Group #20295788 — subgroups

Subgroups — C₂×#16812705 #20295788

Order 6976.

266 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 151 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×#13182029	C₂	1	1	yes	maximal, characteristic
2	#16812705	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13182012	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₁₀₉	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#13182029	C₄	1	4	yes	—
8	#13182029	C₂²	1	4	yes	—
9	C₂×Dic₂₁₈	C₄	1	1	yes	characteristic
10	C₂×D₄₃₆	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₀₉	C₄	1	2	yes	—
12	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
13	#13182013	C₂²	1	4	yes	complemented
14	#13182012	C₂²	1	8	yes	complemented
15	C₂×#11724784	C₂²	1	2	yes	—
16	C₈×D₁₀₉	C₂²	1	8	yes	complemented
17	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂²	1	4	yes	complemented
21	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
27	C₂²⋊D₁₀₉	C₂×C₄	1	8	yes	—
28	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
29	D₄₃₆	C₂×C₄	1	4	yes	—
30	C₄×D₁₀₉	C₂×C₄	1	4	yes	—
31	C₄×D₁₀₉	C₂³	1	4	yes	—
32	Dic₂₁₈	C₂×C₄	1	4	yes	—
33	C₈×C₁₀₉	C₂³	1	2	yes	—
34	C₈×C₁₀₉	C₂³	1	2	yes	—
35	#11724784	C₂³	1	4	yes	—
36	C₂²×C₁₀₉	—	2	2	no	—
37	C₂²×C₁₀₉	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	4	no	—
40	C₂×D₁₀₉	C₂²×C₄	1	4	yes	—
41	C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
42	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
44	Dic₁₀₉	C₂²×C₄	1	4	yes	—
45	C₂×C₁₀₉	—	2	2	no	—
46	C₂×C₁₀₉	#131	1	2	yes	complemented
47	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
48	D₁₀₉	—	2	4	no	—
49	C₁₀₉	C₂×#131	1	1	yes	characteristic, complemented, Sylow
50	C₂×#131	—	109	1	no	maximal, Sylow
51	C₂×#41	—	109	1	no	—
52	#131	—	109	8	no	—
53	C₂×M₁₆	—	109	1	no	—
54	C₂×M₁₆	—	109	2	no	—
55	C₂²×C₈	—	109	2	no	—
56	C₂²×C₈	D₁₀₉	1	1	yes	characteristic, complemented
57	#41	—	109	4	no	—
58	#41	—	109	4	no	—
59	C₂×Q₈	—	109	1	no	—
60	C₂×D₄	—	109	2	no	—
61	C₂×D₄	—	109	1	no	—
62	C₂²×C₄	—	109	2	no	—
63	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
64	M₁₆	—	109	4	no	—
65	M₁₆	—	109	8	no	—
66	C₂×C₈	—	109	2	no	—
67	C₂×C₈	—	109	8	no	—
68	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
69	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
70	C₂×C₈	C₂×D₁₀₉	1	4	yes	complemented
71	C₂³	—	109	2	no	—
72	C₂³	C₄×D₁₀₉	1	1	yes	characteristic
73	Q₈	—	109	4	no	—
74	D₄	—	109	8	no	—
75	D₄	—	109	4	no	—
76	C₂×C₄	—	109	2	no	—
77	C₂×C₄	—	109	4	no	—
78	C₂×C₄	—	109	4	no	—
79	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
81	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
83	C₈	—	109	4	no	—
84	C₈	C₂²×D₁₀₉	1	2	yes	—
85	C₈	C₂²×D₁₀₉	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	109	4	no	—
88	C₂²	—	218	4	no	—
89	C₂²	C₂×C₄×D₁₀₉	1	2	yes	—
90	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
91	C₄	—	109	4	no	—
92	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
93	C₄	C₂×C₄×D₁₀₉	1	2	yes	—
94	C₄	C₂³×D₁₀₉	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	218	4	no	—
97	C₂	#16812705	1	2	yes	direct product factor
98	C₂	C₂²×C₄×D₁₀₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182029

