Group #28379400 — subgroups

Subgroups — #28379400

Order 13952.

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₄⋊#13182032	C₂	1	2	yes	maximal, complemented
2	#20295921	C₂	1	2	yes	maximal
3	#20295915	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295912	C₂	1	1	yes	maximal, characteristic, complemented
5	#20294983	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182032	C₂²	1	2	yes	—
7	C₂×#13182032	C₂²	1	2	yes	—
8	C₂×#13182012	C₄	1	1	yes	characteristic
9	C₂×C₈×D₁₀₉	C₂²	1	1	yes	characteristic
10	#16812656	C₄	1	1	yes	characteristic
11	#16812609	C₄	1	1	yes	characteristic
12	C₄×#11724784	C₂²	1	1	yes	characteristic
13	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	#97×C₁₀₉	C₄	1	1	yes	characteristic
15	#13182032	—	2	2	no	—
16	#13182032	—	2	2	no	—
17	C₂×C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
18	#13182012	—	2	2	no	—
19	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₂×#11724784	C₂³	1	1	yes	characteristic
21	C₈×D₁₀₉	—	2	2	no	—
22	#13182008	C₈	1	2	yes	—
23	#13182006	C₈	1	2	yes	—
24	C₄×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×D₁₀₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂×C₈	1	2	yes	—
32	C₂×Dic₁₀₉	C₂×C₈	1	1	yes	characteristic
33	C₄×D₁₀₉	—	2	1	no	—
34	C₄×D₁₀₉	—	2	1	no	—
35	C₈×C₁₀₉	—	2	1	no	—
36	C₈×C₁₀₉	M₁₆	1	2	yes	—
37	#11724784	—	2	1	no	—
38	#11724784	#41	1	2	yes	—
39	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
40	C₂×D₁₀₉	—	2	1	no	—
41	C₂×D₁₀₉	—	2	2	no	—
42	C₄×C₁₀₉	—	2	2	no	—
43	C₄×C₁₀₉	#117	1	1	yes	characteristic
44	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
45	Dic₁₀₉	—	2	1	no	—
46	Dic₁₀₉	—	2	2	no	—
47	C₂×C₁₀₉	#431	1	1	yes	characteristic
48	C₂×C₁₀₉	#504	1	2	yes	—
49	D₁₀₉	—	4	1	no	—
50	#2164	—	109	1	no	maximal, Sylow
51	C₁₀₉	#2164	1	1	yes	characteristic, complemented, Sylow
52	#405	—	109	1	no	—
53	C₄⋊C₁₆	—	109	2	no	—
54	C₂²⋊C₁₆	—	109	2	no	—
55	#346	—	109	1	no	—
56	C₄×C₁₆	—	109	1	no	—
57	C₂×M₁₆	—	109	1	no	—
58	C₂²×C₈	—	109	1	no	—
59	#117	—	109	1	no	—
60	C₂×C₁₆	—	109	2	no	—
61	C₂×C₁₆	—	109	2	no	—
62	#97	—	109	1	no	—
63	#97	#89137	1	1	yes	characteristic
64	C₄×C₈	—	109	1	no	—
65	C₄×C₈	—	109	1	no	—
66	C₂²×C₄	—	109	1	no	—
67	M₁₆	—	218	2	no	—
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	218	2	no	—
71	C₂×C₈	C₂×#89137	1	1	yes	characteristic
72	C₂×C₈	C₂×#89137	1	1	yes	characteristic
73	C₄⋊C₄	—	109	2	no	—
74	C₂²⋊C₄	—	109	2	no	—
75	C₄²	—	109	1	no	—
76	C₄²	C₂×#89137	1	1	yes	characteristic
77	C₁₆	—	218	2	no	—
78	C₁₆	—	218	2	no	—
79	C₂³	—	109	1	no	—
80	C₂×C₄	—	109	2	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	218	1	no	—
83	C₂×C₄	—	218	1	no	—
84	C₂×C₄	D₁₀₉⋊C₈	1	2	yes	—
85	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	109	2	no	—
88	C₈	—	218	1	no	—
89	C₈	#13182039	1	2	yes	—
90	C₂²	—	218	1	no	—
91	C₂²	—	218	2	no	—
92	C₂²	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	218	1	no	—
95	C₄	—	218	2	no	—
96	C₄	#16812778	1	1	yes	characteristic
97	C₄	C₂×#13182039	1	1	yes	characteristic
98	C₂	—	436	1	no	—
99	C₂	#20296045	1	1	yes	characteristic
100	C₂	#20296085	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#13182032

