Group #28362113 — subgroups

Subgroups — #28362113

Order 13952.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20295742	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295218	C₂	1	2	yes	maximal, complemented
3	#20294775	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×#13182014	—	2	2	no	—
7	#16812688	—	2	2	no	—
8	#16812674	—	2	2	no	—
9	C₄×D₄₃₆	C₂²	1	2	yes	—
10	#16812657	—	2	2	no	—
11	C₄⋊#11724784	C₂²	1	2	yes	—
12	#16812620	—	2	2	no	—
13	C₂×SD₁₆×C₁₀₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	#127×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₄₃₆	—	2	2	no	—
18	C₂×D₄₃₆	D₄	1	2	yes	—
19	C₂×C₄×D₁₀₉	—	2	2	no	—
20	#13182014	—	4	4	no	—
21	C₂×#11724784	—	2	2	no	—
22	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
23	#13182006	—	2	4	no	—
24	#13182006	—	2	2	no	—
25	C₄×Dic₁₀₉	—	2	2	no	—
26	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
27	C₂×D₄×C₁₀₉	—	2	1	no	—
28	C₂×D₄×C₁₀₉	—	2	2	no	—
29	SD₁₆×C₁₀₉	—	4	4	no	—
30	C₂×C₈×C₁₀₉	—	2	1	no	—
31	C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₉	—	4	1	no	—
34	C₂²×D₁₀₉	—	2	2	no	—
35	Q₈×C₁₀₉	—	4	2	no	—
36	D₄×C₁₀₉	—	4	2	no	—
37	D₄×C₁₀₉	—	4	2	no	—
38	D₄×C₁₀₉	—	4	2	no	—
39	C₂×C₄×C₁₀₉	—	2	2	no	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₁₀₉	—	2	2	no	—
43	D₄₃₆	—	2	2	no	—
44	D₄₃₆	—	2	2	no	—
45	D₄₃₆	—	8	2	no	—
46	C₄×D₁₀₉	—	4	2	no	—
47	C₄×D₁₀₉	—	4	2	no	—
48	C₈×C₁₀₉	—	2	2	no	—
49	#11724784	—	4	2	no	—
50	C₂²×C₁₀₉	—	8	2	no	—
51	C₂²×C₁₀₉	—	8	1	no	—
52	C₂²×C₁₀₉	#120	1	1	yes	characteristic
53	C₂×D₁₀₉	—	4	2	no	—
54	C₂×D₁₀₉	—	4	2	no	—
55	C₂×D₁₀₉	—	4	2	no	—
56	C₄×C₁₀₉	—	2	1	no	—
57	C₄×C₁₀₉	—	4	2	no	—
58	C₄×C₁₀₉	#135	1	2	yes	—
59	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₁₀₉	—	4	2	no	—
61	C₂×C₁₀₉	—	16	1	no	—
62	C₂×C₁₀₉	#449	1	2	yes	—
63	C₂×C₁₀₉	#453	1	1	yes	characteristic
64	D₁₀₉	—	8	2	no	—
65	#1624	—	109	1	no	maximal, Sylow
66	C₁₀₉	#1624	1	1	yes	characteristic, complemented, Sylow
67	#555	—	109	1	no	—
68	#492	D₁₀₉	1	1	yes	characteristic, complemented
69	#459	—	109	2	no	—
70	#333	—	109	1	no	—
71	D₄⋊C₈	—	109	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	218	2	no	—
74	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
75	#127	C₂×D₁₀₉	1	1	yes	characteristic
76	#124	—	218	2	no	—
77	#121	—	218	2	no	—
78	C₄×D₄	—	109	2	no	—
79	#117	—	218	2	no	—
80	C₄⋊C₈	—	109	2	no	—
81	#102	—	218	2	no	—
82	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	109	2	no	—
87	C₂×D₄	—	218	2	no	—
88	C₂²×C₄	—	218	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	436	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	218	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	109	2	no	—
95	C₂²⋊C₄	—	218	4	no	—
96	C₂²⋊C₄	—	218	2	no	—
97	C₄²	—	218	2	no	—
98	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	218	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	872	2	no	—
106	D₄	—	218	2	no	—
107	D₄	—	218	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	436	2	no	—
110	C₂×C₄	—	436	2	no	—
111	C₂×C₄	—	218	2	no	—
112	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	436	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	436	2	no	—
119	C₂²	—	436	2	no	—
120	C₂²	—	436	2	no	—
121	C₂²	#16812667	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	436	2	no	—
125	C₄	#16812720	1	2	yes	—
126	C₄	#16812726	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	872	2	no	—
129	C₂	#20295170	1	2	yes	—
130	C₂	#20295193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295742

C₂

yes

maximal, characteristic, complemented

#20295218

C₂

yes

maximal, complemented

#20294775

C₂

yes

maximal, characteristic, complemented

#20294755

C₂

yes

maximal, complemented

#492×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182014

—

#16812688

—

#16812674

—

C₄×D₄₃₆

C₂²

yes

—

#16812657

—

C₄⋊#11724784

C₂²

yes

—

#16812620

—

C₂×SD₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#127×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×D₄₃₆

—

C₂×D₄₃₆

D₄

yes

—

C₂×C₄×D₁₀₉

—

#13182014

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

D₄

yes

—

#13182006

—

#13182006

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

SD₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

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

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#120

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#135

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#449

yes

—

C₂×C₁₀₉

#453

yes

characteristic

D₁₀₉

—

#1624

—

109

maximal, Sylow

C₁₀₉

#1624

yes

characteristic, complemented, Sylow

#555

—

109

—

#492

D₁₀₉

yes

characteristic, complemented

#459

—

109

—

#333

—

109

—

D₄⋊C₈

—

109

—

C₂×SD₁₆

—

C₂×D₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#127

C₂×D₁₀₉

yes

characteristic

#124

—

218

—

#121

—

218

—

C₄×D₄

—

109

—

#117

—

218

—

C₄⋊C₈

—

109

—

#102

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

218

—

SD₁₆

—

D₈

—

436

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

100

C₂³

—

218

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

872

—

106

D₄

—

218

—

107

D₄

—

218

—

108

C₂×C₄

—

109

C₂×C₄

—

436

—

110

C₂×C₄

—

436

—

111

C₂×C₄

—

218

—

112

C₂×C₄

D₄×D₁₀₉

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

114

C₈

—

115

C₈

—

436

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

436

—

119

C₂²

—

436

—

120

C₂²

—

436

—

121

C₂²

#16812667

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

436

—

125

C₄

#16812720

yes

—

126

C₄

#16812726

yes

—

127

C₂

—

128

C₂

—

872

—

129

C₂

#20295170

yes

—

130

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
hypercenter	#492	D₁₀₉
hypocenter	C₁₀₉	#1624
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

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #20295742	C₂	semidirect
2. #20295218	C₂	semidirect
3. #20294775	C₂	semidirect
4. #20294755	C₂	semidirect
5. #492×C₁₀₉	C₂	semidirect
66. C₁₀₉	#1624	semidirect
68. #492	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295742

C₂

semidirect

2. #20295218

C₂

semidirect

3. #20294775

C₂

semidirect

4. #20294755

C₂

semidirect

5. #492×C₁₀₉

C₂

semidirect

66. C₁₀₉

#1624

semidirect

68. #492

D₁₀₉

semidirect

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

subgroup	index
#20295742	2
#20295218	2
#20294775	2
#20294755	2
#492×C₁₀₉	2
#1624	109

subgroup

index

#492×C₁₀₉

109

Subgroups — #28362113

Conjugacy classes of subgroups

Complements

Maximal subgroups