Group #20295207 — subgroups

Subgroups — #20295207

Order 6976.

142 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 39 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	#16812746	C₂	1	1	yes	maximal, characteristic
2	#16812730	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812697	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812626	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812625	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812617	C₂	1	1	yes	maximal, characteristic, complemented
7	#137×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182029	—	2	1	no	—
9	#13182029	C₂²	1	1	yes	characteristic
10	#13182028	—	2	1	no	—
11	#13182028	—	2	2	no	—
12	C₂×Dic₂₁₈	—	2	1	no	—
13	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
14	Q₈×D₁₀₉	—	2	1	no	—
15	#13182019	—	2	1	no	—
16	#13182016	—	2	1	no	—
17	#13182013	C₂²	1	1	yes	characteristic
18	#13182008	—	2	2	no	—
19	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
20	#41×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
22	Q₁₆×C₁₀₉	—	2	1	no	—
23	SD₁₆×C₁₀₉	—	2	1	no	—
24	M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
25	Q₈×C₁₀₉	—	2	1	no	—
26	Q₈×C₁₀₉	—	2	1	no	—
27	Q₈×C₁₀₉	D₄	1	1	yes	characteristic
28	D₄×C₁₀₉	—	2	1	no	—
29	D₄×C₁₀₉	D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₀₉	—	2	1	no	—
31	C₂×C₄×C₁₀₉	—	2	1	no	—
32	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₀₉	—	2	2	no	—
34	C₂²⋊D₁₀₉	—	2	1	no	—
35	C₂×Dic₁₀₉	—	2	2	no	—
36	C₂×Dic₁₀₉	—	2	1	no	—
37	C₂×Dic₁₀₉	D₄	1	2	yes	—
38	D₄₃₆	D₄	1	1	yes	characteristic
39	C₄×D₁₀₉	—	2	2	no	—
40	C₄×D₁₀₉	—	2	1	no	—
41	Dic₂₁₈	—	2	2	no	—
42	Dic₂₁₈	—	2	1	no	—
43	Dic₂₁₈	—	2	1	no	—
44	Dic₂₁₈	D₄	1	1	yes	characteristic
45	C₈×C₁₀₉	—	2	1	no	—
46	#11724784	—	2	1	no	—
47	C₂²×C₁₀₉	—	2	1	no	—
48	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×D₁₀₉	—	2	1	no	—
50	C₄×C₁₀₉	—	2	1	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	2	no	—
55	Dic₁₀₉	—	2	2	no	—
56	Dic₁₀₉	—	2	1	no	—
57	C₂×C₁₀₉	—	2	1	no	—
58	C₂×C₁₀₉	—	4	1	no	—
59	C₂×C₁₀₉	#120	1	1	yes	characteristic
60	D₁₀₉	—	4	1	no	—
61	C₁₀₉	#456	1	1	yes	characteristic, complemented, Sylow
62	#456	—	109	1	no	maximal, Sylow
63	2₋¹⁺⁴	—	109	1	no	—
64	#137	—	109	1	no	—
65	#137	D₁₀₉	1	1	yes	characteristic, complemented
66	C₄⋊Q₈	—	109	1	no	—
67	#104	—	109	1	no	—
68	#104	—	109	1	no	—
69	#101	—	109	1	no	—
70	#41	—	109	1	no	—
71	#41	—	218	1	no	—
72	#41	—	218	2	no	—
73	#41	—	218	1	no	—
74	#41	C₂×D₁₀₉	1	1	yes	characteristic
75	C₂×Q₈	—	109	1	no	—
76	C₂×Q₈	—	218	1	no	—
77	C₂×Q₈	—	218	1	no	—
78	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
79	Q₁₆	—	2	1	no	—
80	Q₁₆	—	218	1	no	—
81	SD₁₆	—	2	1	no	—
82	SD₁₆	—	218	1	no	—
83	M₁₆	—	109	1	no	—
84	M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
85	C₄⋊C₄	—	218	2	no	—
86	C₄²	—	109	1	no	—
87	Q₈	—	2	1	no	—
88	Q₈	—	2	1	no	—
89	Q₈	—	109	1	no	—
90	Q₈	—	218	2	no	—
91	Q₈	—	218	1	no	—
92	Q₈	—	218	1	no	—
93	Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
94	D₄	—	2	1	no	—
95	D₄	—	109	1	no	—
96	D₄	—	218	2	no	—
97	D₄	—	218	1	no	—
98	D₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	109	2	no	—
102	C₂×C₄	—	218	2	no	—
103	C₂×C₄	—	218	2	no	—
104	C₂×C₄	—	218	1	no	—
105	C₂×C₄	—	218	1	no	—
106	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	218	1	no	—
109	C₂²	—	2	1	no	—
110	C₂²	—	218	1	no	—
111	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	1	no	—
114	C₄	—	218	2	no	—
115	C₄	—	218	2	no	—
116	C₄	—	218	1	no	—
117	C₄	D₄×D₁₀₉	1	1	yes	characteristic
118	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	4	1	no	—
121	C₂	—	436	1	no	—
122	C₂	#16812667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16812746

