Group #28379150 — subgroups

Subgroups — #28379150

Order 13952.

196 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 68 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	#20296037	C₂	1	2	yes	maximal
2	C₂×#16812768	C₂	1	1	yes	maximal, characteristic
3	#20295989	C₂	1	1	yes	maximal, characteristic
4	#20295905	C₂	1	2	yes	maximal
5	#108×D₁₀₉	C₂	1	1	yes	maximal, characteristic
6	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
7	#16812778	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊#89137	C₂²	1	2	yes	—
9	#16812768	—	2	2	no	—
10	#16812768	—	2	1	no	—
11	#16812768	—	2	1	no	—
12	#16812767	—	2	2	no	—
13	#16812753	—	2	1	no	—
14	C₈×#89137	—	2	1	no	—
15	M₁₆×D₁₀₉	C₂²	1	2	yes	—
16	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
17	#16812635	C₄	1	1	yes	characteristic, complemented
18	#16812634	C₄	1	2	yes	complemented
19	#108×C₁₀₉	C₄	1	1	yes	characteristic, complemented
20	C₂²×#89137	—	2	2	no	—
21	#13182039	—	2	1	no	—
22	#13182038	Q₈	1	2	yes	—
23	D₁₀₉⋊C₈	—	2	1	no	—
24	C₄⋊#89137	—	2	2	no	—
25	C₄⋊#89137	—	2	2	no	—
26	C₄⋊#89137	D₄	1	1	yes	characteristic
27	C₄⋊#89137	D₄	1	1	yes	characteristic
28	C₂²⋊#89137	—	2	1	no	—
29	C₄×#89137	—	2	1	no	—
30	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
31	#13182013	C₂×C₄	1	2	yes	—
32	#13182012	—	2	2	no	—
33	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
34	C₈×D₁₀₉	—	2	2	no	—
35	C₈×D₁₀₉	D₄	1	1	yes	characteristic
36	C₈×D₁₀₉	D₄	1	1	yes	characteristic
37	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
38	C₈×D₁₀₉	Q₈	1	1	yes	characteristic
39	M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
40	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₁₀₉	#41	1	1	yes	characteristic
42	C₂×#89137	—	2	1	no	—
43	C₂×#89137	—	2	1	no	—
44	C₂×#89137	—	4	2	no	—
45	C₂×#89137	—	8	2	no	—
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₁₀₉	C₂×Q₈	1	1	yes	characteristic
48	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₄×D₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	C₄×D₁₀₉	#41	1	1	yes	characteristic
52	#11724786	—	2	2	no	—
53	C₈×C₁₀₉	—	2	2	no	—
54	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
55	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
56	#11724784	—	2	2	no	—
57	#11724784	C₄⋊C₄	1	1	yes	characteristic
58	#11724784	C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
60	C₂×D₁₀₉	—	2	1	no	—
61	C₂×D₁₀₉	—	2	1	no	—
62	C₂×D₁₀₉	#121	1	1	yes	characteristic
63	C₂×D₁₀₉	#125	1	1	yes	characteristic
64	#89137	—	4	1	no	—
65	#89137	—	4	1	no	—
66	#89137	—	8	2	no	—
67	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
69	Dic₁₀₉	C₂²⋊Q₈	1	1	yes	characteristic
70	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₁₀₉	—	2	1	no	—
72	C₂×C₁₀₉	#389	1	1	yes	characteristic
73	D₁₀₉	—	2	1	no	—
74	D₁₀₉	#472	1	1	yes	characteristic
75	D₁₀₉	#473	1	1	yes	characteristic
76	#1952	—	109	1	no	maximal, Sylow
77	C₁₀₉	#1952	1	1	yes	characteristic, complemented, Sylow
78	C₂×#108	—	109	1	no	—
79	#428	—	109	2	no	—
80	C₂×#107	—	109	1	no	—
81	#405	—	109	1	no	—
82	#343	—	109	2	no	—
83	C₂×M₁₆	—	109	1	no	—
84	C₂×M₁₆	—	109	2	no	—
85	C₂²×C₈	—	109	1	no	—
86	#117	—	109	1	no	—
87	C₂×C₄⋊C₄	—	109	2	no	—
88	#108	—	109	2	no	—
89	#108	—	109	1	no	—
90	#108	#89137	1	1	yes	characteristic, complemented
91	#107	—	218	2	no	—
92	#107	—	218	1	no	—
93	#107	—	218	1	no	—
94	#106	—	218	2	no	—
95	#97	—	218	1	no	—
96	C₄×C₈	—	218	1	no	—
97	C₂²×C₄	—	109	1	no	—
98	C₂²×C₄	—	218	2	no	—
99	M₁₆	—	109	2	no	—
100	M₁₆	—	109	2	no	—
101	M₁₆	—	218	2	no	—
102	M₁₆	—	218	1	no	—
103	M₁₆	C₂×#89137	1	2	yes	—
104	C₂×C₈	—	109	1	no	—
105	C₂×C₈	—	109	1	no	—
106	C₂×C₈	—	109	1	no	—
107	C₂×C₈	—	109	1	no	—
108	C₂×C₈	—	109	1	no	—
109	C₂×C₈	—	218	2	no	—
110	C₂×C₈	—	218	1	no	—
111	C₂×C₈	C₂×#89137	1	1	yes	characteristic
112	C₄⋊C₄	—	109	1	no	—
113	C₄⋊C₄	—	109	1	no	—
114	C₄⋊C₄	—	218	2	no	—
115	C₄⋊C₄	—	218	2	no	—
116	C₂²⋊C₄	—	218	1	no	—
117	C₄²	—	218	1	no	—
118	C₂³	—	109	1	no	—
119	C₂×C₄	—	872	2	no	—
120	C₂×C₄	—	109	1	no	—
121	C₂×C₄	—	109	1	no	—
122	C₂×C₄	—	109	1	no	—
123	C₂×C₄	—	109	1	no	—
124	C₂×C₄	—	109	1	no	—
125	C₂×C₄	—	436	2	no	—
126	C₂×C₄	—	218	1	no	—
127	C₂×C₄	—	218	1	no	—
128	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	109	1	no	—
131	C₈	—	109	1	no	—
132	C₈	—	218	2	no	—
133	C₈	—	218	2	no	—
134	C₈	C₄⋊#89137	1	1	yes	characteristic
135	C₈	C₄⋊#89137	1	1	yes	characteristic
136	C₂²	—	109	1	no	—
137	C₂²	—	109	1	no	—
138	C₂²	—	218	1	no	—
139	C₂²	—	218	1	no	—
140	C₂²	Q₈×#89137	1	1	yes	characteristic
141	C₄	—	872	2	no	—
142	C₄	—	109	1	no	—
143	C₄	—	109	1	no	—
144	C₄	—	436	1	no	—
145	C₄	—	436	1	no	—
146	C₄	C₂×C₄⋊#89137	1	1	yes	characteristic
147	C₄	D₄×#89137	1	1	yes	characteristic
148	C₂	—	2	1	no	—
149	C₂	—	109	1	no	—
150	C₂	—	109	1	no	—
151	C₂	—	218	1	no	—
152	C₂	#20295977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296037

