Group #20295577 — subgroups

Subgroups — #20295577

Order 6976.

310 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 111 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×#13182030	C₂	1	2	yes	maximal, complemented
2	#16812696	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812688	C₂	1	2	yes	maximal, complemented
4	#16812674	C₂	1	4	yes	maximal, complemented
5	#16812661	C₂	1	4	yes	maximal, complemented
6	C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	—	2	8	no	—
9	C₂×D₄₃₆	C₂²	1	4	yes	—
10	C₂×D₄₃₆	C₂²	1	4	yes	—
11	C₂×D₄₃₆	C₂²	1	2	yes	—
12	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄×D₁₀₉	C₂²	1	4	yes	—
14	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	#13182006	C₂²	1	8	yes	—
16	C₄×Dic₁₀₉	C₂²	1	2	yes	—
17	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₁₀₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₁₀₉	C₂²	1	4	yes	—
20	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×D₁₀₉	C₂³	1	4	yes	—
22	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
23	Q₈×C₁₀₉	—	2	4	no	—
24	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
25	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
27	C₂×Dic₁₀₉	C₂³	1	2	yes	—
28	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
29	D₄₃₆	—	2	8	no	—
30	D₄₃₆	—	2	4	no	—
31	D₄₃₆	—	2	8	no	—
32	C₄×D₁₀₉	—	2	4	no	—
33	C₄×D₁₀₉	—	2	8	no	—
34	C₄×D₁₀₉	D₄	1	4	yes	complemented
35	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
36	C₂×D₁₀₉	—	2	2	no	—
37	C₂×D₁₀₉	—	2	4	no	—
38	C₂×D₁₀₉	—	2	8	no	—
39	C₂×D₁₀₉	C₂×D₄	1	2	yes	—
40	C₄×C₁₀₉	—	2	4	no	—
41	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
42	C₄×C₁₀₉	#41	1	4	yes	complemented
43	Dic₁₀₉	—	2	2	no	—
44	Dic₁₀₉	C₂×D₄	1	2	yes	—
45	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
46	C₂×C₁₀₉	C₂×#41	1	2	yes	—
47	D₁₀₉	—	2	2	no	—
48	D₁₀₉	—	4	4	no	—
49	C₁₀₉	#532	1	1	yes	characteristic, complemented, Sylow
50	#532	—	109	1	no	maximal, Sylow
51	C₂×#41	—	109	2	no	—
52	C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
53	#127	—	109	1	no	—
54	#124	—	109	2	no	—
55	#121	—	109	4	no	—
56	C₄×D₄	—	109	4	no	—
57	C₂×C₄²	—	109	1	no	—
58	#41	—	218	8	no	—
59	C₂×Q₈	C₂×D₁₀₉	1	2	yes	—
60	C₂×D₄	—	109	4	no	—
61	C₂×D₄	—	109	4	no	—
62	C₂×D₄	—	109	2	no	—
63	C₂²×C₄	—	109	1	no	—
64	C₂²×C₄	—	109	2	no	—
65	C₂²×C₄	—	109	4	no	—
66	C₄⋊C₄	C₂×D₁₀₉	1	4	yes	—
67	C₂²⋊C₄	—	109	8	no	—
68	C₄²	—	109	2	no	—
69	C₄²	—	109	1	no	—
70	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂³	—	109	1	no	—
72	C₂³	—	109	4	no	—
73	Q₈	—	2	4	no	—
74	D₄	—	218	8	no	—
75	D₄	—	218	4	no	—
76	D₄	—	218	8	no	—
77	C₂×C₄	—	109	1	no	—
78	C₂×C₄	—	109	4	no	—
79	C₂×C₄	—	109	2	no	—
80	C₂×C₄	—	218	8	no	—
81	C₂×C₄	—	218	4	no	—
82	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
83	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
84	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
85	C₂²	—	109	2	no	—
86	C₂²	—	218	4	no	—
87	C₂²	—	218	8	no	—
88	C₂²	—	218	2	no	—
89	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
90	C₄	—	2	4	no	—
91	C₄	—	109	2	no	—
92	C₄	—	218	2	no	—
93	C₄	D₄×D₁₀₉	1	2	yes	—
94	C₄	#13182030	1	4	yes	complemented
95	C₂	—	436	4	no	—
96	C₂	—	218	2	no	—
97	C₂	C₂×D₄×D₁₀₉	1	1	yes	characteristic
98	C₂	C₂×#13182030	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182030

C₂

yes

maximal, complemented

#16812696

C₂

yes

maximal, characteristic, complemented

#16812688

C₂

yes

maximal, complemented

#16812674

C₂

yes

maximal, complemented

#16812661

C₂

yes

maximal, complemented

C₄²×D₁₀₉

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182030

—

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

—

C₂²×D₁₀₉

C₂³

yes

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

characteristic

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₄

yes

complemented

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

#41

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

C₂×D₄

yes

—

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

—

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#532

yes

characteristic, complemented, Sylow

#532

—

109

maximal, Sylow

C₂×#41

—

109

—

C₄⋊Q₈

D₁₀₉

yes

characteristic, complemented

#127

—

109

—

#124

—

109

—

#121

—

109

—

C₄×D₄

—

109

—

C₂×C₄²

—

109

—

#41

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

C₂³

—

109

—

Q₈

—

D₄

—

218

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

109

—

C₄

—

218

—

C₄

D₄×D₁₀₉

yes

—

C₄

#13182030

yes

complemented

C₂

—

436

—

C₂

—

218

—

C₂

C₂×D₄×D₁₀₉

yes

characteristic

C₂

C₂×#13182030

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₉	C₂
hypercenter	C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#532
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

C₄⋊Q₈×C₁₀₉

C₂

hypercenter

C₄⋊Q₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×#13182030	C₂	semidirect
2. #16812696	C₂	semidirect
3. #16812688	C₂	semidirect
4. #16812674	C₂	semidirect
5. #16812661	C₂	semidirect
6. C₄²×D₁₀₉	C₂	semidirect
7. C₄⋊Q₈×C₁₀₉	C₂	semidirect
34. C₄×D₁₀₉	D₄	semidirect
42. C₄×C₁₀₉	#41	semidirect
49. C₁₀₉	#532	semidirect
52. C₄⋊Q₈	D₁₀₉	semidirect
94. C₄	#13182030	semidirect

H ≅ G/N

type

1. C₂×#13182030

C₂

semidirect

2. #16812696

C₂

semidirect

3. #16812688

C₂

semidirect

4. #16812674

C₂

semidirect

5. #16812661

C₂

semidirect

6. C₄²×D₁₀₉

C₂

semidirect

7. C₄⋊Q₈×C₁₀₉

C₂

semidirect

34. C₄×D₁₀₉

D₄

semidirect

42. C₄×C₁₀₉

#41

semidirect

49. C₁₀₉

#532

semidirect

52. C₄⋊Q₈

D₁₀₉

semidirect

94. C₄

#13182030

semidirect

Subgroups — #20295577

Conjugacy classes of subgroups

Complements