Group #28376245 — subgroups

Subgroups — 2₋¹⁺⁴×Dic₁₀₉ #28376245

Order 13952.

1492 conjugacy classes of subgroups; only the 1063 normal subgroups are shown.

normal subgroups only

This group has 1492 conjugacy classes of subgroups — too many to draw. Only its 1063 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	#20295509	C₂	1	10	yes	maximal, complemented
2	#20295504	C₂	1	5	yes	maximal, complemented
3	#41×Dic₁₀₉	C₂	1	10	yes	maximal, complemented
4	C₂×Q₈×Dic₁₀₉	C₂	1	5	yes	maximal, complemented
5	C₂×2₋¹⁺⁴×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₀₉	C₂²	1	10	yes	—
7	Q₈×Dic₁₀₉	C₂²	1	20	yes	—
8	Q₈×Dic₁₀₉	C₂²	1	10	yes	—
9	D₄×Dic₁₀₉	C₂²	1	10	yes	—
10	D₄×Dic₁₀₉	C₂²	1	30	yes	—
11	#16812658	C₂²	1	30	yes	—
12	C₂×C₄⋊Dic₁₀₉	C₂²	1	15	yes	—
13	C₂×C₄×Dic₁₀₉	C₂²	1	15	yes	—
14	2₋¹⁺⁴×C₁₀₉	C₄	1	16	yes	complemented
15	C₂×#41×C₁₀₉	C₂²	1	10	yes	—
16	C₂²×Q₈×C₁₀₉	C₂²	1	5	yes	—
17	C₂²×Dic₁₀₉	C₂³	1	5	yes	—
18	C₂²×Dic₁₀₉	C₂³	1	15	yes	—
19	C₄⋊Dic₁₀₉	C₂³	1	30	yes	—
20	C₄⋊Dic₁₀₉	C₂³	1	10	yes	—
21	C₂²⋊Dic₁₀₉	C₂³	1	20	yes	—
22	C₄×Dic₁₀₉	C₂³	1	10	yes	—
23	C₄×Dic₁₀₉	C₂³	1	30	yes	—
24	#41×C₁₀₉	C₂×C₄	1	80	yes	complemented
25	C₂×Q₈×C₁₀₉	C₂×C₄	1	40	yes	complemented
26	C₂×Q₈×C₁₀₉	C₂³	1	10	yes	—
27	C₂×D₄×C₁₀₉	C₂³	1	10	yes	—
28	C₂²×C₄×C₁₀₉	C₂³	1	15	yes	—
29	C₂³×C₁₀₉	C₂⁴	1	5	yes	—
30	Q₈×C₁₀₉	C₂²×C₄	1	40	yes	—
31	D₄×C₁₀₉	C₂²×C₄	1	40	yes	—
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	60	yes	—
33	C₂×C₄×C₁₀₉	C₂⁴	1	10	yes	—
34	C₂×Dic₁₀₉	—	2	10	no	—
35	C₂×Dic₁₀₉	—	4	15	no	—
36	C₂×Dic₁₀₉	C₂⁴	1	5	yes	—
37	C₂×Dic₁₀₉	C₂⁴	1	10	yes	—
38	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
39	C₂²×C₁₀₉	—	2	5	no	—
40	C₂²×C₁₀₉	—	2	5	no	—
41	C₂²×C₁₀₉	C₂³×C₄	1	10	yes	—
42	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂³×C₄	1	20	yes	—
