Group #28379238 — subgroups

Subgroups — #28379238

Order 13952.

196 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 118 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×#16812773	C₂	1	2	yes	maximal
2	#20295915	C₂	1	2	yes	maximal, complemented
3	#20295912	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812773	C₄	1	8	yes	complemented
6	C₂×#13182032	C₂²	1	4	yes	—
7	C₂²×#11724784	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₉	C₄	1	1	yes	characteristic
9	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
10	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
11	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₄	1	1	yes	characteristic
14	#13182032	C₂×C₄	1	8	yes	complemented
15	C₂²×Dic₁₀₉	C₈	1	2	yes	—
16	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
17	C₂×#11724784	C₂×C₄	1	4	yes	—
18	C₂×#11724784	C₂³	1	1	yes	characteristic
19	C₄×Dic₁₀₉	C₈	1	2	yes	—
20	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₈	1	4	yes	—
24	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₁₀₉	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
28	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	2	4	no	—
32	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
33	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
34	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
35	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
36	#11724784	C₄²	1	2	yes	—
37	#11724784	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₀₉	—	2	2	no	—
39	C₂²×C₁₀₉	C₄×C₈	1	2	yes	—
40	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
41	C₄×C₁₀₉	C₄×C₈	1	2	yes	—
42	C₄×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	M₃₂	1	4	yes	complemented
46	C₂×C₁₀₉	—	2	2	no	—
47	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×C₁₀₉	C₂×M₃₂	1	2	yes	—
49	C₄×M₃₂	—	109	1	no	maximal, Sylow
50	C₁₀₉	C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₃₂	—	109	2	no	—
52	C₂×C₄×C₈	—	109	1	no	—
53	#346	—	109	2	no	—
54	C₄×C₁₆	—	109	2	no	—
55	C₂²×C₈	—	109	1	no	—
56	C₂²×C₈	#89137	1	1	yes	characteristic
57	C₂×C₄²	—	109	1	no	—
58	M₃₂	—	109	8	no	—
59	C₂×C₁₆	—	109	4	no	—
60	C₄×C₈	—	109	1	no	—
61	C₄×C₈	—	109	1	no	—
62	C₄×C₈	—	109	1	no	—
63	C₄×C₈	—	109	1	no	—
64	C₂²×C₄	—	109	2	no	—
65	C₂²×C₄	C₂×#89137	1	1	yes	characteristic
66	C₂×C₈	—	109	4	no	—
67	C₂×C₈	—	109	1	no	—
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	#11724786	1	4	yes	—
70	C₂×C₈	C₂×#89137	1	1	yes	characteristic
71	C₂×C₈	C₂×#89137	1	1	yes	characteristic
72	C₄²	—	109	2	no	—
73	C₄²	—	109	1	no	—
74	C₄²	—	109	1	no	—
75	C₁₆	—	109	8	no	—
76	C₂³	D₁₀₉⋊C₈	1	1	yes	characteristic
77	C₂×C₄	—	109	2	no	—
78	C₂×C₄	—	109	2	no	—
79	C₂×C₄	—	218	4	no	—
80	C₂×C₄	C₄×#89137	1	2	yes	—
81	C₂×C₄	C₂×#11724786	1	2	yes	—
82	C₂×C₄	D₁₀₉⋊C₈	1	1	yes	characteristic
83	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
84	C₈	—	109	2	no	—
85	C₈	—	109	2	no	—
86	C₈	C₂×#11724786	1	2	yes	—
87	C₈	C₂×#11724786	1	2	yes	—
88	C₂²	—	2	2	no	—
89	C₂²	C₄×#11724786	1	2	yes	—
90	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
91	C₄	—	109	4	no	—
92	C₄	—	218	2	no	—
93	C₄	C₄×#11724786	1	2	yes	—
94	C₄	C₂×C₄×#89137	1	1	yes	characteristic
95	C₄	C₂²×#11724786	1	1	yes	characteristic
96	C₂	—	2	2	no	—
97	C₂	C₂×C₄×#11724786	1	1	yes	characteristic
98	C₂	D₁₀₉⋊M₃₂	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812773

C₂

yes

maximal

#20295915

C₂

yes

maximal, complemented

#20295912

C₂

yes

maximal, complemented

C₂×C₈×Dic₁₀₉

C₂

yes

maximal, characteristic

#16812773

C₄

yes

complemented

C₂×#13182032

C₂²

yes

—

C₂²×#11724784

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₄

yes

characteristic

#13182032

C₂×C₄

yes

complemented

C₂²×Dic₁₀₉

C₈

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₂×#11724784

C₂³

yes

characteristic

C₄×Dic₁₀₉

C₈

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₈

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₂×C₈

yes

—

#11724784

C₄²

yes

—

#11724784

C₂²×C₄

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄×C₈

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₄×C₈

yes

—

C₄×C₁₀₉

C₂×C₄²

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

M₃₂

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

C₂×M₃₂

yes

—

C₄×M₃₂

—

109

maximal, Sylow

C₁₀₉

C₄×M₃₂

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

109

—

C₂×C₄×C₈

—

109

—

#346

—

109

—

C₄×C₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

#89137

yes

characteristic

C₂×C₄²

—

109

—

M₃₂

—

109

—

C₂×C₁₆

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×#89137

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

#11724786

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₁₆

—

109

—

C₂³

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₄×#89137

yes

—

C₂×C₄

C₂×#11724786

yes

—

C₂×C₄

D₁₀₉⋊C₈

yes

characteristic

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

C₂×#11724786

yes

—

C₈

C₂×#11724786

yes

—

C₂²

—

C₂²

C₄×#11724786

yes

—

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

C₄

—

109

—

C₄

—

218

—

C₄

C₄×#11724786

yes

—

C₄

C₂×C₄×#89137

yes

characteristic

C₄

C₂²×#11724786

yes

characteristic

C₂

—

C₂

C₂×C₄×#11724786

yes

characteristic

C₂

D₁₀₉⋊M₃₂

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#89137
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂²×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂³×C₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	C₂²×C₈	#89137
hypocenter	C₁₀₉	C₄×M₃₂
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

subgroup

G/H

center

C₂×C₈

C₂×#89137

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂²×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²×C₈

prime radical

C₂³×C₁₀₉

C₂×C₈

C₂³

C₂²×C₈

C₁₀₉

C₄×M₃₂

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈

N	H ≅ G/N	type
2. #20295915	C₂	semidirect
3. #20295912	C₂	semidirect
5. #16812773	C₄	semidirect
14. #13182032	C₂×C₄	semidirect
45. Dic₁₀₉	M₃₂	semidirect
50. C₁₀₉	C₄×M₃₂	semidirect

H ≅ G/N

type

2. #20295915

C₂

semidirect

3. #20295912

C₂

semidirect

5. #16812773

C₄

semidirect

14. #13182032

C₂×C₄

semidirect

45. Dic₁₀₉

M₃₂

semidirect

50. C₁₀₉

C₄×M₃₂

semidirect

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

subgroup	index
C₂×#16812773	2
#20295915	2
#20295912	2
C₂×C₈×Dic₁₀₉	2
C₄×M₃₂	109

subgroup

index

C₂×#16812773

#20295915

#20295912

C₂×C₈×Dic₁₀₉

C₄×M₃₂

109

Subgroups — #28379238

Conjugacy classes of subgroups

Complements

Maximal subgroups