Group #28364789 — subgroups

Subgroups — #28364789

Order 13952.

160 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16812638	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#20294980	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#13182004	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294844	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294844	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13182012	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
10	#16812656	C₄	1	1	yes	characteristic
11	#16812638	—	2	2	no	—
12	C₂×#13182004	C₂²	1	1	yes	characteristic
13	C₂×#13182004	C₂²	1	1	yes	characteristic
14	C₁₆×D₁₀₉	—	2	2	no	—
15	#16812610	C₄	1	1	yes	characteristic
16	#16812609	C₄	1	1	yes	characteristic
17	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	#13182012	—	2	2	no	—
23	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
24	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
25	C₈×D₁₀₉	D₄	1	2	yes	—
26	C₈×D₁₀₉	Q₈	1	2	yes	—
27	#13182008	C₈	1	1	yes	characteristic
28	#13182008	C₈	1	1	yes	characteristic
29	#13182006	C₈	1	1	yes	characteristic
30	#13182006	C₈	1	1	yes	characteristic
31	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
32	#13182004	—	2	1	no	—
33	#13182004	—	2	1	no	—
34	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
37	C₁₆×C₁₀₉	—	2	1	no	—
38	C₁₆×C₁₀₉	—	2	1	no	—
39	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
44	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
45	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
46	C₄×D₁₀₉	C₄⋊C₄	1	2	yes	—
47	C₄×D₁₀₉	M₁₆	1	2	yes	—
48	C₈×C₁₀₉	—	2	1	no	—
49	C₈×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₈×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
51	#11724784	—	2	1	no	—
52	#11724784	C₄⋊C₄	1	2	yes	—
53	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
54	C₂×D₁₀₉	—	2	1	no	—
55	C₂×D₁₀₉	—	2	1	no	—
56	C₂×D₁₀₉	C₄⋊C₈	1	2	yes	—
57	C₄×C₁₀₉	—	2	1	no	—
58	C₄×C₁₀₉	—	2	1	no	—
59	C₄×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
61	Dic₁₀₉	—	2	1	no	—
62	Dic₁₀₉	—	2	1	no	—
63	Dic₁₀₉	C₄⋊C₈	1	2	yes	—
64	C₂×C₁₀₉	C₂×C₄⋊C₈	1	1	yes	characteristic
65	C₂×C₁₀₉	#504	1	1	yes	characteristic
66	C₂×C₁₀₉	#504	1	1	yes	characteristic
67	D₁₀₉	—	2	2	no	—
68	#2151	—	109	1	no	maximal, Sylow
69	C₁₀₉	#2151	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₃₂	—	109	1	no	—
71	C₂²×C₁₆	—	109	1	no	—
72	#405	—	109	1	no	—
73	C₄⋊C₁₆	—	109	1	no	—
74	C₄⋊C₁₆	—	109	1	no	—
75	C₄⋊C₁₆	—	109	1	no	—
76	C₄⋊C₁₆	D₁₀₉	1	1	yes	characteristic, complemented
77	C₂×M₁₆	—	109	1	no	—
78	C₂²×C₈	—	109	1	no	—
79	#117	—	109	1	no	—
80	M₃₂	—	218	2	no	—
81	C₂×C₁₆	—	109	1	no	—
82	C₂×C₁₆	—	109	1	no	—
83	C₂×C₁₆	—	218	2	no	—
84	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
85	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
86	#97	—	109	1	no	—
87	#97	—	109	1	no	—
88	C₄×C₈	—	109	1	no	—
89	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
90	C₂²×C₄	—	109	1	no	—
91	M₁₆	—	218	2	no	—
92	C₂×C₈	—	109	1	no	—
93	C₂×C₈	—	109	2	no	—
94	C₂×C₈	—	109	2	no	—
95	C₂×C₈	—	109	1	no	—
96	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
97	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₄⋊C₄	—	109	1	no	—
99	C₄⋊C₄	—	109	1	no	—
100	C₂²⋊C₄	—	109	1	no	—
101	C₂²⋊C₄	—	109	1	no	—
102	C₄²	—	109	1	no	—
103	C₄²	C₄×D₁₀₉	1	1	yes	characteristic
104	C₁₆	—	2	1	no	—
105	C₁₆	—	2	1	no	—
106	C₁₆	—	218	1	no	—
107	C₁₆	—	218	1	no	—
108	C₂³	—	109	1	no	—
109	C₂×C₄	—	109	1	no	—
110	C₂×C₄	—	109	2	no	—
111	C₂×C₄	—	109	2	no	—
112	C₂×C₄	—	109	1	no	—
113	C₂×C₄	—	109	1	no	—
114	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
115	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
117	C₈	—	2	1	no	—
118	C₈	—	109	2	no	—
119	C₈	—	218	1	no	—
120	C₈	D₄×D₁₀₉	1	1	yes	characteristic
121	C₈	Q₈×D₁₀₉	1	1	yes	characteristic
122	C₂²	—	109	2	no	—
123	C₂²	—	218	1	no	—
124	C₂²	—	218	1	no	—
125	C₂²	C₂×C₈×D₁₀₉	1	1	yes	characteristic
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	109	2	no	—
129	C₄	—	218	1	no	—
130	C₄	—	218	1	no	—
131	C₄	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
132	C₄	M₁₆×D₁₀₉	1	1	yes	characteristic
133	C₂	—	218	2	no	—
134	C₂	C₄⋊C₈×D₁₀₉	1	1	yes	characteristic
135	C₂	#20295442	1	1	yes	characteristic
136	C₂	#20295444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812638

