Group #28366241 — subgroups

Subgroups — C₂×#20294945 #28366241

Order 13952.

908 conjugacy classes of subgroups; only the 285 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 285 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	C₂²×#13182008	C₂	1	3	yes	maximal
2	#20294945	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16812606	C₂	1	3	yes	maximal
4	C₂³×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic
6	C₂³×Dic₁₀₉	C₂²	1	3	yes	—
7	C₂×#13182008	—	2	12	no	—
8	C₂×#13182008	C₂²	1	12	yes	—
9	#16812606	C₂²	1	12	yes	—
10	C₂²×Q₈×C₁₀₉	—	2	12	no	—
11	C₂²×Q₈×C₁₀₉	C₂²	1	4	yes	—
12	C₂³×C₄×C₁₀₉	C₂²	1	3	yes	—
13	C₂²×Dic₁₀₉	—	2	6	no	—
14	C₂²×Dic₁₀₉	—	2	6	no	—
15	C₂²×Dic₁₀₉	—	2	6	no	—
16	C₂²×Dic₁₀₉	Q₈	1	12	yes	complemented
17	C₂²×Dic₁₀₉	C₂³	1	2	yes	—
18	C₂²×Dic₁₀₉	C₂³	1	6	yes	—
19	#13182008	—	2	24	no	—
20	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
21	C₂×Q₈×C₁₀₉	—	2	32	no	—
22	C₂×Q₈×C₁₀₉	—	2	24	no	—
23	C₂²×C₄×C₁₀₉	—	2	12	no	—
24	C₂²×C₄×C₁₀₉	D₄	1	12	yes	—
25	C₂²×C₄×C₁₀₉	C₂³	1	6	yes	—
26	C₂³×C₁₀₉	C₂×D₄	1	6	yes	—
27	C₂³×C₁₀₉	C₂×Q₈	1	6	yes	—
28	C₂³×C₁₀₉	#41	1	2	yes	—
29	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
30	Q₈×C₁₀₉	—	2	32	no	—
31	C₂×C₄×C₁₀₉	—	2	24	no	—
32	C₂×C₄×C₁₀₉	—	2	12	no	—
33	C₂×C₄×C₁₀₉	C₂×D₄	1	12	yes	—
34	C₂×Dic₁₀₉	—	2	24	no	—
35	C₂×Dic₁₀₉	—	2	6	no	—
36	C₂×Dic₁₀₉	—	2	6	no	—
37	C₂×Dic₁₀₉	—	2	6	no	—
38	C₂×Dic₁₀₉	—	4	4	no	—
39	C₂×Dic₁₀₉	C₂×Q₈	1	12	yes	complemented
40	C₂²×C₁₀₉	#120	1	4	yes	—
41	C₂²×C₁₀₉	C₂²⋊Q₈	1	12	yes	—
42	C₂²×C₁₀₉	C₄⋊Q₈	1	12	yes	—
43	C₂²×C₁₀₉	C₂²×D₄	1	3	yes	—
44	C₂²×C₁₀₉	C₂²×Q₈	1	3	yes	—
45	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
46	C₄×C₁₀₉	—	2	12	no	—
47	Dic₁₀₉	—	2	12	no	—
48	Dic₁₀₉	—	4	2	no	—
49	C₂×C₁₀₉	#395	1	8	yes	complemented
50	C₂×C₁₀₉	C₂×#120	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂×C₂²⋊Q₈	1	3	yes	—
52	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	3	yes	—
53	C₂×#395	—	109	1	no	maximal, Sylow
54	C₁₀₉	C₂×#395	1	1	yes	characteristic, complemented, Sylow
55	C₂³×Q₈	D₁₀₉	1	1	yes	characteristic
56	C₂²×C₄⋊C₄	—	109	3	no	—
57	#395	—	109	8	no	—
58	C₂×#95	—	109	3	no	—
59	C₂²×Q₈	—	2	12	no	—
60	C₂²×Q₈	C₂×D₁₀₉	1	4	yes	—
61	C₂³×C₄	—	109	3	no	—
62	C₂³×C₄	—	109	1	no	—
63	C₂³×C₄	C₂×D₁₀₉	1	3	yes	—
64	C₂×C₄⋊C₄	—	109	12	no	—
65	C₂×C₄⋊C₄	—	218	12	no	—
66	#95	—	109	12	no	—
67	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
68	C₂×Q₈	—	2	32	no	—
69	C₂×Q₈	—	2	24	no	—
70	C₂²×C₄	—	2	12	no	—
71	C₂²×C₄	—	109	12	no	—
72	C₂²×C₄	—	109	6	no	—
73	C₂²×C₄	—	109	2	no	—
74	C₂²×C₄	—	218	6	no	—
75	C₂²×C₄	—	218	6	no	—
76	C₂²×C₄	—	218	6	no	—
77	C₂²×C₄	C₂²⋊D₁₀₉	1	12	yes	—
78	C₂²×C₄	C₂²×D₁₀₉	1	6	yes	—
79	C₄⋊C₄	—	218	24	no	—
80	C₂³	C₂×C₂²⋊D₁₀₉	1	6	yes	—
81	C₂³	Q₈×D₁₀₉	1	6	yes	—
82	C₂³	#13182030	1	2	yes	—
83	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
84	Q₈	—	2	32	no	—
85	C₂×C₄	—	2	24	no	—
86	C₂×C₄	—	2	12	no	—
87	C₂×C₄	—	109	12	no	—
88	C₂×C₄	—	436	4	no	—
89	C₂×C₄	—	218	6	no	—
90	C₂×C₄	—	218	24	no	—
91	C₂×C₄	—	218	6	no	—
92	C₂×C₄	—	218	6	no	—
93	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	12	yes	—
94	C₂²	#16812668	1	4	yes	—
95	C₂²	#16812675	1	12	yes	—
96	C₂²	#16812697	1	12	yes	—
97	C₂²	C₂²×C₂²⋊D₁₀₉	1	3	yes	—
98	C₂²	C₂×Q₈×D₁₀₉	1	3	yes	—
99	C₂²	C₂×#13182030	1	1	yes	characteristic
100	C₄	—	2	12	no	—
101	C₄	—	436	2	no	—
102	C₄	—	218	12	no	—
103	C₂	#20294945	1	8	yes	direct product factor
104	C₂	C₂×#16812668	1	1	yes	characteristic
105	C₂	C₂×#16812675	1	3	yes	—
106	C₂	C₂×#16812697	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13182008

