Group #20295026 — subgroups

Subgroups — #20295026

Order 6976.

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

normal subgroups only

This group has 162 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×#13182030	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812623	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812621	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812619	C₂	1	1	yes	maximal, characteristic, complemented
7	#103×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#13182030	—	2	2	no	—
9	#13182030	C₄	1	4	yes	complemented
10	C₂×D₄₃₆	—	2	1	no	—
11	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₀₉	—	2	1	no	—
13	C₂×C₄×D₁₀₉	—	2	1	no	—
14	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
15	#13182012	—	2	2	no	—
16	C₂×#11724784	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
18	#13182008	—	2	1	no	—
19	C₂×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×D₁₀₉	—	2	1	no	—
23	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
24	Q₈×C₁₀₉	—	2	1	no	—
25	Q₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
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	D₄	1	1	yes	characteristic
31	D₄₃₆	—	2	4	no	—
32	D₄₃₆	—	2	1	no	—
33	D₄₃₆	C₂×C₄	1	2	yes	complemented
34	C₄×D₁₀₉	—	2	4	no	—
35	C₄×D₁₀₉	—	4	2	no	—
36	C₄×D₁₀₉	C₂×C₄	1	2	yes	—
37	C₄×D₁₀₉	D₄	1	2	yes	—
38	C₈×C₁₀₉	—	2	1	no	—
39	#11724784	—	2	1	no	—
40	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	1	no	—
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	—	2	2	no	—
44	C₂×D₁₀₉	—	4	1	no	—
45	C₂×D₁₀₉	—	4	1	no	—
46	C₂×D₁₀₉	C₂²⋊C₄	1	2	yes	—
47	C₄×C₁₀₉	—	2	2	no	—
48	C₄×C₁₀₉	—	4	1	no	—
49	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
50	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
51	Dic₁₀₉	—	4	1	no	—
52	Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
53	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₁₀₉	AGL(1,ℤ/8)	1	1	yes	characteristic
55	C₂×C₁₀₉	#137	1	1	yes	characteristic
56	D₁₀₉	—	2	2	no	—
57	D₁₀₉	—	4	2	no	—
58	C₁₀₉	#417	1	1	yes	characteristic, complemented, Sylow
59	#417	—	109	1	no	maximal, Sylow
60	C₂×#41	—	109	1	no	—
61	C₂×M₁₆	—	109	1	no	—
62	C₂×C₄⋊C₄	—	109	1	no	—
63	#103	—	109	1	no	—
64	#103	D₁₀₉	1	1	yes	characteristic, complemented
65	#102	—	109	1	no	—
66	#102	—	109	1	no	—
67	#41	—	109	4	no	—
68	#41	—	218	2	no	—
69	C₂×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
70	C₂×D₄	—	109	1	no	—
71	C₂×D₄	—	218	1	no	—
72	C₂²×C₄	—	109	1	no	—
73	C₂²×C₄	—	218	1	no	—
74	C₂²×C₄	—	218	1	no	—
75	M₁₆	—	218	2	no	—
76	C₂×C₈	—	109	1	no	—
77	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
78	C₄⋊C₄	—	109	1	no	—
79	C₄⋊C₄	—	218	1	no	—
80	C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
81	C₂³	—	109	1	no	—
82	C₂³	—	218	1	no	—
83	Q₈	—	2	1	no	—
84	Q₈	C₄×D₁₀₉	1	2	yes	complemented
85	D₄	—	109	2	no	—
86	D₄	—	218	4	no	—
87	D₄	—	218	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	109	1	no	—
91	C₂×C₄	—	109	2	no	—
92	C₂×C₄	—	109	2	no	—
93	C₂×C₄	—	436	2	no	—
94	C₂×C₄	—	218	4	no	—
95	C₂×C₄	—	218	1	no	—
96	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	218	1	no	—
99	C₂²	—	109	2	no	—
100	C₂²	—	436	1	no	—
101	C₂²	—	436	1	no	—
102	C₂²	—	218	2	no	—
103	C₂²	—	218	1	no	—
104	C₂²	—	218	1	no	—
105	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	4	1	no	—
108	C₄	—	109	2	no	—
109	C₄	—	436	1	no	—
110	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
111	C₄	D₄×D₁₀₉	1	1	yes	characteristic
112	C₂	—	436	2	no	—
113	C₂	—	218	2	no	—
114	C₂	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
115	C₂	#16812726	1	1	yes	characteristic
116	C₂	#16812731	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182030

