Group #28362108 — subgroups

Subgroups — #28362108

Order 13952.

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

normal subgroups only

This group has 228 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	#20295756	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295223	C₂	1	2	yes	maximal
3	#20294775	C₂	1	1	yes	maximal, characteristic, complemented
4	#20294758	C₂	1	2	yes	maximal, complemented
5	#492×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182017	—	2	2	no	—
7	#16812698	—	2	2	no	—
8	#16812697	—	2	2	no	—
9	C₄×Dic₂₁₈	C₂²	1	2	yes	—
10	#16812664	—	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×Dic₂₁₈	—	2	2	no	—
18	C₂×Dic₂₁₈	D₄	1	2	yes	—
19	#13182017	—	4	4	no	—
20	C₂×#11724784	—	2	2	no	—
21	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
22	#13182008	—	2	4	no	—
23	#13182008	—	2	2	no	—
24	C₄×Dic₁₀₉	—	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	Q₈×C₁₀₉	—	4	2	no	—
35	D₄×C₁₀₉	—	4	2	no	—
36	D₄×C₁₀₉	—	4	2	no	—
37	D₄×C₁₀₉	—	4	2	no	—
38	C₂×C₄×C₁₀₉	—	2	2	no	—
39	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	—	2	2	no	—
42	C₂×Dic₁₀₉	—	2	2	no	—
43	Dic₂₁₈	—	2	2	no	—
44	Dic₂₁₈	—	8	2	no	—
45	Dic₂₁₈	SD₁₆	1	4	yes	complemented
46	C₈×C₁₀₉	—	2	2	no	—
47	#11724784	—	4	2	no	—
48	C₂²×C₁₀₉	—	8	2	no	—
49	C₂²×C₁₀₉	—	8	1	no	—
50	C₂²×C₁₀₉	#120	1	1	yes	characteristic
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	—	4	2	no	—
53	C₄×C₁₀₉	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₀₉	#137	1	2	yes	—
55	Dic₁₀₉	—	2	4	no	—
56	Dic₁₀₉	—	4	2	no	—
57	C₂×C₁₀₉	—	16	1	no	—
58	C₂×C₁₀₉	#448	1	2	yes	—
59	C₂×C₁₀₉	#453	1	1	yes	characteristic
60	#1623	—	109	1	no	maximal, Sylow
61	C₁₀₉	#1623	1	1	yes	characteristic, complemented, Sylow
62	Q₈²	—	109	1	no	—
63	#492	D₁₀₉	1	1	yes	characteristic
64	#460	—	109	2	no	—
65	#333	—	109	1	no	—
66	Q₈⋊C₈	—	109	2	no	—
67	C₂×SD₁₆	—	2	2	no	—
68	C₂×SD₁₆	—	218	2	no	—
69	C₄⋊Q₈	—	218	2	no	—
70	C₄⋊Q₈	—	218	2	no	—
71	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
72	#127	C₂×D₁₀₉	1	1	yes	characteristic
73	C₄×Q₈	—	109	2	no	—
74	C₄×Q₈	—	218	2	no	—
75	C₄⋊C₈	—	109	2	no	—
76	#102	—	218	2	no	—
77	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂×Q₈	—	109	2	no	—
79	C₂×Q₈	—	218	2	no	—
80	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	SD₁₆	—	4	4	no	—
84	SD₁₆	—	436	4	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	218	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	109	2	no	—
89	C₄⋊C₄	—	218	4	no	—
90	C₄⋊C₄	—	218	2	no	—
91	C₄²	—	218	2	no	—
92	C₄²	—	218	2	no	—
93	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₂³	—	4	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	872	2	no	—
97	Q₈	—	109	4	no	—
98	Q₈	—	218	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	4	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	—	218	2	no	—
105	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
107	C₈	—	2	2	no	—
108	C₈	—	436	2	no	—
109	C₂²	—	8	1	no	—
110	C₂²	—	8	2	no	—
111	C₂²	#16812667	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	436	2	no	—
115	C₄	—	218	4	no	—
116	C₄	SD₁₆×D₁₀₉	1	2	yes	—
117	C₄	#16812728	1	2	yes	—
118	C₂	—	16	1	no	—
119	C₂	#20295163	1	2	yes	—
120	C₂	#20295193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295756

C₂

yes

maximal, characteristic, complemented

#20295223

C₂

yes

maximal

#20294775

C₂

yes

maximal, characteristic, complemented

#20294758

C₂

yes

maximal, complemented

#492×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182017

—

#16812698

—

#16812697

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812664

—

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×Dic₂₁₈

—

C₂×Dic₂₁₈

D₄

yes

—

#13182017

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

D₄

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

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

—

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

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

SD₁₆

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×SD₁₆

yes

—

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#448

yes

—

C₂×C₁₀₉

#453

yes

characteristic

#1623

—

109

maximal, Sylow

C₁₀₉

#1623

yes

characteristic, complemented, Sylow

Q₈²

—

109

—

#492

D₁₀₉

yes

characteristic

#460

—

109

—

#333

—

109

—

Q₈⋊C₈

—

109

—

C₂×SD₁₆

—

C₂×SD₁₆

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

#127

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

C₄⋊C₈

—

109

—

#102

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

436

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

872

—

Q₈

—

109

—

Q₈

—

218

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

D₄×D₁₀₉

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

107

C₈

—

108

C₈

—

436

—

109

C₂²

—

110

C₂²

—

111

C₂²

#16812667

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

436

—

115

C₄

—

218

—

116

C₄

SD₁₆×D₁₀₉

yes

—

117

C₄

#16812728

yes

—

118

C₂

—

119

C₂

#20295163

yes

—

120

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
prime radical	#127×C₁₀₉	C₂²
hypercenter	#492	D₁₀₉
hypocenter	C₁₀₉	#1623
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

socle

C₂²×C₁₀₉

#120

prime radical

#127×C₁₀₉

C₂²

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #20295756	C₂	semidirect
3. #20294775	C₂	semidirect
4. #20294758	C₂	semidirect
45. Dic₂₁₈	SD₁₆	semidirect
61. C₁₀₉	#1623	semidirect

H ≅ G/N

type

1. #20295756

C₂

semidirect

3. #20294775

C₂

semidirect

4. #20294758

C₂

semidirect

45. Dic₂₁₈

SD₁₆

semidirect

61. C₁₀₉

#1623

semidirect

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

subgroup	index
#20295756	2
#20295223	2
#20294775	2
#20294758	2
#492×C₁₀₉	2
#1623	109

subgroup

index

#492×C₁₀₉

109

Subgroups — #28362108

Conjugacy classes of subgroups

Complements

Maximal subgroups