Group #28362146 — subgroups

Subgroups — #28362146

Order 13952.

314 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20295661	C₂	1	1	yes	maximal, characteristic
2	#20295313	C₂	1	2	yes	maximal, complemented
3	#20294770	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294755	C₂	1	2	yes	maximal, complemented
5	#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₁₀₉	—	2	2	no	—
7	#16812670	—	2	2	no	—
8	#16812667	—	2	2	no	—
9	C₄×D₄₃₆	C₂²	1	2	yes	—
10	#16812630	—	2	2	no	—
11	C₄⋊#11724784	C₂²	1	2	yes	complemented
12	#16812619	—	2	2	no	—
13	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
15	#127×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₁₀₉	—	2	2	no	—
18	C₂×D₄₃₆	D₄	1	2	yes	—
19	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
20	D₄×D₁₀₉	—	4	4	no	—
21	D₄×D₁₀₉	—	4	4	no	—
22	C₂×C₄×D₁₀₉	—	2	2	no	—
23	C₂×#11724784	—	2	2	no	—
24	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
25	C₂²⋊Dic₁₀₉	—	2	2	no	—
26	#13182006	—	2	2	no	—
27	C₂×Q₈×C₁₀₉	—	2	1	no	—
28	C₂×D₄×C₁₀₉	—	2	2	no	—
29	C₂×D₄×C₁₀₉	D₄	1	2	yes	—
30	SD₁₆×C₁₀₉	—	4	4	no	—
31	C₂×C₈×C₁₀₉	—	2	1	no	—
32	C₄⋊C₄×C₁₀₉	—	2	2	no	—
33	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₂³×C₁₀₉	—	2	2	no	—
35	C₂²×D₁₀₉	—	2	2	no	—
36	C₂²×D₁₀₉	—	4	8	no	—
37	C₂²×D₁₀₉	—	8	2	no	—
38	Q₈×C₁₀₉	—	4	2	no	—
39	D₄×C₁₀₉	—	4	2	no	—
40	D₄×C₁₀₉	—	4	2	no	—
41	D₄×C₁₀₉	—	4	2	no	—
42	C₂×C₄×C₁₀₉	—	4	1	no	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂²⋊D₁₀₉	—	4	8	no	—
46	C₂×Dic₁₀₉	—	2	2	no	—
47	D₄₃₆	—	2	2	no	—
48	D₄₃₆	SD₁₆	1	4	yes	complemented
49	C₄×D₁₀₉	—	4	2	no	—
50	C₄×D₁₀₉	—	4	2	no	—
51	C₈×C₁₀₉	—	2	2	no	—
52	#11724784	—	4	2	no	—
53	C₂²×C₁₀₉	—	4	4	no	—
54	C₂²×C₁₀₉	—	4	2	no	—
55	C₂²×C₁₀₉	#120	1	1	yes	characteristic
56	C₂×D₁₀₉	—	2	4	no	—
57	C₂×D₁₀₉	—	4	2	no	—
58	C₂×D₁₀₉	—	4	2	no	—
59	C₂×D₁₀₉	—	8	8	no	—
60	C₄×C₁₀₉	—	2	1	no	—
61	C₄×C₁₀₉	—	8	1	no	—
62	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
63	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
64	Dic₁₀₉	—	4	2	no	—
65	C₂×C₁₀₉	—	8	2	no	—
66	C₂×C₁₀₉	#450	1	2	yes	—
67	C₂×C₁₀₉	#453	1	1	yes	characteristic
68	D₁₀₉	—	4	4	no	—
69	#1629	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1629	1	1	yes	characteristic, complemented, Sylow
71	D₄²	—	109	1	no	—
72	#492	D₁₀₉	1	1	yes	characteristic, complemented
73	#476	—	109	2	no	—
74	#331	—	109	1	no	—
75	D₄⋊C₈	—	109	2	no	—
76	C₂²×D₄	—	218	2	no	—
77	C₂×SD₁₆	—	2	2	no	—
78	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
79	#127	C₂×D₁₀₉	1	1	yes	characteristic
80	#121	—	218	2	no	—
81	#120	—	218	2	no	—
82	C₄×D₄	—	109	2	no	—
83	#106	—	218	2	no	—
84	C₄⋊C₈	—	109	2	no	—
85	#102	—	218	2	no	—
86	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
87	C₂⁴	—	218	2	no	—
88	C₂×Q₈	—	2	1	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	109	2	no	—
91	C₂×D₄	—	436	4	no	—
92	C₂×D₄	—	436	4	no	—
93	C₂×D₄	—	218	4	no	—
94	C₂×D₄	C₂²⋊D₁₀₉	1	2	yes	—
95	C₂²×C₄	—	218	2	no	—
96	SD₁₆	—	4	4	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	218	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	109	2	no	—
101	C₂²⋊C₄	—	218	2	no	—
102	C₂²⋊C₄	—	218	2	no	—
103	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
104	C₂³	—	2	2	no	—
105	C₂³	—	872	2	no	—
106	C₂³	—	436	8	no	—
107	C₂³	—	218	2	no	—
108	Q₈	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	D₄	—	109	4	no	—
113	D₄	—	436	8	no	—
114	D₄	—	218	2	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	436	2	no	—
117	C₂×C₄	—	436	2	no	—
118	C₂×C₄	—	218	2	no	—
119	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	436	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	—	4	2	no	—
125	C₂²	—	872	8	no	—
126	C₂²	—	436	2	no	—
127	C₂²	—	436	2	no	—
128	C₂²	—	218	4	no	—
129	C₂²	#16812667	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	436	2	no	—
133	C₄	SD₁₆×D₁₀₉	1	2	yes	—
134	C₄	#16812726	1	2	yes	—
135	C₂	—	8	2	no	—
136	C₂	—	436	4	no	—
137	C₂	#20295175	1	2	yes	—
138	C₂	#20295193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295661

