Group #28362653 — subgroups

Subgroups — #28362653

Order 13952.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#20295429	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295424	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295398	C₂	1	1	yes	maximal, characteristic
4	#20295398	C₂	1	1	yes	maximal, characteristic
5	#20295376	C₂	1	1	yes	maximal, characteristic, complemented
6	#20294748	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182017	—	2	2	no	—
9	C₂×#13182017	—	2	2	no	—
10	#16812699	C₂²	1	1	yes	characteristic
11	#16812699	C₂²	1	1	yes	characteristic
12	#16812632	—	2	2	no	—
13	#16812631	—	2	2	no	—
14	#16812623	—	2	2	no	—
15	C₄×#11724784	C₂²	1	1	yes	characteristic
16	C₄×#11724784	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
18	#127×C₁₀₉	C₂²	1	1	yes	characteristic
19	#102×C₁₀₉	—	2	2	no	—
20	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₁₈	—	2	1	no	—
22	C₂×Dic₂₁₈	—	2	1	no	—
23	#13182017	—	4	4	no	—
24	#13182017	—	4	4	no	—
25	C₂×#11724784	D₄	1	2	yes	—
26	C₂×#11724784	D₄	1	2	yes	—
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₄⋊Dic₁₀₉	—	2	1	no	—
31	C₂×Q₈×C₁₀₉	—	2	1	no	—
32	C₂×D₄×C₁₀₉	—	2	1	no	—
33	C₂×D₄×C₁₀₉	—	2	1	no	—
34	C₂×D₄×C₁₀₉	—	2	1	no	—
35	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
36	C₄⋊C₄×C₁₀₉	—	2	1	no	—
37	C₄⋊C₄×C₁₀₉	—	2	1	no	—
38	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
39	C₂³×C₁₀₉	—	4	1	no	—
40	Q₈×C₁₀₉	—	4	2	no	—
41	D₄×C₁₀₉	—	4	2	no	—
42	D₄×C₁₀₉	—	4	2	no	—
43	D₄×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	—	4	1	no	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₁₀₉	—	4	1	no	—
49	C₂×Dic₁₀₉	—	4	1	no	—
50	Dic₂₁₈	—	4	2	no	—
51	Dic₂₁₈	—	4	2	no	—
52	C₈×C₁₀₉	—	2	2	no	—
53	#11724784	—	2	2	no	—
54	#11724784	SD₁₆	1	4	yes	complemented
55	C₂²×C₁₀₉	—	8	1	no	—
56	C₂²×C₁₀₉	—	8	1	no	—
57	C₂²×C₁₀₉	—	8	1	no	—
58	C₂²×C₁₀₉	#127	1	1	yes	characteristic
59	C₄×C₁₀₉	—	8	1	no	—
60	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
61	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₁₀₉	#137	1	2	yes	—
63	Dic₁₀₉	—	8	1	no	—
64	Dic₁₀₉	—	8	1	no	—
65	C₂×C₁₀₉	—	16	1	no	—
66	C₂×C₁₀₉	#492	1	1	yes	characteristic
67	C₂×C₁₀₉	#496	1	1	yes	characteristic
68	C₂×C₁₀₉	#497	1	1	yes	characteristic
69	#1714	—	109	1	no	maximal, Sylow
70	C₁₀₉	#1714	1	1	yes	characteristic, complemented, Sylow
71	#500	—	109	1	no	—
72	#498	—	109	1	no	—
73	#492	—	109	1	no	—
74	#492	—	109	1	no	—
75	#487	—	109	1	no	—
76	#486	D₁₀₉	1	1	yes	characteristic
77	#322	—	109	1	no	—
78	C₂×SD₁₆	—	218	2	no	—
79	C₂×SD₁₆	—	218	2	no	—
80	C₄⋊Q₈	—	109	1	no	—
81	C₄⋊Q₈	—	109	1	no	—
82	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
83	#127	C₂×D₁₀₉	1	1	yes	characteristic
84	#107	—	218	2	no	—
85	#106	—	218	2	no	—
86	#103	—	218	2	no	—
87	#102	—	2	2	no	—
88	C₄×C₈	—	109	1	no	—
89	C₄×C₈	—	109	1	no	—
90	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	218	1	no	—
93	C₂×Q₈	—	218	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	436	4	no	—
98	SD₁₆	—	436	4	no	—
99	C₂×C₈	—	109	2	no	—
100	C₂×C₈	—	109	2	no	—
101	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	218	1	no	—
105	C₄⋊C₄	—	218	1	no	—
106	C₄⋊C₄	—	218	1	no	—
107	C₄⋊C₄	—	218	1	no	—
108	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	436	2	no	—
112	Q₈	—	436	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	436	1	no	—
118	C₂×C₄	—	436	1	no	—
119	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
120	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	109	4	no	—
124	C₈	—	218	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#16812695	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	872	1	no	—
131	C₄	—	872	1	no	—
132	C₄	SD₁₆×D₁₀₉	1	2	yes	—
133	C₄	#16812723	1	2	yes	—
134	C₄	#16812729	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#20295397	1	1	yes	characteristic
137	C₂	#20295414	1	1	yes	characteristic
138	C₂	#20295421	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295429

