Group #28362887 — subgroups

Subgroups — #28362887

Order 13952.

292 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 153 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⋊#11724784	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812609	C₂	1	2	yes	maximal
3	C₄²×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#20294780	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11724784	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₀₉	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₄⋊#11724784	C₄	1	4	yes	complemented
12	#16812609	—	2	4	no	—
13	C₂²×C₈×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₉	C₄	1	4	yes	complemented
16	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₀₉	C₂×C₄	1	2	yes	—
18	C₂×#11724784	—	2	2	no	—
19	C₂×#11724784	—	2	2	no	—
20	C₂×#11724784	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₀₉	—	2	4	no	—
22	C₄×Dic₁₀₉	—	2	1	no	—
23	C₄×Dic₁₀₉	—	2	1	no	—
24	C₄×Dic₁₀₉	—	2	4	no	—
25	C₄×Dic₁₀₉	D₄	1	2	yes	—
26	C₄×Dic₁₀₉	Q₈	1	2	yes	—
27	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₀₉	—	2	2	no	—
31	C₂×C₈×C₁₀₉	—	2	2	no	—
32	C₂×C₈×C₁₀₉	C₂×C₄	1	4	yes	—
33	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
34	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
35	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	—	2	2	no	—
37	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
38	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
39	C₂×C₄×C₁₀₉	M₁₆	1	4	yes	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
45	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
46	C₂×Dic₁₀₉	—	2	2	no	—
47	C₂×Dic₁₀₉	—	2	2	no	—
48	C₂×Dic₁₀₉	—	2	2	no	—
49	C₂×Dic₁₀₉	—	2	2	no	—
50	C₂×Dic₁₀₉	C₄⋊C₄	1	4	yes	—
51	C₂×Dic₁₀₉	M₁₆	1	4	yes	—
52	C₈×C₁₀₉	—	2	4	no	—
53	#11724784	—	2	4	no	—
54	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₉	#117	1	1	yes	characteristic
57	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
58	C₂²×C₁₀₉	C₂×M₁₆	1	2	yes	—
59	C₄×C₁₀₉	—	2	2	no	—
60	C₄×C₁₀₉	#97	1	4	yes	complemented
61	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
62	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
63	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
64	C₄×C₁₀₉	C₄×Q₈	1	1	yes	characteristic
65	Dic₁₀₉	—	2	4	no	—
66	Dic₁₀₉	—	2	4	no	—
67	C₂×C₁₀₉	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₁₀₉	C₂×#97	1	1	yes	characteristic
69	C₂×C₁₀₉	C₄×M₁₆	1	1	yes	characteristic
70	C₂×C₁₀₉	#423	1	2	yes	—
71	C₂×C₁₀₉	#432	1	2	yes	—
72	#1767	—	109	1	no	maximal, Sylow
73	C₁₀₉	#1767	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊C₈	—	109	1	no	—
75	C₂×C₄⋊C₈	D₁₀₉	1	1	yes	characteristic
76	C₂×#97	—	109	2	no	—
77	C₄³	—	109	1	no	—
78	#336	—	109	2	no	—
79	C₂²×C₈	—	109	2	no	—
80	C₂²×C₈	C₂×D₁₀₉	1	2	yes	—
81	C₂×C₄²	—	109	1	no	—
82	C₂×C₄²	—	109	2	no	—
83	C₂×C₄²	—	109	2	no	—
84	C₂×C₄²	—	109	1	no	—
85	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
86	C₄⋊C₈	—	109	4	no	—
87	C₄⋊C₈	Dic₁₀₉	1	4	yes	complemented
88	#97	—	218	4	no	—
89	C₂²×C₄	—	109	2	no	—
90	C₂²×C₄	—	109	2	no	—
91	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
92	C₂²×C₄	C₄×D₁₀₉	1	1	yes	characteristic
93	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	109	4	no	—
97	C₂×C₈	—	218	2	no	—
98	C₂×C₈	—	218	2	no	—
99	C₂×C₈	C₂×Dic₁₀₉	1	4	yes	—
100	C₄²	—	109	2	no	—
101	C₄²	—	109	2	no	—
102	C₄²	—	218	4	no	—
103	C₄²	—	218	4	no	—
104	C₄²	—	218	1	no	—
105	C₄²	—	218	1	no	—
106	C₄²	C₄×D₁₀₉	1	2	yes	—
107	C₄²	C₂×Dic₁₀₉	1	2	yes	—
108	C₂³	C₂×C₄×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	109	4	no	—
111	C₂×C₄	—	109	4	no	—
112	C₂×C₄	—	218	2	no	—
113	C₂×C₄	—	218	2	no	—
114	C₂×C₄	—	218	2	no	—
115	C₂×C₄	—	218	2	no	—
116	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
117	C₂×C₄	C₄×Dic₁₀₉	1	2	yes	—
118	C₂×C₄	#13182012	1	4	yes	—
119	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
120	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
121	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₁₀₉	1	1	yes	characteristic
123	C₂×C₄	#13182028	1	1	yes	characteristic
124	C₂×C₄	#13182030	1	1	yes	characteristic
125	C₈	—	2	4	no	—
126	C₈	—	218	4	no	—
127	C₂²	C₂×C₄×Dic₁₀₉	1	1	yes	characteristic
128	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
129	C₂²	#16812657	1	1	yes	characteristic
130	C₂²	M₁₆×D₁₀₉	1	2	yes	—
131	C₂²	C₂×#13182012	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	218	4	no	—
134	C₄	—	218	4	no	—
135	C₄	#16812609	1	4	yes	complemented
136	C₄	#16812661	1	1	yes	characteristic
137	C₄	D₄×Dic₁₀₉	1	1	yes	characteristic
138	C₄	#16812664	1	1	yes	characteristic
139	C₄	Q₈×Dic₁₀₉	1	1	yes	characteristic
140	C₂	C₄⋊C₄×Dic₁₀₉	1	1	yes	characteristic
141	C₂	C₂×#16812609	1	1	yes	characteristic
142	C₂	M₁₆×Dic₁₀₉	1	1	yes	characteristic
143	C₂	#20295048	1	2	yes	—
144	C₂	#20295082	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11724784

