Group #28362400 — subgroups

Subgroups — #28362400

Order 13952.

156 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20295405	C₂	1	1	yes	maximal, characteristic
2	#20295311	C₂	1	1	yes	maximal, characteristic
3	Q₈×#11724784	C₂	1	1	yes	maximal, characteristic
4	#20294778	C₂	1	1	yes	maximal, characteristic
5	#20294772	C₂	1	1	yes	maximal, characteristic
6	#20294771	C₂	1	1	yes	maximal, characteristic
7	#475×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182019	—	2	1	no	—
9	C₂×#13182019	—	2	1	no	—
10	#16812699	C₂²	1	1	yes	characteristic
11	#16812630	—	2	1	no	—
12	C₄⋊#11724784	—	2	1	no	—
13	C₄⋊#11724784	C₂²	1	1	yes	characteristic
14	C₄⋊#11724784	C₂²	1	1	yes	characteristic
15	#16812623	—	2	1	no	—
16	C₄×#11724784	—	2	1	no	—
17	C₄×#11724784	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	#106×C₁₀₉	—	2	1	no	—
21	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	#103×C₁₀₉	—	2	1	no	—
23	C₂×Dic₂₁₈	—	2	1	no	—
24	#13182019	—	4	2	no	—
25	#13182019	—	4	2	no	—
26	C₂×#11724784	—	2	1	no	—
27	C₂×#11724784	—	2	1	no	—
28	C₂×#11724784	D₄	1	1	yes	characteristic
29	C₂×#11724784	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₄⋊Dic₁₀₉	—	2	1	no	—
32	C₂×Q₈×C₁₀₉	—	2	1	no	—
33	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₀₉	—	2	1	no	—
35	C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₄⋊C₄×C₁₀₉	—	2	1	no	—
37	C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	—	2	1	no	—
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	Q₈×C₁₀₉	—	2	1	no	—
42	Q₈×C₁₀₉	—	4	1	no	—
43	Q₈×C₁₀₉	—	4	1	no	—
44	Q₈×C₁₀₉	Q₁₆	1	2	yes	complemented
45	C₂×C₄×C₁₀₉	—	2	1	no	—
46	C₂×C₄×C₁₀₉	—	2	1	no	—
47	C₂×C₄×C₁₀₉	—	4	1	no	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
51	C₂×Dic₁₀₉	—	4	1	no	—
52	Dic₂₁₈	—	4	1	no	—
53	Dic₂₁₈	—	4	1	no	—
54	C₈×C₁₀₉	—	4	1	no	—
55	#11724784	—	2	1	no	—
56	#11724784	—	4	1	no	—
57	#11724784	—	4	1	no	—
58	#11724784	SD₁₆	1	2	yes	—
59	C₂²×C₁₀₉	#121	1	1	yes	characteristic
60	C₄×C₁₀₉	—	2	2	no	—
61	C₄×C₁₀₉	—	2	1	no	—
62	C₄×C₁₀₉	—	4	1	no	—
63	C₄×C₁₀₉	—	8	1	no	—
64	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
65	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
66	C₄×C₁₀₉	#135	1	1	yes	characteristic
67	C₄×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
68	Dic₁₀₉	—	8	1	no	—
69	C₂×C₁₀₉	#460	1	1	yes	characteristic
70	C₂×C₁₀₉	#467	1	1	yes	characteristic
71	C₂×C₁₀₉	#473	1	1	yes	characteristic
72	#1670	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1670	1	1	yes	characteristic, complemented, Sylow
74	#494	—	109	1	no	—
75	#475	—	109	1	no	—
76	#475	D₁₀₉	1	1	yes	characteristic
77	C₈×Q₈	—	109	1	no	—
78	#335	—	109	1	no	—
79	#332	—	109	1	no	—
80	#332	—	109	1	no	—
81	C₂×Q₁₆	—	218	1	no	—
82	C₂×Q₁₆	—	218	1	no	—
83	C₄⋊Q₈	—	109	1	no	—
84	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
85	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
86	#106	—	2	1	no	—
87	#106	—	218	1	no	—
88	C₄⋊C₈	—	109	1	no	—
89	C₄⋊C₈	—	109	1	no	—
90	C₄⋊C₈	—	218	1	no	—
91	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
92	#103	—	2	1	no	—
93	#103	—	218	1	no	—
94	C₄×C₈	—	109	1	no	—
95	C₄×C₈	—	218	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	218	1	no	—
98	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
99	Q₁₆	—	436	2	no	—
100	Q₁₆	—	436	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	109	1	no	—
103	C₂×C₈	—	109	1	no	—
104	C₂×C₈	—	218	1	no	—
105	C₂×C₈	—	218	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	218	1	no	—
110	C₄⋊C₄	—	218	1	no	—
111	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
112	C₄²	—	2	1	no	—
113	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	4	1	no	—
117	Q₈	—	436	1	no	—
118	Q₈	—	436	1	no	—
119	Q₈	#13182019	1	2	yes	complemented
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	436	1	no	—
124	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
126	C₂×C₄	#13182028	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	109	2	no	—
129	C₈	—	436	1	no	—
130	C₈	—	436	1	no	—
131	C₈	—	218	1	no	—
132	C₂²	#16812671	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	8	1	no	—
137	C₄	—	872	1	no	—
138	C₄	SD₁₆×D₁₀₉	1	1	yes	characteristic
139	C₄	C₂×#13182019	1	1	yes	characteristic
140	C₄	#16812719	1	1	yes	characteristic
141	C₄	#16812726	1	1	yes	characteristic
142	C₂	#20295224	1	1	yes	characteristic
143	C₂	#20295265	1	1	yes	characteristic
144	C₂	#20295299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295405