C₂

yes

maximal, complemented

#20295921

C₂

yes

maximal

#20295915

C₂

yes

maximal, characteristic, complemented

#20295912

C₂

yes

maximal, characteristic, complemented

#20294983

C₂

yes

maximal, characteristic

C₂×#13182032

C₂²

yes

—

C₂×#13182032

C₂²

yes

—

C₂×#13182012

C₄

yes

characteristic

C₂×C₈×D₁₀₉

C₂²

yes

characteristic

#16812656

C₄

yes

characteristic

#16812609

C₄

yes

characteristic

C₄×#11724784

C₂²

yes

characteristic

C₈×Dic₁₀₉

C₂²

yes

characteristic

#97×C₁₀₉

C₄

yes

characteristic

#13182032

—

#13182032

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

characteristic

#13182012

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₂×#11724784

C₂³

yes

characteristic

C₈×D₁₀₉

—

#13182008

C₈

yes

—

#13182006

C₈

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄²×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₈

yes

—

C₂×Dic₁₀₉

C₂×C₈

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

M₁₆

yes

—

#11724784

—

#11724784

#41

yes

—

C₂²×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

#117

yes

characteristic

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#431

yes

characteristic

C₂×C₁₀₉

#504

yes

—

D₁₀₉

—

#2164

—

109

maximal, Sylow

C₁₀₉

#2164

yes

characteristic, complemented, Sylow

#405

—

109

—

C₄⋊C₁₆

—

109

—

C₂²⋊C₁₆

—

109

—

#346

—

109

—

C₄×C₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

#117

—

109

—

C₂×C₁₆

—

109

—

C₂×C₁₆

—

109

—

#97

—

109

—

#97

#89137

yes

characteristic

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

characteristic

C₁₆

—

218

—

C₁₆

—

218

—

C₂³

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

D₁₀₉⋊C₈

yes

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

C₈

—

109

—

C₈

—

218

—

C₈

#13182039

yes

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×D₁₀₉⋊C₈

yes

characteristic

C₄

—

C₄

—

218

—

C₄

—

218

—

C₄

yes

C₄

yes

C₂

—

436

—

C₂

#20296045

yes

characteristic

100

C₂

#20296085

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₈
Fitting	#97×C₁₀₉	C₄
prime radical	C₂²×D₁₀₉	C₂×C₈
cosocle	C₂×#11724784	C₂³
hypercenter	#97	#89137
hypocenter	C₁₀₉	#2164
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₈

Fitting

#97×C₁₀₉

C₄

prime radical

C₂²×D₁₀₉

C₂×C₈

C₂³

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
1. C₄⋊#13182032	C₂	semidirect
3. #20295915	C₂	semidirect
4. #20295912	C₂	semidirect
51. C₁₀₉	#2164	semidirect

H ≅ G/N

type

1. C₄⋊#13182032

C₂

semidirect

3. #20295915

C₂

semidirect

4. #20295912

C₂

semidirect

51. C₁₀₉

#2164

semidirect

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

subgroup	index
C₄⋊#13182032	2
#20295921	2
#20295915	2
#20295912	2
#20294983	2
#2164	109

subgroup

index

109

Subgroups — #28379400

Conjugacy classes of subgroups

Complements

Maximal subgroups