Group #28379422 — subgroups

Subgroups — #28379422

Order 13952.

144 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20296084	C₂	1	2	yes	maximal, complemented
2	#20295948	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295924	C₂	1	2	yes	maximal, complemented
4	#20295915	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295143	C₂	1	1	yes	maximal, characteristic
6	#16812773	—	2	2	no	—
7	#16812773	C₂²	1	2	yes	—
8	C₂×#13182032	—	2	2	no	—
9	C₂×#13182032	C₂²	1	1	yes	characteristic
10	C₂×#13182032	C₂²	1	1	yes	characteristic
11	#16812706	C₂²	1	2	yes	—
12	#16812635	C₄	1	1	yes	characteristic
13	#16812626	C₄	1	2	yes	—
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#135×C₁₀₉	C₄	1	1	yes	characteristic
16	#13182032	—	2	2	no	—
17	#13182032	—	2	1	no	—
18	#13182032	D₄	1	2	yes	—
19	#13182013	—	2	2	no	—
20	#13182013	C₂×C₄	1	2	yes	—
21	C₂×#11724784	—	2	2	no	—
22	C₂×#11724784	C₂³	1	1	yes	characteristic
23	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
24	#41×C₁₀₉	C₂×C₄	1	2	yes	—
25	Q₁₆×C₁₀₉	C₈	1	1	yes	characteristic
26	SD₁₆×C₁₀₉	C₈	1	2	yes	—
27	D₈×C₁₀₉	C₈	1	1	yes	characteristic
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	Q₈×C₁₀₉	C₂×C₈	1	2	yes	—
30	D₄×C₁₀₉	—	2	2	no	—
31	D₄×C₁₀₉	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₀₉	—	2	2	no	—
33	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₁₀₉	—	2	1	no	—
35	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
36	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
37	#11724784	—	2	2	no	—
38	#11724784	C₂×D₄	1	1	yes	characteristic
39	#11724784	#41	1	1	yes	characteristic
40	C₂²×C₁₀₉	—	2	2	no	—
41	C₂²×C₁₀₉	#131	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	2	no	—
43	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
44	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
45	Dic₁₀₉	—	4	1	no	—
46	C₂×C₁₀₉	—	2	1	no	—
47	C₂×C₁₀₉	—	4	2	no	—
48	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
49	#2171	—	109	1	no	maximal, Sylow
50	C₁₀₉	#2171	1	1	yes	characteristic, complemented, Sylow
51	#504	—	109	2	no	—
52	#443	—	109	1	no	—
53	#364	—	109	1	no	—
54	#350	—	109	2	no	—
55	#346	—	109	1	no	—
56	#135	#89137	1	1	yes	characteristic
57	#131	—	109	2	no	—
58	M₃₂	—	109	2	no	—
59	M₃₂	—	218	2	no	—
60	C₂×C₁₆	—	109	1	no	—
61	C₂×C₁₆	—	109	1	no	—
62	C₂×C₁₆	—	218	2	no	—
63	#108	—	109	1	no	—
64	#104	—	109	2	no	—
65	C₄×C₈	—	109	1	no	—
66	#41	C₂×#89137	1	2	yes	—
67	Q₁₆	#11724786	1	1	yes	characteristic
68	SD₁₆	#11724786	1	2	yes	—
69	D₈	#11724786	1	1	yes	characteristic
70	M₁₆	—	109	2	no	—
71	M₁₆	—	218	2	no	—
72	C₂×C₈	—	109	1	no	—
73	C₂×C₈	—	218	2	no	—
74	C₂×C₈	C₂×#89137	1	1	yes	characteristic
75	C₄²	—	109	1	no	—
76	C₁₆	—	109	2	no	—
77	C₁₆	—	218	1	no	—
78	C₁₆	—	218	2	no	—
79	Q₈	C₂×#11724786	1	2	yes	—
80	D₄	—	2	2	no	—
81	D₄	C₂×#11724786	1	2	yes	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	218	1	no	—
84	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
85	C₈	—	109	1	no	—
86	C₈	—	109	1	no	—
87	C₈	—	218	2	no	—
88	C₈	C₂×#11724786	1	1	yes	characteristic
89	C₈	C₂×#11724786	1	1	yes	characteristic
90	C₂²	—	2	2	no	—
91	C₂²	#16812786	1	1	yes	characteristic
92	C₄	—	2	2	no	—
93	C₄	—	436	1	no	—
94	C₄	D₄×#89137	1	1	yes	characteristic
95	C₄	C₂²×#11724786	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	4	2	no	—
98	C₂	D₄×#11724786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296084