C₂

yes

maximal, characteristic

#20295311

C₂

yes

maximal, characteristic

Q₈×#11724784

C₂

yes

maximal, characteristic

#20294778

C₂

yes

maximal, characteristic

#20294772

C₂

yes

maximal, characteristic

#20294771

C₂

yes

maximal, characteristic

#475×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

—

C₂×#13182019

—

#16812699

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

#106×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

characteristic

#103×C₁₀₉

—

C₂×Dic₂₁₈

—

—

—

—

—

D₄

yes

characteristic

C₂×#11724784

D₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

Q₁₆

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

—

—

—

SD₁₆

yes

—

C₂²×C₁₀₉

#121

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₉

C₂×Q₁₆

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1670

yes

characteristic, complemented, Sylow

#494

—

109

—

#475

—

109

—

#475

D₁₀₉

yes

characteristic

C₈×Q₈

—

109

—

#335

—

109

—

#332

—

109

—

#332

—

109

—

C₂×Q₁₆

—

218

—

C₂×Q₁₆

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

#106

—

#106

—

218

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

C₄⋊C₈

C₂×D₁₀₉

yes

characteristic

#103

—

#103

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

characteristic

Q₁₆

—

436

—

100

Q₁₆

—

436

—

101

C₂×C₈

—

102

C₂×C₈

—

109

—

103

C₂×C₈

—

109

—

104

C₂×C₈

—

218

—

105

C₂×C₈

—

218

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

218

—

110

C₄⋊C₄

—

218

—

111

C₄⋊C₄

C₂²⋊D₁₀₉

yes

characteristic

112

C₄²

—

113

C₄²

C₂²×D₁₀₉

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

436

—

118

Q₈

—

436

—

119

Q₈

#13182019

yes

complemented

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

436

—

124

C₂×C₄

D₄×D₁₀₉

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

126

C₂×C₄

#13182028

yes

characteristic

127

C₈

—

128

C₈

—

109

—

129

C₈

—

436

—

130

C₈

—

436

—

131

C₈

—

218

—

132

C₂²

#16812671

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

872

—

138

C₄

SD₁₆×D₁₀₉

yes

characteristic

139

C₄

C₂×#13182019

yes

characteristic

140

C₄

#16812719

yes

characteristic

141

C₄

#16812726

yes

characteristic

142

C₂

#20295224

yes

characteristic

143

C₂

#20295265

yes

characteristic

144

C₂

#20295299

yes

characteristic

	subgroup	G/H
center	C₂²	#16812671
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#475×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#121
hypercenter	#475	D₁₀₉
hypocenter	C₁₀₉	#1670
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#16812671

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#475×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
44. Q₈×C₁₀₉	Q₁₆	semidirect
73. C₁₀₉	#1670	semidirect
119. Q₈	#13182019	semidirect

H ≅ G/N

type

44. Q₈×C₁₀₉

Q₁₆

semidirect

73. C₁₀₉

#1670

semidirect

119. Q₈

#13182019

semidirect

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

subgroup	index
#20295405	2
#20295311	2
Q₈×#11724784	2
#20294778	2
#20294772	2
#20294771	2
#475×C₁₀₉	2
#1670	109

subgroup

index

#475×C₁₀₉

109

Subgroups — #28362400

Conjugacy classes of subgroups

Complements

Maximal subgroups