Group #28371749 — subgroups

Subgroups — #28371749

Order 13952.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#16812707	C₂	1	2	yes	maximal, complemented
2	#20295576	C₂	1	1	yes	maximal, characteristic
3	#20295096	C₂	1	8	yes	maximal, complemented
4	#20294994	C₂	1	2	yes	maximal, complemented
5	C₂×#16812610	C₂	1	1	yes	maximal, characteristic, complemented
6	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13182029	C₂²	1	2	yes	—
8	#16812707	—	2	8	no	—
9	C₂×#13182012	C₂²	1	4	yes	—
10	C₂²×#11724784	C₂²	1	1	yes	characteristic
11	C₂²×#11724784	C₂²	1	1	yes	characteristic
12	C₂×C₈×D₁₀₉	C₂²	1	4	yes	—
13	#16812699	C₄	1	1	yes	characteristic
14	#16812696	C₄	1	1	yes	characteristic
15	#16812687	C₄	1	2	yes	—
16	#16812673	C₄	1	4	yes	—
17	C₄×D₄₃₆	C₂²	1	4	yes	—
18	#16812611	C₂²	1	8	yes	—
19	#16812610	C₂²	1	4	yes	—
20	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
23	#13182029	—	2	4	no	—
24	#13182029	—	2	4	no	—
25	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
26	C₂×D₄₃₆	C₂×C₄	1	2	yes	—
27	C₂×D₄₃₆	C₂×C₄	1	4	yes	—
28	C₂×C₂²⋊D₁₀₉	C₂×C₄	1	4	yes	—
29	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
30	#13182012	—	2	8	no	—
31	C₂×#11724784	—	2	2	no	—
32	C₂×#11724784	D₄	1	4	yes	complemented
33	C₂×#11724784	C₂³	1	2	yes	—
34	C₂×#11724784	C₂³	1	2	yes	—
35	C₈×D₁₀₉	—	2	8	no	—
36	C₄⋊Dic₁₀₉	C₂×C₄	1	4	yes	—
37	#13182006	C₂×C₄	1	8	yes	—
38	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
40	M₁₆×C₁₀₉	—	2	4	no	—
41	C₂×C₈×C₁₀₉	C₂³	1	4	yes	—
42	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
44	C₄²×C₁₀₉	C₂³	1	2	yes	—
45	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₁₀₉	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₁₀₉	—	4	2	no	—
48	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₀₉	#41	1	2	yes	—
53	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
54	C₂²⋊D₁₀₉	—	2	8	no	—
55	C₂×Dic₁₀₉	C₂²×C₄	1	4	yes	—
56	D₄₃₆	—	2	4	no	—
57	D₄₃₆	—	4	4	no	—
58	C₄×D₁₀₉	—	2	4	no	—
59	C₄×D₁₀₉	—	2	4	no	—
60	Dic₂₁₈	—	2	4	no	—
61	C₈×C₁₀₉	—	2	4	no	—
62	#11724784	—	2	2	no	—
63	#11724784	C₂×D₄	1	4	yes	—
64	C₂²×C₁₀₉	—	2	2	no	—
65	C₂²×C₁₀₉	C₄×D₄	1	2	yes	—
66	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
67	C₂×D₁₀₉	—	2	4	no	—
68	C₂×D₁₀₉	—	2	8	no	—
69	C₄×C₁₀₉	—	2	2	no	—
70	C₄×C₁₀₉	—	2	2	no	—
71	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
72	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
73	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
74	Dic₁₀₉	—	2	4	no	—
75	C₂×C₁₀₉	—	2	2	no	—
76	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
77	C₂×C₁₀₉	#568	1	2	yes	—
78	D₁₀₉	—	4	4	no	—
79	#2932	—	109	1	no	maximal, Sylow
80	C₁₀₉	#2932	1	1	yes	characteristic, complemented, Sylow
81	C₂×#131	—	109	2	no	—
82	#532	—	109	1	no	—
83	#435	—	109	8	no	—
84	#423	D₁₀₉	1	1	yes	characteristic, complemented
85	#408	—	109	2	no	—
86	C₂×#97	—	109	1	no	—
87	C₂×#41	—	109	2	no	—
88	#131	—	218	8	no	—
89	C₂×M₁₆	—	109	4	no	—
90	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
91	C₂²×C₈	—	109	1	no	—
92	C₂²×C₈	—	109	1	no	—
93	C₂²×C₈	—	109	4	no	—
94	C₄⋊Q₈	—	109	1	no	—
95	#127	—	109	1	no	—
96	#124	—	109	2	no	—
97	#121	—	109	4	no	—
98	C₄×D₄	—	109	4	no	—
99	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
100	C₄⋊C₈	C₂×D₁₀₉	1	4	yes	—
101	C₂²⋊C₈	—	109	8	no	—
102	#97	—	109	4	no	—
103	#41	—	218	4	no	—
104	#41	—	218	4	no	—
105	C₂×Q₈	—	109	2	no	—
106	C₂×D₄	—	109	4	no	—
107	C₂×D₄	—	109	4	no	—
108	C₂×D₄	—	109	2	no	—
109	C₂²×C₄	—	109	4	no	—
110	C₂²×C₄	C₄×D₁₀₉	1	2	yes	—
111	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	218	8	no	—
114	C₂×C₈	—	109	4	no	—
115	C₂×C₈	—	109	2	no	—
116	C₂×C₈	—	109	2	no	—
117	C₂×C₈	—	218	8	no	—
118	C₂×C₈	—	218	2	no	—
119	C₂×C₈	C₂²×D₁₀₉	1	4	yes	—
120	C₄⋊C₄	—	109	4	no	—
121	C₂²⋊C₄	—	109	8	no	—
122	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
123	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
124	C₄²	C₂²×D₁₀₉	1	2	yes	—
125	C₂³	—	109	4	no	—
126	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
127	Q₈	—	218	4	no	—
128	D₄	—	436	4	no	—
129	D₄	—	218	8	no	—
130	D₄	—	218	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	109	4	no	—
133	C₂×C₄	—	218	4	no	—
134	C₂×C₄	—	218	4	no	—
135	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₁₀₉	1	2	yes	—
137	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
139	C₂×C₄	#13182030	1	2	yes	—
140	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	109	4	no	—
143	C₈	—	218	2	no	—
144	C₂²	—	2	2	no	—
145	C₂²	—	218	4	no	—
146	C₂²	—	218	8	no	—
147	C₂²	#16812661	1	2	yes	—
148	C₂²	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	218	4	no	—
152	C₄	#16812661	1	2	yes	—
153	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
154	C₄	C₂×#13182030	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	436	4	no	—
157	C₂	C₂×#16812661	1	1	yes	characteristic
158	C₂	#20295792	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812707