C₂

yes

maximal, characteristic

#16812705

C₂

yes

maximal, direct product factor

C₂×#13182013

C₂

yes

maximal, characteristic, complemented

C₂×#13182012

C₂

yes

maximal, complemented

C₂×C₈×D₁₀₉

C₂

yes

maximal, complemented

C₂²×C₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

C₄

yes

—

#13182029

C₂²

yes

—

C₂×Dic₂₁₈

C₄

yes

characteristic

C₂×D₄₃₆

C₄

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₄

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

#13182013

C₂²

yes

complemented

#13182012

C₂²

yes

complemented

C₂×#11724784

C₂²

yes

—

C₈×D₁₀₉

C₂²

yes

complemented

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂²⋊D₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

D₄₃₆

C₂×C₄

yes

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

C₂³

yes

—

Dic₂₁₈

C₂×C₄

yes

—

C₈×C₁₀₉

C₂³

yes

—

C₈×C₁₀₉

C₂³

yes

—

#11724784

C₂³

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²×C₄

yes

—

C₄×C₁₀₉

C₂²×C₄

yes

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂⁴

yes

characteristic

Dic₁₀₉

C₂²×C₄

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#131

yes

complemented

C₂×C₁₀₉

C₂³×C₄

yes

characteristic

D₁₀₉

—

C₁₀₉

C₂×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

109

maximal, Sylow

C₂×#41

—

109

—

#131

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₂²×C₈

D₁₀₉

yes

characteristic, complemented

#41

—

109

—

#41

—

109

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

M₁₆

—

109

—

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₈

C₂×D₁₀₉

yes

complemented

C₂³

—

109

—

C₂³

C₄×D₁₀₉

yes

characteristic

Q₈

—

109

—

D₄

—

109

—

D₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₄×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

—

C₈

—

109

—

C₈

C₂²×D₁₀₉

yes

—

C₈

C₂²×D₁₀₉

yes

—

C₂²

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₂×C₄×D₁₀₉

yes

—

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

C₄

—

109

—

C₄

C₂×C₄×D₁₀₉

yes

characteristic

C₄

C₂×C₄×D₁₀₉

yes

—

C₄

C₂³×D₁₀₉

yes

characteristic

C₂

—

C₂

—

218

—

C₂

#16812705

yes

direct product factor

C₂

C₂²×C₄×D₁₀₉

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×D₁₀₉
Fitting	C₂²×C₈×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×C₄
hypercenter	C₂²×C₈	D₁₀₉
hypocenter	C₁₀₉	C₂×#131
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₁₀₉

commutator

C₂×C₁₀₉

C₂³×C₄

Frattini

C₄

C₂³×D₁₀₉

Fitting

C₂²×C₈×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×C₄

hypercenter

C₂²×C₈

D₁₀₉

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₈ ◁ C₂²×C₈

N	H ≅ G/N	type
2. #16812705	C₂	direct (N × H)
3. C₂×#13182013	C₂	semidirect
4. C₂×#13182012	C₂	semidirect
5. C₂×C₈×D₁₀₉	C₂	semidirect
6. C₂²×C₈×C₁₀₉	C₂	semidirect
13. #13182013	C₂²	semidirect
14. #13182012	C₂²	semidirect
16. C₈×D₁₀₉	C₂²	semidirect
20. C₂×C₈×C₁₀₉	C₂²	semidirect
46. C₂×C₁₀₉	#131	semidirect
49. C₁₀₉	C₂×#131	semidirect
56. C₂²×C₈	D₁₀₉	semidirect
70. C₂×C₈	C₂×D₁₀₉	semidirect
97. C₂	#16812705	direct (N × H)

H ≅ G/N

type

2. #16812705

C₂

direct (N × H)

3. C₂×#13182013

C₂

semidirect

4. C₂×#13182012

C₂

semidirect

5. C₂×C₈×D₁₀₉

C₂

semidirect

6. C₂²×C₈×C₁₀₉

C₂

semidirect

13. #13182013

C₂²

semidirect

14. #13182012

C₂²

semidirect

16. C₈×D₁₀₉

C₂²

semidirect

20. C₂×C₈×C₁₀₉

C₂²

semidirect

46. C₂×C₁₀₉

#131

semidirect

49. C₁₀₉

C₂×#131

semidirect

56. C₂²×C₈

D₁₀₉

semidirect

70. C₂×C₈

C₂×D₁₀₉

semidirect

97. C₂

#16812705

direct (N × H)

Subgroups — C₂×#16812705 #20295788

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812705 #20295788

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812705 #20295788