Group #28361486 — subgroups

Subgroups — #28361486

Order 13952.

232 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20295583	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11724784	C₂	1	1	yes	maximal, characteristic
3	#20294756	C₂	1	4	yes	maximal, complemented
4	#423×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	—	2	1	no	—
6	#16812699	C₄	1	1	yes	characteristic
7	#16812699	C₄	1	1	yes	characteristic
8	#16812691	—	2	1	no	—
9	#16812678	—	2	1	no	—
10	#16812678	C₄	1	2	yes	—
11	C₄×Dic₂₁₈	—	2	1	no	—
12	C₄×Dic₂₁₈	C₂²	1	2	yes	—
13	#16812658	—	2	1	no	—
14	C₄×#11724784	—	2	1	no	—
15	C₄×#11724784	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₀₉	—	2	1	no	—
17	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₈×C₁₀₉	—	2	1	no	—
19	C₄⋊C₈×C₁₀₉	C₂²	1	2	yes	—
20	C₂×Dic₂₁₈	—	2	1	no	—
21	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
22	C₂×#11724784	—	2	4	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₄⋊Dic₁₀₉	—	2	1	no	—
25	C₄⋊Dic₁₀₉	—	2	1	no	—
26	C₄⋊Dic₁₀₉	—	2	1	no	—
27	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
28	#13182008	—	2	2	no	—
29	#13182008	—	2	2	no	—
30	C₂²⋊Dic₁₀₉	—	2	2	no	—
31	C₄×Dic₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
35	M₁₆×C₁₀₉	—	4	2	no	—
36	C₂×C₈×C₁₀₉	—	2	2	no	—
37	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	2	2	no	—
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	SD₁₆	1	2	yes	—
48	C₂×C₄×C₁₀₉	Q₁₆	1	2	yes	—
49	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₁₀₉	—	2	2	no	—
53	C₂×Dic₁₀₉	—	2	2	no	—
54	Dic₂₁₈	—	2	4	no	—
55	Dic₂₁₈	—	4	2	no	—
56	C₈×C₁₀₉	—	4	2	no	—
57	#11724784	—	2	4	no	—
58	C₂²×C₁₀₉	—	2	1	no	—
59	C₂²×C₁₀₉	—	2	1	no	—
60	C₂²×C₁₀₉	#103	1	2	yes	—
61	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	2	1	no	—
64	C₄×C₁₀₉	#103	1	2	yes	—
65	C₄×C₁₀₉	#104	1	2	yes	—
66	C₄×C₁₀₉	#131	1	2	yes	—
67	C₄×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
69	Dic₁₀₉	—	4	2	no	—
70	Dic₁₀₉	—	4	2	no	—
71	C₂×C₁₀₉	—	2	2	no	—
72	C₂×C₁₀₉	#408	1	1	yes	characteristic
73	C₂×C₁₀₉	C₂×#103	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×#104	1	1	yes	characteristic
75	#1493	—	109	1	no	maximal, Sylow
76	C₁₀₉	#1493	1	1	yes	characteristic, complemented, Sylow
77	#533	—	109	1	no	—
78	#423	D₁₀₉	1	1	yes	characteristic
79	C₂×C₄×C₈	—	109	1	no	—
80	Q₈⋊C₈	—	109	4	no	—
81	C₂×M₁₆	—	2	1	no	—
82	C₂²×C₈	—	218	1	no	—
83	C₄⋊Q₈	—	109	1	no	—
84	C₄⋊Q₈	—	109	1	no	—
85	#125	—	218	1	no	—
86	C₂²⋊Q₈	—	109	2	no	—
87	C₂²⋊Q₈	—	218	1	no	—
88	C₄×Q₈	—	109	2	no	—
89	C₄×Q₈	—	218	1	no	—
90	#117	—	218	1	no	—
91	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
92	C₄⋊C₈	—	2	1	no	—
93	C₄⋊C₈	C₂×D₁₀₉	1	2	yes	—
94	C₄×C₈	—	109	2	no	—
95	C₄×C₈	—	218	1	no	—
96	C₂×Q₈	—	109	2	no	—
97	C₂×Q₈	—	218	1	no	—
98	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
99	C₂²×C₄	D₄₃₆	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
101	M₁₆	—	4	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	218	4	no	—
104	C₂×C₈	—	218	2	no	—
105	C₄⋊C₄	—	109	2	no	—
106	C₄⋊C₄	—	218	2	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄⋊C₄	—	218	1	no	—
109	C₄⋊C₄	—	218	1	no	—
110	C₄⋊C₄	—	218	2	no	—
111	C₂²⋊C₄	—	218	2	no	—
112	C₄²	—	218	2	no	—
113	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
114	C₄²	D₄₃₆	1	1	yes	characteristic
115	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
117	C₂³	#13182006	1	1	yes	characteristic
118	Q₈	—	436	2	no	—
119	Q₈	—	218	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	218	2	no	—
123	C₂×C₄	—	218	2	no	—
124	C₂×C₄	#13182006	1	1	yes	characteristic
125	C₂×C₄	#13182006	1	1	yes	characteristic
126	C₂×C₄	#13182006	1	1	yes	characteristic
127	C₂×C₄	#13182017	1	2	yes	—
128	C₂×C₄	#13182019	1	2	yes	—
129	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
131	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	218	4	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#16812622	1	2	yes	—
137	C₂²	C₂×#13182006	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	436	2	no	—
141	C₄	—	436	2	no	—
142	C₄	#16812622	1	2	yes	—
143	C₄	#16812625	1	2	yes	—
144	C₄	#16812707	1	2	yes	—
145	C₄	C₂×#13182017	1	1	yes	characteristic
146	C₄	C₂×#13182019	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	#20294994	1	1	yes	characteristic
149	C₂	C₂×#16812622	1	1	yes	characteristic
150	C₂	C₂×#16812625	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295583

