Group #28378566 — subgroups

Subgroups — #28378566

Order 13952.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#20296049	C₂	1	2	yes	maximal
2	#20295886	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812766	C₂²	1	4	yes	—
5	#16812756	—	2	2	no	—
6	#16812754	—	2	2	no	—
7	C₂×Q₈×D₁₀₉	C₄	1	2	yes	—
8	#16812699	C₄	1	1	yes	characteristic
9	#16812698	—	2	2	no	—
10	#16812697	C₂²	1	2	yes	—
11	C₄⋊C₄×D₁₀₉	—	2	2	no	—
12	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₁₀₉	C₄	1	1	yes	characteristic
14	C₂×#11724786	—	2	4	no	—
15	C₂×Dic₂₁₈	—	2	2	no	—
16	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
17	Q₈×D₁₀₉	—	4	4	no	—
18	C₂×C₄×D₁₀₉	—	2	2	no	—
19	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₉	D₄	1	2	yes	—
21	C₄⋊Dic₁₀₉	—	2	2	no	—
22	#13182008	—	2	4	no	—
23	C₄×Dic₁₀₉	D₄	1	2	yes	—
24	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₄⋊C₄×C₁₀₉	—	2	2	no	—
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₁₀₉	—	4	2	no	—
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	—	2	2	no	—
34	C₂×Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂×D₄	1	2	yes	—
36	C₄×D₁₀₉	—	2	2	no	—
37	C₄×D₁₀₉	—	2	4	no	—
38	C₄×D₁₀₉	—	4	4	no	—
39	Dic₂₁₈	—	4	4	no	—
40	Dic₂₁₈	—	4	2	no	—
41	#11724786	—	4	4	no	—
42	C₂²×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×D₁₀₉	—	2	2	no	—
44	C₂×D₁₀₉	#100	1	2	yes	—
45	C₄×C₁₀₉	—	2	1	no	—
46	C₄×C₁₀₉	—	2	2	no	—
47	C₄×C₁₀₉	—	4	2	no	—
48	Dic₁₀₉	—	2	1	no	—
49	Dic₁₀₉	—	4	2	no	—
50	Dic₁₀₉	#137	1	4	yes	—
51	C₂×C₁₀₉	C₂×#100	1	1	yes	characteristic
52	C₂×C₁₀₉	#419	1	2	yes	—
53	D₁₀₉	—	2	2	no	—
54	#1558	—	109	1	no	maximal, Sylow
55	C₁₀₉	#1558	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₄⋊Q₈	—	109	1	no	—
57	#432	—	109	2	no	—
58	#333	—	109	4	no	—
59	C₂²×Q₈	—	109	2	no	—
60	C₄⋊Q₈	—	109	2	no	—
61	C₄⋊Q₈	—	109	1	no	—
62	C₄⋊Q₈	—	218	2	no	—
63	C₄⋊Q₈	#89137	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	218	2	no	—
65	C₂×C₄²	—	109	1	no	—
66	C₄⋊C₈	—	109	4	no	—
67	C₂²⋊C₈	—	218	2	no	—
68	#97	—	218	2	no	—
69	C₂×Q₈	—	109	2	no	—
70	C₂×Q₈	—	436	4	no	—
71	C₂×Q₈	—	218	2	no	—
72	C₂×Q₈	C₂×#89137	1	2	yes	—
73	C₂²×C₄	—	109	2	no	—
74	C₂²×C₄	—	109	1	no	—
75	C₂²×C₄	—	218	2	no	—
76	C₂×C₈	—	218	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	218	4	no	—
79	C₄⋊C₄	—	218	2	no	—
80	C₄²	—	109	2	no	—
81	C₄²	—	109	1	no	—
82	C₄²	C₂×#89137	1	1	yes	characteristic
83	C₂³	—	109	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	436	2	no	—
86	Q₈	—	436	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	109	2	no	—
89	C₂×C₄	—	109	1	no	—
90	C₂×C₄	—	436	4	no	—
91	C₂×C₄	—	218	4	no	—
92	C₂×C₄	—	218	2	no	—
93	C₂×C₄	—	218	2	no	—
94	C₂×C₄	C₂²⋊#89137	1	2	yes	—
95	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
96	C₈	—	436	4	no	—
97	C₂²	—	109	2	no	—
98	C₂²	—	218	2	no	—
99	C₂²	C₂×C₂²⋊#89137	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	109	4	no	—
104	C₄	—	436	2	no	—
105	C₄	—	218	1	no	—
106	C₂	—	218	2	no	—
107	C₂	D₁₀₉⋊#100	1	1	yes	characteristic
108	C₂	#20296019	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20296049

C₂

yes

maximal

#20295886

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812766

C₂²

yes

—

#16812756

—

#16812754

—

C₂×Q₈×D₁₀₉

C₄

yes

—

#16812699

C₄

yes

characteristic

#16812698

—

#16812697

C₂²

yes

—

C₄⋊C₄×D₁₀₉

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₉

D₄

yes

—

C₄⋊Dic₁₀₉

—

#13182008

—

C₄×Dic₁₀₉

D₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²⋊C₄

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×D₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

C₂²×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

#100

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#137

yes

—

C₂×C₁₀₉

C₂×#100

yes

characteristic

C₂×C₁₀₉

#419

yes

—

D₁₀₉

—

#1558

—

109

maximal, Sylow

C₁₀₉

#1558

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

109

—

#432

—

109

—

#333

—

109

—

C₂²×Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

—

218

—

C₄⋊Q₈

#89137

yes

characteristic

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

218

—

#97

—

218

—

C₂×Q₈

—

109

—

C₂×Q₈

—

436

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×#89137

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

Q₈

—

Q₈

—

436

—

Q₈

—

436

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²⋊#89137

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

436

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₂×C₂²⋊#89137

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

109

—

104

C₄

—

436

—

105

C₄

—

218

—

106

C₂

—

218

—

107

C₂

D₁₀₉⋊#100

yes

characteristic

108

C₂

#20296019

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄⋊Q₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₀₉	C₂²⋊C₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄⋊Q₈	#89137
hypocenter	C₁₀₉	#1558
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄⋊Q₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₀₉

C₂²⋊C₄

cosocle

C₄×Dic₁₀₉

C₂³

C₄⋊Q₈

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #20295886	C₂	semidirect
55. C₁₀₉	#1558	semidirect

H ≅ G/N

type

2. #20295886

C₂

semidirect

55. C₁₀₉

#1558

semidirect

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

subgroup	index
#20296049	2
#20295886	2
C₄⋊Q₈×D₁₀₉	2
#1558	109

subgroup

index

#20296049

#20295886

C₄⋊Q₈×D₁₀₉

#1558

109

Subgroups — #28378566

Conjugacy classes of subgroups

Complements

Maximal subgroups