×C₈

C₂×#88324

yes

characteristic

118

C₄⋊C₄

C₂×#88324

yes

—

119

C₂²⋊C₄

—

120

C₄²

C₂²×#6157

yes

characteristic

121

C₁₆

—

106

—

122

C₁₆

—

212

—

123

C₁₆

—

212

—

124

C₂³

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

D₄×#6157

yes

characteristic

131

C₂×C₄

C₂²×#88324

yes

characteristic

132

C₂×C₄

#13133304

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

106

—

136

C₈

—

212

—

137

C₈

—

212

—

138

C₈

#13133291

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

D₄×#88324

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#16811664

yes

characteristic

146

C₄

#16811670

yes

characteristic

147

C₄

C₂×#13133291

yes

characteristic

148

C₄

#16811681

yes

characteristic

149

C₂

—

150

C₂

#20290311

yes

characteristic

151

C₂

#20291099

yes

characteristic

152

C₂

#20291105

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₄×D₈×C₅₃	C₄
socle	C₂²×C₅₃	C₈×D₄
prime radical	C₂×D₈×C₅₃	C₈
cosocle	C₄×#88322	C₂³
hypercenter	C₄×D₈	#6157
hypocenter	C₅₃	#12064
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×#6157

Fitting

C₄×D₈×C₅₃

C₄

socle

C₂²×C₅₃

C₈×D₄

prime radical

C₂×D₈×C₅₃

C₈

C₂³

C₄×D₈

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

N	H ≅ G/N	type
1. #20291099	C₂	semidirect
2. #20289737	C₂	semidirect
3. D₄⋊#11719910	C₂	semidirect
4. #20289628	C₂	semidirect
57. C₈×C₅₃	M₃₂	semidirect
92. C₅₃	#12064	semidirect
138. C₈	#13133291	semidirect

H ≅ G/N

type

1. #20291099

C₂

semidirect

2. #20289737

C₂

semidirect

3. D₄⋊#11719910

C₂

semidirect

4. #20289628

C₂

semidirect

57. C₈×C₅₃

M₃₂

semidirect

92. C₅₃

#12064

semidirect

138. C₈

#13133291

semidirect

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

subgroup	index
#20291099	2
#20289737	2
D₄⋊#11719910	2
#20289628	2
D₈×#88322	2
#12064	53

subgroup

index

Subgroups — #28310049

Conjugacy classes of subgroups

Complements

Maximal subgroups