Group #28363853 — subgroups

Subgroups — #28363853

Order 13952.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×#16812623	C₂	1	4	yes	maximal
2	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
3	#20294892	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11724784	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
7	#16812623	—	2	8	no	—
8	#16812610	—	2	2	no	—
9	#16812606	—	2	2	no	—
10	C₂²×Q₈×C₁₀₉	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₀₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₀₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	—	2	2	no	—
16	C₂×#11724784	—	2	2	no	—
17	C₂×#11724784	—	2	2	no	—
18	C₂×#11724784	D₄	1	4	yes	—
19	C₄⋊Dic₁₀₉	—	2	4	no	—
20	C₂×Q₈×C₁₀₉	—	2	2	no	—
21	C₂×Q₈×C₁₀₉	—	2	4	no	—
22	C₂×Q₈×C₁₀₉	—	2	4	no	—
23	C₂×Q₈×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₀₉	—	2	2	no	—
25	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₀₉	—	2	8	no	—
28	C₄²×C₁₀₉	C₂×C₄	1	2	yes	—
29	C₄²×C₁₀₉	D₄	1	2	yes	—
30	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₁₀₉	—	2	8	no	—
32	Q₈×C₁₀₉	—	4	4	no	—
33	C₂×C₄×C₁₀₉	—	2	4	no	—
34	C₂×C₄×C₁₀₉	—	2	4	no	—
35	C₂×C₄×C₁₀₉	—	4	2	no	—
36	C₂×C₄×C₁₀₉	—	4	2	no	—
37	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	#41	1	2	yes	—
42	C₂×Dic₁₀₉	—	4	2	no	—
43	C₂×Dic₁₀₉	—	4	4	no	—
44	#11724784	—	2	4	no	—
45	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₀₉	#124	1	1	yes	characteristic
47	C₂²×C₁₀₉	#127	1	1	yes	characteristic
48	C₂²×C₁₀₉	#137	1	4	yes	—
49	C₄×C₁₀₉	—	2	4	no	—
50	C₄×C₁₀₉	—	4	4	no	—
51	C₄×C₁₀₉	C₄×D₄	1	2	yes	—
52	C₄×C₁₀₉	#121	1	2	yes	—
53	Dic₁₀₉	—	8	2	no	—
54	C₂×C₁₀₉	#390	1	1	yes	characteristic
55	C₂×C₁₀₉	#419	1	2	yes	—
56	C₂×C₁₀₉	#491	1	2	yes	—
57	C₂×C₁₀₉	#497	1	2	yes	—
58	#1960	—	109	1	no	maximal, Sylow
59	C₁₀₉	#1960	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic
61	C₂×#103	—	109	4	no	—
62	C₂×#97	—	109	1	no	—
63	#382	—	109	1	no	—
64	C₂²×Q₈	C₂×D₁₀₉	1	2	yes	—
65	C₂²×C₈	—	109	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₁₀₉	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	109	2	no	—
70	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
71	#103	—	218	8	no	—
72	#97	—	218	2	no	—
73	#95	—	218	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₀₉	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	218	2	no	—
80	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
82	C₂×C₈	—	109	4	no	—
83	C₂×C₈	—	218	2	no	—
84	C₂×C₈	—	218	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	218	4	no	—
87	C₄²	C₂×Dic₁₀₉	1	2	yes	—
88	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	436	2	no	—
97	C₂×C₄	—	436	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₀₉	1	4	yes	—
99	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
102	C₂×C₄	#13182028	1	2	yes	—
103	C₈	—	218	4	no	—
104	C₂²	C₂×C₂²⋊Dic₁₀₉	1	1	yes	characteristic
105	C₂²	#16812686	1	1	yes	characteristic
106	C₂²	#16812695	1	1	yes	characteristic
107	C₂²	#16812730	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	872	2	no	—
111	C₄	D₄×Dic₁₀₉	1	2	yes	—
112	C₄	#16812670	1	2	yes	—
113	C₂	#20294929	1	1	yes	characteristic
114	C₂	#20295035	1	2	yes	—
115	C₂	#20295394	1	2	yes	—
116	C₂	#20295418	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812623

C₂

yes

maximal

C₂×#16812610

C₂

yes

maximal, characteristic

#20294892

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812623

—

#16812610

—

#16812606

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

—

C₂×#11724784

—

C₂×#11724784

—

C₂×#11724784

D₄

yes

—

C₄⋊Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

C₂²×C₁₀₉

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

—

C₄×C₁₀₉

#121

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

#390

yes

characteristic

C₂×C₁₀₉

#419

yes

—

C₂×C₁₀₉

#491

yes

—

C₂×C₁₀₉

#497

yes

—

#1960

—

109

maximal, Sylow

C₁₀₉

#1960

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₁₀₉

yes

characteristic

C₂×#103

—

109

—

C₂×#97

—

109

—

#382

—

109

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₀₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#103

—

218

—

#97

—

218

—

#95

—

218

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₀₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄²

C₂×Dic₁₀₉

yes

—

C₄²

C₂²⋊D₁₀₉

yes

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

436

—

C₂×C₄

C₂²⋊Dic₁₀₉

yes

—

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

100

C₂×C₄

D₄×D₁₀₉

yes

—

101

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

102

C₂×C₄

#13182028

yes

—

103

C₈

—

218

—

104

C₂²

C₂×C₂²⋊Dic₁₀₉

yes

characteristic

105

C₂²

#16812686

yes

characteristic

106

C₂²

#16812695

yes

characteristic

107

C₂²

#16812730

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

872

—

111

C₄

D₄×Dic₁₀₉

yes

—

112

C₄

#16812670

yes

—

113

C₂

#20294929

yes

characteristic

114

C₂

#20295035

yes

—

115

C₂

#20295394

yes

—

116

C₂

#20295418

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	C₂×C₄⋊Q₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
cosocle	C₂²×C₄×C₁₀₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#1960
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

C₂×C₄⋊Q₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

cosocle

C₂²×C₄×C₁₀₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₀₉

hypocenter

C₁₀₉

#1960

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₀₉	C₄	semidirect
59. C₁₀₉	#1960	semidirect
67. C₄⋊Q₈	Dic₁₀₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₀₉

C₄

semidirect

59. C₁₀₉

#1960

semidirect

67. C₄⋊Q₈

Dic₁₀₉

semidirect

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

subgroup	index
C₂×#16812623	2
C₂×#16812610	2
#20294892	2
C₂×C₄⋊Q₈×C₁₀₉	2
#1960	109

subgroup

index

C₂×C₄⋊Q₈×C₁₀₉

109

Subgroups — #28363853

Conjugacy classes of subgroups

Complements

Maximal subgroups