Group #20295521 — subgroups

Subgroups — C₂×#16812675 #20295521

Order 6976.

322 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 135 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×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#16812675	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13182008	C₂	1	2	yes	maximal, complemented
5	C₂×#13182006	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
9	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₀₉	—	2	4	no	—
11	C₂×C₄×D₁₀₉	C₂²	1	4	yes	—
12	C₄⋊Dic₁₀₉	C₂²	1	4	yes	complemented
13	#13182008	C₂²	1	8	yes	complemented
14	#13182006	C₂²	1	8	yes	—
15	C₂×Q₈×C₁₀₉	—	2	4	no	—
16	C₂×Q₈×C₁₀₉	C₂²	1	4	yes	complemented
17	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
18	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
20	C₂²×D₁₀₉	—	2	2	no	—
21	C₂²×D₁₀₉	—	2	2	no	—
22	C₂²×D₁₀₉	Q₈	1	4	yes	complemented
23	C₂²×D₁₀₉	C₂³	1	2	yes	—
24	Q₈×C₁₀₉	—	2	8	no	—
25	C₂×C₄×C₁₀₉	—	2	4	no	—
26	C₂×C₄×C₁₀₉	D₄	1	4	yes	—
27	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
28	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
29	C₂×Dic₁₀₉	—	2	2	no	—
30	C₂×Dic₁₀₉	—	2	4	no	—
31	C₂×Dic₁₀₉	C₂³	1	4	yes	—
32	C₂×Dic₁₀₉	C₂³	1	2	yes	—
33	C₄×D₁₀₉	—	2	8	no	—
34	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
35	C₂²×C₁₀₉	C₂×Q₈	1	2	yes	—
36	C₂²×C₁₀₉	#41	1	2	yes	—
37	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₀₉	—	2	8	no	—
39	C₂×D₁₀₉	—	2	2	no	—
40	C₂×D₁₀₉	—	2	2	no	—
41	C₂×D₁₀₉	C₂×Q₈	1	4	yes	complemented
42	C₄×C₁₀₉	—	2	4	no	—
43	C₄×C₁₀₉	C₂×D₄	1	4	yes	—
44	Dic₁₀₉	—	2	2	no	—
45	Dic₁₀₉	—	2	4	no	—
46	C₂×C₁₀₉	C₂²⋊Q₈	1	4	yes	complemented
47	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
50	D₁₀₉	—	2	4	no	—
51	C₁₀₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₂²⋊Q₈	—	109	1	no	maximal, Sylow
53	C₂²×Q₈	D₁₀₉	1	1	yes	characteristic, complemented
54	C₂³×C₄	—	109	1	no	—
55	C₂²⋊Q₈	—	109	8	no	—
56	C₂×C₄⋊C₄	—	109	2	no	—
57	C₂×C₄⋊C₄	—	109	1	no	—
58	C₂×C₂²⋊C₄	—	109	2	no	—
59	C₂⁴	—	109	1	no	—
60	C₂×Q₈	—	2	4	no	—
61	C₂×Q₈	C₂×D₁₀₉	1	4	yes	complemented
62	C₂²×C₄	—	109	2	no	—
63	C₂²×C₄	—	109	1	no	—
64	C₂²×C₄	—	109	4	no	—
65	C₂²×C₄	—	218	4	no	—
66	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
67	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
68	C₄⋊C₄	—	109	8	no	—
69	C₄⋊C₄	—	109	4	no	—
70	C₂²⋊C₄	—	109	8	no	—
71	C₂³	—	109	4	no	—
72	C₂³	—	109	2	no	—
73	C₂³	—	218	2	no	—
74	C₂³	—	218	2	no	—
75	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
76	Q₈	—	2	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	109	4	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	218	2	no	—
81	C₂×C₄	—	218	4	no	—
82	C₂×C₄	—	218	8	no	—
83	C₂×C₄	C₂²⋊D₁₀₉	1	4	yes	—
84	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
85	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
86	C₂²	—	109	4	no	—
87	C₂²	—	218	8	no	—
88	C₂²	—	218	2	no	—
89	C₂²	—	218	2	no	—
90	C₂²	C₂×C₂²⋊D₁₀₉	1	2	yes	—
91	C₂²	Q₈×D₁₀₉	1	2	yes	—
92	C₂²	#13182030	1	2	yes	—
93	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	218	2	no	—
96	C₄	—	218	4	no	—
97	C₄	C₂×C₂²⋊D₁₀₉	1	4	yes	—
98	C₂	—	218	4	no	—
99	C₂	#16812675	1	4	yes	direct product factor
100	C₂	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
101	C₂	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
102	C₂	C₂×#13182030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₀₉

