Group #28379824 — subgroups

Subgroups — #28379824

Order 13952.

348 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 162 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	#20296078	C₂	1	4	yes	maximal, complemented
2	#20296076	C₂	1	4	yes	maximal, complemented
3	#20296033	C₂	1	2	yes	maximal
4	#20296030	C₂	1	1	yes	maximal, characteristic
5	#20295991	C₂	1	2	yes	maximal
6	#20295985	C₂	1	1	yes	maximal, characteristic
7	#20295582	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182039	C₂²	1	2	yes	—
9	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
11	#16812766	C₂²	1	4	yes	—
12	C₄⋊#11724786	C₂²	1	4	yes	—
13	C₄⋊#11724786	C₂²	1	4	yes	—
14	#16812754	C₂²	1	2	yes	—
15	#16812754	C₂²	1	2	yes	—
16	#16812754	C₂²	1	4	yes	—
17	C₄×#11724786	C₂²	1	4	yes	—
18	#16812698	C₄	1	2	yes	—
19	#16812691	C₄	1	1	yes	characteristic
20	#16812677	C₄	1	4	yes	—
21	#16812664	C₂²	1	4	yes	—
22	#16812657	C₂²	1	2	yes	—
23	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
24	#125×C₁₀₉	C₄	1	1	yes	characteristic
25	#13182039	—	2	4	no	—
26	D₁₀₉⋊C₈	—	2	2	no	—
27	D₁₀₉⋊C₈	Q₈	1	4	yes	complemented
28	C₂×#11724786	C₂³	1	4	yes	—
29	C₂×#11724786	C₂³	1	1	yes	characteristic
30	C₂×#11724786	C₂³	1	1	yes	characteristic
31	C₂×#11724786	C₂³	1	2	yes	—
32	C₂×Dic₂₁₈	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
36	#13182008	C₂×C₄	1	4	yes	—
37	#13182008	C₂×C₄	1	4	yes	—
38	#13182006	C₂×C₄	1	4	yes	—
39	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₀₉	C₂³	1	2	yes	—
41	C₄×Dic₁₀₉	C₂³	1	4	yes	—
42	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
44	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
50	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
51	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
52	C₄×D₁₀₉	—	4	2	no	—
53	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
54	C₄×D₁₀₉	#41	1	2	yes	—
55	Dic₂₁₈	—	4	4	no	—
56	#11724786	—	2	1	no	—
57	#11724786	—	2	1	no	—
58	#11724786	—	2	4	no	—
59	#11724786	C₂×Q₈	1	4	yes	—
60	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
61	C₂×D₁₀₉	—	2	2	no	—
62	C₂×D₁₀₉	C₄×Q₈	1	2	yes	—
63	C₄×C₁₀₉	—	2	4	no	—
64	C₄×C₁₀₉	—	2	2	no	—
65	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
66	Dic₁₀₉	—	2	2	no	—
67	Dic₁₀₉	—	2	4	no	—
68	Dic₁₀₉	C₂²×Q₈	1	1	yes	characteristic
69	Dic₁₀₉	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₀₉	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₁₀₉	#568	1	2	yes	—
72	D₁₀₉	—	2	2	no	—
73	#2961	—	109	1	no	maximal, Sylow
74	C₁₀₉	#2961	1	1	yes	characteristic, complemented, Sylow
75	#533	—	109	1	no	—
76	#446	—	109	4	no	—
77	#446	—	109	4	no	—
78	#424	—	109	2	no	—
79	#423	—	109	1	no	—
80	#405	—	109	2	no	—
81	C₂×#97	—	109	1	no	—
82	C₂×M₁₆	—	109	2	no	—
83	C₂²×C₈	—	109	1	no	—
84	C₂²×C₈	—	109	1	no	—
85	C₄⋊Q₈	—	109	2	no	—
86	#125	—	109	1	no	—
87	#125	#89137	1	1	yes	characteristic
88	C₂²⋊Q₈	—	109	4	no	—
89	C₄×Q₈	—	109	4	no	—
90	#117	—	109	2	no	—
91	C₂×C₄²	—	109	1	no	—
92	C₄⋊C₈	—	109	4	no	—
93	C₄⋊C₈	—	109	4	no	—
94	C₄⋊C₈	—	109	4	no	—
95	#97	—	109	2	no	—
96	#97	—	109	2	no	—
97	#97	—	109	4	no	—
98	C₄×C₈	—	109	4	no	—
99	C₂×Q₈	—	109	4	no	—
100	C₂²×C₄	—	109	1	no	—
101	C₂²×C₄	—	109	2	no	—
102	M₁₆	—	218	4	no	—
103	C₂×C₈	—	109	2	no	—
104	C₂×C₈	—	109	4	no	—
105	C₂×C₈	—	109	4	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	109	1	no	—
108	C₂×C₈	—	218	2	no	—
109	C₄⋊C₄	—	109	4	no	—
110	C₄⋊C₄	—	109	4	no	—
111	C₄⋊C₄	—	109	2	no	—
112	C₄⋊C₄	C₂×#89137	1	2	yes	—
113	C₄⋊C₄	C₂×#89137	1	4	yes	—
114	C₂²⋊C₄	—	109	4	no	—
115	C₄²	—	109	1	no	—
116	C₄²	—	109	4	no	—
117	C₄²	—	109	2	no	—
118	C₄²	C₂×#89137	1	1	yes	characteristic
119	C₂³	—	109	1	no	—
120	Q₈	—	436	4	no	—
121	C₂×C₄	—	109	1	no	—
122	C₂×C₄	—	109	2	no	—
123	C₂×C₄	—	109	2	no	—
124	C₂×C₄	—	109	2	no	—
125	C₂×C₄	—	109	4	no	—
126	C₂×C₄	—	436	2	no	—
127	C₂×C₄	C₂²×#89137	1	2	yes	—
128	C₂×C₄	C₂²×#89137	1	4	yes	—
129	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
130	C₈	—	109	4	no	—
131	C₈	—	218	1	no	—
132	C₈	—	218	1	no	—
133	C₈	—	218	4	no	—
134	C₂²	—	109	2	no	—
135	C₂²	—	218	2	no	—
136	C₂²	C₂³×#89137	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	109	1	no	—
140	C₄	—	109	1	no	—
141	C₄	—	218	2	no	—
142	C₄	—	218	4	no	—
143	C₄	Q₈×#89137	1	2	yes	—
144	C₂	—	218	2	no	—
145	C₂	C₂×Q₈×#89137	1	1	yes	characteristic
146	C₂	#20296105	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20296078

