Group #28364542 — subgroups

Subgroups — #28364542

Order 13952.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20295069	C₂	1	2	yes	maximal
2	C₂×#16812631	C₂	1	1	yes	maximal, characteristic
3	#20294794	C₂	1	2	yes	maximal
4	#20294791	C₂	1	1	yes	maximal, characteristic
5	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
8	#16812632	—	2	2	no	—
9	#16812631	—	2	2	no	—
10	#16812630	—	2	2	no	—
11	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	#117×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₀₉	—	2	2	no	—
15	C₂²×Dic₁₀₉	—	2	2	no	—
16	#13182013	—	2	2	no	—
17	#13182013	Q₈	1	4	yes	—
18	C₂×#11724784	—	2	2	no	—
19	C₄⋊Dic₁₀₉	—	2	2	no	—
20	C₄⋊Dic₁₀₉	—	2	2	no	—
21	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₀₉	—	2	2	no	—
23	C₂×C₈×C₁₀₉	—	2	2	no	—
24	C₂×C₈×C₁₀₉	—	2	1	no	—
25	C₂×C₈×C₁₀₉	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
27	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
28	C₄²×C₁₀₉	—	2	1	no	—
29	C₂³×C₁₀₉	#41	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	2	2	no	—
31	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₀₉	#41	1	2	yes	—
35	C₂×Dic₁₀₉	—	4	2	no	—
36	C₂×Dic₁₀₉	—	8	2	no	—
37	C₈×C₁₀₉	—	2	2	no	—
38	C₈×C₁₀₉	—	4	1	no	—
39	#11724784	—	2	4	no	—
40	C₂²×C₁₀₉	—	2	2	no	—
41	C₂²×C₁₀₉	#121	1	1	yes	characteristic
42	C₂²×C₁₀₉	#125	1	2	yes	—
43	C₄×C₁₀₉	—	4	2	no	—
44	C₄×C₁₀₉	C₂²⋊Q₈	1	2	yes	—
45	C₄×C₁₀₉	#123	1	1	yes	characteristic
46	C₄×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
47	Dic₁₀₉	—	8	2	no	—
48	C₂×C₁₀₉	—	2	2	no	—
49	C₂×C₁₀₉	#398	1	1	yes	characteristic
50	C₂×C₁₀₉	#471	1	2	yes	—
51	#2090	—	109	1	no	maximal, Sylow
52	C₁₀₉	#2090	1	1	yes	characteristic, complemented, Sylow
53	#428	—	109	2	no	—
54	C₂×#106	—	109	1	no	—
55	#424	D₁₀₉	1	1	yes	characteristic
56	#343	—	109	2	no	—
57	#341	—	109	1	no	—
58	C₂×M₁₆	—	109	2	no	—
59	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
61	#117	C₂×D₁₀₉	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	109	2	no	—
63	#107	—	218	2	no	—
64	#106	—	218	2	no	—
65	#106	—	218	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₂²×C₄	—	218	2	no	—
68	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	109	4	no	—
71	M₁₆	—	218	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	218	2	no	—
75	C₂×C₈	C₂²⋊D₁₀₉	1	2	yes	—
76	C₄⋊C₄	—	218	2	no	—
77	C₄⋊C₄	—	218	2	no	—
78	C₄⋊C₄	D₄₃₆	1	2	yes	—
79	C₂²⋊C₄	—	2	1	no	—
80	C₄²	—	2	1	no	—
81	C₂³	#13182029	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	872	2	no	—
84	C₂×C₄	—	436	2	no	—
85	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
86	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₀₉	1	2	yes	—
88	C₂×C₄	#13182028	1	2	yes	—
89	C₈	—	2	2	no	—
90	C₈	—	4	1	no	—
91	C₈	—	218	4	no	—
92	C₂²	—	2	2	no	—
93	C₂²	#16812673	1	1	yes	characteristic
94	C₂²	#16812689	1	2	yes	—
95	C₄	—	4	2	no	—
96	C₄	—	872	2	no	—
97	C₄	#16812679	1	2	yes	—
98	C₄	#16812680	1	1	yes	characteristic
99	C₄	#16812697	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#20294956	1	1	yes	characteristic
102	C₂	#20295289	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295069

C₂

yes

maximal

C₂×#16812631

C₂

yes

maximal, characteristic

#20294794

C₂

yes

maximal

#20294791

C₂

yes

maximal, characteristic

#424×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

#16812632

—

#16812631

—

#16812630

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

—

C₂²×Dic₁₀₉

—

#13182013

—

#13182013

Q₈

yes

—

C₂×#11724784

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

D₄

yes

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₂³×C₁₀₉

#41

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

#41

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

#121

yes

characteristic

C₂²×C₁₀₉

#125

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊Q₈

yes

—

C₄×C₁₀₉

#123

yes

characteristic

C₄×C₁₀₉

C₄⋊Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#398

yes

characteristic

C₂×C₁₀₉

#471

yes

—

#2090

—

109

maximal, Sylow

C₁₀₉

#2090

yes

characteristic, complemented, Sylow

#428

—

109

—

C₂×#106

—

109

—

#424

D₁₀₉

yes

characteristic

#343

—

109

—

#341

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#117

C₂×D₁₀₉

yes

characteristic

C₂×C₄⋊C₄

—

109

—

#107

—

218

—

#106

—

218

—

#106

—

218

—

C₄⋊C₈

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

218

—

C₂×C₈

C₂²⋊D₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄⋊C₄

D₄₃₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#13182029

yes

characteristic

C₂×C₄

—

C₂×C₄

—

872

—

C₂×C₄

—

436

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

Q₈×D₁₀₉

yes

—

C₂×C₄

#13182028

yes

—

C₈

—

C₈

—

C₈

—

218

—

C₂²

—

C₂²

#16812673

yes

characteristic

C₂²

#16812689

yes

—

C₄

—

C₄

—

872

—

C₄

#16812679

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#20294956

yes

characteristic

102

C₂

#20295289

yes

—

	subgroup	G/H
center	C₂²	#16812673
commutator, cosocle	C₂²×C₄×C₁₀₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₀₉
Fitting	#424×C₁₀₉	C₂
socle	C₂²×C₁₀₉	#121
prime radical	C₂³×C₁₀₉	#41
hypercenter	#424	D₁₀₉
hypocenter	C₁₀₉	#2090
LCS	G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#16812673

commutator, cosocle

C₂²×C₄×C₁₀₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₀₉

Fitting

#424×C₁₀₉

C₂

C₂²×C₁₀₉

C₂³×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂²×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
52. C₁₀₉	#2090	semidirect

H ≅ G/N

type

52. C₁₀₉

#2090

semidirect

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

subgroup	index
#20295069	2
C₂×#16812631	2
#20294794	2
#20294791	2
#424×C₁₀₉	2
#2090	109

subgroup

index

#424×C₁₀₉

109

Subgroups — #28364542

Conjugacy classes of subgroups

Complements

Maximal subgroups