Group #28310039 — subgroups

Subgroups — #28310039

Order 13568.

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

normal subgroups only

This group has 200 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	#20291130	C₂	1	2	yes	maximal
2	#20289737	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11719910	C₂	1	2	yes	maximal
4	#20289628	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#88322	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11719910	—	2	2	no	—
7	C₄⋊#11719910	C₂²	1	2	yes	—
8	#16811511	—	2	2	no	—
9	C₄×#11719910	C₂²	1	1	yes	characteristic
10	C₄×#11719910	C₂²	1	1	yes	characteristic
11	Q₈×#88322	C₂²	1	2	yes	—
12	#16810376	C₄	1	1	yes	characteristic
13	Q₈⋊#88322	C₄	1	2	yes	—
14	C₈×#88322	C₂²	1	1	yes	characteristic
15	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#11719910	—	2	2	no	—
17	C₂×#11719910	—	2	1	no	—
18	C₂×#11719910	D₄	1	2	yes	—
19	C₄⋊#88322	—	2	2	no	—
20	C₄⋊#88322	C₂×C₄	1	2	yes	—
21	C₄×#88322	—	2	2	no	—
22	C₄×#88322	C₂×C₄	1	1	yes	characteristic
23	C₄×#88322	C₂³	1	1	yes	characteristic
24	C₂×Q₁₆×C₅₃	C₈	1	1	yes	characteristic
25	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
26	#107×C₅₃	C₈	1	1	yes	characteristic
27	#103×C₅₃	C₈	1	2	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	C₂×C₈	1	2	yes	—
30	Q₁₆×C₅₃	—	2	2	no	—
31	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
35	C₄²×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×#88322	—	2	2	no	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	C₂×D₄	1	1	yes	characteristic
40	C₂×#88322	#41	1	1	yes	characteristic
41	#11719910	—	2	2	no	—
42	#11719910	—	4	1	no	—
43	#11719910	—	4	2	no	—
44	Q₈×C₅₃	—	2	2	no	—
45	Q₈×C₅₃	—	2	2	no	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	2	no	—
48	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
51	C₈×C₅₃	—	2	1	no	—
52	C₈×C₅₃	M₃₂	1	2	yes	—
53	#88322	—	2	1	no	—
54	#88322	—	4	1	no	—
55	#88322	—	4	2	no	—
56	#88322	#137	1	2	yes	—
57	#12061	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	4	2	no	—
61	C₄×C₅₃	—	4	2	no	—
62	C₄×C₅₃	#441	1	1	yes	characteristic
63	C₄×C₅₃	#444	1	1	yes	characteristic
64	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
65	C₄×C₅₃	#504	1	1	yes	characteristic
66	#2183	—	53	2	no	—
67	C₈×Q₁₆	—	53	1	no	—
68	#1367	—	53	1	no	—
69	Q₈⋊C₁₆	—	53	2	no	—
70	#1312	—	53	1	no	—
71	C₂×C₅₃	#1579	1	1	yes	characteristic
72	C₂×C₅₃	#2169	1	1	yes	characteristic
73	C₂×C₅₃	#2171	1	1	yes	characteristic
74	C₈×Q₈	—	53	2	no	—
75	C₄×Q₁₆	#6157	1	1	yes	characteristic
76	C₄⋊C₁₆	—	53	2	no	—
77	C₄⋊C₁₆	—	106	2	no	—
78	#346	—	106	2	no	—
79	C₄×C₁₆	—	53	1	no	—
80	C₄×C₁₆	—	53	1	no	—
81	#335	—	53	1	no	—
82	Q₈⋊C₈	—	53	2	no	—
83	C₈²	—	53	1	no	—
84	C₅₃	#12061	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	#88324	1	1	yes	characteristic
86	C₄×Q₈	C₂×#6157	1	2	yes	—
87	C₂×C₁₆	—	53	2	no	—
88	C₂×C₁₆	—	106	1	no	—
89	C₂×C₁₆	—	106	2	no	—
90	#107	#88324	1	1	yes	characteristic
91	C₄⋊C₈	—	53	2	no	—
92	C₄⋊C₈	—	106	2	no	—
93	#103	#88324	1	2	yes	—
94	C₄×C₈	—	53	1	no	—
95	C₄×C₈	—	53	1	no	—
96	C₄×C₈	—	106	2	no	—
97	C₄×C₈	C₂×#6157	1	1	yes	characteristic
98	C₂×Q₈	C₂×#88324	1	2	yes	—
99	Q₁₆	—	2	2	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	106	2	no	—
103	C₂×C₈	—	106	1	no	—
104	C₂×C₈	C₂×#88324	1	1	yes	characteristic
105	C₂×C₈	C₂×#88324	1	1	yes	characteristic
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	C₂×#88324	1	2	yes	—
108	C₄²	—	2	2	no	—
109	C₄²	C₂²×#6157	1	1	yes	characteristic
110	C₁₆	—	106	2	no	—
111	C₁₆	—	212	2	no	—
112	C₁₆	—	212	1	no	—
113	Q₈	—	2	2	no	—
114	Q₈	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	D₄×#6157	1	1	yes	characteristic
118	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
119	C₂×C₄	#13133304	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	53	2	no	—
122	C₈	—	106	1	no	—
123	C₈	—	212	1	no	—
124	C₈	—	212	2	no	—
125	C₈	#13133291	1	2	yes	—
126	C₂²	D₄×#88324	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	—	4	2	no	—
130	C₄	#16811663	1	1	yes	characteristic
131	C₄	#16811669	1	1	yes	characteristic
132	C₄	C₂×#13133291	1	1	yes	characteristic
133	C₄	#16811681	1	1	yes	characteristic
134	C₂	#20290308	1	1	yes	characteristic
135	C₂	#20291099	1	1	yes	characteristic
136	C₂	#20291105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291130

