Group #28362170 — subgroups

Subgroups — #28362170

Order 13952.

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

normal subgroups only

This group has 206 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
2	#20295318	C₂	1	2	yes	maximal
3	#20294775	C₂	1	1	yes	maximal, characteristic
4	#20294758	C₂	1	2	yes	maximal
5	#494×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812698	—	2	2	no	—
7	#16812697	—	2	2	no	—
8	C₄×Dic₂₁₈	C₂²	1	2	yes	—
9	#16812664	—	2	2	no	—
10	#16812632	—	2	2	no	—
11	C₄⋊#11724784	C₂²	1	2	yes	—
12	#16812622	—	2	2	no	—
13	C₂×Q₁₆×C₁₀₉	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×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	C₂×#11724784	—	2	2	no	—
20	C₄⋊Dic₁₀₉	D₄	1	2	yes	—
21	#13182008	—	2	4	no	—
22	#13182008	—	2	2	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	—	2	2	no	—
25	C₂×Q₈×C₁₀₉	—	2	1	no	—
26	C₂×Q₈×C₁₀₉	D₄	1	2	yes	—
27	Q₁₆×C₁₀₉	—	4	4	no	—
28	C₂×C₈×C₁₀₉	—	2	1	no	—
29	C₄⋊C₄×C₁₀₉	—	2	2	no	—
30	C₄⋊C₄×C₁₀₉	—	2	2	no	—
31	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
32	Q₈×C₁₀₉	—	4	2	no	—
33	Q₈×C₁₀₉	—	4	2	no	—
34	C₂×C₄×C₁₀₉	—	2	2	no	—
35	C₂×C₄×C₁₀₉	—	4	1	no	—
36	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₁₀₉	—	2	2	no	—
39	C₂×Dic₁₀₉	—	2	2	no	—
40	Dic₂₁₈	—	2	2	no	—
41	Dic₂₁₈	—	8	2	no	—
42	Dic₂₁₈	Q₁₆	1	4	yes	complemented
43	C₈×C₁₀₉	—	2	2	no	—
44	#11724784	—	4	2	no	—
45	C₂²×C₁₀₉	#120	1	1	yes	characteristic
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	—	4	2	no	—
48	C₄×C₁₀₉	—	8	1	no	—
49	C₄×C₁₀₉	C₂×Q₁₆	1	2	yes	—
50	C₄×C₁₀₉	#137	1	2	yes	—
51	Dic₁₀₉	—	2	4	no	—
52	Dic₁₀₉	—	4	2	no	—
53	C₂×C₁₀₉	#451	1	2	yes	—
54	C₂×C₁₀₉	#453	1	1	yes	characteristic
55	#1633	—	109	1	no	maximal, Sylow
56	C₁₀₉	#1633	1	1	yes	characteristic, complemented, Sylow
57	Q₈²	—	109	1	no	—
58	#494	D₁₀₉	1	1	yes	characteristic
59	#477	—	109	2	no	—
60	#333	—	109	1	no	—
61	Q₈⋊C₈	—	109	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	218	2	no	—
64	C₄⋊Q₈	—	218	2	no	—
65	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₁₀₉	1	1	yes	characteristic
67	C₄×Q₈	—	109	2	no	—
68	C₄×Q₈	—	218	2	no	—
69	#107	—	218	2	no	—
70	C₄⋊C₈	—	109	2	no	—
71	#103	—	218	2	no	—
72	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	109	2	no	—
75	C₂×Q₈	—	218	2	no	—
76	C₂×Q₈	C₂²⋊D₁₀₉	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	218	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	109	2	no	—
83	C₄⋊C₄	—	218	4	no	—
84	C₄⋊C₄	—	218	2	no	—
85	C₄²	—	218	2	no	—
86	C₄²	—	218	2	no	—
87	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	872	2	no	—
91	Q₈	—	109	4	no	—
92	Q₈	—	218	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	218	2	no	—
96	C₂×C₄	—	218	2	no	—
97	C₂×C₄	D₄×D₁₀₉	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	436	2	no	—
101	C₂²	#16812667	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	436	2	no	—
106	C₄	—	218	4	no	—
107	C₄	Q₁₆×D₁₀₉	1	2	yes	—
108	C₄	#16812731	1	2	yes	—
109	C₂	#20295180	1	2	yes	—
110	C₂	#20295193	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295756

C₂

yes

maximal, characteristic

#20295318

C₂

yes

maximal

#20294775

C₂

yes

maximal, characteristic

#20294758

C₂

yes

maximal

#494×C₁₀₉

C₂

yes

maximal, characteristic

#16812698

—

#16812697

—

C₄×Dic₂₁₈

C₂²

yes

—

#16812664

—

#16812632

—

C₄⋊#11724784

C₂²

yes

—

#16812622

—

C₂×Q₁₆×C₁₀₉

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

D₄

yes

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

D₄

yes

—

#13182008

—

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

D₄

yes

—

Q₁₆×C₁₀₉

—

C₂×C₈×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

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

Q₁₆

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

#120

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₁₆

yes

—

C₄×C₁₀₉

#137

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#451

yes

—

C₂×C₁₀₉

#453

yes

characteristic

#1633

—

109

maximal, Sylow

C₁₀₉

#1633

yes

characteristic, complemented, Sylow

Q₈²

—

109

—

#494

D₁₀₉

yes

characteristic

#477

—

109

—

#333

—

109

—

Q₈⋊C₈

—

109

—

C₂×Q₁₆

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₉

yes

characteristic

C₄×Q₈

—

109

—

C₄×Q₈

—

218

—

#107

—

218

—

C₄⋊C₈

—

109

—

#103

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂²⋊D₁₀₉

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

218

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

872

—

Q₈

—

109

—

Q₈

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₄×D₁₀₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₈

—

100

C₈

—

436

—

101

C₂²

#16812667

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

436

—

106

C₄

—

218

—

107

C₄

Q₁₆×D₁₀₉

yes

—

108

C₄

#16812731

yes

—

109

C₂

#20295180

yes

—

110

C₂

#20295193

yes

characteristic

	subgroup	G/H
center	C₂²	#16812667
commutator, cosocle	C₄²×C₁₀₉	C₂³
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#494×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	#120
hypercenter	#494	D₁₀₉
hypocenter	C₁₀₉	#1633
LCS	G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#16812667

commutator, cosocle

C₄²×C₁₀₉

C₂³

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#494×C₁₀₉

C₂

C₂²×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₄²×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
42. Dic₂₁₈	Q₁₆	semidirect
56. C₁₀₉	#1633	semidirect

H ≅ G/N

type

42. Dic₂₁₈

Q₁₆

semidirect

56. C₁₀₉

#1633

semidirect

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

subgroup	index
#20295756	2
#20295318	2
#20294775	2
#20294758	2
#494×C₁₀₉	2
#1633	109

subgroup

index

#494×C₁₀₉

109

Subgroups — #28362170

Conjugacy classes of subgroups

Complements

Maximal subgroups