Group #20295029 — subgroups

Subgroups — #20295029

Order 6976.

190 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 55 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×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812657	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812621	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812620	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812619	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₀₉	—	2	1	no	—
9	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
11	D₄×D₁₀₉	—	2	2	no	—
12	D₄×D₁₀₉	C₄	1	4	yes	complemented
13	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
14	#13182012	—	2	2	no	—
15	C₂×#11724784	C₂²	1	1	yes	characteristic, complemented
16	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
17	#13182006	—	2	1	no	—
18	C₄×Dic₁₀₉	—	2	1	no	—
19	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
21	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
22	C₂³×C₁₀₉	—	2	1	no	—
23	C₂²×D₁₀₉	—	2	1	no	—
24	C₂²×D₁₀₉	—	2	4	no	—
25	C₂²×D₁₀₉	—	4	2	no	—
26	C₂²×D₁₀₉	—	4	2	no	—
27	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
28	D₄×C₁₀₉	—	2	1	no	—
29	D₄×C₁₀₉	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂²⋊D₁₀₉	—	2	4	no	—
33	C₂×Dic₁₀₉	—	2	1	no	—
34	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
35	D₄₃₆	—	2	1	no	—
36	D₄₃₆	C₂×C₄	1	2	yes	—
37	C₄×D₁₀₉	C₂×C₄	1	2	yes	—
38	C₄×D₁₀₉	D₄	1	2	yes	—
39	C₈×C₁₀₉	—	2	1	no	—
40	#11724784	—	2	1	no	—
41	C₂²×C₁₀₉	—	2	2	no	—
42	C₂²×C₁₀₉	—	4	1	no	—
43	C₂²×C₁₀₉	—	4	1	no	—
44	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
45	C₂×D₁₀₉	—	2	1	no	—
46	C₂×D₁₀₉	—	2	1	no	—
47	C₂×D₁₀₉	—	2	2	no	—
48	C₂×D₁₀₉	—	4	8	no	—
49	C₂×D₁₀₉	—	4	1	no	—
50	C₂×D₁₀₉	—	4	1	no	—
51	C₂×D₁₀₉	C₂²⋊C₄	1	2	yes	—
52	C₄×C₁₀₉	—	4	1	no	—
53	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
55	Dic₁₀₉	—	4	1	no	—
56	Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
57	C₂×C₁₀₉	—	4	2	no	—
58	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
61	D₁₀₉	—	2	2	no	—
62	D₁₀₉	—	4	2	no	—
63	C₁₀₉	#418	1	1	yes	characteristic, complemented, Sylow
64	#418	—	109	1	no	maximal, Sylow
65	C₂²×D₄	—	109	1	no	—
66	C₂×M₁₆	—	109	1	no	—
67	#117	—	109	1	no	—
68	#102	—	109	1	no	—
69	#102	—	109	1	no	—
70	#102	—	109	1	no	—
71	#102	D₁₀₉	1	1	yes	characteristic, complemented
72	C₂⁴	—	218	1	no	—
73	C₂×D₄	—	109	4	no	—
74	C₂×D₄	—	109	1	no	—
75	C₂×D₄	—	218	2	no	—
76	C₂×D₄	—	218	1	no	—
77	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
78	C₂²×C₄	—	109	1	no	—
79	M₁₆	—	218	2	no	—
80	C₂×C₈	—	109	1	no	—
81	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
82	C₄⋊C₄	—	109	1	no	—
83	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
84	C₂²⋊C₄	—	218	1	no	—
85	C₄²	—	218	1	no	—
86	C₂³	—	2	1	no	—
87	C₂³	—	109	1	no	—
88	C₂³	—	436	2	no	—
89	C₂³	—	436	2	no	—
90	C₂³	—	218	4	no	—
91	C₂³	—	218	1	no	—
92	D₄	—	2	1	no	—
93	D₄	—	109	2	no	—
94	D₄	—	218	4	no	—
95	D₄	—	218	1	no	—
96	D₄	C₄×D₁₀₉	1	2	yes	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	109	1	no	—
99	C₂×C₄	—	109	2	no	—
100	C₂×C₄	—	109	2	no	—
101	C₂×C₄	—	218	1	no	—
102	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	218	1	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	1	no	—
107	C₂²	—	4	1	no	—
108	C₂²	—	109	2	no	—
109	C₂²	—	436	8	no	—
110	C₂²	—	436	1	no	—
111	C₂²	—	436	1	no	—
112	C₂²	—	218	2	no	—
113	C₂²	—	218	1	no	—
114	C₂²	—	218	1	no	—
115	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
116	C₄	—	4	1	no	—
117	C₄	—	109	2	no	—
118	C₄	—	436	1	no	—
119	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
120	C₄	D₄×D₁₀₉	1	1	yes	characteristic
121	C₂	—	4	2	no	—
122	C₂	—	436	2	no	—
123	C₂	—	218	2	no	—
124	C₂	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
125	C₂	#16812723	1	1	yes	characteristic
126	C₂	#16812726	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₄×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182012

