Group #28364014 — subgroups

Subgroups — #28364014

Order 13952.

244 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20295050	C₂	1	1	yes	maximal, characteristic
2	#20294872	C₂	1	1	yes	maximal, characteristic
3	#20294785	C₂	1	4	yes	maximal
4	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₀₉	C₂²	1	2	yes	—
7	C₄⋊#11724784	—	2	2	no	—
8	#16812606	—	2	2	no	—
9	C₄⋊Q₈×C₁₀₉	C₄	1	2	yes	complemented
10	#125×C₁₀₉	C₄	1	2	yes	complemented
11	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
12	C₄×Q₈×C₁₀₉	—	2	2	no	—
13	#117×C₁₀₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	—	2	2	no	—
16	#13182013	—	2	4	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₄×Dic₁₀₉	—	2	2	no	—
19	C₄×Dic₁₀₉	Q₈	1	4	yes	complemented
20	C₂×Q₈×C₁₀₉	—	2	2	no	—
21	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₀₉	—	2	4	no	—
24	C₄⋊C₄×C₁₀₉	—	2	2	no	—
25	C₄⋊C₄×C₁₀₉	C₂×C₄	1	4	yes	—
26	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄²×C₁₀₉	—	2	2	no	—
28	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₁₀₉	D₄	1	2	yes	—
31	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₁₀₉	—	4	4	no	—
33	C₂×C₄×C₁₀₉	—	2	4	no	—
34	C₂×C₄×C₁₀₉	—	4	2	no	—
35	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
36	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₁₀₉	#41	1	2	yes	—
41	C₂×Dic₁₀₉	—	2	4	no	—
42	C₂×Dic₁₀₉	—	4	4	no	—
43	#11724784	—	4	2	no	—
44	C₂²×C₁₀₉	—	2	2	no	—
45	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	—	2	2	no	—
49	C₄×C₁₀₉	—	4	4	no	—
50	C₄×C₁₀₉	C₄×Q₈	1	2	yes	—
51	C₄×C₁₀₉	#124	1	1	yes	characteristic
52	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₀₉	—	4	4	no	—
54	C₂×C₁₀₉	—	2	2	no	—
55	C₂×C₁₀₉	#391	1	1	yes	characteristic
56	C₂×C₁₀₉	#421	1	2	yes	—
57	#1995	—	109	1	no	maximal, Sylow
58	C₁₀₉	#1995	1	1	yes	characteristic, complemented, Sylow
59	#533	D₁₀₉	1	1	yes	characteristic
60	#423	—	109	1	no	—
61	#376	—	109	1	no	—
62	#339	—	109	4	no	—
63	C₂×M₁₆	—	109	2	no	—
64	C₄⋊Q₈	Dic₁₀₉	1	2	yes	complemented
65	#125	Dic₁₀₉	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₁₀₉	1	2	yes	—
69	C₂×C₄²	—	109	2	no	—
70	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
71	C₄⋊C₈	—	218	2	no	—
72	#95	—	218	2	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂²×C₄	—	218	2	no	—
75	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
76	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
77	M₁₆	—	218	4	no	—
78	C₂×C₈	—	218	2	no	—
79	C₄⋊C₄	—	2	4	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	C₂×Dic₁₀₉	1	4	yes	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	109	4	no	—
85	C₄²	—	218	2	no	—
86	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
87	C₄²	C₂×Dic₁₀₉	1	1	yes	characteristic
88	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
90	Q₈	—	4	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	436	4	no	—
94	C₂×C₄	—	218	4	no	—
95	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
96	C₂×C₄	C₂²⋊Dic₁₀₉	1	2	yes	—
97	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
100	C₂×C₄	#13182030	1	2	yes	—
101	C₈	—	436	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
104	C₂²	#16812675	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	436	4	no	—
109	C₄	Q₈×Dic₁₀₉	1	2	yes	—
110	C₄	#16812688	1	1	yes	characteristic
111	C₄	#16812697	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#20294933	1	1	yes	characteristic
114	C₂	#20295043	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295050

C₂

yes

maximal, characteristic

#20294872

C₂

yes

maximal, characteristic

#20294785

C₂

yes

maximal

#533×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

C₄⋊#11724784

—

#16812606

—

C₄⋊Q₈×C₁₀₉

C₄

yes

complemented

#125×C₁₀₉

C₄

yes

complemented

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

#13182013

—

C₂×#11724784

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

Q₈

yes

complemented

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×Q₈

yes

—

C₄×C₁₀₉

#124

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#391

yes

characteristic

C₂×C₁₀₉

#421

yes

—

#1995

—

109

maximal, Sylow

C₁₀₉

#1995

yes

characteristic, complemented, Sylow

#533

D₁₀₉

yes

characteristic

#423

—

109

—

#376

—

109

—

#339

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

Dic₁₀₉

yes

complemented

#125

Dic₁₀₉

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

C₄⋊C₈

—

218

—

#95

—

218

—

C₂×Q₈

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₄²

C₂×Dic₁₀₉

yes

characteristic

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

Q₈×D₁₀₉

yes

—

100

C₂×C₄

#13182030

yes

—

101

C₈

—

436

—

102

C₂²

—

103

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

104

C₂²

#16812675

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

436

—

109

C₄

Q₈×Dic₁₀₉

yes

—

110

C₄

#16812688

yes

characteristic

111

C₄

#16812697

yes

characteristic

112

C₂

—

113

C₂

#20294933

yes

characteristic

114

C₂

#20295043

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#533×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#1995
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#533×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
9. C₄⋊Q₈×C₁₀₉	C₄	semidirect
10. #125×C₁₀₉	C₄	semidirect
19. C₄×Dic₁₀₉	Q₈	semidirect
58. C₁₀₉	#1995	semidirect
64. C₄⋊Q₈	Dic₁₀₉	semidirect
65. #125	Dic₁₀₉	semidirect

H ≅ G/N

type

9. C₄⋊Q₈×C₁₀₉

C₄

semidirect

10. #125×C₁₀₉

C₄

semidirect

19. C₄×Dic₁₀₉

Q₈

semidirect

58. C₁₀₉

#1995

semidirect

64. C₄⋊Q₈

Dic₁₀₉

semidirect

65. #125

Dic₁₀₉

semidirect

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

subgroup	index
#20295050	2
#20294872	2
#20294785	2
#533×C₁₀₉	2
#1995	109

subgroup

index

#533×C₁₀₉

109

Subgroups — #28364014

Conjugacy classes of subgroups

Complements

Maximal subgroups