C₂

yes

maximal, complemented

#20295948

C₂

yes

maximal, characteristic, complemented

#20295924

C₂

yes

maximal, complemented

#20295915

C₂

yes

maximal, characteristic, complemented

#20295143

C₂

yes

maximal, characteristic

#16812773

—

#16812773

C₂²

yes

—

C₂×#13182032

—

C₂×#13182032

C₂²

yes

characteristic

C₂×#13182032

C₂²

yes

characteristic

#16812706

C₂²

yes

—

#16812635

C₄

yes

characteristic

#16812626

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

#135×C₁₀₉

C₄

yes

—

—

D₄

yes

—

#13182013

—

#13182013

C₂×C₄

yes

—

C₂×#11724784

—

C₂×#11724784

C₂³

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

#41×C₁₀₉

C₂×C₄

yes

—

Q₁₆×C₁₀₉

C₈

yes

characteristic

SD₁₆×C₁₀₉

C₈

yes

—

D₈×C₁₀₉

C₈

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

Q₈×C₁₀₉

C₂×C₈

yes

—

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₂×C₈

yes

characteristic

C₈×C₁₀₉

C₂×C₈

yes

characteristic

#11724784

—

#11724784

C₂×D₄

yes

yes

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#131

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₈×D₄

yes

characteristic

#2171

—

109

maximal, Sylow

C₁₀₉

#2171

yes

characteristic, complemented, Sylow

#504

—

109

—

#443

—

109

—

#364

—

109

—

#350

—

109

—

#346

—

109

—

yes

—

109

—

M₃₂

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

#108

—

109

—

#104

—

109

—

C₄×C₈

—

109

—

#41

C₂×#89137

yes

—

Q₁₆

#11724786

yes

characteristic

SD₁₆

#11724786

yes

—

D₈

#11724786

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄²

—

109

—

C₁₆

—

109

—

C₁₆

—

218

—

C₁₆

—

218

—

Q₈

C₂×#11724786

yes

—

D₄

—

D₄

C₂×#11724786

yes

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₈

yes

C₈

yes

C₂²

—

C₂²

#16812786

yes

characteristic

C₄

—

C₄

—

436

—

C₄

yes

C₄

yes

C₂

—

C₂

—

C₂

D₄×#11724786

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting, prime radical	#135×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₈×D₄
cosocle	C₂×#11724784	C₂³
hypercenter	#135	#89137
hypocenter	C₁₀₉	#2171
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₄

D₄×#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting, prime radical

#135×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₈×D₄

C₂³

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #20296084	C₂	semidirect
2. #20295948	C₂	semidirect
3. #20295924	C₂	semidirect
4. #20295915	C₂	semidirect
50. C₁₀₉	#2171	semidirect

H ≅ G/N

type

1. #20296084

C₂

semidirect

2. #20295948

C₂

semidirect

3. #20295924

C₂

semidirect

4. #20295915

C₂

semidirect

50. C₁₀₉

#2171

semidirect

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

subgroup	index
#20296084	2
#20295948	2
#20295924	2
#20295915	2
#20295143	2
#2171	109

subgroup

index

109

Subgroups — #28379422

Conjugacy classes of subgroups

Complements

Maximal subgroups