Group #28309978 — subgroups

Subgroups — #28309978

Order 13568.

222 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 82 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	Q₁₆×#88324	C₂	1	2	yes	maximal
2	#20289748	C₂	1	1	yes	maximal, characteristic
3	#20289671	C₂	1	2	yes	maximal
4	#20289626	C₂	1	1	yes	maximal, characteristic
5	Q₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88324	—	2	2	no	—
7	#16811509	C₂²	1	1	yes	characteristic
8	#16811486	C₂²	1	2	yes	—
9	Q₈⋊#88324	—	2	2	no	—
10	C₈×#88324	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
12	#16810449	C₄	1	1	yes	characteristic
13	#16810427	C₄	1	2	yes	—
14	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#11719910	—	2	1	no	—
17	C₄⋊#88324	—	2	2	no	—
18	C₄⋊#88324	—	4	2	no	—
19	C₄×#88324	—	4	2	no	—
20	C₄×#88324	D₄	1	1	yes	characteristic
21	C₄×#88324	D₄	1	1	yes	characteristic
22	Q₈×Dic₅₃	—	2	2	no	—
23	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
24	#13133151	C₂×C₄	1	2	yes	—
25	#13133141	—	2	2	no	—
26	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
27	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
28	Q₃₂×C₅₃	C₈	1	4	yes	complemented
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	—	2	2	no	—
31	C₂×#88324	—	2	1	no	—
32	C₂×#88324	—	4	2	no	—
33	C₂×#88324	D₈	1	2	yes	—
34	Q₁₆×C₅₃	—	2	2	no	—
35	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	2	2	no	—
38	C₄⋊Dic₅₃	—	4	2	no	—
39	C₄×Dic₅₃	—	4	2	no	—
40	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×#88322	#41	1	1	yes	characteristic
42	#11719910	—	2	2	no	—
43	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
44	#11719908	—	2	1	no	—
45	Q₈×C₅₃	—	2	4	no	—
46	Q₈×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
49	C₂×Dic₅₃	—	4	2	no	—
50	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₅₃	#135	1	1	yes	characteristic
52	#88324	—	2	2	no	—
53	#88324	—	4	1	no	—
54	#88324	—	8	2	no	—
55	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₅₃	#131	1	1	yes	characteristic
57	#88322	—	2	1	no	—
58	#12044	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
60	C₄×C₅₃	—	4	4	no	—
61	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₅₃	#443	1	1	yes	characteristic
63	Dic₅₃	—	2	1	no	—
64	Dic₅₃	—	8	2	no	—
65	Dic₅₃	#510	1	2	yes	—
66	C₄×Q₃₂	—	53	1	no	—
67	C₈×Q₁₆	—	53	2	no	—
68	#1372	—	53	1	no	—
69	#1332	—	53	2	no	—
70	#1311	—	53	1	no	—
71	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
72	C₂×C₅₃	#2176	1	1	yes	characteristic
73	C₂×C₅₃	#2179	1	1	yes	characteristic
74	C₂×Q₃₂	#6157	1	1	yes	characteristic
75	C₈×Q₈	—	106	2	no	—
76	C₄×Q₁₆	—	53	2	no	—
77	#366	—	53	1	no	—
78	#358	—	53	2	no	—
79	C₄×C₁₆	—	53	1	no	—
80	C₄×C₁₆	—	53	1	no	—
81	#335	—	53	2	no	—
82	Q₈⋊C₈	—	106	2	no	—
83	C₈²	—	53	1	no	—
84	C₅₃	#12044	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₁₆	C₂×#6157	1	2	yes	—
86	C₄×Q₈	—	106	2	no	—
87	Q₃₂	#88324	1	4	yes	complemented
88	C₂×C₁₆	—	53	1	no	—
89	C₂×C₁₆	—	106	1	no	—
90	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
91	#107	—	53	2	no	—
92	C₄⋊C₈	—	106	2	no	—
93	C₄⋊C₈	—	212	2	no	—
94	#103	—	106	2	no	—
95	C₄×C₈	—	53	1	no	—
96	C₄×C₈	—	53	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	—	212	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	Q₁₆	—	2	2	no	—
101	Q₁₆	C₂×#88324	1	4	yes	—
102	C₂×C₈	—	53	2	no	—
103	C₂×C₈	—	53	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	212	2	no	—
106	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
107	C₄⋊C₄	—	106	2	no	—
108	C₄⋊C₄	—	212	2	no	—
109	C₄²	—	53	1	no	—
110	C₄²	—	212	2	no	—
111	C₁₆	—	106	2	no	—
112	C₁₆	—	106	1	no	—
113	C₁₆	C₂×#88324	1	2	yes	—
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	D₄×#6157	1	1	yes	characteristic
121	C₈	—	106	2	no	—
122	C₈	—	106	1	no	—
123	C₈	—	424	2	no	—
124	C₈	—	212	1	no	—
125	C₈	C₂²×#88324	1	1	yes	characteristic
126	C₈	#13133304	1	1	yes	characteristic
127	C₂²	D₈×#6157	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	53	2	no	—
130	C₄	—	106	1	no	—
131	C₄	—	424	2	no	—
132	C₄	D₄×#88324	1	1	yes	characteristic
133	C₄	#16811665	1	1	yes	characteristic
134	C₂	D₈×#88324	1	1	yes	characteristic
135	C₂	#20291112	1	1	yes	characteristic
136	C₂	#20291118	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₁₆×#88324