C₂

yes

maximal, characteristic, complemented

#20295424

C₂

yes

maximal, characteristic, complemented

#20295398

C₂

yes

maximal, characteristic

#20295398

C₂

yes

maximal, characteristic

#20295376

C₂

yes

maximal, characteristic, complemented

#20294748

C₂

yes

maximal, characteristic, complemented

#486×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182017

—

C₂×#13182017

—

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#127×C₁₀₉

C₂²

yes

characteristic

#102×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

#13182017

—

#13182017

—

C₂×#11724784

D₄

yes

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×D₄×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

—

Q₈×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

C₈×C₁₀₉

—

#11724784

—

#11724784

SD₁₆

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1714

yes

characteristic, complemented, Sylow

#500

—

109

—

#498

—

109

—

#492

—

109

—

#492

—

109

—

#487

—

109

—

#486

D₁₀₉

yes

characteristic

#322

—

109

—

C₂×SD₁₆

—

218

—

C₂×SD₁₆

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#127

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#106

—

218

—

#103

—

218

—

#102

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

436

—

SD₁₆

—

436

—

C₂×C₈

—

109

—

100

C₂×C₈

—

109

—

101

C₂×C₈

C₂²⋊D₁₀₉

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

218

—

105

C₄⋊C₄

—

218

—

106

C₄⋊C₄

—

218

—

107

C₄⋊C₄

—

218

—

108

C₄²

C₂²×D₁₀₉

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

436

—

112

Q₈

—

436

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

436

—

118

C₂×C₄

—

436

—

119

C₂×C₄

D₄×D₁₀₉

yes

characteristic

120

C₂×C₄

D₄×D₁₀₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

122

C₈

—

123

C₈

—

109

—

124

C₈

—

218

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#16812695

yes

characteristic

129

C₄

—

130

C₄

—

872

—

131

C₄

—

872

—

132

C₄

SD₁₆×D₁₀₉

yes

—

133

C₄

#16812723

yes

—

134

C₄

#16812729

yes

—

135

C₂

—

136

C₂

#20295397

yes

characteristic

137

C₂

#20295414

yes

characteristic

138

C₂

#20295421

yes

characteristic

	subgroup	G/H
center	C₂²	#16812695
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#486×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#127
prime radical	#127×C₁₀₉	C₂²
hypercenter	#486	D₁₀₉
hypocenter	C₁₀₉	#1714
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#16812695

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#486×C₁₀₉

C₂

socle

C₂²×C₁₀₉

#127

prime radical

#127×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #20295429	C₂	semidirect
2. #20295424	C₂	semidirect
5. #20295376	C₂	semidirect
6. #20294748	C₂	semidirect
54. #11724784	SD₁₆	semidirect
70. C₁₀₉	#1714	semidirect

H ≅ G/N

type

1. #20295429

C₂

semidirect

2. #20295424

C₂

semidirect

5. #20295376

C₂

semidirect

6. #20294748

C₂

semidirect

54. #11724784

SD₁₆

semidirect

70. C₁₀₉

#1714

semidirect

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

subgroup	index
#20295429	2
#20295424	2
#20295398	2
#20295376	2
#20294748	2
#486×C₁₀₉	2
#1714	109

subgroup

index

#486×C₁₀₉

109

Subgroups — #28362653

Conjugacy classes of subgroups

Complements

Maximal subgroups