Group #20295166 — subgroups

Subgroups — #20295166

Order 6976.

158 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 45 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×Q₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13182018	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13182016	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812674	C₂	1	1	yes	maximal, characteristic
5	#16812622	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812611	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₁₈	—	2	1	no	—
9	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
10	Q₈×D₁₀₉	—	2	2	no	—
11	Q₈×D₁₀₉	—	2	2	no	—
12	C₂×D₄₃₆	—	2	1	no	—
13	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₁₀₉	—	2	1	no	—
15	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
16	#13182018	—	2	2	no	—
17	#13182016	—	2	2	no	—
18	C₂×#11724784	C₂²	1	1	yes	characteristic
19	#13182006	—	2	1	no	—
20	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×D₁₀₉	—	2	1	no	—
24	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
25	Q₈×C₁₀₉	—	2	1	no	—
26	Q₈×C₁₀₉	D₄	1	2	yes	complemented
27	C₂×C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄×C₁₀₉	—	2	1	no	—
29	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₀₉	—	2	1	no	—
31	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
32	D₄₃₆	—	2	1	no	—
33	D₄₃₆	—	2	1	no	—
34	D₄₃₆	—	4	2	no	—
35	C₄×D₁₀₉	—	2	4	no	—
36	C₄×D₁₀₉	—	2	1	no	—
37	C₄×D₁₀₉	—	2	1	no	—
38	Dic₂₁₈	—	2	4	no	—
39	Dic₂₁₈	—	2	1	no	—
40	Dic₂₁₈	D₄	1	2	yes	complemented
41	C₈×C₁₀₉	—	2	1	no	—
42	#11724784	—	2	1	no	—
43	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
44	C₂×D₁₀₉	—	2	1	no	—
45	C₂×D₁₀₉	—	2	1	no	—
46	C₂×D₁₀₉	—	4	1	no	—
47	C₂×D₁₀₉	—	4	1	no	—
48	C₂×D₁₀₉	—	4	1	no	—
49	C₂×D₁₀₉	SD₁₆	1	2	yes	complemented
50	C₄×C₁₀₉	—	2	2	no	—
51	C₄×C₁₀₉	—	4	1	no	—
52	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
53	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
54	Dic₁₀₉	—	2	1	no	—
55	Dic₁₀₉	—	2	2	no	—
56	C₂×C₁₀₉	#120	1	1	yes	characteristic
57	C₂×C₁₀₉	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₁₀₉	#137	1	1	yes	characteristic
59	D₁₀₉	—	2	2	no	—
60	D₁₀₉	—	8	1	no	—
61	C₁₀₉	#448	1	1	yes	characteristic, complemented, Sylow
62	#448	—	109	1	no	maximal, Sylow
63	C₂²×Q₈	—	109	1	no	—
64	C₂×SD₁₆	—	109	1	no	—
65	C₂×SD₁₆	—	109	1	no	—
66	#121	—	109	1	no	—
67	#103	—	109	1	no	—
68	#103	D₁₀₉	1	1	yes	characteristic, complemented
69	C₂²⋊C₈	—	109	1	no	—
70	C₂×Q₈	—	109	1	no	—
71	C₂×Q₈	—	218	1	no	—
72	C₂×Q₈	—	218	2	no	—
73	C₂×Q₈	—	218	2	no	—
74	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
75	C₂×D₄	—	109	1	no	—
76	C₂×D₄	—	218	1	no	—
77	C₂²×C₄	—	109	1	no	—
78	C₂²×C₄	—	218	1	no	—
79	SD₁₆	—	218	2	no	—
80	SD₁₆	—	218	2	no	—
81	C₂×C₈	—	109	1	no	—
82	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
83	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
84	C₂²⋊C₄	—	218	1	no	—
85	C₂³	—	109	1	no	—
86	C₂³	—	218	1	no	—
87	Q₈	—	2	1	no	—
88	Q₈	—	109	2	no	—
89	Q₈	—	218	4	no	—
90	Q₈	—	218	1	no	—
91	Q₈	D₄₃₆	1	2	yes	complemented
92	D₄	—	436	2	no	—
93	D₄	—	218	1	no	—
94	D₄	—	218	1	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	109	1	no	—
98	C₂×C₄	—	218	4	no	—
99	C₂×C₄	—	218	1	no	—
100	C₂×C₄	—	218	1	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	—	109	2	no	—
106	C₂²	—	436	1	no	—
107	C₂²	—	436	1	no	—
108	C₂²	—	436	1	no	—
109	C₂²	—	218	1	no	—
110	C₂²	—	218	1	no	—
111	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	4	1	no	—
114	C₄	—	218	1	no	—
115	C₄	—	218	2	no	—
116	C₄	D₄×D₁₀₉	1	1	yes	characteristic
117	C₄	C₂×D₄₃₆	1	1	yes	characteristic
118	C₂	—	872	1	no	—
119	C₂	—	218	2	no	—
120	C₂	#16812669	1	1	yes	characteristic
121	C₂	SD₁₆×D₁₀₉	1	1	yes	characteristic
122	C₂	#16812731	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×#13182018

