Group #20295581 — subgroups

Subgroups — #20295581

Order 6976.

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

normal subgroups only

This group has 222 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	#16812699	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812690	C₂	1	2	yes	maximal, complemented
3	#16812679	C₂	1	2	yes	maximal
4	#16812675	C₂	1	2	yes	maximal
5	Q₈×Dic₁₀₉	C₂	1	2	yes	maximal, complemented
6	#16812664	C₂	1	2	yes	maximal, complemented
7	#16812657	C₂	1	2	yes	maximal
8	C₄²×D₁₀₉	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₈	C₂²	1	2	yes	—
11	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
15	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
16	C₄⋊Dic₁₀₉	C₂²	1	4	yes	—
17	#13182008	C₂²	1	4	yes	—
18	#13182006	C₂²	1	4	yes	—
19	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
20	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
21	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
22	C₄×Dic₁₀₉	C₂²	1	4	yes	—
23	C₂×Q₈×C₁₀₉	C₂²	1	2	yes	—
24	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²×D₁₀₉	C₂³	1	1	yes	characteristic
28	Q₈×C₁₀₉	—	2	4	no	—
29	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
30	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
34	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
35	C₂×Dic₁₀₉	C₂³	1	4	yes	—
36	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
37	C₄×D₁₀₉	—	2	2	no	—
38	C₄×D₁₀₉	—	2	2	no	—
39	C₄×D₁₀₉	Q₈	1	4	yes	complemented
40	Dic₂₁₈	—	2	4	no	—
41	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	—	2	1	no	—
44	C₂×D₁₀₉	C₂×Q₈	1	2	yes	—
45	C₄×C₁₀₉	—	2	4	no	—
46	C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
47	C₄×C₁₀₉	#41	1	2	yes	—
48	C₄×C₁₀₉	#41	1	2	yes	—
49	Dic₁₀₉	—	2	4	no	—
50	Dic₁₀₉	—	2	1	no	—
51	Dic₁₀₉	—	2	1	no	—
52	Dic₁₀₉	C₂×Q₈	1	2	yes	—
53	C₂×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₀₉	C₂×#41	1	1	yes	characteristic
56	D₁₀₉	—	2	2	no	—
57	C₁₀₉	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	109	1	no	maximal, Sylow
59	C₄⋊Q₈	—	109	1	no	—
60	C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
61	#125	—	109	2	no	—
62	C₂²⋊Q₈	—	109	2	no	—
63	C₂²⋊Q₈	—	109	2	no	—
64	C₄×Q₈	—	109	2	no	—
65	C₄×Q₈	—	109	2	no	—
66	#117	—	109	2	no	—
67	C₂×C₄²	—	109	1	no	—
68	C₂×Q₈	—	109	2	no	—
69	C₂×Q₈	C₂×D₁₀₉	1	2	yes	—
70	C₂²×C₄	—	109	1	no	—
71	C₂²×C₄	—	109	1	no	—
72	C₂²×C₄	—	109	1	no	—
73	C₄⋊C₄	—	109	4	no	—
74	C₄⋊C₄	—	109	2	no	—
75	C₄⋊C₄	—	109	4	no	—
76	C₄⋊C₄	—	109	2	no	—
77	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
78	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
79	C₂²⋊C₄	—	109	4	no	—
80	C₄²	—	109	1	no	—
81	C₄²	—	109	1	no	—
82	C₄²	—	109	1	no	—
83	C₄²	—	109	4	no	—
84	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
85	C₂³	—	109	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	218	4	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	109	4	no	—
90	C₂×C₄	—	109	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	109	4	no	—
93	C₂×C₄	—	218	2	no	—
94	C₂×C₄	—	218	2	no	—
95	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
96	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₂²	—	109	2	no	—
100	C₂²	—	218	1	no	—
101	C₂²	—	218	1	no	—
102	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	109	2	no	—
105	C₄	—	218	1	no	—
106	C₄	—	218	1	no	—
107	C₄	—	218	4	no	—
108	C₄	Q₈×D₁₀₉	1	2	yes	—
109	C₄	#13182028	1	2	yes	—
110	C₄	#13182030	1	2	yes	—
111	C₂	—	218	2	no	—
112	C₂	C₂×Q₈×D₁₀₉	1	1	yes	characteristic
113	C₂	C₂×#13182028	1	1	yes	characteristic
114	C₂	C₂×#13182030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812699

C₂

yes

maximal, characteristic, complemented

#16812690

C₂

yes

maximal, complemented

#16812679

C₂

yes

maximal

#16812675

C₂

yes

maximal

Q₈×Dic₁₀₉

C₂

yes

maximal, complemented

#16812664

C₂

yes

maximal, complemented

#16812657

C₂

yes

maximal

C₄²×D₁₀₉

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

—

#13182008

C₂²

yes

—

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

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

characteristic

Q₈×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

Q₈

yes

complemented

Dic₂₁₈

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×Q₈

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×Q₈

yes

—

C₄×C₁₀₉

#41

yes

—

C₄×C₁₀₉

#41

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×Q₈

yes

—

C₂×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

C₂×C₁₀₉

C₂×#41

yes

characteristic

D₁₀₉

—

C₁₀₉

#533

yes

characteristic, complemented, Sylow

#533

—

109

maximal, Sylow

C₄⋊Q₈

—

109

—

C₄⋊Q₈

D₁₀₉

yes

characteristic, complemented

#125

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×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

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

Q₈

—

Q₈

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

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

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²

—

109

—

100

C₂²

—

218

—

101

C₂²

—

218

—

102

C₂²

C₂³×D₁₀₉

yes

characteristic

103

C₄

—

104

C₄

—

109

—

105

C₄

—

218

—

106

C₄

—

218

—

107

C₄

—

218

—

108

C₄

Q₈×D₁₀₉

yes

—

109

C₄

#13182028

yes

—

110

C₄

#13182030

yes

—

111

C₂

—

218

—

112

C₂

C₂×Q₈×D₁₀₉

yes

characteristic

113

C₂

C₂×#13182028

yes

characteristic

114

C₂

C₂×#13182030

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	#533
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. #16812699	C₂	semidirect
2. #16812690	C₂	semidirect
5. Q₈×Dic₁₀₉	C₂	semidirect
6. #16812664	C₂	semidirect
9. C₄⋊Q₈×C₁₀₉	C₂	semidirect
39. C₄×D₁₀₉	Q₈	semidirect
57. C₁₀₉	#533	semidirect
60. C₄⋊Q₈	D₁₀₉	semidirect

H ≅ G/N

type

1. #16812699

C₂

semidirect

2. #16812690

C₂

semidirect

5. Q₈×Dic₁₀₉

C₂

semidirect

6. #16812664

C₂

semidirect

9. C₄⋊Q₈×C₁₀₉

C₂

semidirect

39. C₄×D₁₀₉

Q₈

semidirect

57. C₁₀₉

#533

semidirect

60. C₄⋊Q₈

D₁₀₉

semidirect

Subgroups — #20295581

Conjugacy classes of subgroups

Complements