Group #28378678 — subgroups

Subgroups — #28378678

Order 13952.

148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 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	#20296076	C₂	1	2	yes	maximal
2	#20295887	C₂	1	1	yes	maximal, characteristic
3	#20295876	C₂	1	2	yes	maximal
4	#20295867	C₂	1	1	yes	maximal, characteristic
5	#20295120	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11724786	—	2	2	no	—
7	C₄⋊#11724786	C₂²	1	2	yes	—
8	#16812754	—	2	2	no	—
9	C₄×#11724786	C₂²	1	1	yes	characteristic
10	C₄×#11724786	C₂²	1	1	yes	characteristic
11	C₂×Dic₄₃₆	C₄	1	1	yes	characteristic
12	#16812664	C₂²	1	2	yes	—
13	#16812622	C₄	1	2	yes	—
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	#107×C₁₀₉	C₄	1	1	yes	characteristic
16	C₂×#11724786	—	2	2	no	—
17	C₂×#11724786	—	2	1	no	—
18	C₂×#11724786	D₄	1	2	yes	—
19	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
20	Dic₄₃₆	—	2	2	no	—
21	C₂×#11724784	C₂×C₄	1	1	yes	characteristic
22	#13182008	—	2	2	no	—
23	C₄×Dic₁₀₉	—	2	2	no	—
24	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₄×C₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₀₉	—	2	2	no	—
28	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	2	2	no	—
30	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
32	Dic₂₁₈	—	2	2	no	—
33	Dic₂₁₈	—	2	2	no	—
34	#11724786	—	2	2	no	—
35	#11724786	—	4	1	no	—
36	#11724786	—	4	2	no	—
37	C₈×C₁₀₉	M₁₆	1	2	yes	—
38	#11724784	—	2	1	no	—
39	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₀₉	—	4	2	no	—
41	C₄×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₁₀₉	#131	1	1	yes	characteristic
43	Dic₁₀₉	—	2	1	no	—
44	Dic₁₀₉	—	4	2	no	—
45	Dic₁₀₉	#137	1	2	yes	—
46	C₂×C₁₀₉	#436	1	1	yes	characteristic
47	C₂×C₁₀₉	#441	1	1	yes	characteristic
48	C₂×C₁₀₉	#444	1	1	yes	characteristic
49	#1593	—	109	1	no	maximal, Sylow
50	C₁₀₉	#1593	1	1	yes	characteristic, complemented, Sylow
51	#446	—	109	2	no	—
52	C₄×Q₁₆	—	109	1	no	—
53	#334	—	109	1	no	—
54	Q₈⋊C₈	—	109	2	no	—
55	#322	—	109	1	no	—
56	C₂×Q₁₆	—	109	1	no	—
57	C₄×Q₈	—	109	2	no	—
58	#107	#89137	1	1	yes	characteristic
59	C₄⋊C₈	—	109	2	no	—
60	C₄⋊C₈	—	218	2	no	—
61	#103	—	109	2	no	—
62	#97	—	218	2	no	—
63	C₄×C₈	—	109	1	no	—
64	C₄×C₈	—	109	1	no	—
65	C₄×C₈	—	109	1	no	—
66	C₂×Q₈	—	109	2	no	—
67	Q₁₆	—	218	2	no	—
68	C₂×C₈	—	109	2	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	218	1	no	—
71	C₂×C₈	—	218	2	no	—
72	C₂×C₈	C₂×#89137	1	1	yes	characteristic
73	C₄⋊C₄	—	218	2	no	—
74	C₄⋊C₄	C₂×#89137	1	2	yes	—
75	C₄²	—	109	1	no	—
76	C₄²	—	218	2	no	—
77	Q₈	—	218	2	no	—
78	Q₈	—	218	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	109	1	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	218	2	no	—
83	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
84	C₈	—	436	2	no	—
85	C₈	—	436	1	no	—
86	C₈	—	218	2	no	—
87	C₈	—	218	1	no	—
88	C₈	#13182039	1	2	yes	—
89	C₂²	D₄×#89137	1	1	yes	characteristic
90	C₄	—	4	2	no	—
91	C₄	—	109	2	no	—
92	C₄	—	436	2	no	—
93	C₄	—	218	1	no	—
94	C₄	C₂×#13182039	1	1	yes	characteristic
95	C₄	#16812787	1	1	yes	characteristic
96	C₂	#20296060	1	1	yes	characteristic
97	C₂	#20296066	1	1	yes	characteristic
98	C₂	#20296073	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20296076

C₂

yes

maximal

#20295887

C₂

yes

maximal, characteristic

#20295876

C₂

yes

maximal

#20295867

C₂

yes

maximal, characteristic

#20295120

C₂

yes

maximal, characteristic

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

#16812754

—

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

C₂×Dic₄₃₆

C₄

yes

characteristic

#16812664

C₂²

yes

—

#16812622

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

#107×C₁₀₉

C₄

yes

—

—

D₄

yes

—

C₂×Dic₂₁₈

C₂×C₄

yes

—

Dic₄₃₆

—

C₂×#11724784

C₂×C₄

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×Dic₁₀₉

#41

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

—

#11724786

—

#11724786

—

#11724786

—

C₈×C₁₀₉

M₁₆

yes

—

#11724784

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#137

yes

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1593

yes

characteristic, complemented, Sylow

#446

—

109

—

C₄×Q₁₆

—

109

—

#334

—

109

—

Q₈⋊C₈

—

109

—

#322

—

109

—

C₂×Q₁₆

—

109

—

C₄×Q₈

—

109

—

#107

#89137

yes

characteristic

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

#103

—

109

—

#97

—

218

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

109

—

Q₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×#89137

yes

—

C₄²

—

109

—

C₄²

—

218

—

Q₈

—

218

—

Q₈

—

218

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×#89137

yes

characteristic

C₈

—

436

—

C₈

—

436

—

C₈

—

218

—

C₈

—

218

—

C₈

#13182039

yes

—

C₂²

D₄×#89137

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

436

—

C₄

—

218

—

C₄

yes

C₄

yes

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×#89137
commutator	C₂×C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×#89137
Fitting	#107×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	#107	#89137
hypocenter	C₁₀₉	#1593
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₂×C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×#89137

Fitting

#107×C₁₀₉

C₄

socle, prime radical

C₂²×C₁₀₉

C₄×D₄

cosocle

C₄×Dic₁₀₉

C₂³

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
50. C₁₀₉	#1593	semidirect

H ≅ G/N

type

50. C₁₀₉

#1593

semidirect

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

subgroup	index
#20296076	2
#20295887	2
#20295876	2
#20295867	2
#20295120	2
#1593	109

subgroup

index

109

Subgroups — #28378678

Conjugacy classes of subgroups

Complements

Maximal subgroups