Group #28361852 — subgroups

Subgroups — #28361852

Order 13952.

160 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₈×Dic₂₁₈	C₂	1	2	yes	maximal
2	C₄×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic
3	#20294777	C₂	1	1	yes	maximal, characteristic
4	#20294758	C₂	1	2	yes	maximal
5	#322×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₃₆	C₄	1	1	yes	characteristic
7	C₄×Dic₂₁₈	C₂²	1	2	yes	—
8	#16812633	C₄	1	1	yes	characteristic
9	C₄⋊#11724784	C₂²	1	2	yes	—
10	#16812628	—	2	2	no	—
11	#16812624	C₄	1	2	yes	—
12	C₈×Dic₁₀₉	—	2	2	no	—
13	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₈	C₂×C₄	1	2	yes	—
17	Dic₄₃₆	C₈	1	4	yes	complemented
18	C₂×#11724784	—	2	2	no	—
19	C₄⋊Dic₁₀₉	C₂×C₄	1	2	yes	—
20	#13182008	—	2	2	no	—
21	C₄×Dic₁₀₉	—	2	2	no	—
22	C₂×C₈×C₁₀₉	—	2	1	no	—
23	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
26	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
30	C₂×Dic₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	—	2	2	no	—
32	Dic₂₁₈	—	2	2	no	—
33	Dic₂₁₈	C₂×C₈	1	4	yes	—
34	C₈×C₁₀₉	—	2	1	no	—
35	C₈×C₁₀₉	—	2	2	no	—
36	C₈×C₁₀₉	—	4	1	no	—
37	C₈×C₁₀₉	C₂×C₈	1	2	yes	—
38	#11724784	—	4	2	no	—
39	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
40	C₄×C₁₀₉	—	2	1	no	—
41	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₀₉	#131	1	1	yes	characteristic
43	C₄×C₁₀₉	#137	1	2	yes	—
44	Dic₁₀₉	—	2	4	no	—
45	Dic₁₀₉	—	4	2	no	—
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	D₁₀₉	1	1	yes	characteristic
56	C₂×Q₁₆	—	109	1	no	—
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	—	218	2	no	—
63	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₀₉	1	1	yes	characteristic
66	C₂×Q₈	—	109	2	no	—
67	Q₁₆	—	109	4	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	218	2	no	—
70	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
71	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
72	C₂×C₈	D₄₃₆	1	2	yes	—
73	C₄⋊C₄	—	109	2	no	—
74	C₄⋊C₄	—	218	2	no	—
75	C₄²	—	218	2	no	—
76	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
77	Q₈	—	109	4	no	—
78	Q₈	—	218	2	no	—
79	C₂×C₄	—	218	2	no	—
80	C₂×C₄	—	218	2	no	—
81	C₂×C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
83	C₂×C₄	#13182029	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	436	2	no	—
88	C₈	C₈×D₁₀₉	1	2	yes	—
89	C₂²	C₄×D₄₃₆	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	436	2	no	—
92	C₄	—	218	4	no	—
93	C₄	C₂×C₈×D₁₀₉	1	1	yes	characteristic
94	C₄	#16812705	1	1	yes	characteristic
95	C₄	#16812732	1	2	yes	—
96	C₂	C₈×D₄₃₆	1	1	yes	characteristic
97	C₂	#20295134	1	1	yes	characteristic
98	C₂	#20295149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₂₁₈

C₂

yes

maximal

C₄×Dic₄₃₆

C₂

yes

maximal, characteristic

#20294777

C₂

yes

maximal, characteristic

#20294758

C₂

yes

maximal

#322×C₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₄

yes

characteristic

C₄×Dic₂₁₈

C₂²

yes

—

#16812633

C₄

yes

characteristic

C₄⋊#11724784

C₂²

yes

—

#16812628

—

#16812624

C₄

yes

—

C₈×Dic₁₀₉

—

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

C₂×C₄

yes

—

Dic₄₃₆

C₈

yes

complemented

C₂×#11724784

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182008

—

C₄×Dic₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

#41

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₈

yes

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

C₂×C₈

yes

—

#11724784

—

C₂²×C₁₀₉

C₄×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₈

yes

C₄×C₁₀₉

yes

C₄×C₁₀₉

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

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

D₁₀₉

yes

characteristic

C₂×Q₁₆

—

109

—

C₄×Q₈

—

109

—

#107

—

109

—

C₄⋊C₈

—

109

—

C₄⋊C₈

—

218

—

#103

—

109

—

C₄×C₈

—

218

—

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₄×C₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

Q₁₆

—

109

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

C₄×D₁₀₉

yes

characteristic

C₂×C₈

D₄₃₆

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄²

—

218

—

C₄²

C₂²×D₁₀₉

yes

characteristic

Q₈

—

109

—

Q₈

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

#13182029

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

436

—

C₈

C₈×D₁₀₉

yes

—

C₂²

C₄×D₄₃₆

yes

characteristic

C₄

—

C₄

—

436

—

C₄

—

218

—

C₄

C₂×C₈×D₁₀₉

yes

C₄

yes

C₄

yes

—

C₂

C₈×D₄₃₆

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₄₃₆
commutator	C₄×C₁₀₉	C₂²×C₈
Frattini	C₄²	C₂²×D₁₀₉
Fitting	#322×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	C₄×D₄
cosocle	C₄²×C₁₀₉	C₂³
hypercenter	#322	D₁₀₉
hypocenter	C₁₀₉	#1579
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₄₃₆

commutator

C₄×C₁₀₉

C₂²×C₈

Frattini

C₄²

C₂²×D₁₀₉

Fitting

#322×C₁₀₉

C₂

socle, prime radical

C₂²×C₁₀₉

C₄×D₄

cosocle

C₄²×C₁₀₉

C₂³

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
17. Dic₄₃₆	C₈	semidirect
50. C₁₀₉	#1579	semidirect

H ≅ G/N

type

17. Dic₄₃₆

C₈

semidirect

50. C₁₀₉

#1579

semidirect

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

subgroup	index
C₈×Dic₂₁₈	2
C₄×Dic₄₃₆	2
#20294777	2
#20294758	2
#322×C₁₀₉	2
#1579	109

subgroup

index

C₈×Dic₂₁₈

C₄×Dic₄₃₆

#20294777

#20294758

#322×C₁₀₉

#1579

109

Subgroups — #28361852

Conjugacy classes of subgroups

Complements

Maximal subgroups