Group #20295261 — subgroups

Subgroups — #20295261

Order 6976.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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×#13182014	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812674	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812670	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812632	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812621	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₃₆	—	2	1	no	—
9	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
11	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	#13182014	—	2	2	no	—
13	C₂×#11724784	C₂²	1	1	yes	characteristic
14	C₈×D₁₀₉	—	2	2	no	—
15	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂²⋊Dic₁₀₉	—	2	1	no	—
17	#13182006	—	2	1	no	—
18	C₂×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₉	—	2	1	no	—
22	C₂²×D₁₀₉	—	2	1	no	—
23	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
24	D₄×C₁₀₉	—	2	1	no	—
25	D₄×C₁₀₉	—	2	1	no	—
26	C₂×C₄×C₁₀₉	—	2	1	no	—
27	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
28	C₂²⋊D₁₀₉	—	4	2	no	—
29	C₂×Dic₁₀₉	—	2	1	no	—
30	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
31	D₄₃₆	—	2	1	no	—
32	D₄₃₆	—	2	1	no	—
33	D₄₃₆	—	4	2	no	—
34	C₄×D₁₀₉	—	2	1	no	—
35	C₄×D₁₀₉	—	2	1	no	—
36	C₈×C₁₀₉	—	2	1	no	—
37	#11724784	D₄	1	2	yes	complemented
38	C₂²×C₁₀₉	—	4	1	no	—
39	C₂²×C₁₀₉	—	4	1	no	—
40	C₂²×C₁₀₉	—	4	1	no	—
41	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	—	2	1	no	—
44	C₂×D₁₀₉	—	4	1	no	—
45	C₂×D₁₀₉	—	4	1	no	—
46	C₂×D₁₀₉	—	4	1	no	—
47	C₂×D₁₀₉	D₈	1	2	yes	complemented
48	C₄×C₁₀₉	—	4	1	no	—
49	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	C₄×C₁₀₉	#41	1	1	yes	characteristic
51	Dic₁₀₉	—	2	1	no	—
52	Dic₁₀₉	—	4	1	no	—
53	C₂×C₁₀₉	—	8	1	no	—
54	C₂×C₁₀₉	#121	1	1	yes	characteristic
55	C₂×C₁₀₉	C₂×D₈	1	1	yes	characteristic
56	C₂×C₁₀₉	#135	1	1	yes	characteristic
57	D₁₀₉	—	2	2	no	—
58	D₁₀₉	—	8	1	no	—
59	C₁₀₉	#466	1	1	yes	characteristic, complemented, Sylow
60	#466	—	109	1	no	maximal, Sylow
61	C₂×D₈	—	109	1	no	—
62	C₂²×C₈	—	109	1	no	—
63	#121	—	109	1	no	—
64	#121	—	109	1	no	—
65	#107	—	109	1	no	—
66	#102	—	109	1	no	—
67	#102	D₁₀₉	1	1	yes	characteristic, complemented
68	C₂×D₄	—	109	1	no	—
69	C₂×D₄	—	218	1	no	—
70	C₂×D₄	—	218	1	no	—
71	C₂×D₄	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂²×C₄	—	109	1	no	—
73	D₈	—	218	2	no	—
74	C₂×C₈	—	109	1	no	—
75	C₂×C₈	—	218	2	no	—
76	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
77	C₄⋊C₄	—	109	1	no	—
78	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
79	C₂²⋊C₄	—	218	1	no	—
80	C₂²⋊C₄	—	218	1	no	—
81	C₂³	—	2	1	no	—
82	C₂³	—	109	1	no	—
83	C₂³	—	218	1	no	—
84	D₄	—	2	1	no	—
85	D₄	—	2	1	no	—
86	D₄	—	436	2	no	—
87	D₄	—	436	2	no	—
88	D₄	—	218	1	no	—
89	D₄	—	218	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	109	1	no	—
92	C₂×C₄	—	218	1	no	—
93	C₂×C₄	—	218	1	no	—
94	C₂×C₄	—	218	1	no	—
95	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	109	2	no	—
98	C₂²	—	4	1	no	—
99	C₂²	—	4	1	no	—
100	C₂²	—	4	1	no	—
101	C₂²	—	109	2	no	—
102	C₂²	—	436	1	no	—
103	C₂²	—	436	1	no	—
104	C₂²	—	436	1	no	—
105	C₂²	—	218	1	no	—
106	C₂²	—	218	1	no	—
107	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
108	C₄	—	4	1	no	—
109	C₄	—	436	1	no	—
110	C₄	—	218	1	no	—
111	C₄	D₄×D₁₀₉	1	1	yes	characteristic
112	C₄	#13182029	1	1	yes	characteristic
113	C₂	—	8	1	no	—
114	C₂	—	872	1	no	—
115	C₂	—	218	2	no	—
116	C₂	#16812672	1	1	yes	characteristic
117	C₂	D₈×D₁₀₉	1	1	yes	characteristic
118	C₂	#16812720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182014