44	Dic₁₀₉	—	2	5	no	—
45	Dic₁₀₉	—	2	10	no	—
46	Dic₁₀₉	2₋¹⁺⁴	1	2	yes	direct product factor
47	C₂×C₁₀₉	—	2	10	no	—
48	C₂×C₁₀₉	C₂⁴×C₄	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
50	C₂×C₁₀₉	#585	1	1	yes	characteristic
51	C₄×2₋¹⁺⁴	—	109	1	no	maximal, Sylow
52	C₁₀₉	C₄×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
53	C₂×2₋¹⁺⁴	D₁₀₉	1	1	yes	characteristic
54	#520	—	109	10	no	—
55	#519	—	109	5	no	—
56	C₄×#41	—	109	10	no	—
57	C₂×C₄×Q₈	—	109	5	no	—
58	2₋¹⁺⁴	Dic₁₀₉	1	16	yes	direct product factor
59	C₂×#41	C₂×D₁₀₉	1	10	yes	—
60	C₂²×Q₈	C₂×D₁₀₉	1	5	yes	—
61	C₄×Q₈	—	109	10	no	—
62	C₄×Q₈	—	109	20	no	—
63	C₄×Q₈	—	109	10	no	—
64	C₄×D₄	—	109	10	no	—
65	C₄×D₄	—	109	30	no	—
66	#117	—	109	30	no	—
67	C₂×C₄⋊C₄	—	109	15	no	—
68	C₂×C₄²	—	109	15	no	—
69	#41	C₂×Dic₁₀₉	1	80	yes	complemented
70	C₂×Q₈	C₂×Dic₁₀₉	1	40	yes	complemented
71	C₂×Q₈	C₂²×D₁₀₉	1	10	yes	—
72	C₂×D₄	C₂²×D₁₀₉	1	10	yes	—
73	C₂²×C₄	—	109	15	no	—
74	C₂²×C₄	—	109	5	no	—
75	C₂²×C₄	C₂²×D₁₀₉	1	15	yes	—
76	C₄⋊C₄	—	109	30	no	—
77	C₄⋊C₄	—	109	10	no	—
78	C₂²⋊C₄	—	109	20	no	—
79	C₄²	—	109	10	no	—
80	C₄²	—	109	30	no	—
81	C₂³	C₂³×D₁₀₉	1	5	yes	—
82	Q₈	C₂²×Dic₁₀₉	1	40	yes	—
83	D₄	C₂²×Dic₁₀₉	1	40	yes	—
84	C₂×C₄	—	109	1	no	—
85	C₂×C₄	—	109	5	no	—
86	C₂×C₄	—	109	10	no	—
87	C₂×C₄	—	436	15	no	—
88	C₂×C₄	—	218	10	no	—
89	C₂×C₄	C₂²×Dic₁₀₉	1	60	yes	—
90	C₂×C₄	C₂³×D₁₀₉	1	10	yes	—
91	C₂²	—	2	5	no	—
92	C₂²	—	2	5	no	—
93	C₂²	C₂³×Dic₁₀₉	1	10	yes	—
94	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
95	C₄	—	109	2	no	—
96	C₄	—	218	5	no	—
97	C₄	—	218	10	no	—
98	C₄	C₂³×Dic₁₀₉	1	20	yes	—
99	C₂	—	2	10	no	—
100	C₂	C₂⁴×Dic₁₀₉	1	1	yes	characteristic
101	C₂	2₋¹⁺⁴×D₁₀₉	1	1	yes	characteristic
102	C₂	#20295863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20295509