C₂

yes

maximal

C₂×#16812768

C₂

yes

maximal, characteristic

#20295989

C₂

yes

maximal, characteristic

#20295905

C₂

yes

maximal

#108×D₁₀₉

C₂

yes

maximal, characteristic

D₁₀₉⋊M₁₆

C₂²

yes

characteristic

#16812778

C₂²

yes

characteristic

C₂×C₄⋊#89137

C₂²

yes

—

—

—

—

—

—

—

M₁₆×D₁₀₉

C₂²

yes

—

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812635

C₄

yes

characteristic, complemented

#16812634

C₄

yes

complemented

#108×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

#13182039

—

#13182038

Q₈

yes

—

D₁₀₉⋊C₈

—

C₄⋊#89137

—

C₄⋊#89137

—

C₄⋊#89137

D₄

yes

characteristic

C₄⋊#89137

D₄

yes

characteristic

C₂²⋊#89137

—

C₄×#89137

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

#13182013

C₂×C₄

yes

—

#13182012

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

—

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

characteristic

C₈×D₁₀₉

Q₈

yes

characteristic

C₈×D₁₀₉

Q₈

yes

characteristic

M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

C₂²×D₁₀₉

yes

—

—

—

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×Q₈

yes

characteristic

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

C₂×Q₈

yes

C₄×D₁₀₉

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

C₈×C₁₀₉

C₄⋊C₄

yes

characteristic

#11724784

—

#11724784

C₄⋊C₄

yes

characteristic

#11724784

C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₄×Q₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

