Group #28361375 — subgroups

Subgroups — #28361375

Order 13952.

236 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 77 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×C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic
2	#20295083	C₂	1	1	yes	maximal, characteristic
3	#20294751	C₂	1	4	yes	maximal
4	#432×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₈	C₄	1	1	yes	characteristic
6	C₄×Dic₂₁₈	—	2	1	no	—
7	C₄×Dic₂₁₈	—	2	1	no	—
8	C₄×Dic₂₁₈	—	2	1	no	—
9	C₄×Dic₂₁₈	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
11	C₂×#13182008	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
13	C₄⋊#11724784	—	2	1	no	—
14	C₂²⋊#11724784	C₂²	1	2	yes	—
15	#16812610	—	2	1	no	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₀₉	—	2	1	no	—
18	C₂²⋊C₈×C₁₀₉	C₂²	1	2	yes	—
19	#97×C₁₀₉	—	2	1	no	—
20	C₂×Dic₂₁₈	—	2	1	no	—
21	C₂×Dic₂₁₈	—	2	1	no	—
22	C₂×Dic₂₁₈	—	2	4	no	—
23	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×#11724784	—	2	2	no	—
26	C₄⋊Dic₁₀₉	—	2	1	no	—
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	#13182008	—	2	2	no	—
29	#13182008	—	2	2	no	—
30	C₄×Dic₁₀₉	—	2	2	no	—
31	C₄×Dic₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₁₀₉	—	2	2	no	—
36	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
37	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
39	C₄²×C₁₀₉	D₄	1	1	yes	characteristic
40	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	—	2	2	no	—
42	C₂×C₄×C₁₀₉	—	2	2	no	—
43	C₂×C₄×C₁₀₉	—	4	1	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	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	2	no	—
51	C₂×Dic₁₀₉	—	2	2	no	—
52	C₂×Dic₁₀₉	—	2	2	no	—
53	C₂×Dic₁₀₉	—	4	2	no	—
54	C₂×Dic₁₀₉	M₁₆	1	4	yes	—
55	Dic₂₁₈	—	2	4	no	—
56	Dic₂₁₈	—	2	4	no	—
57	C₈×C₁₀₉	—	4	2	no	—
58	#11724784	—	4	2	no	—
59	C₂²×C₁₀₉	—	2	1	no	—
60	C₂²×C₁₀₉	—	2	1	no	—
61	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
63	C₄×C₁₀₉	—	2	1	no	—
64	C₄×C₁₀₉	—	2	1	no	—
65	C₄×C₁₀₉	—	2	1	no	—
66	C₄×C₁₀₉	—	2	1	no	—
67	C₄×C₁₀₉	—	2	1	no	—
68	C₄×C₁₀₉	#101	1	2	yes	—
69	Dic₁₀₉	—	2	4	no	—
70	Dic₁₀₉	—	4	2	no	—
71	C₂×C₁₀₉	—	2	2	no	—
72	C₂×C₁₀₉	#407	1	1	yes	characteristic
73	C₂×C₁₀₉	#410	1	1	yes	characteristic
74	C₂×C₁₀₉	C₂×#101	1	1	yes	characteristic
75	#1467	—	109	1	no	maximal, Sylow
76	C₁₀₉	#1467	1	1	yes	characteristic, complemented, Sylow
77	C₂×C₄×Q₈	—	109	1	no	—
78	#432	—	109	1	no	—
79	#432	D₁₀₉	1	1	yes	characteristic
80	#324	—	109	4	no	—
81	C₂²×Q₈	—	109	1	no	—
82	C₄×Q₈	—	218	1	no	—
83	C₄×Q₈	—	218	1	no	—
84	C₄×Q₈	—	218	1	no	—
85	C₄×Q₈	—	218	1	no	—
86	C₂×C₄⋊C₄	—	109	1	no	—
87	C₂×C₄⋊C₄	—	109	2	no	—
88	C₂×C₄²	—	109	2	no	—
89	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	—	218	1	no	—
92	C₂²⋊C₈	—	109	2	no	—
93	C₂²⋊C₈	C₂×D₁₀₉	1	2	yes	—
94	#97	—	2	1	no	—
95	#97	—	218	1	no	—
96	C₂×Q₈	—	218	4	no	—
97	C₂×Q₈	—	218	1	no	—
98	C₂×Q₈	—	218	1	no	—
99	C₂²×C₄	—	109	2	no	—
100	C₂²×C₄	—	109	2	no	—
101	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	218	2	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄⋊C₄	—	218	2	no	—
109	C₄⋊C₄	—	218	2	no	—
110	C₄²	—	218	2	no	—
111	C₄²	—	218	2	no	—
112	C₄²	D₄₃₆	1	1	yes	characteristic
113	C₄²	D₄₃₆	1	1	yes	characteristic
114	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
115	C₄²	C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
117	Q₈	—	218	4	no	—
118	Q₈	—	218	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	109	4	no	—
123	C₂×C₄	—	436	2	no	—
124	C₂×C₄	—	218	2	no	—
125	C₂×C₄	—	218	2	no	—
126	C₂×C₄	—	218	2	no	—
127	C₂×C₄	#13182006	1	1	yes	characteristic
128	C₂×C₄	#13182006	1	1	yes	characteristic
129	C₂×C₄	#13182006	1	1	yes	characteristic
130	C₂×C₄	#13182006	1	1	yes	characteristic
131	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
132	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
133	C₈	—	4	2	no	—
134	C₈	—	436	2	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	C₂×#13182006	1	1	yes	characteristic
138	C₂²	M₁₆×D₁₀₉	1	2	yes	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	436	2	no	—
145	C₄	—	218	4	no	—
146	C₄	#16812617	1	2	yes	—
147	C₂	—	2	2	no	—
148	C₂	#20294989	1	1	yes	characteristic
149	C₂	#20294999	1	1	yes	characteristic
150	C₂	C₂×#16812617	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₁₈