C₂

yes

maximal, characteristic, complemented

C₂×#13182016

C₂

yes

maximal, characteristic, complemented

#16812674

C₂

yes

maximal, characteristic

#16812622

C₂

yes

maximal, characteristic, complemented

#16812611

C₂

yes

maximal, characteristic, complemented

#103×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

characteristic

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

—

—

C₂²

yes

characteristic

#13182006

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

D₄

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

D₄

yes

complemented

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

SD₁₆

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#120

yes

characteristic

C₂×C₁₀₉

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₉

#137

yes

characteristic

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#448

yes

characteristic, complemented, Sylow

#448

—

109

maximal, Sylow

C₂²×Q₈

—

109

—

C₂×SD₁₆

—

109

—

C₂×SD₁₆

—

109

—

#121

—

109

—

#103

—

109

—

#103

D₁₀₉

yes

characteristic, complemented

C₂²⋊C₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

SD₁₆

—

218

—

SD₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₂³

—

109

—

C₂³

—

218

—

Q₈

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

D₄₃₆

yes

complemented

D₄

—

436

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

100

C₂×C₄

—

218

—

101

C₂×C₄

—

218

—

102

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

103

C₈

—

104

C₈

—

218

—

105

C₂²

—

109

—

106

C₂²

—

436

—

107

C₂²

—

436

—

108

C₂²

—

436

—

109

C₂²

—

218

—

110

C₂²

—

218

—

111

C₂²

D₄×D₁₀₉

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

218

—

115

C₄

—

218

—

116

C₄

D₄×D₁₀₉

yes

characteristic

117

C₄

C₂×D₄₃₆

yes

characteristic

118

C₂

—

872

—

119

C₂

—

218

—

120

C₂

#16812669

yes

characteristic

121

C₂

SD₁₆×D₁₀₉

yes

characteristic

122

C₂

#16812731

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#103×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#103	D₁₀₉
hypocenter	C₁₀₉	#448
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#103×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
1. C₂×Q₈×D₁₀₉	C₂	semidirect
2. C₂×#13182018	C₂	semidirect
3. C₂×#13182016	C₂	semidirect
5. #16812622	C₂	semidirect
6. #16812611	C₂	semidirect
7. #103×C₁₀₉	C₂	semidirect
26. Q₈×C₁₀₉	D₄	semidirect
40. Dic₂₁₈	D₄	semidirect
49. C₂×D₁₀₉	SD₁₆	semidirect
61. C₁₀₉	#448	semidirect
68. #103	D₁₀₉	semidirect
91. Q₈	D₄₃₆	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₁₀₉

C₂

semidirect

2. C₂×#13182018

C₂

semidirect

3. C₂×#13182016

C₂

semidirect

5. #16812622

C₂

semidirect

6. #16812611

C₂

semidirect

7. #103×C₁₀₉

C₂

semidirect

26. Q₈×C₁₀₉

D₄

semidirect

40. Dic₂₁₈

D₄

semidirect

49. C₂×D₁₀₉

SD₁₆

semidirect

61. C₁₀₉

#448

semidirect

68. #103

D₁₀₉

semidirect

91. Q₈

D₄₃₆

semidirect

Subgroups — #20295166

Conjugacy classes of subgroups

Complements