Group #20295183 — subgroups

Subgroups — #20295183

Order 6976.

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

normal subgroups only

This group has 148 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
2	C₂×Dic₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13182019	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812679	C₂	1	1	yes	maximal, characteristic
5	#16812622	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812611	C₂	1	1	yes	maximal, characteristic
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	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	Q₈×D₁₀₉	—	2	2	no	—
12	Q₈×D₁₀₉	—	2	2	no	—
13	C₂×C₄×D₁₀₉	—	2	1	no	—
14	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	Dic₄₃₆	—	2	2	no	—
16	#13182019	—	2	2	no	—
17	C₂×#11724784	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₀₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
24	Q₈×C₁₀₉	—	2	1	no	—
25	Q₈×C₁₀₉	D₄	1	2	yes	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	—	2	1	no	—
28	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	—	2	1	no	—
31	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
32	C₄×D₁₀₉	—	2	4	no	—
33	C₄×D₁₀₉	—	2	1	no	—
34	C₄×D₁₀₉	—	2	1	no	—
35	Dic₂₁₈	—	2	4	no	—
36	Dic₂₁₈	—	2	1	no	—
37	Dic₂₁₈	—	2	1	no	—
38	Dic₂₁₈	—	2	1	no	—
39	Dic₂₁₈	D₄	1	2	yes	—
40	C₈×C₁₀₉	—	2	1	no	—
41	#11724784	—	2	1	no	—
42	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×D₁₀₉	—	2	1	no	—
44	C₂×D₁₀₉	—	2	1	no	—
45	C₂×D₁₀₉	Q₁₆	1	2	yes	complemented
46	C₄×C₁₀₉	—	2	2	no	—
47	C₄×C₁₀₉	—	4	1	no	—
48	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	Dic₁₀₉	—	2	1	no	—
51	Dic₁₀₉	—	2	2	no	—
52	Dic₁₀₉	—	4	1	no	—
53	C₂×C₁₀₉	#120	1	1	yes	characteristic
54	C₂×C₁₀₉	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₁₀₉	#137	1	1	yes	characteristic
56	D₁₀₉	—	2	2	no	—
57	C₁₀₉	#451	1	1	yes	characteristic, complemented, Sylow
58	#451	—	109	1	no	maximal, Sylow
59	C₂²×Q₈	—	109	1	no	—
60	C₂×Q₁₆	—	109	1	no	—
61	C₂×Q₁₆	—	109	1	no	—
62	C₂²⋊Q₈	—	109	1	no	—
63	#103	—	109	1	no	—
64	#103	D₁₀₉	1	1	yes	characteristic, complemented
65	C₂²⋊C₈	—	109	1	no	—
66	C₂×Q₈	—	109	1	no	—
67	C₂×Q₈	—	109	1	no	—
68	C₂×Q₈	—	218	1	no	—
69	C₂×Q₈	—	218	2	no	—
70	C₂×Q₈	—	218	2	no	—
71	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂²×C₄	—	109	1	no	—
73	C₂²×C₄	—	218	1	no	—
74	Q₁₆	—	218	2	no	—
75	Q₁₆	—	218	2	no	—
76	C₂×C₈	—	109	1	no	—
77	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	C₄⋊C₄	—	218	1	no	—
79	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
80	C₂²⋊C₄	—	218	1	no	—
81	C₂³	—	109	1	no	—
82	Q₈	—	2	1	no	—
83	Q₈	—	109	2	no	—
84	Q₈	—	218	4	no	—
85	Q₈	—	218	1	no	—
86	Q₈	—	218	1	no	—
87	Q₈	—	218	1	no	—
88	Q₈	D₄₃₆	1	2	yes	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	218	4	no	—
93	C₂×C₄	—	218	1	no	—
94	C₂×C₄	—	218	1	no	—
95	C₂×C₄	—	218	1	no	—
96	C₂×C₄	—	218	1	no	—
97	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	—	218	1	no	—
100	C₂²	—	109	2	no	—
101	C₂²	—	218	1	no	—
102	C₂²	—	218	1	no	—
103	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	4	1	no	—
106	C₄	—	436	1	no	—
107	C₄	—	218	1	no	—
108	C₄	—	218	2	no	—
109	C₄	D₄×D₁₀₉	1	1	yes	characteristic
110	C₄	C₂×D₄₃₆	1	1	yes	characteristic
111	C₂	—	218	2	no	—
112	C₂	#16812669	1	1	yes	characteristic
113	C₂	Q₁₆×D₁₀₉	1	1	yes	characteristic
114	C₂	#16812728	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₁₀₉

C₂

yes

maximal, characteristic

C₂×Dic₄₃₆

C₂

yes

maximal, characteristic, complemented

C₂×#13182019

C₂

yes

maximal, characteristic, complemented

#16812679

C₂

yes

maximal, characteristic

#16812622

C₂

yes

maximal, characteristic, complemented

#16812611

C₂

yes

maximal, characteristic

#103×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×Dic₂₁₈

C₂²

yes

characteristic

Q₈×D₁₀₉

—

Q₈×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

Dic₄₃₆

—

#13182019

—

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

—

#13182006

—

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂²×D₁₀₉

D₄

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

D₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

Q₁₆

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

#120

yes

characteristic

C₂×C₁₀₉

C₂×Q₁₆

yes

characteristic

C₂×C₁₀₉

#137

yes

characteristic

D₁₀₉

—

C₁₀₉

#451

yes

characteristic, complemented, Sylow

#451

—

109

maximal, Sylow

C₂²×Q₈

—

109

—

C₂×Q₁₆

—

109

—

C₂×Q₁₆

—

109

—

C₂²⋊Q₈

—

109

—

#103

—

109

—

#103

D₁₀₉

yes

characteristic, complemented

C₂²⋊C₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

Q₁₆

—

218

—

Q₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₂³

—

109

—

Q₈

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

D₄₃₆

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

100

C₂²

—

109

—

101

C₂²

—

218

—

102

C₂²

—

218

—

103

C₂²

D₄×D₁₀₉

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

436

—

107

C₄

—

218

—

108

C₄

—

218

—

109

C₄

D₄×D₁₀₉

yes

characteristic

110

C₄

C₂×D₄₃₆

yes

characteristic

111

C₂

—

218

—

112

C₂

#16812669

yes

characteristic

113

C₂

Q₁₆×D₁₀₉

yes

characteristic

114

C₂

#16812728

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	#451
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
2. C₂×Dic₄₃₆	C₂	semidirect
3. C₂×#13182019	C₂	semidirect
5. #16812622	C₂	semidirect
7. #103×C₁₀₉	C₂	semidirect
45. C₂×D₁₀₉	Q₁₆	semidirect
57. C₁₀₉	#451	semidirect
64. #103	D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×Dic₄₃₆

C₂

semidirect

3. C₂×#13182019

C₂

semidirect

5. #16812622

C₂

semidirect

7. #103×C₁₀₉

C₂

semidirect

45. C₂×D₁₀₉

Q₁₆

semidirect

57. C₁₀₉

#451

semidirect

64. #103

D₁₀₉

semidirect

Subgroups — #20295183

Conjugacy classes of subgroups

Complements