Group #28379219 — subgroups

Subgroups — #28379219

Order 13952.

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20296030	C₂	1	1	yes	maximal, characteristic
2	#20295962	C₂	1	1	yes	maximal, characteristic
3	#20295896	C₂	1	4	yes	maximal
4	#20295582	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182039	C₂²	1	2	yes	—
6	C₂×C₄×#89137	C₂²	1	2	yes	—
7	C₄⋊#11724786	—	2	2	no	—
8	(C₂×#89137)⋊C₄	—	2	2	no	—
9	#16812698	C₄	1	2	yes	complemented
10	#16812691	C₄	1	1	yes	characteristic, complemented
11	#16812677	—	2	2	no	—
12	#16812664	—	2	2	no	—
13	#16812657	C₂²	1	2	yes	—
14	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
15	#125×C₁₀₉	C₄	1	1	yes	characteristic, complemented
16	C₂²×#89137	—	2	2	no	—
17	#13182039	—	2	4	no	—
18	C₂×#11724786	—	2	2	no	—
19	C₄×#89137	—	2	2	no	—
20	C₄×#89137	Q₈	1	4	yes	complemented
21	C₂×Dic₂₁₈	—	2	2	no	—
22	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
23	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
25	#13182008	—	2	2	no	—
26	#13182008	—	2	2	no	—
27	#13182006	—	2	2	no	—
28	C₄×Dic₁₀₉	—	2	2	no	—
29	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₁₀₉	D₄	1	2	yes	—
31	C₄⋊C₄×C₁₀₉	—	2	2	no	—
32	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
33	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
35	C₂×#89137	—	2	4	no	—
36	C₂×#89137	—	4	4	no	—
37	C₂×C₄×C₁₀₉	—	2	2	no	—
38	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×Dic₁₀₉	—	2	2	no	—
41	C₂×Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
42	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₄×D₁₀₉	—	4	2	no	—
44	C₄×D₁₀₉	C₂×Q₈	1	2	yes	—
45	C₄×D₁₀₉	#41	1	2	yes	—
46	Dic₂₁₈	—	4	4	no	—
47	#11724786	—	4	2	no	—
48	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	2	no	—
50	C₂×D₁₀₉	C₂²⋊Q₈	1	2	yes	—
51	#89137	—	4	4	no	—
52	C₄×C₁₀₉	—	2	2	no	—
53	C₄×C₁₀₉	—	4	2	no	—
54	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
55	Dic₁₀₉	—	2	2	no	—
56	Dic₁₀₉	—	4	2	no	—
57	Dic₁₀₉	#124	1	1	yes	characteristic
58	Dic₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₁₀₉	#391	1	1	yes	characteristic
60	C₂×C₁₀₉	#421	1	2	yes	—
61	D₁₀₉	—	2	2	no	—
62	#1995	—	109	1	no	maximal, Sylow
63	C₁₀₉	#1995	1	1	yes	characteristic, complemented, Sylow
64	#533	—	109	1	no	—
65	#423	—	109	1	no	—
66	#376	—	109	1	no	—
67	#339	—	109	4	no	—
68	C₂×M₁₆	—	109	2	no	—
69	C₄⋊Q₈	—	109	2	no	—
70	#125	—	109	1	no	—
71	#125	#89137	1	1	yes	characteristic, complemented
72	C₂²⋊Q₈	—	218	2	no	—
73	C₄×Q₈	—	218	2	no	—
74	#117	—	109	2	no	—
75	C₂×C₄²	—	109	1	no	—
76	C₂×C₄²	—	109	2	no	—
77	C₄⋊C₈	—	218	2	no	—
78	#95	—	218	2	no	—
79	C₂×Q₈	—	218	2	no	—
80	C₂²×C₄	—	109	1	no	—
81	C₂²×C₄	—	109	2	no	—
82	C₂²×C₄	—	218	2	no	—
83	M₁₆	—	218	4	no	—
84	C₂×C₈	—	218	2	no	—
85	C₄⋊C₄	—	2	2	no	—
86	C₄⋊C₄	—	109	2	no	—
87	C₄⋊C₄	—	218	2	no	—
88	C₄⋊C₄	—	218	2	no	—
89	C₄⋊C₄	C₂×#89137	1	2	yes	—
90	C₂²⋊C₄	—	218	2	no	—
91	C₄²	—	109	2	no	—
92	C₄²	—	109	1	no	—
93	C₄²	—	109	4	no	—
94	C₄²	—	218	2	no	—
95	C₄²	—	218	2	no	—
96	C₄²	C₂×#89137	1	1	yes	characteristic
97	C₂³	—	109	1	no	—
98	Q₈	—	436	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	109	1	no	—
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	109	2	no	—
103	C₂×C₄	—	109	2	no	—
104	C₂×C₄	—	436	2	no	—
105	C₂×C₄	—	436	4	no	—
106	C₂×C₄	—	218	4	no	—
107	C₂×C₄	—	218	2	no	—
108	C₂×C₄	C₂²⋊#89137	1	2	yes	—
109	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
110	C₈	—	436	2	no	—
111	C₂²	—	109	2	no	—
112	C₂²	—	218	2	no	—
113	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	4	2	no	—
116	C₄	—	109	1	no	—
117	C₄	—	109	1	no	—
118	C₄	—	436	4	no	—
119	C₄	—	436	2	no	—
120	C₄	—	218	2	no	—
121	C₄	Q₈×#89137	1	2	yes	—
122	C₂	—	218	2	no	—
123	C₂	(C₂×#89137)⋊Q₈	1	1	yes	characteristic
124	C₂	#20296023	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20296030