C₂

yes

maximal, characteristic

C₂×C₁₆×D₁₀₉

C₂

yes

maximal, characteristic

#20294980

C₂

yes

maximal, characteristic

C₄⋊#13182004

C₂

yes

maximal, characteristic, complemented

#20294844

C₂

yes

maximal, characteristic, complemented

#20294844

C₂

yes

maximal, characteristic, complemented

C₄⋊C₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182012

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812656

C₄

yes

characteristic

#16812638

—

C₂×#13182004

C₂²

yes

characteristic

C₂×#13182004

C₂²

yes

characteristic

C₁₆×D₁₀₉

—

#16812610

C₄

yes

characteristic

#16812609

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₂×C₁₆×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

#13182012

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

D₄

yes

—

C₈×D₁₀₉

Q₈

yes

—

#13182008

C₈

yes

characteristic

#13182008

C₈

yes

characteristic

#13182006

C₈

yes

characteristic

#13182006

C₈

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#13182004

—

#13182004

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₁₆×C₁₀₉

—

C₁₆×C₁₀₉

—

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

C₄⋊C₄

yes

—

C₄×D₁₀₉

M₁₆

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×D₄

yes

characteristic

C₈×C₁₀₉

C₂×Q₈

yes

characteristic

#11724784

—

#11724784

C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄⋊C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₄⋊C₈

yes

—

C₂×C₁₀₉

C₂×C₄⋊C₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

#2151

—

109

maximal, Sylow

C₁₀₉

#2151

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂²×C₁₆

—

109

—

#405

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

—

109

—

C₄⋊C₁₆

D₁₀₉

yes

characteristic, complemented

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

C₂×C₁₆

C₂×D₁₀₉

yes

characteristic

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₂²×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

100

C₂²⋊C₄

—

109

—

101

C₂²⋊C₄

—

109

—

102

C₄²

—

109

—

103

C₄²

C₄×D₁₀₉

yes

characteristic

104

C₁₆

—

105

C₁₆

—

106

C₁₆

—

218

—

107

C₁₆

—

218

—

108

C₂³

—

109

—

109

C₂×C₄

—

109

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

109

—

112

C₂×C₄

—

109

—

113

C₂×C₄

—

109

—

114

C₂×C₄

C₈×D₁₀₉

yes

characteristic

115

C₂×C₄

C₈×D₁₀₉

yes

characteristic

116

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

117

C₈

—

118

C₈

—

109

—

119

C₈

—

218

—

120

C₈

D₄×D₁₀₉

yes

characteristic

121

C₈

Q₈×D₁₀₉

yes

characteristic

122

C₂²

—

109

—

123

C₂²

—

218

—

124

C₂²

—

218

—

125

C₂²

C₂×C₈×D₁₀₉

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

109

—

129

C₄

—

218

—

130

C₄

—

218

—

131

C₄

C₄⋊C₄×D₁₀₉

yes

characteristic

132

C₄

M₁₆×D₁₀₉

yes

characteristic

133

C₂

—

218

—

134

C₂

C₄⋊C₈×D₁₀₉

yes

characteristic

135

C₂

#20295442

yes

characteristic

136

C₂

#20295444

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	C₄⋊C₁₆×C₁₀₉	C₂
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×C₈×C₁₀₉	C₂³
hypercenter	C₄⋊C₁₆	D₁₀₉
hypocenter	C₁₀₉	#2151
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

C₄⋊C₁₆×C₁₀₉

C₂

prime radical

C₂²×D₁₀₉

C₂×C₈

cosocle

C₂×C₈×C₁₀₉

C₂³

hypercenter

C₄⋊C₁₆

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
4. C₄⋊#13182004	C₂	semidirect
5. #20294844	C₂	semidirect
6. #20294844	C₂	semidirect
7. C₄⋊C₁₆×C₁₀₉	C₂	semidirect
69. C₁₀₉	#2151	semidirect
76. C₄⋊C₁₆	D₁₀₉	semidirect

H ≅ G/N

type

4. C₄⋊#13182004

C₂

semidirect

5. #20294844

C₂

semidirect

6. #20294844

C₂

semidirect

7. C₄⋊C₁₆×C₁₀₉

C₂

semidirect

69. C₁₀₉

#2151

semidirect

76. C₄⋊C₁₆

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812638	2
C₂×C₁₆×D₁₀₉	2
#20294980	2
C₄⋊#13182004	2
#20294844	2
C₄⋊C₁₆×C₁₀₉	2
#2151	109

subgroup

index

C₂×C₁₆×D₁₀₉

C₄⋊C₁₆×C₁₀₉

109

Subgroups — #28364789

Conjugacy classes of subgroups

Complements

Maximal subgroups