C₂

yes

maximal, characteristic

C₂×C₄×#11724784

C₂

yes

maximal, characteristic

#20294756

C₂

yes

maximal, complemented

#423×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

—

#16812699

C₄

yes

characteristic

#16812699

C₄

yes

—

—

C₄

yes

—

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812658

—

C₄×#11724784

—

C₄×#11724784

C₂²

yes

—

C₂×M₁₆×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

SD₁₆

yes

—

C₂×C₄×C₁₀₉

Q₁₆

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#103

yes

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#103

yes

—

C₄×C₁₀₉

#104

yes

—

C₄×C₁₀₉

#131

yes

—

C₄×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₀₉

C₂×Q₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1493

yes

characteristic, complemented, Sylow

#533

—

109

—

#423

D₁₀₉

yes

characteristic

C₂×C₄×C₈

—

109

—

Q₈⋊C₈

—

109

—

C₂×M₁₆

—

C₂²×C₈

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

#125

—

218

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

#117

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

D₄₃₆

yes

characteristic

100

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

218

—

104

C₂×C₈

—

218

—

105

C₄⋊C₄

—

109

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

—

218

—

108

C₄⋊C₄

—

218

—

109

C₄⋊C₄

—

218

—

110

C₄⋊C₄

—

218

—

111

C₂²⋊C₄

—

218

—

112

C₄²

—

218

—

113

C₄²

C₄×D₁₀₉

yes

characteristic

114

C₄²

D₄₃₆

yes

characteristic

115

C₄²

C₂²⋊D₁₀₉

yes

characteristic

116

C₄²

C₂²×D₁₀₉

yes

characteristic

117

C₂³

#13182006

yes

characteristic

118

Q₈

—

436

—

119

Q₈

—

218

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

218

—

123

C₂×C₄

—

218

—

124

C₂×C₄

#13182006

yes

characteristic

125

C₂×C₄

#13182006

yes

characteristic

126

C₂×C₄

#13182006

yes

characteristic

127

C₂×C₄

#13182017

yes

—

128

C₂×C₄

#13182019

yes

—

129

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

131

C₂×C₄

C₂×D₄₃₆

yes

characteristic

132

C₈

—

133

C₈

—

218

—

134

C₂²

—

135

C₂²

—

136

C₂²

#16812622

yes

—

137

C₂²

C₂×#13182006

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

436

—

141

C₄

—

436

—

142

C₄

#16812622

yes

—

143

C₄

#16812625

yes

—

144

C₄

#16812707

yes

—

145

C₄

C₂×#13182017

yes

characteristic

146

C₄

C₂×#13182019

yes

characteristic

147

C₂

—

148

C₂

#20294994

yes

characteristic

149

C₂

C₂×#16812622

yes

characteristic

150

C₂

C₂×#16812625

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#423×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₉	C₂²⋊C₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#423	D₁₀₉
hypocenter	C₁₀₉	#1493
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#423×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₉

C₂²⋊C₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #20294756	C₂	semidirect
76. C₁₀₉	#1493	semidirect

H ≅ G/N

type

3. #20294756

C₂

semidirect

76. C₁₀₉

#1493

semidirect

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

subgroup	index
#20295583	2
C₂×C₄×#11724784	2
#20294756	2
#423×C₁₀₉	2
#1493	109

subgroup

index

#423×C₁₀₉

109

Subgroups — #28361486

Conjugacy classes of subgroups

Complements

Maximal subgroups