C₂

yes

maximal, characteristic

#20295962

C₂

yes

maximal, characteristic

#20295896

C₂

yes

maximal

#20295582

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₂×C₄×#89137

C₂²

yes

—

C₄⋊#11724786

—

(C₂×#89137)⋊C₄

—

#16812698

C₄

yes

complemented

#16812691

C₄

yes

characteristic, complemented

#16812677

—

#16812664

—

#16812657

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

#125×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

—

—

—

—

Q₈

yes

complemented

C₂×Dic₂₁₈

—

C₂×C₄×D₁₀₉

D₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

#13182008

—

#13182006

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×D₄

yes

characteristic

C₂×#89137

—

C₂×#89137

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×Q₈

yes

—

C₄×D₁₀₉

#41

yes

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊Q₈

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#124

yes

characteristic

Dic₁₀₉

C₄⋊Q₈

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

D₁₀₉

—

#1995

—

109

maximal, Sylow

C₁₀₉

#1995

yes

characteristic, complemented, Sylow

#533

—

109

—

#423

—

109

—

#376

—

109

—

#339

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

109

—

#125

—

109

—

#125

#89137

yes

characteristic, complemented

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

218

—

#95

—

218

—

C₂×Q₈

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×#89137

yes

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

Q₈

—

436

—

C₂×C₄

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

436

—

105

C₂×C₄

—

436

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

C₂²⋊#89137

yes

—

109

C₂×C₄

C₂²×#89137

yes

characteristic

110

C₈

—

436

—

111

C₂²

—

109

—

112

C₂²

—

218

—

113

C₂²

C₂×C₂²⋊#89137

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

109

—

117

C₄

—

109

—

118

C₄

—

436

—

119

C₄

—

436

—

120

C₄

—

218

—

121

C₄

Q₈×#89137

yes

—

122

C₂

—

218

—

123

C₂

(C₂×#89137)⋊Q₈

yes

characteristic

124

C₂

#20296023

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#125×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂×D₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	#125	#89137
hypocenter	C₁₀₉	#1995
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

#125×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₉

C₂×D₄

cosocle

C₂×C₄×D₁₀₉

C₂³

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
9. #16812698	C₄	semidirect
10. #16812691	C₄	semidirect
15. #125×C₁₀₉	C₄	semidirect
20. C₄×#89137	Q₈	semidirect
63. C₁₀₉	#1995	semidirect
71. #125	#89137	semidirect

H ≅ G/N

type

9. #16812698

C₄

semidirect

10. #16812691

C₄

semidirect

15. #125×C₁₀₉

C₄

semidirect

20. C₄×#89137

Q₈

semidirect

63. C₁₀₉

#1995

semidirect

71. #125

#89137

semidirect

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

subgroup	index
#20296030	2
#20295962	2
#20295896	2
#20295582	2
#1995	109

subgroup

index

109

Subgroups — #28379219

Conjugacy classes of subgroups

Complements

Maximal subgroups