C₂

yes

maximal, characteristic, complemented

#16812657

C₂

yes

maximal, characteristic, complemented

#16812621

C₂

yes

maximal, characteristic, complemented

#16812620

C₂

yes

maximal, characteristic, complemented

#16812619

C₂

yes

maximal, characteristic, complemented

#102×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

—

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

D₄×D₁₀₉

—

D₄×D₁₀₉

C₄

yes

complemented

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182012

—

C₂×#11724784

C₂²

yes

characteristic, complemented

C₄⋊Dic₁₀₉

C₂²

yes

characteristic, complemented

#13182006

—

C₄×Dic₁₀₉

—

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic, complemented

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

D₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

D₄₃₆

—

D₄₃₆

C₂×C₄

yes

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

D₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#418

yes

characteristic, complemented, Sylow

#418

—

109

maximal, Sylow

C₂²×D₄

—

109

—

C₂×M₁₆

—

109

—

#117

—

109

—

#102

—

109

—

#102

—

109

—

#102

—

109

—

#102

D₁₀₉

yes

characteristic, complemented

C₂⁴

—

218

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic, complemented

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic, complemented

C₂²⋊C₄

—

218

—

C₄²

—

218

—

C₂³

—

C₂³

—

109

—

C₂³

—

436

—

C₂³

—

436

—

C₂³

—

218

—

C₂³

—

218

—

D₄

—

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

D₄

C₄×D₁₀₉

yes

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

100

C₂×C₄

—

109

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

103

C₈

—

104

C₈

—

218

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

—

109

C₂²

—

436

—

110

C₂²

—

436

—

111

C₂²

—

436

—

112

C₂²

—

218

—

113

C₂²

—

218

—

114

C₂²

—

218

—

115

C₂²

D₄×D₁₀₉

yes

characteristic

116

C₄

—

117

C₄

—

109

—

118

C₄

—

436

—

119

C₄

C₂×C₄×D₁₀₉

yes

characteristic

120

C₄

D₄×D₁₀₉

yes

characteristic

121

C₂

—

122

C₂

—

436

—

123

C₂

—

218

—

124

C₂

C₂²⋊C₄×D₁₀₉

yes

characteristic

125

C₂

#16812723

yes

characteristic

126

C₂

#16812726

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#102×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#102	D₁₀₉
hypocenter	C₁₀₉	#418
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#102×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
2. C₂×#13182012	C₂	semidirect
3. #16812657	C₂	semidirect
4. #16812621	C₂	semidirect
5. #16812620	C₂	semidirect
6. #16812619	C₂	semidirect
7. #102×C₁₀₉	C₂	semidirect
12. D₄×D₁₀₉	C₄	semidirect
15. C₂×#11724784	C₂²	semidirect
16. C₄⋊Dic₁₀₉	C₂²	semidirect
20. C₂×C₈×C₁₀₉	C₂²	semidirect
21. C₄⋊C₄×C₁₀₉	C₂²	semidirect
63. C₁₀₉	#418	semidirect
71. #102	D₁₀₉	semidirect
81. C₂×C₈	C₂×D₁₀₉	semidirect
83. C₄⋊C₄	C₂×D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#13182012

C₂

semidirect

3. #16812657

C₂

semidirect

4. #16812621

C₂

semidirect

5. #16812620

C₂

semidirect

6. #16812619

C₂

semidirect

7. #102×C₁₀₉

C₂

semidirect

12. D₄×D₁₀₉

C₄

semidirect

15. C₂×#11724784

C₂²

semidirect

16. C₄⋊Dic₁₀₉

C₂²

semidirect

20. C₂×C₈×C₁₀₉

C₂²

semidirect

21. C₄⋊C₄×C₁₀₉

C₂²

semidirect

63. C₁₀₉

#418

semidirect

71. #102

D₁₀₉

semidirect

81. C₂×C₈

C₂×D₁₀₉

semidirect

83. C₄⋊C₄

C₂×D₁₀₉

semidirect

Subgroups — #20295029

Conjugacy classes of subgroups

Complements