C₂

yes

maximal, characteristic

C₂×#13182012

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812623

C₂

yes

maximal, characteristic, complemented

#16812621

C₂

yes

maximal, characteristic, complemented

#16812619

C₂

yes

maximal, characteristic, complemented

#103×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182030

—

#13182030

C₄

yes

complemented

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182012

—

C₂×#11724784

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182008

—

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

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

D₄₃₆

—

D₄₃₆

—

D₄₃₆

C₂×C₄

yes

complemented

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₄

yes

—

C₄×D₁₀₉

D₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

C₂×C₁₀₉

yes

C₂×C₁₀₉

yes

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#417

yes

characteristic, complemented, Sylow

#417

—

109

maximal, Sylow

C₂×#41

—

109

—

C₂×M₁₆

—

109

—

C₂×C₄⋊C₄

—

109

—

#103

—

109

—

#103

D₁₀₉

yes

characteristic, complemented

#102

—

109

—

#102

—

109

—

#41

—

109

—

#41

—

218

—

C₂×Q₈

C₂×D₁₀₉

yes

characteristic

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

C₄⋊C₄

C₂×D₁₀₉

yes

characteristic

C₂³

—

109

—

C₂³

—

218

—

Q₈

—

Q₈

C₄×D₁₀₉

yes

complemented

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

—

109

—

100

C₂²

—

436

—

101

C₂²

—

436

—

102

C₂²

—

218

—

103

C₂²

—

218

—

104

C₂²

—

218

—

105

C₂²

D₄×D₁₀₉

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

—

109

C₄

—

436

—

110

C₄

C₂×C₄×D₁₀₉

yes

characteristic

111

C₄

D₄×D₁₀₉

yes

characteristic

112

C₂

—

436

—

113

C₂

—

218

—

114

C₂

C₂²⋊C₄×D₁₀₉

yes

characteristic

115

C₂

#16812726

yes

characteristic

116

C₂

#16812731

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#103×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#103	D₁₀₉
hypocenter	C₁₀₉	#417
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

subgroup

G/H

center

C₂²

D₄×D₁₀₉

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#103×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

N	H ≅ G/N	type
2. C₂×#13182012	C₂	semidirect
3. C₄⋊C₄×D₁₀₉	C₂	semidirect
4. #16812623	C₂	semidirect
5. #16812621	C₂	semidirect
6. #16812619	C₂	semidirect
7. #103×C₁₀₉	C₂	semidirect
9. #13182030	C₄	semidirect
25. Q₈×C₁₀₉	C₂×C₄	semidirect
33. D₄₃₆	C₂×C₄	semidirect
58. C₁₀₉	#417	semidirect
64. #103	D₁₀₉	semidirect
84. Q₈	C₄×D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#13182012

C₂

semidirect

3. C₄⋊C₄×D₁₀₉

C₂

semidirect

4. #16812623

C₂

semidirect

5. #16812621

C₂

semidirect

6. #16812619

C₂

semidirect

7. #103×C₁₀₉

C₂

semidirect

9. #13182030

C₄

semidirect

25. Q₈×C₁₀₉

C₂×C₄

semidirect

33. D₄₃₆

C₂×C₄

semidirect

58. C₁₀₉

#417

semidirect

64. #103

D₁₀₉

semidirect

84. Q₈

C₄×D₁₀₉

semidirect

Subgroups — #20295026

Conjugacy classes of subgroups

Complements