C₂

yes

maximal, characteristic

C₂×#16812609

C₂

yes

maximal

C₄²×Dic₁₀₉

C₂

yes

maximal, characteristic

#20294780

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₄

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₂²

yes

characteristic

C₄⋊#11724784

C₄

yes

complemented

#16812609

—

C₂²×C₈×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₄

yes

complemented

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂²×Dic₁₀₉

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

M₁₆

yes

—

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×Q₈

yes

C₂×C₄×C₁₀₉

yes

C₂×C₄×C₁₀₉

yes

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

C₂×Dic₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

#117

yes

characteristic

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₂²×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#97

yes

complemented

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

C₄×C₁₀₉

C₄×Q₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×#97

yes

characteristic

C₂×C₁₀₉

C₄×M₁₆

yes

characteristic

C₂×C₁₀₉

#423

yes

—

C₂×C₁₀₉

#432

yes

—

#1767

—

109

maximal, Sylow

C₁₀₉

#1767

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

109

—

C₂×C₄⋊C₈

D₁₀₉

yes

characteristic

C₂×#97

—

109

—

C₄³

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

Dic₁₀₉

yes

complemented

#97

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₄×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×Dic₁₀₉

yes

—

100

C₄²

—

109

—

101

C₄²

—

109

—

102

C₄²

—

218

—

103

C₄²

—

218

—

104

C₄²

—

218

—

105

C₄²

—

218

—

106

C₄²

C₄×D₁₀₉

yes

—

107

C₄²

C₂×Dic₁₀₉

yes

—

108

C₂³

C₂×C₄×D₁₀₉

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

218

—

113

C₂×C₄

—

218

—

114

C₂×C₄

—

218

—

115

C₂×C₄

—

218

—

116

C₂×C₄

C₄×Dic₁₀₉

yes

—

117

C₂×C₄

C₄×Dic₁₀₉

yes

—

118

C₂×C₄

#13182012

yes

—

119

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

120

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

121

C₂×C₄

D₄×D₁₀₉

yes

characteristic

122

C₂×C₄

Q₈×D₁₀₉

yes

characteristic

123

C₂×C₄

#13182028

yes

characteristic

124

C₂×C₄

#13182030

yes

characteristic

125

C₈

—

126

C₈

—

218

—

127

C₂²

C₂×C₄×Dic₁₀₉

yes

characteristic

128

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

129

C₂²

#16812657

yes

characteristic

130

C₂²

M₁₆×D₁₀₉

yes

—

131

C₂²

C₂×#13182012

yes

—

132

C₄

—

133

C₄

—

218

—

134

C₄

—

218

—

135

C₄

#16812609

yes

complemented

136

C₄

#16812661

yes

characteristic

137

C₄

D₄×Dic₁₀₉

yes

characteristic

138

C₄

#16812664

yes

characteristic

139

C₄

Q₈×Dic₁₀₉

yes

characteristic

140

C₂

C₄⋊C₄×Dic₁₀₉

yes

characteristic

141

C₂

C₂×#16812609

yes

characteristic

142

C₂

M₁₆×Dic₁₀₉

yes

characteristic

143

C₂

#20295048

yes

—

144

C₂

#20295082

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₂×C₄⋊C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#1767
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂×C₄²

Fitting

C₂×C₄⋊C₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
11. C₄⋊#11724784	C₄	semidirect
15. C₄⋊C₈×C₁₀₉	C₄	semidirect
60. C₄×C₁₀₉	#97	semidirect
73. C₁₀₉	#1767	semidirect
87. C₄⋊C₈	Dic₁₀₉	semidirect
135. C₄	#16812609	semidirect

H ≅ G/N

type

11. C₄⋊#11724784

C₄

semidirect

15. C₄⋊C₈×C₁₀₉

C₄

semidirect

60. C₄×C₁₀₉

#97

semidirect

73. C₁₀₉

#1767

semidirect

87. C₄⋊C₈

Dic₁₀₉

semidirect

135. C₄

#16812609

semidirect

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

subgroup	index
C₂×C₄⋊#11724784	2
C₂×#16812609	2
C₄²×Dic₁₀₉	2
#20294780	2
C₂×C₄⋊C₈×C₁₀₉	2
#1767	109

subgroup

index

C₄²×Dic₁₀₉

C₂×C₄⋊C₈×C₁₀₉

109

Subgroups — #28362887

Conjugacy classes of subgroups

Complements

Maximal subgroups