C₂

yes

maximal

#20289737

C₂

yes

maximal, characteristic

Q₈⋊#11719910

C₂

yes

maximal

#20289628

C₂

yes

maximal, characteristic

Q₁₆×#88322

C₂

yes

maximal, characteristic

C₄⋊#11719910

—

C₄⋊#11719910

C₂²

yes

—

#16811511

—

C₄×#11719910

C₂²

yes

characteristic

C₄×#11719910

C₂²

yes

characteristic

Q₈×#88322

C₂²

yes

—

#16810376

C₄

yes

characteristic

Q₈⋊#88322

C₄

yes

—

C₈×#88322

C₂²

yes

characteristic

C₄×Q₁₆×C₅₃

C₄

yes

—

—

D₄

yes

—

C₄⋊#88322

—

C₄⋊#88322

C₂×C₄

yes

—

C₄×#88322

—

C₄×#88322

C₂×C₄

yes

characteristic

C₄×#88322

C₂³

yes

characteristic

C₂×Q₁₆×C₅₃

C₈

yes

characteristic

C₄×Q₈×C₅₃

C₂×C₄

yes

—

#107×C₅₃

C₈

yes

characteristic

#103×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₈

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

—

—

—

Q₈×C₅₃

—

Q₈×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

—

#12061

—

maximal, Sylow

C₂²×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×M₃₂

yes

C₄×C₅₃

yes

—

C₈×Q₁₆

—

#1367

—

Q₈⋊C₁₆

—

#1312

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₈×Q₈

—

C₄×Q₁₆

#6157

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

#346

—

106

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

—

C₅₃

#12061

yes

characteristic, complemented, Sylow

C₂×Q₁₆

#88324

yes

characteristic

C₄×Q₈

C₂×#6157

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

#107

#88324

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

106

—

#103

#88324

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

106

—

C₄×C₈

C₂×#6157

yes

characteristic

C₂×Q₈

C₂×#88324

yes

—

Q₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

106

—

103

C₂×C₈

—

106

—

104

C₂×C₈

C₂×#88324

yes

characteristic

105

C₂×C₈

C₂×#88324

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂×#88324

yes

—

108

C₄²

—

109

C₄²

C₂²×#6157

yes

characteristic

110

C₁₆

—

106

—

111

C₁₆

—

212

—

112

C₁₆

—

212

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

D₄×#6157

yes

characteristic

118

C₂×C₄

C₂²×#88324

yes

characteristic

119

C₂×C₄

#13133304

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

106

—

123

C₈

—

212

—

124

C₈

—

212

—

125

C₈

#13133291

yes

—

126

C₂²

D₄×#88324

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#16811663

yes

characteristic

131

C₄

#16811669

yes

characteristic

132

C₄

C₂×#13133291

yes

characteristic

133

C₄

#16811681

yes

characteristic

134

C₂

#20290308

yes

characteristic

135

C₂

#20291099

yes

characteristic

136

C₂

#20291105

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#6157
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₄×Q₁₆×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₈×D₄
cosocle	C₄×#88322	C₂³
hypercenter	C₄×Q₁₆	#6157
hypocenter	C₅₃	#12061
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂×C₄

D₄×#6157

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₄²

C₂²×#6157

Fitting

C₄×Q₁₆×C₅₃

C₄

socle, prime radical

C₂²×C₅₃

C₈×D₄

C₂³

C₄×Q₁₆

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
84. C₅₃	#12061	semidirect

H ≅ G/N

type

84. C₅₃

#12061

semidirect

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

subgroup	index
#20291130	2
#20289737	2
Q₈⋊#11719910	2
#20289628	2
Q₁₆×#88322	2
#12061	53

subgroup

index

Q₁₆×#88322

Subgroups — #28310039

Conjugacy classes of subgroups

Complements

Maximal subgroups