Group #20296101 — subgroups

Subgroups — C₂×#16812786 #20296101

Order 6976.

266 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 140 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	#16812786	C₂	1	8	yes	maximal, direct product factor
2	C₂×#13182039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13182038	C₂	1	2	yes	maximal, complemented
4	C₂×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#11724786	C₂	1	2	yes	maximal, complemented
6	C₂×#13182028	C₂	1	1	yes	maximal, characteristic
7	#13182039	C₂²	1	4	yes	complemented
8	#13182038	C₂²	1	8	yes	complemented
9	D₁₀₉⋊C₈	C₂²	1	4	yes	complemented
10	C₂×#11724786	C₂²	1	1	yes	characteristic
11	C₂×#11724786	C₂²	1	2	yes	—
12	C₂×#11724786	C₂²	1	1	yes	characteristic
13	C₂×#11724786	C₂²	1	8	yes	complemented
14	#13182028	C₄	1	4	yes	—
15	#13182028	C₂²	1	4	yes	—
16	C₂×Dic₂₁₈	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₁₀₉	C₄	1	2	yes	—
18	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
19	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₁₀₉	C₄	1	1	yes	characteristic
21	C₂³×C₁₀₉	C₂×C₄	1	2	yes	—
22	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	D₄×C₁₀₉	C₂×C₄	1	4	yes	—
24	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊D₁₀₉	C₂×C₄	1	8	yes	—
26	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
27	C₂×Dic₁₀₉	C₂×C₄	1	4	yes	—
28	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀₉	C₂³	1	4	yes	—
30	C₄×D₁₀₉	C₂×C₄	1	2	yes	—
31	C₄×D₁₀₉	C₂³	1	2	yes	—
32	Dic₂₁₈	C₂×C₄	1	4	yes	—
33	#11724786	C₂³	1	2	yes	—
34	#11724786	C₂³	1	4	yes	—
35	#11724786	C₂³	1	2	yes	—
36	C₂²×C₁₀₉	—	2	4	no	—
37	C₂²×C₁₀₉	C₂²×C₄	1	4	yes	—
38	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	2	no	—
40	C₂×D₁₀₉	C₂²×C₄	1	2	yes	—
41	C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
42	Dic₁₀₉	C₂²×C₄	1	4	yes	—
43	Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	Dic₁₀₉	C₂⁴	1	1	yes	characteristic
45	C₂×C₁₀₉	—	2	4	no	—
46	C₂×C₁₀₉	#131	1	2	yes	complemented
47	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
48	D₁₀₉	—	2	2	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	2	no	—
54	C₂×M₁₆	—	109	1	no	—
55	C₂²×C₈	—	109	2	no	—
56	C₂²×C₈	—	109	1	no	—
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	#89137	1	1	yes	characteristic
62	C₂²×C₄	—	109	2	no	—
63	C₂²×C₄	—	109	1	no	—
64	M₁₆	—	109	8	no	—
65	M₁₆	—	109	4	no	—
66	C₂×C₈	—	109	1	no	—
67	C₂×C₈	—	109	2	no	—
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	—	109	8	no	—
70	C₂×C₈	—	109	4	no	—
71	C₂³	—	109	1	no	—
72	C₂³	C₂×#89137	1	2	yes	—
73	Q₈	—	109	4	no	—
74	D₄	—	109	8	no	—
75	D₄	C₂×#89137	1	4	yes	—
76	C₂×C₄	—	109	4	no	—
77	C₂×C₄	—	109	4	no	—
78	C₂×C₄	—	109	1	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	109	2	no	—
81	C₂×C₄	—	109	2	no	—
82	C₂×C₄	C₂×#89137	1	1	yes	characteristic
83	C₈	—	109	2	no	—
84	C₈	—	109	4	no	—
85	C₈	—	109	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	—	109	2	no	—
88	C₂²	—	218	2	no	—
89	C₂²	C₂²×#89137	1	4	yes	—
90	C₂²	C₂²×#89137	1	1	yes	characteristic
91	C₄	—	109	1	no	—
92	C₄	—	109	1	no	—
93	C₄	—	109	4	no	—
94	C₄	C₂²×#89137	1	2	yes	—
95	C₂	—	2	4	no	—
96	C₂	—	218	2	no	—
97	C₂	#16812786	1	2	yes	direct product factor
98	C₂	C₂³×#89137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812786