C₂

yes

maximal, characteristic

#16812675

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182008

C₂

yes

maximal, complemented

C₂×#13182006

C₂

yes

maximal

C₂²×Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

complemented

#13182008

C₂²

yes

complemented

#13182006

C₂²

yes

—

C₂×Q₈×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂²

yes

complemented

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

Q₈

yes

complemented

C₂²×D₁₀₉

C₂³

yes

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₄×D₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂×Q₈

yes

—

C₂²×C₁₀₉

#41

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×Q₈

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂²⋊Q₈

yes

complemented

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

D₁₀₉

—

C₁₀₉

C₂×C₂²⋊Q₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

109

maximal, Sylow

C₂²×Q₈

D₁₀₉

yes

characteristic, complemented

C₂³×C₄

—

109

—

C₂²⋊Q₈

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂⁴

—

109

—

C₂×Q₈

—

C₂×Q₈

C₂×D₁₀₉

yes

complemented

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²⋊D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₂²⋊D₁₀₉

yes

—

C₂²

Q₈×D₁₀₉

yes

—

C₂²

#13182030

yes

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

218

—

C₄

—

218

—

C₄

C₂×C₂²⋊D₁₀₉

yes

—

C₂

—

218

—

C₂

#16812675

yes

direct product factor

100

C₂

C₂²×C₂²⋊D₁₀₉

yes

characteristic

101

C₂

C₂×Q₈×D₁₀₉

yes

characteristic

102

C₂

C₂×#13182030

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₀₉
commutator	C₂²×C₁₀₉	C₂⁴
Frattini	C₂²	C₂³×D₁₀₉
Fitting	C₂²×Q₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂²×Q₈	D₁₀₉
hypocenter	C₁₀₉	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²×D₁₀₉

commutator

C₂²×C₁₀₉

C₂⁴

Frattini

C₂²

C₂³×D₁₀₉

Fitting

C₂²×Q₈×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂²×Q₈

D₁₀₉

hypocenter

C₁₀₉

C₂×C₂²⋊Q₈

LCS

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #16812675	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₀₉	C₂	semidirect
4. C₂×#13182008	C₂	semidirect
6. C₂²×Q₈×C₁₀₉	C₂	semidirect
12. C₄⋊Dic₁₀₉	C₂²	semidirect
13. #13182008	C₂²	semidirect
16. C₂×Q₈×C₁₀₉	C₂²	semidirect
22. C₂²×D₁₀₉	Q₈	semidirect
41. C₂×D₁₀₉	C₂×Q₈	semidirect
46. C₂×C₁₀₉	C₂²⋊Q₈	semidirect
51. C₁₀₉	C₂×C₂²⋊Q₈	semidirect
53. C₂²×Q₈	D₁₀₉	semidirect
61. C₂×Q₈	C₂×D₁₀₉	semidirect
99. C₂	#16812675	direct (N × H)

H ≅ G/N

type

2. #16812675

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₀₉

C₂

semidirect

4. C₂×#13182008

C₂

semidirect

6. C₂²×Q₈×C₁₀₉

C₂

semidirect

12. C₄⋊Dic₁₀₉

C₂²

semidirect

13. #13182008

C₂²

semidirect

16. C₂×Q₈×C₁₀₉

C₂²

semidirect

22. C₂²×D₁₀₉

Q₈

semidirect

41. C₂×D₁₀₉

C₂×Q₈

semidirect

46. C₂×C₁₀₉

C₂²⋊Q₈

semidirect

51. C₁₀₉

C₂×C₂²⋊Q₈

semidirect

53. C₂²×Q₈

D₁₀₉

semidirect

61. C₂×Q₈

C₂×D₁₀₉

semidirect

99. C₂

#16812675

direct (N × H)

Subgroups — C₂×#16812675 #20295521

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812675 #20295521

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812675 #20295521