C₂

yes

maximal, complemented

#20296076

C₂

yes

maximal, complemented

#20296033

C₂

yes

maximal

#20296030

C₂

yes

maximal, characteristic

#20295991

C₂

yes

maximal

#20295985

C₂

yes

maximal, characteristic

#20295582

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×D₁₀₉⋊C₈

C₂²

yes

characteristic

C₂×D₁₀₉⋊C₈

C₂²

yes

characteristic

#16812766

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

#16812754

C₂²

yes

—

#16812754

C₂²

yes

—

#16812754

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

#16812698

C₄

yes

—

#16812691

C₄

yes

characteristic

#16812677

C₄

yes

—

#16812664

C₂²

yes

—

#16812657

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#125×C₁₀₉

C₄

yes

characteristic

#13182039

—

D₁₀₉⋊C₈

—

D₁₀₉⋊C₈

Q₈

yes

complemented

C₂×#11724786

C₂³

yes

—

C₂×#11724786

C₂³

yes

characteristic

C₂×#11724786

C₂³

yes

characteristic

C₂×#11724786

C₂³

yes

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

C₂×C₄

yes

—

#13182008

C₂×C₄

yes

—

#13182006

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×Q₈

yes

—

C₄×D₁₀₉

#41

yes

—

Dic₂₁₈

—

—

—

—

C₂×Q₈

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₀₉

#568

yes

—

D₁₀₉

—

#2961

—

109

maximal, Sylow

C₁₀₉

#2961

yes

characteristic, complemented, Sylow

#533

—

109

—

#446

—

109

—

#446

—

109

—

#424

—

109

—

#423

—

109

—

#405

—

109

—

C₂×#97

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

#125

—

109

—

#125

#89137

yes

characteristic

C₂²⋊Q₈

—

109

—

C₄×Q₈

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

109

—

#97

—

109

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

—

109

—

102

M₁₆

—

218

—

103

C₂×C₈

—

109

—

104

C₂×C₈

—

109

—

105

C₂×C₈

—

109

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

109

—

108

C₂×C₈

—

218

—

109

C₄⋊C₄

—

109

—

110

C₄⋊C₄

—

109

—

111

C₄⋊C₄

—

109

—

112

C₄⋊C₄

C₂×#89137

yes

—

113

C₄⋊C₄

C₂×#89137

yes

—

114

C₂²⋊C₄

—

109

—

115

C₄²

—

109

—

116

C₄²

—

109

—

117

C₄²

—

109

—

118

C₄²

C₂×#89137

yes

characteristic

119

C₂³

—

109

—

120

Q₈

—

436

—

121

C₂×C₄

—

109

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

109

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

109

—

126

C₂×C₄

—

436

—

127

C₂×C₄

C₂²×#89137

yes

—

128

C₂×C₄

C₂²×#89137

yes

—

129

C₂×C₄

C₂²×#89137

yes

characteristic

130

C₈

—

109

—

131

C₈

—

218

—

132

C₈

—

218

—

133

C₈

—

218

—

134

C₂²

—

109

—

135

C₂²

—

218

—

136

C₂²

C₂³×#89137

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

109

—

140

C₄

—

109

—

141

C₄

—

218

—

142

C₄

—

218

—

143

C₄

Q₈×#89137

yes

—

144

C₂

—

218

—

145

C₂

C₂×Q₈×#89137

yes

characteristic

146

C₂

#20296105

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#125×C₁₀₉	C₄
prime radical	C₂²×D₁₀₉	C₂²×C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	#125	#89137
hypocenter	C₁₀₉	#2961
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#89137

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

#125×C₁₀₉

C₄

prime radical

C₂²×D₁₀₉

C₂²×C₄

cosocle

C₂×Dic₁₀₉

C₂⁴

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #20296078	C₂	semidirect
2. #20296076	C₂	semidirect
27. D₁₀₉⋊C₈	Q₈	semidirect
74. C₁₀₉	#2961	semidirect

H ≅ G/N

type

1. #20296078

C₂

semidirect

2. #20296076

C₂

semidirect

27. D₁₀₉⋊C₈

Q₈

semidirect

74. C₁₀₉

#2961

semidirect

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

subgroup	index
#20296078	2
#20296076	2
#20296033	2
#20296030	2
#20295991	2
#20295985	2
#20295582	2
#2961	109

subgroup

index

109

Subgroups — #28379824

Conjugacy classes of subgroups

Complements

Maximal subgroups