C₂

yes

maximal, characteristic

#20295313

C₂

yes

maximal, complemented

#20294770

C₂

yes

maximal, characteristic, complemented

#20294755

C₂

yes

maximal, complemented

#492×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₁₀₉

—

#16812670

—

#16812667

—

C₄×D₄₃₆

C₂²

yes

—

#16812630

—

C₄⋊#11724784

C₂²

yes

complemented

#16812619

—

C₂×SD₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic, complemented

#127×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic, complemented

C₂³×D₁₀₉

—

C₂×D₄₃₆

D₄

yes

—

C₂×C₂²⋊D₁₀₉

—

D₄×D₁₀₉

—

D₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

D₄

yes

—

C₂²⋊Dic₁₀₉

—

#13182006

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

D₄

yes

—

SD₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

SD₁₆

yes

complemented

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#450

yes

—

C₂×C₁₀₉

#453

yes

characteristic

D₁₀₉

—

#1629

—

109

maximal, Sylow

C₁₀₉

#1629

yes

characteristic, complemented, Sylow

D₄²

—

109

—

#492

D₁₀₉

yes

characteristic, complemented

#476

—

109

—

#331

—

109

—

D₄⋊C₈

—

109

—

C₂²×D₄

—

218

—

C₂×SD₁₆

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic, complemented

#127

C₂×D₁₀₉

yes

characteristic

#121

—

218

—

#120

—

218

—

C₄×D₄

—

109

—

#106

—

218

—

C₄⋊C₈

—

109

—

#102

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic, complemented

C₂⁴

—

218

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

109

—

C₂×D₄

—

436

—

C₂×D₄

—

436

—

C₂×D₄

—

218

—

C₂×D₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

—

218

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

100

C₄⋊C₄

—

109

—

101

C₂²⋊C₄

—

218

—

102

C₂²⋊C₄

—

218

—

103

C₄²

C₂²×D₁₀₉

yes

characteristic

104

C₂³

—

105

C₂³

—

872

—

106

C₂³

—

436

—

107

C₂³

—

218

—

108

Q₈

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

109

—

113

D₄

—

436

—

114

D₄

—

218

—

115

C₂×C₄

—

116

C₂×C₄

—

436

—

117

C₂×C₄

—

436

—

118

C₂×C₄

—

218

—

119

C₂×C₄

D₄×D₁₀₉

yes

—

120

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

121

C₈

—

122

C₈

—

436

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

872

—

126

C₂²

—

436

—

127

C₂²

—

436

—

128

C₂²

—

218

—

129

C₂²

#16812667

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

436

—

133

C₄

SD₁₆×D₁₀₉

yes

—

134

C₄

#16812726

yes

—

135

C₂

—

136

C₂

—

436

—

137

C₂

#20295175

yes

—

138

C₂

#20295193

yes

characteristic

	subgroup	G/H
center	C₂²	#16812667
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#492×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#120
prime radical	#20295661	C₂
hypercenter	#492	D₁₀₉
hypocenter	C₁₀₉	#1629
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#16812667

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#492×C₁₀₉

C₂

C₂²×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
2. #20295313	C₂	semidirect
3. #20294770	C₂	semidirect
4. #20294755	C₂	semidirect
5. #492×C₁₀₉	C₂	semidirect
11. C₄⋊#11724784	C₂²	semidirect
14. C₄⋊Q₈×C₁₀₉	C₂²	semidirect
16. C₄×C₈×C₁₀₉	C₂²	semidirect
48. D₄₃₆	SD₁₆	semidirect
70. C₁₀₉	#1629	semidirect
72. #492	D₁₀₉	semidirect
78. C₄⋊Q₈	C₂×D₁₀₉	semidirect
86. C₄×C₈	C₂×D₁₀₉	semidirect

H ≅ G/N

type

2. #20295313

C₂

semidirect

3. #20294770

C₂

semidirect

4. #20294755

C₂

semidirect

5. #492×C₁₀₉

C₂

semidirect

11. C₄⋊#11724784

C₂²

semidirect

14. C₄⋊Q₈×C₁₀₉

C₂²

semidirect

16. C₄×C₈×C₁₀₉

C₂²

semidirect

48. D₄₃₆

SD₁₆

semidirect

70. C₁₀₉

#1629

semidirect

72. #492

D₁₀₉

semidirect

78. C₄⋊Q₈

C₂×D₁₀₉

semidirect

86. C₄×C₈

C₂×D₁₀₉

semidirect

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

subgroup	index
#20295661	2
#20295313	2
#20294770	2
#20294755	2
#492×C₁₀₉	2
#1629	109

subgroup

index

#492×C₁₀₉

109

Subgroups — #28362146

Conjugacy classes of subgroups

Complements

Maximal subgroups