C₂

yes

maximal

#20289748

C₂

yes

maximal, characteristic

#20289671

C₂

yes

maximal

#20289626

C₂

yes

maximal, characteristic

Q₃₂×Dic₅₃

C₂

yes

maximal, characteristic

Q₈×#88324

—

#16811509

C₂²

yes

characteristic

#16811486

C₂²

yes

—

Q₈⋊#88324

—

C₈×#88324

C₂²

yes

characteristic

Q₁₆×Dic₅₃

C₂²

yes

—

#16810449

C₄

yes

characteristic

#16810427

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

C₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#88324

D₄

yes

characteristic

Q₈×Dic₅₃

—

C₂×#11719908

C₂×C₄

yes

characteristic

#13133151

C₂×C₄

yes

—

#13133141

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×Q₁₆×C₅₃

C₂×C₄

yes

—

Q₃₂×C₅₃

C₈

yes

complemented

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×#88324

—

C₂×#88324

—

C₂×#88324

D₈

yes

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×D₄

yes

yes

—

C₁₆×C₅₃

C₂×C₈

yes

—

#11719908

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₈

yes

C₂×Dic₅₃

yes

—

—

—

C₈×C₅₃

C₂²×C₈

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₄×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#443

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#510

yes

—

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

—

C₂×C₅₃

C₈×D₈

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

yes

C₈×Q₈

—

106

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#335

—

Q₈⋊C₈

—

106

—

C₈²

—

C₅₃

#12044

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×#6157

yes

—

C₄×Q₈

—

106

—

Q₃₂

#88324

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

characteristic

#107

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

#103

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

212

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

C₂×#88324

yes

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

—

212

—

106

C₂×C₈

C₂²×#6157

yes

characteristic

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

212

—

109

C₄²

—

110

C₄²

—

212

—

111

C₁₆

—

106

—

112

C₁₆

—

106

—

113

C₁₆

C₂×#88324

yes

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

212

—

120

C₂×C₄

D₄×#6157

yes

characteristic

121

C₈

—

106

—

122

C₈

—

106

—

123

C₈

—

424

—

124

C₈

—

212

—

125

C₈

C₂²×#88324

yes

characteristic

126

C₈

#13133304

yes

characteristic

127

C₂²

D₈×#6157

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

106

—

131

C₄

—

424

—

132

C₄

D₄×#88324

yes

characteristic

133

C₄

#16811665

yes

characteristic

134

C₂

D₈×#88324

yes

characteristic

135

C₂

#20291112

yes

characteristic

136

C₂

#20291118

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×#6157
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×Q₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	C₄×D₈
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂×Q₃₂	#6157
hypocenter	C₅₃	#12044
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×#6157

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₂×Q₃₂×C₅₃

C₄

socle, prime radical

C₂²×C₅₃

C₄×D₈

cosocle

C₈×Dic₅₃

C₂³

C₂×Q₃₂

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
28. Q₃₂×C₅₃	C₈	semidirect
84. C₅₃	#12044	semidirect
87. Q₃₂	#88324	semidirect

H ≅ G/N

type

28. Q₃₂×C₅₃

C₈

semidirect

84. C₅₃

#12044

semidirect

87. Q₃₂

#88324

semidirect

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

subgroup	index
Q₁₆×#88324	2
#20289748	2
#20289671	2
#20289626	2
Q₃₂×Dic₅₃	2
#12044	53

subgroup

index

Q₁₆×#88324

Q₃₂×Dic₅₃

Subgroups — #28309978

Conjugacy classes of subgroups

Complements

Maximal subgroups