C₂

yes

maximal, characteristic

#16812730

C₂

yes

maximal, characteristic, complemented

#16812697

C₂

yes

maximal, characteristic, complemented

#16812626

C₂

yes

maximal, characteristic, complemented

#16812625

C₂

yes

maximal, characteristic, complemented

#16812617

C₂

yes

maximal, characteristic, complemented

#137×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

—

#13182029

C₂²

yes

characteristic

#13182028

—

#13182028

—

C₂×Dic₂₁₈

—

C₂×Dic₂₁₈

C₂²

yes

characteristic

Q₈×D₁₀₉

—

#13182019

—

#13182016

—

#13182013

C₂²

yes

characteristic

#13182008

—

C₄×Dic₁₀₉

C₂²

yes

characteristic

#41×C₁₀₉

C₂²

yes

characteristic

C₂×Q₈×C₁₀₉

C₂²

yes

characteristic

Q₁₆×C₁₀₉

—

SD₁₆×C₁₀₉

—

M₁₆×C₁₀₉

C₂²

yes

characteristic

Q₈×C₁₀₉

—

Q₈×C₁₀₉

—

Q₈×C₁₀₉

D₄

yes

characteristic

D₄×C₁₀₉

—

D₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

—

D₄₃₆

D₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

—

Dic₂₁₈

D₄

yes

characteristic

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#120

yes

characteristic

D₁₀₉

—

C₁₀₉

#456

yes

characteristic, complemented, Sylow

#456

—

109

maximal, Sylow

2₋¹⁺⁴

—

109

—

#137

—

109

—

#137

D₁₀₉

yes

characteristic, complemented

C₄⋊Q₈

—

109

—

#104

—

109

—

#104

—

109

—

#101

—

109

—

#41

—

109

—

#41

—

218

—

#41

—

218

—

#41

—

218

—

#41

C₂×D₁₀₉

yes

characteristic

C₂×Q₈

—

109

—

C₂×Q₈

—

218

—

C₂×Q₈

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

Q₁₆

—

Q₁₆

—

218

—

SD₁₆

—

SD₁₆

—

218

—

M₁₆

—

109

—

M₁₆

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

218

—

C₄²

—

109

—

Q₈

—

Q₈

—

Q₈

—

109

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

—

218

—

Q₈

C₂²⋊D₁₀₉

yes

characteristic

D₄

—

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

D₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

109

—

102

C₂×C₄

—

218

—

103

C₂×C₄

—

218

—

104

C₂×C₄

—

218

—

105

C₂×C₄

—

218

—

106

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

107

C₈

—

108

C₈

—

218

—

109

C₂²

—

110

C₂²

—

218

—

111

C₂²

D₄×D₁₀₉

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

218

—

115

C₄

—

218

—

116

C₄

—

218

—

117

C₄

D₄×D₁₀₉

yes

characteristic

118

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

436

—

122

C₂

#16812667

yes

characteristic

	subgroup	G/H
center	C₂	#16812667
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#137×C₁₀₉	C₂
socle	C₂×C₁₀₉	#120
hypercenter	#137	D₁₀₉
hypocenter	C₁₀₉	#456
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

subgroup

G/H

center

C₂

#16812667

commutator

C₂×C₄×C₁₀₉

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#137×C₁₀₉

C₂

C₂×C₁₀₉

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

N	H ≅ G/N	type
2. #16812730	C₂	semidirect
3. #16812697	C₂	semidirect
4. #16812626	C₂	semidirect
5. #16812625	C₂	semidirect
6. #16812617	C₂	semidirect
7. #137×C₁₀₉	C₂	semidirect
61. C₁₀₉	#456	semidirect
65. #137	D₁₀₉	semidirect

H ≅ G/N

type

2. #16812730

C₂

semidirect

3. #16812697

C₂

semidirect

4. #16812626

C₂

semidirect

5. #16812625

C₂

semidirect

6. #16812617

C₂

semidirect

7. #137×C₁₀₉

C₂

semidirect

61. C₁₀₉

#456

semidirect

65. #137

D₁₀₉

semidirect

Subgroups — #20295207

Conjugacy classes of subgroups

Complements