C₂

yes

maximal

#20294945

C₂

yes

maximal, direct product factor

C₂×#16812606

C₂

yes

maximal

C₂³×Q₈×C₁₀₉

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₉

C₂²

yes

characteristic

C₂³×Dic₁₀₉

C₂²

yes

—

C₂×#13182008

—

C₂×#13182008

C₂²

yes

—

#16812606

C₂²

yes

—

C₂²×Q₈×C₁₀₉

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₂³×C₄×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

—

C₂²×Dic₁₀₉

Q₈

yes

complemented

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

#13182008

—

C₂⁴×C₁₀₉

C₂³

yes

characteristic

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

—

C₂³×C₁₀₉

C₂×Q₈

yes

—

C₂³×C₁₀₉

#41

yes

—

C₂³×C₁₀₉

C₂⁴

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×Q₈

yes

complemented

C₂²×C₁₀₉

#120

yes

—

C₂²×C₁₀₉

C₂²⋊Q₈

yes

—

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

—

C₂²×C₁₀₉

C₂²×Q₈

yes

—

C₂²×C₁₀₉

C₂×#41

yes

characteristic

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

C₂×C₂²⋊Q₈

yes

—

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

—

C₂×#395

—

109

maximal, Sylow

C₁₀₉

C₂×#395

yes

characteristic, complemented, Sylow

C₂³×Q₈

D₁₀₉

yes

characteristic

C₂²×C₄⋊C₄

—

109

—

#395

—

109

—

C₂×#95

—

109

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₂³×C₄

—

109

—

C₂³×C₄

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

218

—

#95

—

109

—

C₂⁴

C₂²×D₁₀₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

218

—

C₂³

C₂×C₂²⋊D₁₀₉

yes

—

C₂³

Q₈×D₁₀₉

yes

—

C₂³

#13182030

yes

—

C₂³

C₂³×D₁₀₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

C₂²

#16812668

yes

—

C₂²

#16812675

yes

—

C₂²

#16812697

yes

—

C₂²

C₂²×C₂²⋊D₁₀₉

yes

—

C₂²

C₂×Q₈×D₁₀₉

yes

—

C₂²

C₂×#13182030

yes

characteristic

100

C₄

—

101

C₄

—

436

—

102

C₄

—

218

—

103

C₂

#20294945

yes

direct product factor

104

C₂

C₂×#16812668

yes

characteristic

105

C₂

C₂×#16812675

yes

—

106

C₂

C₂×#16812697

yes

—

	subgroup	G/H
center	C₂⁴	C₂²×D₁₀₉
commutator, cosocle	C₂³×C₁₀₉	C₂⁴
Frattini	C₂³	C₂³×D₁₀₉
Fitting	C₂³×Q₈×C₁₀₉	C₂
socle, prime radical	C₂⁴×C₁₀₉	C₂³
hypercenter	C₂³×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#395
LCS	G ▷ C₂³×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

subgroup

G/H

center

C₂⁴

C₂²×D₁₀₉

commutator, cosocle

C₂³×C₁₀₉

C₂⁴

Frattini

C₂³

C₂³×D₁₀₉

Fitting

C₂³×Q₈×C₁₀₉

C₂

socle, prime radical

C₂⁴×C₁₀₉

C₂³

hypercenter

C₂³×Q₈

D₁₀₉

C₁₀₉

G ▷ C₂³×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂⁴ ◁ C₂³×Q₈

N	H ≅ G/N	type
2. #20294945	C₂	direct (N × H)
16. C₂²×Dic₁₀₉	Q₈	semidirect
39. C₂×Dic₁₀₉	C₂×Q₈	semidirect
49. C₂×C₁₀₉	#395	semidirect
54. C₁₀₉	C₂×#395	semidirect
103. C₂	#20294945	direct (N × H)

H ≅ G/N

type

2. #20294945

C₂

direct (N × H)

16. C₂²×Dic₁₀₉

Q₈

semidirect

39. C₂×Dic₁₀₉

C₂×Q₈

semidirect

49. C₂×C₁₀₉

#395

semidirect

54. C₁₀₉

C₂×#395

semidirect

103. C₂

#20294945

direct (N × H)

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

subgroup	index
C₂²×#13182008	2
#20294945	2
C₂×#16812606	2
C₂³×Q₈×C₁₀₉	2
C₂×#395	109

subgroup

index

C₂³×Q₈×C₁₀₉

109

Subgroups — C₂×#20294945 #28366241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#20294945 #28366241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20294945 #28366241