Group #20295775 — subgroups

Subgroups — #20295775

Order 6976.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#16812697	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812694	C₂	1	2	yes	maximal, complemented
3	#16812693	C₂	1	2	yes	maximal, complemented
4	#16812689	C₂	1	2	yes	maximal, complemented
5	#16812688	C₂	1	1	yes	maximal, characteristic
6	#16812684	C₂	1	2	yes	maximal
7	#16812677	C₂	1	2	yes	maximal, complemented
8	#16812675	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
13	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
14	#13182008	C₂²	1	2	yes	—
15	#13182008	C₂²	1	2	yes	—
16	#13182008	C₂²	1	2	yes	—
17	#13182008	C₂²	1	2	yes	—
18	#13182008	C₂²	1	2	yes	—
19	#13182006	C₂²	1	2	yes	—
20	#13182006	C₂²	1	2	yes	—
21	#13182006	C₂²	1	2	yes	—
22	#13182006	C₂²	1	2	yes	—
23	#13182006	C₂²	1	2	yes	—
24	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
30	C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
31	C₂²×D₁₀₉	C₂³	1	2	yes	—
32	Q₈×C₁₀₉	—	4	1	no	—
33	Q₈×C₁₀₉	—	4	1	no	—
34	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
35	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂³	1	2	yes	—
39	C₂×Dic₁₀₉	C₂³	1	2	yes	—
40	C₂×Dic₁₀₉	C₂³	1	2	yes	—
41	C₂×Dic₁₀₉	C₂³	1	2	yes	—
42	D₄₃₆	—	4	1	no	—
43	C₄×D₁₀₉	—	4	2	no	—
44	Dic₂₁₈	—	4	1	no	—
45	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
46	C₂×D₁₀₉	—	2	2	no	—
47	C₂×D₁₀₉	—	2	4	no	—
48	C₄×C₁₀₉	—	2	1	no	—
49	C₄×C₁₀₉	—	2	2	no	—
50	C₄×C₁₀₉	—	2	1	no	—
51	C₄×C₁₀₉	—	2	1	no	—
52	C₄×C₁₀₉	—	2	2	no	—
53	Dic₁₀₉	—	2	2	no	—
54	Dic₁₀₉	—	2	2	no	—
55	Dic₁₀₉	—	2	2	no	—
56	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₀₉	2₋¹⁺⁴	1	2	yes	—
58	D₁₀₉	—	4	2	no	—
59	C₁₀₉	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	109	1	no	maximal, Sylow
61	C₄⋊Q₈	—	109	1	no	—
62	C₄⋊Q₈	D₁₀₉	1	1	yes	characteristic, complemented
63	#126	—	109	2	no	—
64	#126	—	109	2	no	—
65	#125	—	109	2	no	—
66	#124	—	109	1	no	—
67	#123	—	109	2	no	—
68	C₂²⋊Q₈	—	109	2	no	—
69	C₂²⋊Q₈	—	109	2	no	—
70	C₂×Q₈	—	109	1	no	—
71	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
73	C₂×D₄	—	109	1	no	—
74	C₂²×C₄	—	109	2	no	—
75	C₄⋊C₄	—	109	2	no	—
76	C₄⋊C₄	—	109	2	no	—
77	C₄⋊C₄	—	109	2	no	—
78	C₄⋊C₄	—	109	2	no	—
79	C₄⋊C₄	—	109	2	no	—
80	C₄⋊C₄	—	109	2	no	—
81	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
82	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
83	C₂²⋊C₄	—	109	2	no	—
84	C₂²⋊C₄	—	109	2	no	—
85	C₂²⋊C₄	—	109	2	no	—
86	C₂²⋊C₄	—	109	2	no	—
87	C₂²⋊C₄	—	109	2	no	—
88	C₄²	—	109	1	no	—
89	C₄²	—	109	1	no	—
90	C₄²	C₂×D₁₀₉	1	1	yes	characteristic
91	C₂³	—	109	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	436	1	no	—
95	D₄	—	436	1	no	—
96	C₂×C₄	—	109	2	no	—
97	C₂×C₄	—	109	2	no	—
98	C₂×C₄	—	109	2	no	—
99	C₂×C₄	—	436	2	no	—
100	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
103	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₀₉	1	2	yes	—
105	C₂²	—	218	2	no	—
106	C₂²	—	218	4	no	—
107	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	218	2	no	—
114	C₄	—	218	2	no	—
115	C₄	—	218	2	no	—
116	C₂	—	436	2	no	—
117	C₂	#16812744	1	1	yes	characteristic
118	C₂	#16812747	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16812697

C₂

yes

maximal, characteristic, complemented

#16812694

C₂

yes

maximal, complemented

#16812693

C₂

yes

maximal, complemented

#16812689

C₂

yes

maximal, complemented

#16812688

C₂

yes

maximal, characteristic

#16812684

C₂

yes

maximal

#16812677

C₂

yes

maximal, complemented

#16812675

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

—

C₄⋊Dic₁₀₉

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182008

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

—

Q₈×C₁₀₉

—

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×C₄×C₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₉

2₋¹⁺⁴

yes

—

D₁₀₉

—

C₁₀₉

#563

yes

characteristic, complemented, Sylow

#563

—

109

maximal, Sylow

C₄⋊Q₈

—

109

—

C₄⋊Q₈

D₁₀₉

yes

characteristic, complemented

#126

—

109

—

#126

—

109

—

#125

—

109

—

#124

—

109

—

#123

—

109

—

C₂²⋊Q₈

—

109

—

C₂²⋊Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₄⋊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

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

Q₈

—

Q₈

—

Q₈

—

436

—

D₄

—

436

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

100

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

101

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

102

C₂×C₄

C₂²×D₁₀₉

yes

—

103

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

104

C₂×C₄

C₂²×D₁₀₉

yes

—

105

C₂²

—

218

—

106

C₂²

—

218

—

107

C₂²

C₂³×D₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

218

—

114

C₄

—

218

—

115

C₄

—

218

—

116

C₂

—

436

—

117

C₂

#16812744

yes

characteristic

118

C₂

#16812747

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	#563
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. #16812697	C₂	semidirect
2. #16812694	C₂	semidirect
3. #16812693	C₂	semidirect
4. #16812689	C₂	semidirect
7. #16812677	C₂	semidirect
8. #16812675	C₂	semidirect
9. C₄⋊Q₈×C₁₀₉	C₂	semidirect
59. C₁₀₉	#563	semidirect
62. C₄⋊Q₈	D₁₀₉	semidirect

H ≅ G/N

type

1. #16812697

C₂

semidirect

2. #16812694

C₂

semidirect

3. #16812693

C₂

semidirect

4. #16812689

C₂

semidirect

7. #16812677

C₂

semidirect

8. #16812675

C₂

semidirect

9. C₄⋊Q₈×C₁₀₉

C₂

semidirect

59. C₁₀₉

#563

semidirect

62. C₄⋊Q₈

D₁₀₉

semidirect

Subgroups — #20295775

Conjugacy classes of subgroups

Complements