Group #28363989 — subgroups

Subgroups — #28363989

Order 13952.

260 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 109 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×#16812691	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#11724784	C₂	1	1	yes	maximal, characteristic
3	#20294788	C₂	1	2	yes	maximal
4	#20294788	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	#16812691	—	2	2	no	—
8	#16812691	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×#13182008	—	2	1	no	—
12	C₂×#13182008	—	2	1	no	—
13	C₄⋊#11724784	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₉	—	2	1	no	—
16	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
17	#95×C₁₀₉	—	2	1	no	—
18	C₂²×Dic₁₀₉	—	2	1	no	—
19	C₂²×Dic₁₀₉	—	2	1	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₂×#11724784	—	2	4	no	—
22	C₄⋊Dic₁₀₉	—	2	1	no	—
23	C₄⋊Dic₁₀₉	—	2	1	no	—
24	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
25	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
26	#13182008	—	2	4	no	—
27	#13182008	—	2	4	no	—
28	C₂²×C₄×C₁₀₉	—	2	2	no	—
29	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₁₀₉	D₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₄⋊C₄×C₁₀₉	Q₈	1	4	yes	—
34	C₄²×C₁₀₉	—	4	2	no	—
35	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
36	C₄²×C₁₀₉	D₄	1	2	yes	—
37	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	—	2	1	no	—
40	C₂×C₄×C₁₀₉	—	2	1	no	—
41	C₂×C₄×C₁₀₉	—	2	4	no	—
42	C₂×C₄×C₁₀₉	—	4	2	no	—
43	C₂×C₄×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₁₀₉	#41	1	2	yes	—
50	C₂×Dic₁₀₉	—	2	2	no	—
51	C₂×Dic₁₀₉	—	2	2	no	—
52	C₂×Dic₁₀₉	—	4	2	no	—
53	C₂×Dic₁₀₉	—	4	1	no	—
54	C₂×Dic₁₀₉	—	4	1	no	—
55	#11724784	—	4	2	no	—
56	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
58	C₂²×C₁₀₉	#135	1	2	yes	—
59	C₂²×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂²×C₁₀₉	#137	1	1	yes	characteristic
61	C₄×C₁₀₉	—	2	2	no	—
62	C₄×C₁₀₉	—	2	2	no	—
63	C₄×C₁₀₉	—	4	4	no	—
64	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
65	C₄×C₁₀₉	#124	1	1	yes	characteristic
66	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₁₀₉	—	4	2	no	—
68	Dic₁₀₉	—	4	2	no	—
69	C₂×C₁₀₉	#391	1	1	yes	characteristic
70	C₂×C₁₀₉	#416	1	1	yes	characteristic
71	C₂×C₁₀₉	#417	1	1	yes	characteristic
72	C₂×C₁₀₉	#478	1	2	yes	—
73	C₂×C₁₀₉	#479	1	2	yes	—
74	#1989	—	109	1	no	maximal, Sylow
75	C₁₀₉	#1989	1	1	yes	characteristic, complemented, Sylow
76	C₂×#125	—	109	1	no	—
77	C₂×C₄⋊C₈	—	109	1	no	—
78	C₄×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic
79	#340	—	109	2	no	—
80	#340	—	109	2	no	—
81	C₂²×C₈	—	109	2	no	—
82	#125	—	109	4	no	—
83	#125	—	218	2	no	—
84	C₂×C₄⋊C₄	—	109	1	no	—
85	C₂×C₄⋊C₄	—	109	1	no	—
86	C₂×C₄⋊C₄	—	218	1	no	—
87	C₂×C₄⋊C₄	—	218	1	no	—
88	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
89	C₂×C₄²	—	2	1	no	—
90	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
91	C₄⋊C₈	—	218	2	no	—
92	#95	—	2	1	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	218	1	no	—
95	C₂²×C₄	—	218	1	no	—
96	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₂×C₈	—	218	2	no	—
100	C₂×C₈	—	218	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	109	2	no	—
103	C₄⋊C₄	—	109	2	no	—
104	C₄⋊C₄	—	218	4	no	—
105	C₄⋊C₄	—	218	4	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄⋊C₄	Dic₂₁₈	1	4	yes	—
109	C₄²	—	4	2	no	—
110	C₄²	C₄×D₁₀₉	1	2	yes	—
111	C₄²	D₄₃₆	1	2	yes	—
112	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	436	2	no	—
120	C₂×C₄	—	436	1	no	—
121	C₂×C₄	—	436	1	no	—
122	C₂×C₄	—	218	2	no	—
123	C₂×C₄	—	218	2	no	—
124	C₂×C₄	#13182006	1	2	yes	—
125	C₂×C₄	#13182006	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
127	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
128	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
129	C₂×C₄	#13182029	1	2	yes	—
130	C₈	—	436	2	no	—
131	C₂²	C₂×#13182006	1	1	yes	characteristic
132	C₂²	#16812678	1	2	yes	—
133	C₂²	#16812719	1	2	yes	—
134	C₂²	#16812725	1	1	yes	characteristic
135	C₂²	#16812730	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	4	4	no	—
139	C₄	—	436	2	no	—
140	C₄	—	436	2	no	—
141	C₄	C₄×Dic₂₁₈	1	2	yes	—
142	C₄	#16812687	1	1	yes	characteristic
143	C₄	#16812699	1	1	yes	characteristic
144	C₂	#20294932	1	1	yes	characteristic
145	C₂	#20295020	1	1	yes	characteristic
146	C₂	#20295025	1	1	yes	characteristic
147	C₂	#20295326	1	2	yes	—
148	C₂	#20295333	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812691

