Group #28362706 — subgroups

Subgroups — #28362706

Order 13952.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20295431	C₂	1	1	yes	maximal, characteristic
2	#20295429	C₂	1	1	yes	maximal, characteristic
3	#20295405	C₂	1	1	yes	maximal, characteristic
4	#20295378	C₂	1	1	yes	maximal, characteristic
5	#20295376	C₂	1	1	yes	maximal, characteristic
6	#20294747	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₀₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#13182019	—	2	2	no	—
9	#16812699	C₂²	1	1	yes	characteristic
10	#16812699	C₂²	1	1	yes	characteristic
11	#16812633	—	2	2	no	—
12	#16812632	—	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	#13182019	—	4	4	no	—
24	C₂×#11724784	D₄	1	2	yes	—
25	C₂×#11724784	D₄	1	2	yes	—
26	C₄⋊Dic₁₀₉	—	2	1	no	—
27	C₄⋊Dic₁₀₉	—	2	1	no	—
28	C₄⋊Dic₁₀₉	—	2	1	no	—
29	C₄⋊Dic₁₀₉	—	2	1	no	—
30	C₂×Q₈×C₁₀₉	—	2	1	no	—
31	C₂×Q₈×C₁₀₉	—	2	1	no	—
32	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₀₉	—	2	1	no	—
34	C₄⋊C₄×C₁₀₉	—	2	1	no	—
35	C₄⋊C₄×C₁₀₉	—	2	1	no	—
36	C₄⋊C₄×C₁₀₉	—	2	1	no	—
37	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
38	Q₈×C₁₀₉	—	4	2	no	—
39	Q₈×C₁₀₉	—	4	2	no	—
40	C₂×C₄×C₁₀₉	—	4	1	no	—
41	C₂×C₄×C₁₀₉	—	4	1	no	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₀₉	—	4	1	no	—
46	C₂×Dic₁₀₉	—	4	1	no	—
47	Dic₂₁₈	—	4	2	no	—
48	Dic₂₁₈	—	4	2	no	—
49	C₈×C₁₀₉	Q₁₆	1	4	yes	complemented
50	#11724784	—	2	2	no	—
51	#11724784	—	2	2	no	—
52	C₂²×C₁₀₉	#127	1	1	yes	characteristic
53	C₄×C₁₀₉	—	8	1	no	—
54	C₄×C₁₀₉	—	8	1	no	—
55	C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
56	C₄×C₁₀₉	AGL(1,ℤ/8)	1	2	yes	—
57	C₄×C₁₀₉	#137	1	2	yes	—
58	Dic₁₀₉	—	8	1	no	—
59	Dic₁₀₉	—	8	1	no	—
60	C₂×C₁₀₉	#494	1	1	yes	characteristic
61	C₂×C₁₀₉	#496	1	1	yes	characteristic
62	C₂×C₁₀₉	#497	1	1	yes	characteristic
63	#1723	—	109	1	no	maximal, Sylow
64	C₁₀₉	#1723	1	1	yes	characteristic, complemented, Sylow
65	#500	—	109	1	no	—
66	#500	—	109	1	no	—
67	#498	D₁₀₉	1	1	yes	characteristic
68	#494	—	109	1	no	—
69	#487	—	109	1	no	—
70	#487	—	109	1	no	—
71	#322	—	109	1	no	—
72	C₂×Q₁₆	—	218	2	no	—
73	C₄⋊Q₈	—	109	1	no	—
74	C₄⋊Q₈	—	109	1	no	—
75	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
76	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
77	#107	—	218	2	no	—
78	#107	—	218	2	no	—
79	#106	—	2	2	no	—
80	#103	—	218	2	no	—
81	#103	—	218	2	no	—
82	C₄×C₈	—	109	1	no	—
83	C₄×C₈	—	109	1	no	—
84	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	218	1	no	—
88	C₂×Q₈	—	218	1	no	—
89	Q₁₆	—	436	4	no	—
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	109	2	no	—
92	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	218	1	no	—
98	C₄⋊C₄	—	218	1	no	—
99	C₄⋊C₄	—	218	1	no	—
100	C₄⋊C₄	—	218	1	no	—
101	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	436	2	no	—
105	Q₈	—	436	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	436	1	no	—
109	C₂×C₄	—	436	1	no	—
110	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
113	C₈	—	218	2	no	—
114	C₈	—	218	2	no	—
115	C₈	#13182019	1	4	yes	complemented
116	C₂²	#16812695	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	872	1	no	—
120	C₄	—	872	1	no	—
121	C₄	C₂×#13182019	1	2	yes	—
122	C₄	#16812726	1	2	yes	—
123	C₄	#16812728	1	2	yes	—
124	C₂	#20295405	1	1	yes	characteristic
125	C₂	#20295412	1	1	yes	characteristic
126	C₂	#20295417	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295431

C₂

yes

maximal, characteristic

#20295429

C₂

yes

maximal, characteristic

#20295405

C₂

yes

maximal, characteristic

#20295378

C₂

yes

maximal, characteristic

#20295376

C₂

yes

maximal, characteristic

#20294747

C₂

yes

maximal, characteristic

#498×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182019

—

#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

—

#13182019

—

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

Q₁₆

yes

complemented

#11724784

—

#11724784

—

C₂²×C₁₀₉

#127

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₁₆

yes

—

C₄×C₁₀₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1723

yes

characteristic, complemented, Sylow

#500

—

109

—

#500

—

109

—

#498

D₁₀₉

yes

characteristic

#494

—

109

—

#487

—

109

—

#487

—

109

—

#322

—

109

—

C₂×Q₁₆

—

218

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#107

—

218

—

#107

—

218

—

#106

—

#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

—

Q₁₆

—

436

—

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

—

100

C₄⋊C₄

—

218

—

101

C₄²

C₂²×D₁₀₉

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

436

—

105

Q₈

—

436

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

436

—

109

C₂×C₄

—

436

—

110

C₂×C₄

D₄×D₁₀₉

yes

characteristic

111

C₂×C₄

D₄×D₁₀₉

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

113

C₈

—

218

—

114

C₈

—

218

—

115

C₈

#13182019

yes

complemented

116

C₂²

#16812695

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

872

—

120

C₄

—

872

—

121

C₄

C₂×#13182019

yes

—

122

C₄

#16812726

yes

—

123

C₄

#16812728

yes

—

124

C₂

#20295405

yes

characteristic

125

C₂

#20295412

yes

characteristic

126

C₂

#20295417

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	#1723
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
49. C₈×C₁₀₉	Q₁₆	semidirect
64. C₁₀₉	#1723	semidirect
115. C₈	#13182019	semidirect

H ≅ G/N

type

49. C₈×C₁₀₉

Q₁₆

semidirect

64. C₁₀₉

#1723

semidirect

115. C₈

#13182019

semidirect

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

subgroup	index
#20295431	2
#20295429	2
#20295405	2
#20295378	2
#20295376	2
#20294747	2
#498×C₁₀₉	2
#1723	109

subgroup

index

#498×C₁₀₉

109

Subgroups — #28362706

Conjugacy classes of subgroups

Complements

Maximal subgroups