Group #28373536 — subgroups

Subgroups — C₂×#20295428 #28373536

Order 13952.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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	C₂×#16812691	C₂	1	2	yes	maximal
2	#20295428	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812633	C₂	1	2	yes	maximal
4	C₂×#16812631	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	#16812691	—	2	4	no	—
7	#16812691	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
9	C₂×#13182008	—	2	4	no	—
10	#16812633	C₂²	1	8	yes	—
11	#16812631	C₂²	1	8	yes	—
12	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₀₉	C₂²	1	4	yes	—
16	C₂²×Dic₁₀₉	—	2	4	no	—
17	C₄⋊Dic₁₀₉	—	2	4	no	—
18	C₄⋊Dic₁₀₉	C₂³	1	8	yes	—
19	#13182008	—	2	16	no	—
20	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
21	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
23	C₂×C₈×C₁₀₉	Q₈	1	4	yes	—
24	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
25	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
26	C₄²×C₁₀₉	D₄	1	2	yes	—
27	C₄²×C₁₀₉	C₂³	1	2	yes	—
28	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₁₀₉	—	2	4	no	—
30	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₁₀₉	—	2	8	no	—
36	C₂×Dic₁₀₉	—	4	8	no	—
37	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
38	C₈×C₁₀₉	C₂×Q₈	1	4	yes	—
39	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₁₀₉	#135	1	4	yes	—
41	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₁₀₉	—	2	4	no	—
43	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
44	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
46	Dic₁₀₉	—	4	8	no	—
47	C₂×C₁₀₉	#499	1	4	yes	complemented
48	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂×#135	1	2	yes	—
50	C₂×#499	—	109	1	no	maximal, Sylow
51	C₁₀₉	C₂×#499	1	1	yes	characteristic, complemented, Sylow
52	C₂×#125	—	109	2	no	—
53	#499	—	109	8	no	—
54	C₂×#107	—	109	2	no	—
55	C₂×#106	—	109	2	no	—
56	C₂×C₄×C₈	D₁₀₉	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
59	#125	—	109	8	no	—
60	#125	—	218	4	no	—
61	C₂×C₄⋊C₄	—	109	4	no	—
62	C₂×C₄⋊C₄	—	218	4	no	—
63	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
64	#107	—	109	8	no	—
65	#106	—	109	8	no	—
66	C₄×C₈	C₂×D₁₀₉	1	4	yes	—
67	C₂²×C₄	—	218	4	no	—
68	C₂²×C₄	D₄₃₆	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
70	C₂×C₈	Dic₂₁₈	1	4	yes	—
71	C₂×C₈	Dic₂₁₈	1	4	yes	—
72	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
73	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
74	C₄⋊C₄	—	109	8	no	—
75	C₄⋊C₄	—	218	16	no	—
76	C₄⋊C₄	—	218	4	no	—
77	C₄²	D₄₃₆	1	2	yes	—
78	C₄²	C₂²×D₁₀₉	1	2	yes	—
79	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	436	8	no	—
82	C₂×C₄	—	218	8	no	—
83	C₂×C₄	C₂×D₄₃₆	1	4	yes	—
84	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
86	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
87	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
88	C₈	C₂×Dic₂₁₈	1	4	yes	—
89	C₈	C₂×Dic₂₁₈	1	4	yes	—
90	C₂²	#16812699	1	2	yes	—
91	C₂²	#16812721	1	4	yes	—
92	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	436	8	no	—
95	C₄	#16812699	1	2	yes	—
96	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
97	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
98	C₂	#20295428	1	4	yes	direct product factor
99	C₂	C₂×#16812699	1	1	yes	characteristic
100	C₂	C₂×#16812721	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812691

C₂

yes

maximal

#20295428

C₂

yes

maximal, direct product factor

C₂×#16812633

C₂

yes

maximal

C₂×#16812631

C₂

yes

maximal

C₂×C₄×C₈×C₁₀₉

C₂

yes

maximal, characteristic

#16812691

—

#16812691

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

#16812633

C₂²

yes

—

#16812631

C₂²

yes

—

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₄×C₈×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

Q₈

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₈×C₁₀₉

C₂×Q₈

yes

—

C₈×C₁₀₉

C₂×Q₈

yes

—

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

#135

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

#499

yes

complemented

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#135

yes

—

C₂×#499

—

109

maximal, Sylow

C₁₀₉

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#125

—

109

—

#499

—

109

—

C₂×#107

—

109

—

C₂×#106

—

109

—

C₂×C₄×C₈

D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#125

—

109

—

#125

—

218

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#106

—

109

—

C₄×C₈

C₂×D₁₀₉

yes

—

C₂²×C₄

—

218

—

C₂²×C₄

D₄₃₆

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

Dic₂₁₈

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

—

C₂×C₈

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

—

218

—

C₄²

D₄₃₆

yes

—

C₄²

C₂²×D₁₀₉

yes

—

C₂³

C₂×D₄₃₆

yes

characteristic

C₂×C₄

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂×D₄₃₆

yes

—

C₂×C₄

C₂×D₄₃₆

yes

characteristic

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂×C₄

C₂×Dic₂₁₈

yes

—

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

C₈

C₂×Dic₂₁₈

yes

—

C₈

C₂×Dic₂₁₈

yes

—

C₂²

#16812699

yes

—

C₂²

#16812721

yes

—

C₂²

C₂²×D₄₃₆

yes

characteristic

C₄

—

C₄

—

436

—

C₄

#16812699

yes

—

C₄

C₂²×Dic₂₁₈

yes

characteristic

C₄

C₂²×Dic₂₁₈

yes

characteristic

C₂

#20295428

yes

direct product factor

C₂

C₂×#16812699

yes

characteristic

100

C₂

C₂×#16812721

yes

—

	subgroup	G/H
center	C₂³	C₂×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×C₄×C₈×C₁₀₉	C₂
socle, prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#499
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₄₃₆

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×C₄×C₈×C₁₀₉

C₂

socle, prime radical

C₂³×C₁₀₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₀₉

hypocenter

C₁₀₉

C₂×#499

LCS

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #20295428	C₂	direct (N × H)
47. C₂×C₁₀₉	#499	semidirect
51. C₁₀₉	C₂×#499	semidirect
98. C₂	#20295428	direct (N × H)

H ≅ G/N

type

2. #20295428

C₂

direct (N × H)

47. C₂×C₁₀₉

#499

semidirect

51. C₁₀₉

C₂×#499

semidirect

98. C₂

#20295428

direct (N × H)

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

subgroup	index
C₂×#16812691	2
#20295428	2
C₂×#16812633	2
C₂×#16812631	2
C₂×C₄×C₈×C₁₀₉	2
C₂×#499	109

subgroup

index

C₂×C₄×C₈×C₁₀₉

109

Subgroups — C₂×#20295428 #28373536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20295428 #28373536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295428 #28373536