Group #28378750 — subgroups

Subgroups — #28378750

Order 13952.

240 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 90 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	#20295985	C₂	1	1	yes	maximal, characteristic
2	#20295962	C₂	1	1	yes	maximal, characteristic
3	(C₂×#89137)⋊C₈	C₂	1	4	yes	maximal
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₀₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×#89137	C₂²	1	2	yes	—
7	#16812754	C₄	1	4	yes	complemented
8	(C₂×#89137)⋊C₄	C₄	1	4	yes	—
9	C₂×C₈×D₁₀₉	C₂²	1	2	yes	—
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄×#11724784	C₄	1	1	yes	characteristic, complemented
12	C₈×Dic₁₀₉	C₄	1	2	yes	complemented
13	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic, complemented
14	C₂²×#89137	C₂×C₄	1	4	yes	—
15	D₁₀₉⋊C₈	—	2	4	no	—
16	C₂×#11724786	C₂×C₄	1	4	yes	—
17	C₄×#89137	—	2	4	no	—
18	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
19	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
20	C₂×#11724784	C₂×C₄	1	2	yes	—
21	C₈×D₁₀₉	—	2	4	no	—
22	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×#89137	—	2	4	no	—
28	C₂×#89137	—	2	8	no	—
29	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₄²	1	2	yes	—
32	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
33	C₄×D₁₀₉	—	2	4	no	—
34	C₄×D₁₀₉	#41	1	4	yes	—
35	#11724786	—	2	4	no	—
36	C₈×C₁₀₉	—	2	2	no	—
37	#11724784	—	2	2	no	—
38	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
39	C₂×D₁₀₉	#117	1	2	yes	—
40	C₂×D₁₀₉	#131	1	4	yes	—
41	#89137	—	4	4	no	—
42	C₄×C₁₀₉	—	2	2	no	—
43	C₄×C₁₀₉	#117	1	2	yes	—
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	#117	1	2	yes	—
46	C₂×C₁₀₉	#376	1	1	yes	characteristic
47	C₂×C₁₀₉	#405	1	2	yes	—
48	D₁₀₉	#433	1	4	yes	—
49	#1746	—	109	1	no	maximal, Sylow
50	C₁₀₉	#1746	1	1	yes	characteristic, complemented, Sylow
51	C₂×#97	—	109	1	no	—
52	C₂×C₄×C₈	—	109	1	no	—
53	#376	—	109	1	no	—
54	#336	—	109	4	no	—
55	C₂²×C₈	—	109	2	no	—
56	C₂²×C₈	—	109	2	no	—
57	C₂×C₄²	—	109	1	no	—
58	C₂×C₄²	—	109	2	no	—
59	#97	—	109	4	no	—
60	C₄×C₈	—	109	2	no	—
61	C₄×C₈	—	109	1	no	—
62	C₄×C₈	#89137	1	1	yes	characteristic, complemented
63	#95	—	109	4	no	—
64	C₂²×C₄	—	109	1	no	—
65	C₂²×C₄	—	109	2	no	—
66	C₂²×C₄	—	109	4	no	—
67	C₂×C₈	—	109	4	no	—
68	C₂×C₈	—	109	2	no	—
69	C₂×C₈	—	218	4	no	—
70	C₂×C₈	—	218	4	no	—
71	C₂×C₈	C₂×#89137	1	2	yes	—
72	C₄²	—	109	2	no	—
73	C₄²	—	109	1	no	—
74	C₄²	—	218	4	no	—
75	C₄²	C₂×#89137	1	1	yes	characteristic
76	C₂³	—	109	1	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	109	4	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	218	4	no	—
81	C₂×C₄	—	218	4	no	—
82	C₂×C₄	—	218	8	no	—
83	C₂×C₄	C₄×#89137	1	2	yes	—
84	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	218	4	no	—
87	C₈	—	218	2	no	—
88	C₂²	—	109	2	no	—
89	C₂²	—	109	4	no	—
90	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	109	2	no	—
93	C₄	—	436	4	no	—
94	C₄	—	218	2	no	—
95	C₄	#16812778	1	2	yes	—
96	C₂	—	109	4	no	—
97	C₂	#20295962	1	1	yes	characteristic
98	C₂	#20295989	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295985

C₂

yes

maximal, characteristic

#20295962

C₂

yes

maximal, characteristic

(C₂×#89137)⋊C₈

C₂

yes

maximal

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×D₁₀₉⋊C₈

C₂²

yes

—

C₂×C₄×#89137

C₂²

yes

—

#16812754

C₄

yes

complemented

(C₂×#89137)⋊C₄

C₄

yes

—

C₂×C₈×D₁₀₉

C₂²

yes

—

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic, complemented

C₈×Dic₁₀₉

C₄

yes

complemented

C₄×C₈×C₁₀₉

C₄

yes

characteristic, complemented

C₂²×#89137

C₂×C₄

yes

—

D₁₀₉⋊C₈

—

C₂×#11724786

C₂×C₄

yes

—

C₄×#89137

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

characteristic

C₂×#11724784

C₂×C₄

yes

—

C₈×D₁₀₉

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂²×C₄

yes

characteristic

C₂×#89137

—

C₂×#89137

—

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

#41

yes

—

#11724786

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×D₁₀₉

#117

yes

—

C₂×D₁₀₉

#131

yes

—

#89137

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

—

Dic₁₀₉

—

Dic₁₀₉

#117

yes

—

C₂×C₁₀₉

#376

yes

characteristic

C₂×C₁₀₉

#405

yes

—

D₁₀₉

#433

yes

—

#1746

—

109

maximal, Sylow

C₁₀₉

#1746

yes

characteristic, complemented, Sylow

C₂×#97

—

109

—

C₂×C₄×C₈

—

109

—

#376

—

109

—

#336

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic, complemented

#95

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

218

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×#89137

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

218

—

C₈

—

218

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

C₂×C₄×#89137

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

436

—

C₄

—

218

—

C₄

#16812778

yes

—

C₂

—

109

—

C₂

#20295962

yes

characteristic

C₂

#20295989

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂×C₄²
Fitting	C₄×C₈×C₁₀₉	C₄
prime radical	C₂²×D₁₀₉	C₂²×C₄
cosocle	C₂×C₄×D₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1746
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂×C₄²

Fitting

C₄×C₈×C₁₀₉

C₄

prime radical

C₂²×D₁₀₉

C₂²×C₄

cosocle

C₂×C₄×D₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₄×C₈

N	H ≅ G/N	type
7. #16812754	C₄	semidirect
11. C₄×#11724784	C₄	semidirect
12. C₈×Dic₁₀₉	C₄	semidirect
13. C₄×C₈×C₁₀₉	C₄	semidirect
50. C₁₀₉	#1746	semidirect
62. C₄×C₈	#89137	semidirect

H ≅ G/N

type

7. #16812754

C₄

semidirect

11. C₄×#11724784

C₄

semidirect

12. C₈×Dic₁₀₉

C₄

semidirect

13. C₄×C₈×C₁₀₉

C₄

semidirect

50. C₁₀₉

#1746

semidirect

62. C₄×C₈

#89137

semidirect

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

subgroup	index
#20295985	2
#20295962	2
(C₂×#89137)⋊C₈	2
C₄×C₈×D₁₀₉	2
#1746	109

subgroup

index

C₄×C₈×D₁₀₉

109

Subgroups — #28378750

Conjugacy classes of subgroups

Complements

Maximal subgroups