yes

C₂×D₁₀₉

yes

—

—

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

Dic₁₀₉

C₂²⋊Q₈

yes

characteristic

Dic₁₀₉

C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

#389

yes

characteristic

D₁₀₉

—

D₁₀₉

yes

D₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1952

yes

characteristic, complemented, Sylow

C₂×#108

—

109

—

#428

—

109

—

C₂×#107

—

109

—

#405

—

109

—

#343

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

#108

—

109

—

#108

—

109

—

#108

#89137

yes

characteristic, complemented

#107

—

218

—

#107

—

218

—

#107

—

218

—

#106

—

218

—

#97

—

218

—

C₄×C₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

109

—

100

M₁₆

—

109

—

101

M₁₆

—

218

—

102

M₁₆

—

218

—

103

M₁₆

C₂×#89137

yes

—

104

C₂×C₈

—

109

—

105

C₂×C₈

—

109

—

106

C₂×C₈

—

109

—

107

C₂×C₈

—

109

—

108

C₂×C₈

—

109

—

109

C₂×C₈

—

218

—

110

C₂×C₈

—

218

—

111

C₂×C₈

C₂×#89137

yes

characteristic

112

C₄⋊C₄

—

109

—

113

C₄⋊C₄

—

109

—

114

C₄⋊C₄

—

218

—

115

C₄⋊C₄

—

218

—

116

C₂²⋊C₄

—

218

—

117

C₄²

—

218

—

118

C₂³

—

109

—

119

C₂×C₄

—

872

—

120

C₂×C₄

—

109

—

121

C₂×C₄

—

109

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

109

—

124

C₂×C₄

—

109

—

125

C₂×C₄

—

436

—

126

C₂×C₄

—

218

—

127

C₂×C₄

—

218

—

128

C₂×C₄

C₂²×#89137

yes

characteristic

129

C₈

—

130

C₈

—

109

—

131

C₈

—

109

—

132

C₈

—

218

—

133

C₈

—

218

—

134

C₈

C₄⋊#89137

yes

characteristic

135

C₈

C₄⋊#89137

yes

characteristic

136

C₂²

—

109

—

137

C₂²

—

109

—

138

C₂²

—

218

—

139

C₂²

—

218

—

140

C₂²

Q₈×#89137

yes

characteristic

141

C₄

—

872

—

142

C₄

—

109

—

143

C₄

—

109

—

144

C₄

—

436

—

145

C₄

—

436

—

146

C₄

C₂×C₄⋊#89137

yes

characteristic

147

C₄

D₄×#89137

yes

characteristic

148

C₂

—

149

C₂

—

109

—

150

C₂

—

109

—

151

C₂

—

218

—

152

C₂

#20295977

yes

characteristic

	subgroup	G/H
center	C₂	#20295977
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#108×C₁₀₉	C₄
socle	C₂×C₁₀₉	#389
prime radical	C₂²×D₁₀₉	#41
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#108	#89137
hypocenter	C₁₀₉	#1952
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₂

#20295977

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

#108×C₁₀₉

C₄

C₂×C₁₀₉

C₂²×D₁₀₉

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
17. #16812635	C₄	semidirect
18. #16812634	C₄	semidirect
19. #108×C₁₀₉	C₄	semidirect
77. C₁₀₉	#1952	semidirect
90. #108	#89137	semidirect

H ≅ G/N

type

17. #16812635

C₄

semidirect

18. #16812634

C₄

semidirect

19. #108×C₁₀₉

C₄

semidirect

77. C₁₀₉

#1952

semidirect

90. #108

#89137

semidirect

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

subgroup	index
#20296037	2
C₂×#16812768	2
#20295989	2
#20295905	2
#108×D₁₀₉	2
#1952	109

subgroup

index

#108×D₁₀₉

109

Subgroups — #28379150

Conjugacy classes of subgroups

Complements

Maximal subgroups