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812674

C₂

yes

maximal, characteristic, complemented

#16812670

C₂

yes

maximal, characteristic, complemented

#16812632

C₂

yes

maximal, characteristic, complemented

#16812621

C₂

yes

maximal, characteristic, complemented

#102×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182014

—

C₂×#11724784

C₂²

yes

characteristic

C₈×D₁₀₉

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₂²⋊Dic₁₀₉

—

#13182006

—

C₂×D₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

D₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

#11724784

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

D₈

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#121

yes

characteristic

C₂×C₁₀₉

C₂×D₈

yes

characteristic

C₂×C₁₀₉

#135

yes

characteristic

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#466

yes

characteristic, complemented, Sylow

#466

—

109

maximal, Sylow

C₂×D₈

—

109

—

C₂²×C₈

—

109

—

#121

—

109

—

#121

—

109

—

#107

—

109

—

#102

—

109

—

#102

D₁₀₉

yes

characteristic, complemented

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂×D₄

C₂×D₁₀₉

yes

characteristic

C₂²×C₄

—

109

—

D₈

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₂³

—

C₂³

—

109

—

C₂³

—

218

—

D₄

—

D₄

—

D₄

—

436

—

D₄

—

436

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

109

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

109

—

102

C₂²

—

436

—

103

C₂²

—

436

—

104

C₂²

—

436

—

105

C₂²

—

218

—

106

C₂²

—

218

—

107

C₂²

D₄×D₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

436

—

110

C₄

—

218

—

111

C₄

D₄×D₁₀₉

yes

characteristic

112

C₄

#13182029

yes

characteristic

113

C₂

—

114

C₂

—

872

—

115

C₂

—

218

—

116

C₂

#16812672

yes

characteristic

117

C₂

D₈×D₁₀₉

yes

characteristic

118

C₂

#16812720

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#102×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#102	D₁₀₉
hypocenter	C₁₀₉	#466
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#102×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. C₂×#13182014	C₂	semidirect
2. C₂×C₈×D₁₀₉	C₂	semidirect
3. #16812674	C₂	semidirect
4. #16812670	C₂	semidirect
5. #16812632	C₂	semidirect
6. #16812621	C₂	semidirect
7. #102×C₁₀₉	C₂	semidirect
37. #11724784	D₄	semidirect
47. C₂×D₁₀₉	D₈	semidirect
59. C₁₀₉	#466	semidirect
67. #102	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182014

C₂

semidirect

2. C₂×C₈×D₁₀₉

C₂

semidirect

3. #16812674

C₂

semidirect

4. #16812670

C₂

semidirect

5. #16812632

C₂

semidirect

6. #16812621

C₂

semidirect

7. #102×C₁₀₉

C₂

semidirect

37. #11724784

D₄

semidirect

47. C₂×D₁₀₉

D₈

semidirect

59. C₁₀₉

#466

semidirect

67. #102

D₁₀₉

semidirect

Subgroups — #20295261

Conjugacy classes of subgroups

Complements