Group #28378625 — subgroups

Subgroups — #28378625

Order 13952.

160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 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	Q₈×#11724786	C₂	1	2	yes	maximal
2	#20295887	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#11724786	C₂	1	2	yes	maximal
4	#20295867	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₄⋊#11724786	—	2	2	no	—
7	C₄⋊#11724786	C₂²	1	2	yes	—
8	C₄×#11724786	—	2	2	no	—
9	C₄×#11724786	C₂²	1	1	yes	characteristic
10	C₄×#11724786	C₂²	1	1	yes	characteristic
11	Q₈×Dic₁₀₉	C₂²	1	2	yes	—
12	#16812633	C₄	1	1	yes	characteristic
13	#16812623	C₄	1	2	yes	—
14	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆×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×#11724784	C₂×C₄	1	1	yes	characteristic
20	C₄⋊Dic₁₀₉	—	2	2	no	—
21	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₀₉	—	2	2	no	—
23	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
24	C₂×Q₈×C₁₀₉	C₂×C₄	1	2	yes	—
25	Q₁₆×C₁₀₉	C₈	1	4	yes	complemented
26	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
27	Q₈×C₁₀₉	—	2	2	no	—
28	Q₈×C₁₀₉	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₁₀₉	—	2	2	no	—
30	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	—	2	2	no	—
32	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₁₀₉	#41	1	1	yes	characteristic
34	#11724786	—	2	2	no	—
35	#11724786	—	4	1	no	—
36	#11724786	—	4	2	no	—
37	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
38	#11724784	—	2	1	no	—
39	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₀₉	—	2	4	no	—
41	C₄×C₁₀₉	C₂²×C₈	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	C₈×D₄	1	1	yes	characteristic
47	C₂×C₁₀₉	#441	1	1	yes	characteristic
48	C₂×C₁₀₉	#444	1	1	yes	characteristic
49	#1579	—	109	1	no	maximal, Sylow
50	C₁₀₉	#1579	1	1	yes	characteristic, complemented, Sylow
51	C₈×Q₈	—	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	#89137	1	1	yes	characteristic
57	C₄×Q₈	—	109	2	no	—
58	#107	—	109	1	no	—
59	C₄⋊C₈	—	109	2	no	—
60	C₄⋊C₈	—	218	2	no	—
61	#103	—	109	2	no	—
62	C₄×C₈	—	109	1	no	—
63	C₄×C₈	—	109	1	no	—
64	C₄×C₈	—	109	1	no	—
65	C₄×C₈	—	218	2	no	—
66	C₂×Q₈	C₂×#89137	1	2	yes	—
67	Q₁₆	#11724786	1	4	yes	complemented
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	—	109	2	no	—
74	C₄⋊C₄	—	218	2	no	—
75	C₄²	—	109	1	no	—
76	C₄²	—	218	2	no	—
77	Q₈	—	2	2	no	—
78	Q₈	C₂×#11724786	1	4	yes	—
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	C₂×#11724786	1	2	yes	—
89	C₂²	D₄×#89137	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	109	2	no	—
92	C₄	—	436	2	no	—
93	C₄	—	218	1	no	—
94	C₄	C₂²×#11724786	1	1	yes	characteristic
95	C₄	#16812786	1	1	yes	characteristic
96	C₂	D₄×#11724786	1	1	yes	characteristic
97	C₂	#20296066	1	1	yes	characteristic
98	C₂	#20296072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11724786

C₂

yes

maximal

#20295887

C₂

yes

maximal, characteristic

Q₈⋊#11724786

C₂

yes

maximal

#20295867

C₂

yes

maximal, characteristic

Q₁₆×Dic₁₀₉

C₂

yes

maximal, characteristic

C₄⋊#11724786

—

C₄⋊#11724786

C₂²

yes

—

C₄×#11724786

—

C₄×#11724786

C₂²

yes

characteristic

C₄×#11724786

C₂²

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

—

#16812633

C₄

yes

characteristic

#16812623

C₄

yes

—

C₈×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₁₆×C₁₀₉

C₄

yes

—

—

D₄

yes

—

C₂×#11724784

C₂×C₄

yes

characteristic

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

C₂×C₄

yes

—

Q₁₆×C₁₀₉

C₈

yes

complemented

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

C₂×C₈

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×D₄

yes

C₂×Dic₁₀₉

yes

—

—

—

C₈×C₁₀₉

C₂×C₈

yes

—

#11724784

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

characteristic

C₄×C₁₀₉

#131

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

#137

yes

—

C₂×C₁₀₉

C₈×D₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#1579

yes

characteristic, complemented, Sylow

C₈×Q₈

—

109

—

C₄×Q₁₆

—

109

—

#334

—

109

—

Q₈⋊C₈

—

109

—

#322

—

109

—

C₂×Q₁₆

#89137

yes

characteristic

C₄×Q₈

—

109

—

#107

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

#103

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

109

—

C₄×C₈

—

218

—

C₂×Q₈

C₂×#89137

yes

—

Q₁₆

#11724786

yes

complemented

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₂×C₈

C₂×#89137

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄²

—

109

—

C₄²

—

218

—

Q₈

—

Q₈

C₂×#11724786

yes

—

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

C₂×#11724786

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	C₄×C₁₀₉	C₂²×C₈
Frattini	C₂×C₄	C₂²×#89137
Fitting	C₂×Q₁₆×C₁₀₉	C₄
socle, prime radical	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄×Dic₁₀₉	C₂³
hypercenter	C₂×Q₁₆	#89137
hypocenter	C₁₀₉	#1579
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×#89137

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₂×C₄

C₂²×#89137

Fitting

C₂×Q₁₆×C₁₀₉

C₄

socle, prime radical

C₂²×C₁₀₉

C₄×D₄

cosocle

C₄×Dic₁₀₉

C₂³

C₂×Q₁₆

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
25. Q₁₆×C₁₀₉	C₈	semidirect
50. C₁₀₉	#1579	semidirect
67. Q₁₆	#11724786	semidirect

H ≅ G/N

type

25. Q₁₆×C₁₀₉

C₈

semidirect

50. C₁₀₉

#1579

semidirect

67. Q₁₆

#11724786

semidirect

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

subgroup	index
Q₈×#11724786	2
#20295887	2
Q₈⋊#11724786	2
#20295867	2
Q₁₆×Dic₁₀₉	2
#1579	109

subgroup

index

Q₁₆×Dic₁₀₉

109

Subgroups — #28378625

Conjugacy classes of subgroups

Complements

Maximal subgroups