C₂

yes

maximal, direct product factor

C₂×#13182039

C₂

yes

maximal, characteristic, complemented

C₂×#13182038

C₂

yes

maximal, complemented

C₂×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂²×#11724786

C₂

yes

maximal, complemented

C₂×#13182028

C₂

yes

maximal, characteristic

#13182039

C₂²

yes

complemented

#13182038

C₂²

yes

complemented

D₁₀₉⋊C₈

C₂²

yes

complemented

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

—

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

complemented

#13182028

C₄

yes

—

#13182028

C₂²

yes

—

C₂×Dic₂₁₈

C₄

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₄

yes

—

C₂²×Dic₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

D₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²⋊D₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

C₂³

yes

—

Dic₂₁₈

C₂×C₄

yes

—

#11724786

C₂³

yes

—

#11724786

C₂³

yes

—

#11724786

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

—

Dic₁₀₉

C₂²×C₄

yes

—

Dic₁₀₉

C₂²×C₄

yes

characteristic

Dic₁₀₉

C₂⁴

yes

characteristic

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

—

109

—

#41

—

109

—

#41

—

109

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

109

—

M₁₆

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂³

—

109

—

C₂³

C₂×#89137

yes

—

Q₈

—

109

—

D₄

—

109

—

D₄

C₂×#89137

yes

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

C₂×#89137

yes

characteristic

C₈

—

109

—

C₈

—

109

—

C₈

—

109

—

C₂²

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

C₂²×#89137

yes

—

C₂²

C₂²×#89137

yes

characteristic

C₄

—

109

—

C₄

—

109

—

C₄

—

109

—

C₄

C₂²×#89137

yes

—

C₂

—

C₂

—

218

—

C₂

#16812786

yes

direct product factor

C₂

C₂³×#89137

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#89137
commutator	C₂×C₁₀₉	C₂³×C₄
Frattini	C₂	C₂³×#89137
Fitting	C₂×D₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²×C₄
hypercenter	C₂×D₄	#89137
hypocenter	C₁₀₉	C₂×#131
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center

C₂²

C₂²×#89137

commutator

C₂×C₁₀₉

C₂³×C₄

Frattini

C₂

C₂³×#89137

Fitting

C₂×D₄×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²×C₄

C₂×D₄

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. #16812786	C₂	direct (N × H)
2. C₂×#13182039	C₂	semidirect
3. C₂×#13182038	C₂	semidirect
4. C₂×D₁₀₉⋊C₈	C₂	semidirect
5. C₂²×#11724786	C₂	semidirect
7. #13182039	C₂²	semidirect
8. #13182038	C₂²	semidirect
9. D₁₀₉⋊C₈	C₂²	semidirect
13. C₂×#11724786	C₂²	semidirect
46. C₂×C₁₀₉	#131	semidirect
49. C₁₀₉	C₂×#131	semidirect
97. C₂	#16812786	direct (N × H)

H ≅ G/N

type

1. #16812786

C₂

direct (N × H)

2. C₂×#13182039

C₂

semidirect

3. C₂×#13182038

C₂

semidirect

4. C₂×D₁₀₉⋊C₈

C₂

semidirect

5. C₂²×#11724786

C₂

semidirect

7. #13182039

C₂²

semidirect

8. #13182038

C₂²

semidirect

9. D₁₀₉⋊C₈

C₂²

semidirect

13. C₂×#11724786

C₂²

semidirect

46. C₂×C₁₀₉

#131

semidirect

49. C₁₀₉

C₂×#131

semidirect

97. C₂

#16812786

direct (N × H)

Subgroups — C₂×#16812786 #20296101

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16812786 #20296101

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16812786 #20296101