C₂

yes

maximal, characteristic

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

#20294788

C₂

yes

maximal

#20294788

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

#16812691

—

#16812691

C₄

yes

complemented

C₂×C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

—

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

#95×C₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

D₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

Q₈

yes

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

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

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

C₂×C₁₀₉

#479

yes

—

#1989

—

109

maximal, Sylow

C₁₀₉

#1989

yes

characteristic, complemented, Sylow

C₂×#125

—

109

—

C₂×C₄⋊C₈

—

109

—

C₄×C₄⋊C₄

D₁₀₉

yes

characteristic

#340

—

109

—

#340

—

109

—

C₂²×C₈

—

109

—

#125

—

109

—

#125

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

218

—

100

C₂×C₈

—

218

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

109

—

103

C₄⋊C₄

—

109

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

—

218

—

108

C₄⋊C₄

Dic₂₁₈

yes

—

109

C₄²

—

110

C₄²

C₄×D₁₀₉

yes

—

111

C₄²

D₄₃₆

yes

—

112

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

436

—

120

C₂×C₄

—

436

—

121

C₂×C₄

—

436

—

122

C₂×C₄

—

218

—

123

C₂×C₄

—

218

—

124

C₂×C₄

#13182006

yes

—

125

C₂×C₄

#13182006

yes

—

126

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

127

C₂×C₄

C₂×D₄₃₆

yes

characteristic

128

C₂×C₄

C₂×Dic₂₁₈

yes

—

129

C₂×C₄

#13182029

yes

—

130

C₈

—

436

—

131

C₂²

C₂×#13182006

yes

characteristic

132

C₂²

#16812678

yes

—

133

C₂²

#16812719

yes

—

134

C₂²

#16812725

yes

characteristic

135

C₂²

#16812730

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

436

—

140

C₄

—

436

—

141

C₄

C₄×Dic₂₁₈

yes

—

142

C₄

#16812687

yes

characteristic

143

C₄

#16812699

yes

characteristic

144

C₂

#20294932

yes

characteristic

145

C₂

#20295020

yes

characteristic

146

C₂

#20295025

yes

characteristic

147

C₂

#20295326

yes

—

148

C₂

#20295333

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₄×C₄⋊C₄×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#1989
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₄×C₄⋊C₄×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₀₉

hypocenter

C₁₀₉

#1989

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
8. #16812691	C₄	semidirect
75. C₁₀₉	#1989	semidirect

H ≅ G/N

type

8. #16812691

C₄

semidirect

75. C₁₀₉

#1989

semidirect

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

subgroup	index
C₂×#16812691	2
C₂×C₄⋊#11724784	2
#20294788	2
C₄×C₄⋊C₄×C₁₀₉	2
#1989	109

subgroup

index

C₄×C₄⋊C₄×C₁₀₉

109

Subgroups — #28363989

Conjugacy classes of subgroups

Complements

Maximal subgroups