Group #28362676 — subgroups

Subgroups — #28362676

Order 13952.

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

normal subgroups only

This group has 164 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	#20295429	C₂	1	1	yes	maximal, characteristic
2	#20295378	C₂	1	1	yes	maximal, characteristic
3	#20295377	C₂	1	1	yes	maximal, characteristic
4	#20295376	C₂	1	1	yes	maximal, characteristic
5	#20295376	C₂	1	1	yes	maximal, characteristic
6	#20294748	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#16812699	C₂²	1	1	yes	characteristic
9	#16812699	C₂²	1	1	yes	characteristic
10	#16812632	—	2	2	no	—
11	#16812624	—	2	2	no	—
12	#16812623	—	2	2	no	—
13	#16812623	—	2	2	no	—
14	#16812622	—	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	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	#106×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	C₂×#11724784	D₄	1	2	yes	—
24	C₂×#11724784	D₄	1	2	yes	—
25	C₄⋊Dic₁₀₉	—	2	1	no	—
26	C₄⋊Dic₁₀₉	—	2	1	no	—
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₂×Q₈×C₁₀₉	—	2	1	no	—
30	C₂×Q₈×C₁₀₉	—	2	1	no	—
31	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
32	C₄⋊C₄×C₁₀₉	—	2	1	no	—
33	C₄⋊C₄×C₁₀₉	—	2	1	no	—
34	C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
37	Q₈×C₁₀₉	—	4	2	no	—
38	Q₈×C₁₀₉	—	4	2	no	—
39	C₂×C₄×C₁₀₉	—	4	1	no	—
40	C₂×C₄×C₁₀₉	—	4	1	no	—
41	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×Dic₁₀₉	—	4	1	no	—
45	C₂×Dic₁₀₉	—	4	1	no	—
46	Dic₂₁₈	—	4	2	no	—
47	Dic₂₁₈	—	4	2	no	—
48	C₈×C₁₀₉	—	2	2	no	—
49	#11724784	—	2	2	no	—
50	#11724784	—	2	2	no	—
51	C₂²×C₁₀₉	#127	1	1	yes	characteristic
52	C₄×C₁₀₉	—	8	1	no	—
53	C₄×C₁₀₉	—	8	1	no	—
54	C₄×C₁₀₉	#135	1	2	yes	—
55	C₄×C₁₀₉	#137	1	2	yes	—
56	C₄×C₁₀₉	#137	1	2	yes	—
57	Dic₁₀₉	—	8	1	no	—
58	Dic₁₀₉	—	8	1	no	—
59	C₂×C₁₀₉	#495	1	1	yes	characteristic
60	C₂×C₁₀₉	#497	1	1	yes	characteristic
61	C₂×C₁₀₉	#497	1	1	yes	characteristic
62	#1718	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1718	1	1	yes	characteristic, complemented, Sylow
64	#500	—	109	1	no	—
65	#498	D₁₀₉	1	1	yes	characteristic
66	#487	—	109	1	no	—
67	#487	—	109	1	no	—
68	#487	—	109	1	no	—
69	#487	—	109	1	no	—
70	#322	—	109	1	no	—
71	C₄⋊Q₈	—	109	1	no	—
72	C₄⋊Q₈	—	109	1	no	—
73	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
74	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
75	#107	—	218	2	no	—
76	#106	—	2	2	no	—
77	#103	—	218	2	no	—
78	#103	—	218	2	no	—
79	#103	—	218	2	no	—
80	#103	—	218	2	no	—
81	C₄×C₈	—	109	1	no	—
82	C₄×C₈	—	109	1	no	—
83	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	218	1	no	—
87	C₂×Q₈	—	218	1	no	—
88	C₂×C₈	—	109	2	no	—
89	C₂×C₈	—	109	2	no	—
90	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	218	1	no	—
96	C₄⋊C₄	—	218	1	no	—
97	C₄⋊C₄	—	218	1	no	—
98	C₄⋊C₄	—	218	1	no	—
99	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
100	Q₈	—	4	2	no	—
101	Q₈	—	4	2	no	—
102	Q₈	—	436	2	no	—
103	Q₈	—	436	2	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	436	1	no	—
107	C₂×C₄	—	436	1	no	—
108	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
109	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	218	2	no	—
113	C₈	—	218	2	no	—
114	C₂²	#16812695	1	1	yes	characteristic
115	C₄	—	8	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	872	1	no	—
118	C₄	—	872	1	no	—
119	C₄	#16812720	1	2	yes	—
120	C₄	#16812728	1	2	yes	—
121	C₄	#16812730	1	2	yes	—
122	C₂	#20295407	1	1	yes	characteristic
123	C₂	#20295417	1	1	yes	characteristic
124	C₂	#20295418	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295429

C₂

yes

maximal, characteristic

#20295378

C₂

yes

maximal, characteristic

#20295377

C₂

yes

maximal, characteristic

#20295376

C₂

yes

maximal, characteristic

#20295376

C₂

yes

maximal, characteristic

#20294748

C₂

yes

maximal, characteristic

#498×C₁₀₉

C₂

yes

maximal, characteristic

#16812699

C₂²

yes

characteristic

#16812699

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

#106×C₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

—

C₂×#11724784

D₄

yes

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×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

—

C₂²×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#135

yes

—

C₄×C₁₀₉

#137

yes

—

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1718

yes

characteristic, complemented, Sylow

#500

—

109

—

#498

D₁₀₉

yes

characteristic

#487

—

109

—

#487

—

109

—

#487

—

109

—

#487

—

109

—

#322

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#106

—

#103

—

218

—

#103

—

218

—

#103

—

218

—

#103

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

436

—

103

Q₈

—

436

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

436

—

107

C₂×C₄

—

436

—

108

C₂×C₄

D₄×D₁₀₉

yes

characteristic

109

C₂×C₄

D₄×D₁₀₉

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

111

C₈

—

112

C₈

—

218

—

113

C₈

—

218

—

114

C₂²

#16812695

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

872

—

118

C₄

—

872

—

119

C₄

#16812720

yes

—

120

C₄

#16812728

yes

—

121

C₄

#16812730

yes

—

122

C₂

#20295407

yes

characteristic

123

C₂

#20295417

yes

characteristic

124

C₂

#20295418

yes

characteristic

	subgroup	G/H
center	C₂²	#16812695
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#498×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#127
hypercenter	#498	D₁₀₉
hypocenter	C₁₀₉	#1718
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#16812695

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#498×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
63. C₁₀₉	#1718	semidirect

H ≅ G/N

type

63. C₁₀₉

#1718

semidirect

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

subgroup	index
#20295429	2
#20295378	2
#20295377	2
#20295376	2
#20294748	2
#498×C₁₀₉	2
#1718	109

subgroup

index

#498×C₁₀₉

109

Subgroups — #28362676

Conjugacy classes of subgroups

Complements

Maximal subgroups