C₂

yes

maximal, characteristic

#20295083

C₂

yes

maximal, characteristic

#20294751

C₂

yes

maximal

#432×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₈

C₄

yes

characteristic

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

—

C₄×Dic₂₁₈

—

C₂×C₄⋊Dic₁₀₉

C₄

yes

characteristic

C₂×#13182008

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₄⋊#11724784

—

C₂²⋊#11724784

C₂²

yes

—

#16812610

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂²⋊C₈×C₁₀₉

C₂²

yes

—

#97×C₁₀₉

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

characteristic

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

M₁₆

yes

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#101

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1467

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

109

—

#432

—

109

—

#432

D₁₀₉

yes

characteristic

#324

—

109

—

C₂²×Q₈

—

109

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

C₄×Q₈

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

218

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

C₂×D₁₀₉

yes

—

#97

—

#97

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

100

C₂²×C₄

—

109

—

101

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

102

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

103

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

—

218

—

108

C₄⋊C₄

—

218

—

109

C₄⋊C₄

—

218

—

110

C₄²

—

218

—

111

C₄²

—

218

—

112

C₄²

D₄₃₆

yes

characteristic

113

C₄²

D₄₃₆

yes

characteristic

114

C₄²

C₂²⋊D₁₀₉

yes

characteristic

115

C₄²

C₂²⋊D₁₀₉

yes

characteristic

116

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

117

Q₈

—

218

—

118

Q₈

—

218

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

109

—

123

C₂×C₄

—

436

—

124

C₂×C₄

—

218

—

125

C₂×C₄

—

218

—

126

C₂×C₄

—

218

—

127

C₂×C₄

#13182006

yes

characteristic

128

C₂×C₄

#13182006

yes

characteristic

129

C₂×C₄

#13182006

yes

characteristic

130

C₂×C₄

#13182006

yes

characteristic

131

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

132

C₂×C₄

C₂×D₄₃₆

yes

characteristic

133

C₈

—

134

C₈

—

436

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂×#13182006

yes

characteristic

138

C₂²

M₁₆×D₁₀₉

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

436

—

145

C₄

—

218

—

146

C₄

#16812617

yes

—

147

C₂

—

148

C₂

#20294989

yes

characteristic

149

C₂

#20294999

yes

characteristic

150

C₂

C₂×#16812617

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13182006
commutator, prime radical	C₂³×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#432×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#432	D₁₀₉
hypocenter	C₁₀₉	#1467
LCS	G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×#13182006

commutator, prime radical

C₂³×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#432×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
76. C₁₀₉	#1467	semidirect

H ≅ G/N

type

76. C₁₀₉

#1467

semidirect

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

subgroup	index
C₂×C₄×Dic₂₁₈	2
#20295083	2
#20294751	2
#432×C₁₀₉	2
#1467	109

subgroup

index

C₂×C₄×Dic₂₁₈

#20295083

#20294751

#432×C₁₀₉

#1467

109

Subgroups — #28361375

Conjugacy classes of subgroups

Complements

Maximal subgroups