C₂

yes

maximal, complemented

#20295504

C₂

yes

maximal, complemented

#41×Dic₁₀₉

C₂

yes

maximal, complemented

C₂×Q₈×Dic₁₀₉

C₂

yes

maximal, complemented

C₂×2₋¹⁺⁴×C₁₀₉

C₂

yes

maximal, characteristic

Q₈×Dic₁₀₉

C₂²

yes

—

Q₈×Dic₁₀₉

C₂²

yes

—

Q₈×Dic₁₀₉

C₂²

yes

—

D₄×Dic₁₀₉

C₂²

yes

—

D₄×Dic₁₀₉

C₂²

yes

—

#16812658

C₂²

yes

—

C₂×C₄⋊Dic₁₀₉

C₂²

yes

—

C₂×C₄×Dic₁₀₉

C₂²

yes

—

2₋¹⁺⁴×C₁₀₉

C₄

yes

complemented

C₂×#41×C₁₀₉

C₂²

yes

—

C₂²×Q₈×C₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₂²×Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂³

yes

—

C₂²⋊Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

C₄×Dic₁₀₉

C₂³

yes

—

#41×C₁₀₉

C₂×C₄

yes

complemented

C₂×Q₈×C₁₀₉

C₂×C₄

yes

complemented

C₂×Q₈×C₁₀₉

C₂³

yes

—

C₂×D₄×C₁₀₉

C₂³

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂⁴

yes

—

Q₈×C₁₀₉

C₂²×C₄

yes

—

D₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂³×C₄

yes

—

C₂²×C₁₀₉

C₂⁵

yes

characteristic

C₄×C₁₀₉

C₂³×C₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

2₋¹⁺⁴

yes

direct product factor

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂⁴×C₄

yes

characteristic

C₂×C₁₀₉

C₂×2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₉

#585

yes

characteristic

C₄×2₋¹⁺⁴

—

109

maximal, Sylow

C₁₀₉

C₄×2₋¹⁺⁴

yes

characteristic, complemented, Sylow

C₂×2₋¹⁺⁴

D₁₀₉

yes

characteristic

#520

—

109

—

#519

—

109

—

C₄×#41

—

109

—

C₂×C₄×Q₈

—

109

—

2₋¹⁺⁴

Dic₁₀₉

yes

direct product factor

C₂×#41

C₂×D₁₀₉

yes

—

C₂²×Q₈

C₂×D₁₀₉

yes

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×D₄

—

109

—

C₄×D₄

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄²

—

109

—

#41

C₂×Dic₁₀₉

yes

complemented

C₂×Q₈

C₂×Dic₁₀₉

yes

complemented

C₂×Q₈

C₂²×D₁₀₉

yes

—

C₂×D₄

C₂²×D₁₀₉

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

C₂³×D₁₀₉

yes

—

Q₈

C₂²×Dic₁₀₉

yes

—

D₄

C₂²×Dic₁₀₉

yes

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×Dic₁₀₉

yes

—

C₂×C₄

C₂³×D₁₀₉

yes

—

C₂²

—

C₂²

—

C₂²

C₂³×Dic₁₀₉

yes

—

C₂²

C₂⁴×D₁₀₉

yes

characteristic

C₄

—

109

—

C₄

—

218

—

C₄

—

218

—

C₄

C₂³×Dic₁₀₉

yes

—

C₂

—

100

C₂

C₂⁴×Dic₁₀₉

yes

characteristic

101

C₂

2₋¹⁺⁴×D₁₀₉

yes

characteristic

102

C₂

#20295863

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator	C₂×C₁₀₉	C₂⁴×C₄
Fitting, prime radical	C₂×2₋¹⁺⁴×C₁₀₉	C₂
socle, cosocle	C₂²×C₁₀₉	C₂⁵
hypercenter	C₂×2₋¹⁺⁴	D₁₀₉
hypocenter	C₁₀₉	C₄×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₁₀₉

commutator

C₂×C₁₀₉

C₂⁴×C₄

Fitting, prime radical

C₂×2₋¹⁺⁴×C₁₀₉

C₂

socle, cosocle

C₂²×C₁₀₉

C₂⁵

hypercenter

C₂×2₋¹⁺⁴

D₁₀₉

hypocenter

C₁₀₉

C₄×2₋¹⁺⁴

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×2₋¹⁺⁴

N	H ≅ G/N	type
1. #20295509	C₂	semidirect
2. #20295504	C₂	semidirect
3. #41×Dic₁₀₉	C₂	semidirect
4. C₂×Q₈×Dic₁₀₉	C₂	semidirect
14. 2₋¹⁺⁴×C₁₀₉	C₄	semidirect
24. #41×C₁₀₉	C₂×C₄	semidirect
25. C₂×Q₈×C₁₀₉	C₂×C₄	semidirect
46. Dic₁₀₉	2₋¹⁺⁴	direct (N × H)
52. C₁₀₉	C₄×2₋¹⁺⁴	semidirect
58. 2₋¹⁺⁴	Dic₁₀₉	direct (N × H)
69. #41	C₂×Dic₁₀₉	semidirect
70. C₂×Q₈	C₂×Dic₁₀₉	semidirect

H ≅ G/N

type

1. #20295509

C₂

semidirect

2. #20295504

C₂

semidirect

3. #41×Dic₁₀₉

C₂

semidirect

4. C₂×Q₈×Dic₁₀₉

C₂

semidirect

14. 2₋¹⁺⁴×C₁₀₉

C₄

semidirect

24. #41×C₁₀₉

C₂×C₄

semidirect

25. C₂×Q₈×C₁₀₉

C₂×C₄

semidirect

46. Dic₁₀₉

2₋¹⁺⁴

direct (N × H)

52. C₁₀₉

C₄×2₋¹⁺⁴

semidirect

58. 2₋¹⁺⁴

Dic₁₀₉

direct (N × H)

69. #41

C₂×Dic₁₀₉

semidirect

70. C₂×Q₈

C₂×Dic₁₀₉

semidirect

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

subgroup	index
#20295509	2
#20295504	2
#41×Dic₁₀₉	2
C₂×Q₈×Dic₁₀₉	2
C₂×2₋¹⁺⁴×C₁₀₉	2
C₄×2₋¹⁺⁴	109

subgroup

index

#20295509

#20295504

#41×Dic₁₀₉

C₂×Q₈×Dic₁₀₉

C₂×2₋¹⁺⁴×C₁₀₉

C₄×2₋¹⁺⁴

109

Subgroups — 2₋¹⁺⁴×Dic₁₀₉ #28376245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2−1+4×Dic109 #28376245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 2₋¹⁺⁴×Dic₁₀₉ #28376245