Group #20295737 — subgroups

Subgroups — #20295737

Order 6976.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	#16812690	C₂	1	2	yes	maximal, complemented
2	#16812679	C₂	1	2	yes	maximal, complemented
3	#16812675	C₂	1	4	yes	maximal, complemented
4	C₄×Dic₂₁₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₄₃₆	C₂	1	1	yes	maximal, characteristic
6	#16812661	C₂	1	2	yes	maximal
7	C₄⋊C₄×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
10	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
12	C₂×C₄×D₁₀₉	C₂²	1	4	yes	—
13	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
16	#13182008	C₂²	1	4	yes	—
17	#13182008	C₂²	1	2	yes	—
18	#13182006	C₂²	1	4	yes	—
19	#13182006	C₂²	1	2	yes	—
20	C₄×Dic₁₀₉	C₂²	1	2	yes	—
21	C₂×Q₈×C₁₀₉	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
24	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
25	C₂²×D₁₀₉	C₂³	1	2	yes	—
26	Q₈×C₁₀₉	—	4	2	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
29	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂³	1	2	yes	—
32	C₂×Dic₁₀₉	C₂³	1	2	yes	—
33	C₂×Dic₁₀₉	C₂³	1	2	yes	—
34	D₄₃₆	Q₈	1	4	yes	complemented
35	C₄×D₁₀₉	—	2	4	no	—
36	C₄×D₁₀₉	—	4	4	no	—
37	Dic₂₁₈	—	2	2	no	—
38	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	2	no	—
40	C₂×D₁₀₉	—	2	2	no	—
41	C₂×D₁₀₉	C₂×Q₈	1	4	yes	—
42	C₄×C₁₀₉	—	2	4	no	—
43	C₄×C₁₀₉	—	2	1	no	—
44	C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
45	C₄×C₁₀₉	#41	1	2	yes	—
46	Dic₁₀₉	—	2	2	no	—
47	Dic₁₀₉	—	2	2	no	—
48	Dic₁₀₉	—	2	2	no	—
49	C₂×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
50	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
52	D₁₀₉	—	2	4	no	—
53	C₁₀₉	#554	1	1	yes	characteristic, complemented, Sylow
54	#554	—	109	1	no	maximal, Sylow
55	C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
56	#125	—	109	2	no	—
57	C₂²⋊Q₈	—	109	4	no	—
58	C₂²⋊Q₈	—	109	2	no	—
59	C₄×Q₈	—	109	1	no	—
60	C₄×D₄	—	109	2	no	—
61	C₄×D₄	—	109	1	no	—
62	C₂×C₄⋊C₄	—	109	2	no	—
63	C₂×Q₈	—	109	1	no	—
64	C₂×Q₈	C₂×D₁₀₉	1	2	yes	—
65	C₂×D₄	—	109	1	no	—
66	C₂²×C₄	—	109	4	no	—
67	C₂²×C₄	—	109	2	no	—
68	C₄⋊C₄	—	109	4	no	—
69	C₄⋊C₄	—	109	2	no	—
70	C₄⋊C₄	—	109	1	no	—
71	C₄⋊C₄	—	109	1	no	—
72	C₄⋊C₄	—	109	4	no	—
73	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
74	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
75	C₂²⋊C₄	—	109	4	no	—
76	C₂²⋊C₄	—	109	2	no	—
77	C₄²	—	109	2	no	—
78	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
79	C₂³	—	109	2	no	—
80	Q₈	—	4	2	no	—
81	Q₈	—	218	2	no	—
82	D₄	—	109	4	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	109	2	no	—
85	C₂×C₄	—	109	2	no	—
86	C₂×C₄	—	436	4	no	—
87	C₂×C₄	—	218	4	no	—
88	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
90	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
92	C₂²	—	109	4	no	—
93	C₂²	—	218	2	no	—
94	C₂²	—	218	2	no	—
95	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	2	1	no	—
98	C₄	—	218	2	no	—
99	C₄	—	218	2	no	—
100	C₄	—	218	2	no	—
101	C₄	Q₈×D₁₀₉	1	2	yes	—
102	C₄	#13182030	1	2	yes	—
103	C₂	—	218	4	no	—
104	C₂	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
105	C₂	C₂×#13182030	1	1	yes	characteristic
106	C₂	#16812744	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812690

C₂

yes

maximal, complemented

#16812679

C₂

yes

maximal, complemented

#16812675

C₂

yes

maximal, complemented

C₄×Dic₂₁₈

C₂

yes

maximal, characteristic, complemented

C₄×D₄₃₆

C₂

yes

maximal, characteristic

#16812661

C₂

yes

maximal

C₄⋊C₄×D₁₀₉

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₂×Q₈×C₁₀₉

C₂²

yes

—

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

—

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

Q₈

yes

complemented

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₈

yes

—

C₄×C₁₀₉

#41

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

H(5,𝔽₂)

yes

characteristic

D₁₀₉

—

C₁₀₉

#554

yes

characteristic, complemented, Sylow

#554

—

109

maximal, Sylow

C₄⋊Q₈

D₁₀₉

yes

characteristic, complemented

#125

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×D₄

—

109

—

C₄×D₄

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

Q₈

—

Q₈

—

218

—

D₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂³×D₁₀₉

yes

characteristic

C₄

—

C₄

—

C₄

—

218

—

C₄

—

218

—

100

C₄

—

218

—

101

C₄

Q₈×D₁₀₉

yes

—

102

C₄

#13182030

yes

—

103

C₂

—

218

—

104

C₂

C₂×Q₈×D₁₀₉

yes

characteristic

105

C₂

C₂×#13182030

yes

characteristic

106

C₂

#16812744

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₉	C₂
hypercenter	C₄⋊Q₈	D₁₀₉
hypocenter	C₁₀₉	#554
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. #16812690	C₂	semidirect
2. #16812679	C₂	semidirect
3. #16812675	C₂	semidirect
4. C₄×Dic₂₁₈	C₂	semidirect
7. C₄⋊C₄×D₁₀₉	C₂	semidirect
8. C₄⋊Q₈×C₁₀₉	C₂	semidirect
34. D₄₃₆	Q₈	semidirect
53. C₁₀₉	#554	semidirect
55. C₄⋊Q₈	D₁₀₉	semidirect

H ≅ G/N

type

1. #16812690

C₂

semidirect

2. #16812679

C₂

semidirect

3. #16812675

C₂

semidirect

4. C₄×Dic₂₁₈

C₂

semidirect

7. C₄⋊C₄×D₁₀₉

C₂

semidirect

8. C₄⋊Q₈×C₁₀₉

C₂

semidirect

34. D₄₃₆

Q₈

semidirect

53. C₁₀₉

#554

semidirect

55. C₄⋊Q₈

D₁₀₉

semidirect

Subgroups — #20295737

Conjugacy classes of subgroups

Complements