Group #28363081 — subgroups

Subgroups — #28363081

Order 13952.

252 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 81 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×#16812707	C₂	1	1	yes	maximal, characteristic
2	C₂×#16812634	C₂	1	2	yes	maximal, complemented
3	#20295008	C₂	1	2	yes	maximal
4	#20294784	C₂	1	1	yes	maximal, characteristic, complemented
5	#338×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13182029	C₂²	1	1	yes	characteristic
7	#16812707	—	2	4	no	—
8	C₂×#13182013	C₂²	1	2	yes	—
9	C₂×#13182012	C₄	1	2	yes	—
10	C₂²×#11724784	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₁₀₉	C₄	1	2	yes	—
12	#16812634	—	2	4	no	—
13	#16812617	—	2	2	no	—
14	#16812615	—	2	2	no	—
15	C₂×M₁₆×C₁₀₉	C₂²	1	2	yes	—
16	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
17	#13182029	—	2	2	no	—
18	#13182029	—	2	2	no	—
19	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
20	C₂×D₄₃₆	D₄	1	1	yes	characteristic
21	C₂×C₂²⋊D₁₀₉	Q₈	1	2	yes	—
22	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
23	#13182013	—	2	2	no	—
24	#13182013	—	2	2	no	—
25	#13182012	—	2	4	no	—
26	C₂×#11724784	—	2	2	no	—
27	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
28	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
29	C₂×#11724784	D₄	1	4	yes	—
30	C₈×D₁₀₉	—	2	4	no	—
31	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₀₉	—	2	2	no	—
33	M₁₆×C₁₀₉	—	2	2	no	—
34	M₁₆×C₁₀₉	—	2	2	no	—
35	C₂×C₈×C₁₀₉	—	2	2	no	—
36	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
37	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
38	C₂²×D₁₀₉	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	#41	1	2	yes	—
43	C₂²⋊D₁₀₉	—	2	4	no	—
44	C₂×Dic₁₀₉	C₄⋊C₄	1	2	yes	—
45	D₄₃₆	—	2	2	no	—
46	C₄×D₁₀₉	—	2	2	no	—
47	C₄×D₁₀₉	—	2	2	no	—
48	Dic₂₁₈	—	2	2	no	—
49	C₈×C₁₀₉	—	2	2	no	—
50	C₈×C₁₀₉	—	4	2	no	—
51	#11724784	—	4	2	no	—
52	#11724784	C₂²⋊C₄	1	2	yes	—
53	#11724784	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₁₀₉	—	4	1	no	—
55	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
57	C₂×D₁₀₉	—	2	2	no	—
58	C₂×D₁₀₉	—	4	2	no	—
59	C₄×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₀₉	#117	1	1	yes	characteristic
61	C₄×C₁₀₉	#121	1	2	yes	—
62	Dic₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	—	2	1	no	—
64	C₂×C₁₀₉	—	2	2	no	—
65	C₂×C₁₀₉	#380	1	1	yes	characteristic
66	D₁₀₉	—	4	2	no	—
67	#1813	—	109	1	no	maximal, Sylow
68	C₁₀₉	#1813	1	1	yes	characteristic, complemented, Sylow
69	C₂×#131	—	109	1	no	—
70	C₂×#108	—	109	2	no	—
71	#413	—	109	2	no	—
72	#338	—	109	1	no	—
73	#338	D₁₀₉	1	1	yes	characteristic, complemented
74	C₂×#41	—	109	1	no	—
75	#131	—	218	4	no	—
76	C₂×M₁₆	—	109	2	no	—
77	C₂×M₁₆	—	109	2	no	—
78	C₂×M₁₆	C₂×D₁₀₉	1	2	yes	—
79	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂²×C₈	—	109	1	no	—
81	C₂²×C₈	—	109	2	no	—
82	#108	—	218	4	no	—
83	#101	—	218	2	no	—
84	#100	—	218	2	no	—
85	#41	—	218	2	no	—
86	#41	—	218	2	no	—
87	C₂×Q₈	—	109	1	no	—
88	C₂×D₄	—	109	2	no	—
89	C₂×D₄	—	109	1	no	—
90	C₂²×C₄	—	109	2	no	—
91	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
92	M₁₆	—	2	2	no	—
93	M₁₆	—	2	2	no	—
94	M₁₆	—	2	2	no	—
95	M₁₆	—	218	4	no	—
96	M₁₆	—	218	2	no	—
97	M₁₆	—	218	2	no	—
98	C₂×C₈	—	2	2	no	—
99	C₂×C₈	—	109	1	no	—
100	C₂×C₈	—	109	1	no	—
101	C₂×C₈	—	109	4	no	—
102	C₂×C₈	—	218	4	no	—
103	C₂×C₈	—	218	2	no	—
104	C₂×C₈	C₄×D₁₀₉	1	2	yes	—
105	C₂³	—	109	2	no	—
106	C₂³	Q₈×D₁₀₉	1	1	yes	characteristic
107	Q₈	—	218	2	no	—
108	D₄	—	218	4	no	—
109	D₄	—	218	2	no	—
110	C₂×C₄	—	109	2	no	—
111	C₂×C₄	—	218	2	no	—
112	C₂×C₄	—	218	2	no	—
113	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
114	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
115	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
116	C₂×C₄	#13182029	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	4	2	no	—
119	C₈	—	109	2	no	—
120	C₈	—	109	2	no	—
121	C₈	—	436	2	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	436	2	no	—
124	C₂²	—	218	2	no	—
125	C₂²	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
126	C₂²	#16812677	1	2	yes	—
127	C₄	—	218	2	no	—
128	C₄	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
129	C₄	#16812656	1	1	yes	characteristic
130	C₄	#16812672	1	2	yes	—
131	C₂	—	2	1	no	—
132	C₂	—	2	2	no	—
133	C₂	—	436	2	no	—
134	C₂	#20294885	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812707

