Group #28378236 — subgroups

Subgroups — #28378236

Order 13952.

212 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 106 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	#20296044	C₂	1	2	yes	maximal
2	#20295867	C₂	1	2	yes	maximal, complemented
3	C₈×#11724786	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#16812766	C₄	1	4	yes	—
6	#16812756	C₄	1	4	yes	—
7	C₄×#11724786	C₂²	1	4	yes	—
8	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	C₄	1	1	yes	characteristic
10	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
11	C₄×#11724784	C₄	1	1	yes	characteristic
12	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
15	C₂×#11724786	C₂×C₄	1	8	yes	—
16	C₂×C₄×D₁₀₉	C₂×C₄	1	2	yes	—
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₈×D₁₀₉	C₈	1	4	yes	complemented
21	C₈×D₁₀₉	C₈	1	4	yes	complemented
22	C₄×Dic₁₀₉	C₂×C₄	1	2	yes	—
23	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₁₀₉	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₄²	1	2	yes	—
29	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	C₄²	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂²×C₄	1	2	yes	—
32	C₄×D₁₀₉	—	2	4	no	—
33	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
34	C₄×D₁₀₉	C₂×C₈	1	2	yes	—
35	#11724786	—	2	8	no	—
36	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
37	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
38	#11724784	C₂×C₈	1	2	yes	—
39	#11724784	C₂×C₈	1	2	yes	—
40	C₂²×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	2	no	—
42	C₂×D₁₀₉	C₄×C₈	1	2	yes	—
43	C₄×C₁₀₉	—	2	2	no	—
44	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
46	Dic₁₀₉	C₄×C₈	1	2	yes	—
47	Dic₁₀₉	#131	1	4	yes	—
48	C₂×C₁₀₉	C₂×C₄×C₈	1	1	yes	characteristic
49	C₂×C₁₀₉	#405	1	2	yes	—
50	D₁₀₉	—	2	2	no	—
51	#1450	—	109	1	no	maximal, Sylow
52	C₁₀₉	#1450	1	1	yes	characteristic, complemented, Sylow
53	#431	—	109	2	no	—
54	C₂×C₄×C₈	—	109	1	no	—
55	#322	—	109	2	no	—
56	C₈²	—	109	2	no	—
57	C₂²×C₈	—	109	1	no	—
58	C₂²×C₈	—	109	1	no	—
59	C₂×C₄²	—	109	1	no	—
60	C₄⋊C₈	—	109	4	no	—
61	C₂²⋊C₈	—	109	4	no	—
62	C₄×C₈	—	109	4	no	—
63	C₄×C₈	—	109	1	no	—
64	C₄×C₈	—	109	1	no	—
65	C₄×C₈	—	109	1	no	—
66	C₄×C₈	#89137	1	1	yes	characteristic
67	C₂²×C₄	—	109	1	no	—
68	C₂²×C₄	—	109	2	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	—	109	4	no	—
72	C₂×C₈	—	109	4	no	—
73	C₂×C₈	—	109	8	no	—
74	C₂×C₈	C₂×#89137	1	1	yes	characteristic
75	C₂×C₈	C₂×#89137	1	1	yes	characteristic
76	C₄²	—	109	2	no	—
77	C₄²	—	109	1	no	—
78	C₄²	C₂×#89137	1	1	yes	characteristic
79	C₂³	—	109	1	no	—
80	C₂×C₄	—	109	2	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	109	2	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	218	4	no	—
85	C₂×C₄	C₄×#89137	1	2	yes	—
86	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
87	C₈	—	109	2	no	—
88	C₈	—	109	2	no	—
89	C₈	—	218	8	no	—
90	C₈	D₁₀₉⋊C₈	1	2	yes	—
91	C₈	D₁₀₉⋊C₈	1	2	yes	—
92	C₂²	—	109	2	no	—
93	C₂²	—	218	2	no	—
94	C₂²	C₂×C₄×#89137	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	109	2	no	—
97	C₄	—	109	4	no	—
98	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
99	C₄	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
100	C₂	—	218	2	no	—
101	C₂	C₄×D₁₀₉⋊C₈	1	1	yes	characteristic
102	C₂	#20295989	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20296044

C₂

yes

maximal

#20295867

C₂

yes

maximal, complemented

C₈×#11724786

C₂

yes

maximal, complemented

C₄×C₈×D₁₀₉

C₂

yes

maximal, characteristic

#16812766

C₄

yes

—

#16812756

C₄

yes

—

C₄×#11724786

C₂²

yes

—

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₄

yes

characteristic

C₄²×D₁₀₉

C₂²

yes

characteristic

C₄×#11724784

C₄

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₄

yes

characteristic

C₂×#11724786

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂×C₄

yes

characteristic

C₈×D₁₀₉

C₈

yes

complemented

C₈×D₁₀₉

C₈

yes

complemented

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₄²

yes

characteristic

C₂×C₄×C₁₀₉

C₄²

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₄²

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₈

yes

—

C₄×D₁₀₉

C₂×C₈

yes

—

#11724786

—

C₈×C₁₀₉

C₂×C₈

yes

—

C₈×C₁₀₉

C₂×C₈

yes

—

#11724784

C₂×C₈

yes

—

#11724784

C₂×C₈

yes

—

C₂²×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄×C₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

Dic₁₀₉

C₄×C₈

yes

—

Dic₁₀₉

#131

yes

—

C₂×C₁₀₉

C₂×C₄×C₈

yes

characteristic

C₂×C₁₀₉

#405

yes

—

D₁₀₉

—

#1450

—

109

maximal, Sylow

C₁₀₉

#1450

yes

characteristic, complemented, Sylow

#431

—

109

—

C₂×C₄×C₈

—

109

—

#322

—

109

—

C₈²

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₄×#89137

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

—

218

—

C₈

D₁₀₉⋊C₈

yes

—

C₈

D₁₀₉⋊C₈

yes

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₂×C₄×#89137

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

109

—

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

C₄

C₂×D₁₀₉⋊C₈

yes

characteristic

100

C₂

—

218

—

101

C₂

C₄×D₁₀₉⋊C₈

yes

characteristic

102

C₂

#20295989

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×#89137
commutator	C₂×C₁₀₉	C₂×C₄×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₄×C₈×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂×C₄²
prime radical	C₂²×D₁₀₉	C₄²
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₄×C₈	#89137
hypocenter	C₁₀₉	#1450
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₂×C₈

C₂×#89137

commutator

C₂×C₁₀₉

C₂×C₄×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₄×C₈×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂×C₄²

prime radical

C₂²×D₁₀₉

C₄²

cosocle

C₄×Dic₁₀₉

C₂³

C₄×C₈

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #20295867	C₂	semidirect
3. C₈×#11724786	C₂	semidirect
20. C₈×D₁₀₉	C₈	semidirect
21. C₈×D₁₀₉	C₈	semidirect
52. C₁₀₉	#1450	semidirect

H ≅ G/N

type

2. #20295867

C₂

semidirect

3. C₈×#11724786

C₂

semidirect

20. C₈×D₁₀₉

C₈

semidirect

21. C₈×D₁₀₉

C₈

semidirect

52. C₁₀₉

#1450

semidirect

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

subgroup	index
#20296044	2
#20295867	2
C₈×#11724786	2
C₄×C₈×D₁₀₉	2
#1450	109

subgroup

index

C₄×C₈×D₁₀₉

109

Subgroups — #28378236

Conjugacy classes of subgroups

Complements

Maximal subgroups