C₂

yes

maximal, complemented

#20295576

C₂

yes

maximal, characteristic

#20295096

C₂

yes

maximal, complemented

#20294994

C₂

yes

maximal, complemented

C₂×#16812610

C₂

yes

maximal, characteristic, complemented

#423×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182029

C₂²

yes

—

#16812707

—

C₂×#13182012

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

—

#16812699

C₄

yes

characteristic

#16812696

C₄

yes

characteristic

#16812687

C₄

yes

—

#16812673

C₄

yes

—

C₄×D₄₃₆

C₂²

yes

—

#16812611

C₂²

yes

—

#16812610

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

#13182029

—

#13182029

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

—

#13182012

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

complemented

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

—

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182006

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂²×C₄

yes

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₂×D₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×D₄

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂×#41

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×D₄

yes

characteristic

C₂×C₁₀₉

#568

yes

—

D₁₀₉

—

#2932

—

109

maximal, Sylow

C₁₀₉

#2932

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

#532

—

109

—

#435

—

109

—

#423

D₁₀₉

yes

characteristic, complemented

#408

—

109

—

C₂×#97

—

109

—

C₂×#41

—

109

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

#127

—

109

—

#124

—

109

—

#121

—

109

—

C₄×D₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

100

C₄⋊C₈

C₂×D₁₀₉

yes

—

101

C₂²⋊C₈

—

109

—

102

#97

—

109

—

103

#41

—

218

—

104

#41

—

218

—

105

C₂×Q₈

—

109

—

106

C₂×D₄

—

109

—

107

C₂×D₄

—

109

—

108

C₂×D₄

—

109

—

109

C₂²×C₄

—

109

—

110

C₂²×C₄

C₄×D₁₀₉

yes

—

111

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

112

M₁₆

—

113

M₁₆

—

218

—

114

C₂×C₈

—

109

—

115

C₂×C₈

—

109

—

116

C₂×C₈

—

109

—

117

C₂×C₈

—

218

—

118

C₂×C₈

—

218

—

119

C₂×C₈

C₂²×D₁₀₉

yes

—

120

C₄⋊C₄

—

109

—

121

C₂²⋊C₄

—

109

—

122

C₄²

C₄×D₁₀₉

yes

characteristic

123

C₄²

C₄×D₁₀₉

yes

characteristic

124

C₄²

C₂²×D₁₀₉

yes

—

125

C₂³

—

109

—

126

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

127

Q₈

—

218

—

128

D₄

—

436

—

129

D₄

—

218

—

130

D₄

—

218

—

131

C₂×C₄

—

132

C₂×C₄

—

109

—

133

C₂×C₄

—

218

—

134

C₂×C₄

—

218

—

135

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

136

C₂×C₄

C₂×C₄×D₁₀₉

yes

—

137

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

138

C₂×C₄

D₄×D₁₀₉

yes

—

139

C₂×C₄

#13182030

yes

—

140

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

141

C₈

—

142

C₈

—

109

—

143

C₈

—

218

—

144

C₂²

—

145

C₂²

—

218

—

146

C₂²

—

218

—

147

C₂²

#16812661

yes

—

148

C₂²

C₂²×C₄×D₁₀₉

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

218

—

152

C₄

#16812661

yes

—

153

C₄

C₂×D₄×D₁₀₉

yes

characteristic

154

C₄

C₂×#13182030

yes

characteristic

155

C₂

—

156

C₂

—

436

—

157

C₂

C₂×#16812661

yes

characteristic

158

C₂

#20295792

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	#423×C₁₀₉	C₂
prime radical	#20295576	C₂
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	#2932
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

#423×C₁₀₉

C₂

prime radical

#20295576

C₂

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. C₂×#16812707	C₂	semidirect
3. #20295096	C₂	semidirect
4. #20294994	C₂	semidirect
5. C₂×#16812610	C₂	semidirect
6. #423×C₁₀₉	C₂	semidirect
32. C₂×#11724784	D₄	semidirect
80. C₁₀₉	#2932	semidirect
84. #423	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#16812707

C₂

semidirect

3. #20295096

C₂

semidirect

4. #20294994

C₂

semidirect

5. C₂×#16812610

C₂

semidirect

6. #423×C₁₀₉

C₂

semidirect

32. C₂×#11724784

D₄

semidirect

80. C₁₀₉

#2932

semidirect

84. #423

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812707	2
#20295576	2
#20295096	2
#20294994	2
C₂×#16812610	2
#423×C₁₀₉	2
#2932	109

subgroup

index

#423×C₁₀₉

109

Subgroups — #28371749

Conjugacy classes of subgroups

Complements

Maximal subgroups