C₂

yes

maximal, characteristic

C₂×#16812634

C₂

yes

maximal, complemented

#20295008

C₂

yes

maximal

#20294784

C₂

yes

maximal, characteristic, complemented

#338×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182029

C₂²

yes

characteristic

#16812707

—

C₂×#13182013

C₂²

yes

—

C₂×#13182012

C₄

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

—

#16812634

—

#16812617

—

#16812615

—

C₂×M₁₆×C₁₀₉

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

#13182029

—

#13182029

—

C₂×Dic₂₁₈

D₄

yes

characteristic

C₂×D₄₃₆

D₄

yes

characteristic

C₂×C₂²⋊D₁₀₉

Q₈

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

—

—

—

—

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

D₄

yes

—

C₈×D₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

characteristic

C₂²×D₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₄⋊C₄

yes

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

#11724784

C₂²⋊C₄

yes

—

#11724784

C₂²⋊C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

C₂×C₂²⋊C₄

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#380

yes

characteristic

D₁₀₉

—

#1813

—

109

maximal, Sylow

C₁₀₉

#1813

yes

characteristic, complemented, Sylow

C₂×#131

—

109

—

C₂×#108

—

109

—

#413

—

109

—

#338

—

109

—

#338

D₁₀₉

yes

characteristic, complemented

C₂×#41

—

109

—

#131

—

218

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

#108

—

218

—

#101

—

218

—

#100

—

218

—

#41

—

218

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

218

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

109

—

100

C₂×C₈

—

109

—

101

C₂×C₈

—

109

—

102

C₂×C₈

—

218

—

103

C₂×C₈

—

218

—

104

C₂×C₈

C₄×D₁₀₉

yes

—

105

C₂³

—

109

—

106

C₂³

Q₈×D₁₀₉

yes

characteristic

107

Q₈

—

218

—

108

D₄

—

218

—

109

D₄

—

218

—

110

C₂×C₄

—

109

—

111

C₂×C₄

—

218

—

112

C₂×C₄

—

218

—

113

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

114

C₂×C₄

D₄×D₁₀₉

yes

—

115

C₂×C₄

D₄×D₁₀₉

yes

characteristic

116

C₂×C₄

#13182029

yes

—

117

C₈

—

118

C₈

—

119

C₈

—

109

—

120

C₈

—

109

—

121

C₈

—

436

—

122

C₂²

—

123

C₂²

—

436

—

124

C₂²

—

218

—

125

C₂²

C₄⋊C₄×D₁₀₉

yes

characteristic

126

C₂²

#16812677

yes

—

127

C₄

—

218

—

128

C₄

C₂²⋊C₄×D₁₀₉

yes

characteristic

129

C₄

#16812656

yes

characteristic

130

C₄

#16812672

yes

—

131

C₂

—

132

C₂

—

133

C₂

—

436

—

134

C₂

#20294885

yes

characteristic

	subgroup	G/H
center	C₄	#16812656
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#338×C₁₀₉	C₂
socle	C₂×C₁₀₉	#380
prime radical	C₂×#13182029	C₂²
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#338	D₁₀₉
hypocenter	C₁₀₉	#1813
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#16812656

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#338×C₁₀₉

C₂

C₂×C₁₀₉

C₂²

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
2. C₂×#16812634	C₂	semidirect
4. #20294784	C₂	semidirect
5. #338×C₁₀₉	C₂	semidirect
68. C₁₀₉	#1813	semidirect
73. #338	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#16812634

C₂

semidirect

4. #20294784

C₂

semidirect

5. #338×C₁₀₉

C₂

semidirect

68. C₁₀₉

#1813

semidirect

73. #338

D₁₀₉

semidirect

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

subgroup	index
C₂×#16812707	2
C₂×#16812634	2
#20295008	2
#20294784	2
#338×C₁₀₉	2
#1813	109

subgroup

index

#338×C₁₀₉

109

Subgroups — #28363081

Conjugacy classes of subgroups

Complements

Maximal subgroups