Group #28379424 — subgroups

Subgroups — #28379424

Order 13952.

144 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 58 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	#20296085	C₂	1	2	yes	maximal, complemented
2	#20295948	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295925	C₂	1	2	yes	maximal, complemented
4	#20295915	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295142	C₂	1	1	yes	maximal, characteristic
6	#16812774	—	2	2	no	—
7	#16812773	C₂²	1	2	yes	—
8	D₁₀₉⋊C₁₆	—	2	2	no	—
9	C₂×#13182032	C₂²	1	1	yes	characteristic
10	C₂×#13182032	C₂²	1	1	yes	characteristic
11	#16812721	C₄	1	1	yes	characteristic
12	#16812707	C₂²	1	2	yes	—
13	#16812625	C₄	1	2	yes	—
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#108×C₁₀₉	C₄	1	1	yes	characteristic
16	#13182032	—	2	2	no	—
17	#13182032	—	2	1	no	—
18	#13182032	D₄	1	2	yes	—
19	#13182029	C₂×C₄	1	2	yes	—
20	Dic₄₃₆	C₈	1	1	yes	characteristic
21	#13182018	C₈	1	2	yes	—
22	D₈₇₂	C₈	1	1	yes	characteristic
23	#13182012	—	2	2	no	—
24	C₂×#11724784	C₂³	1	1	yes	characteristic
25	C₈×D₁₀₉	—	2	2	no	—
26	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
27	M₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂²⋊D₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	—	2	1	no	—
32	D₄₃₆	C₂×C₈	1	2	yes	—
33	C₄×D₁₀₉	—	2	2	no	—
34	Dic₂₁₈	C₂×C₈	1	2	yes	—
35	C₈×C₁₀₉	—	2	2	no	—
36	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
37	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
38	#11724784	C₂×D₄	1	1	yes	characteristic
39	#11724784	#41	1	1	yes	characteristic
40	C₂²×C₁₀₉	#131	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	2	no	—
42	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	—	4	1	no	—
46	C₂×C₁₀₉	—	2	1	no	—
47	C₂×C₁₀₉	C₈×D₄	1	1	yes	characteristic
48	D₁₀₉	—	4	2	no	—
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	—	109	1	no	—
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	#89137	1	1	yes	characteristic
64	#104	—	109	2	no	—
65	C₄×C₈	—	109	1	no	—
66	#41	—	109	2	no	—
67	Q₁₆	—	109	1	no	—
68	SD₁₆	—	109	2	no	—
69	D₈	—	109	1	no	—
70	M₁₆	—	218	2	no	—
71	M₁₆	C₂×#89137	1	2	yes	—
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	—	109	2	no	—
80	D₄	—	109	2	no	—
81	D₄	—	218	2	no	—
82	C₂×C₄	—	218	1	no	—
83	C₂×C₄	—	218	2	no	—
84	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	109	1	no	—
87	C₈	—	109	1	no	—
88	C₈	D₁₀₉⋊C₈	1	1	yes	characteristic
89	C₈	D₁₀₉⋊C₈	1	1	yes	characteristic
90	C₂²	—	218	2	no	—
91	C₂²	#16812787	1	1	yes	characteristic
92	C₄	—	436	1	no	—
93	C₄	—	218	2	no	—
94	C₄	D₄×#89137	1	1	yes	characteristic
95	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	436	2	no	—
98	C₂	#20296054	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296085

C₂

yes

maximal, complemented

#20295948

C₂

yes

maximal, characteristic, complemented

#20295925

C₂

yes

maximal, complemented

#20295915

C₂

yes

maximal, characteristic, complemented

#20295142

C₂

yes

maximal, characteristic

#16812774

—

#16812773

C₂²

yes

—

D₁₀₉⋊C₁₆

—

C₂×#13182032

C₂²

yes

characteristic

C₂×#13182032

C₂²

yes

characteristic

#16812721

C₄

yes

characteristic

#16812707

C₂²

yes

—

#16812625

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

#108×C₁₀₉

C₄

yes

—

—

D₄

yes

—

#13182029

C₂×C₄

yes

—

Dic₄₃₆

C₈

yes

characteristic

#13182018

C₈

yes

—

D₈₇₂

C₈

yes

characteristic

#13182012

—

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

M₁₆×C₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

C₂×C₈

yes

—

C₄×D₁₀₉

—

Dic₂₁₈

C₂×C₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×C₈

yes

characteristic

C₈×C₁₀₉

C₂×C₈

yes

characteristic

#11724784

C₂×D₄

yes

yes

C₂²×C₁₀₉

yes

C₂×D₁₀₉

—

C₄×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₈×D₄

yes

characteristic

D₁₀₉

—

#2171

—

109

maximal, Sylow

C₁₀₉

#2171

yes

characteristic, complemented, Sylow

#504

—

109

—

#443

—

109

—

#364

—

109

—

#350

—

109

—

#346

—

109

—

#135

—

109

—

#131

—

109

—

M₃₂

—

109

—

M₃₂

—

218

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

218

—

yes

—

109

—

C₄×C₈

—

109

—

#41

—

109

—

Q₁₆

—

109

—

SD₁₆

—

109

—

D₈

—

109

—

M₁₆

—

218

—

M₁₆

C₂×#89137

yes

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄²

—

109

—

C₁₆

—

109

—

C₁₆

—

218

—

C₁₆

—

218

—

Q₈

—

109

—

D₄

—

109

—

D₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

109

—

C₈

D₁₀₉⋊C₈

yes

characteristic

C₈

D₁₀₉⋊C₈

yes

characteristic

C₂²

—

218

—

C₂²

#16812787

yes

characteristic

C₄

—

436

—

C₄

—

218

—

C₄

D₄×#89137

yes

characteristic

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

C₂

—

C₂

—

436

—

C₂

#20296054

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#89137
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	#108×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₈×D₄
prime radical	#16812721	C₄
cosocle	C₂×#11724784	C₂³
hypercenter	#108	#89137
hypocenter	C₁₀₉	#2171
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

#108×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₈×D₄

C₄

C₂³

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
1. #20296085	C₂	semidirect
2. #20295948	C₂	semidirect
3. #20295925	C₂	semidirect
4. #20295915	C₂	semidirect
50. C₁₀₉	#2171	semidirect

H ≅ G/N

type

1. #20296085

C₂

semidirect

2. #20295948

C₂

semidirect

3. #20295925

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
#20296085	2
#20295948	2
#20295925	2
#20295915	2
#20295142	2
#2171	109

subgroup

index

109

Subgroups — #28379424

Conjugacy classes of subgroups

Complements

Maximal subgroups