Group #20295102 — subgroups

Subgroups — #20295102

Order 6976.

122 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 122 conjugacy classes of subgroups — too many to draw. Only its 53 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×#13182012	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#16812628	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812611	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₁₀₉	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
10	#13182012	—	2	2	no	—
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₂×#11724784	C₂²	1	1	yes	characteristic
13	#13182008	C₄	1	1	yes	characteristic
14	C₂²⋊Dic₁₀₉	C₄	1	1	yes	characteristic
15	#13182006	C₄	1	1	yes	characteristic
16	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
18	M₁₆×C₁₀₉	—	2	2	no	—
19	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	—	2	1	no	—
24	C₂×C₄×C₁₀₉	—	2	1	no	—
25	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₀₉	—	2	2	no	—
28	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	1	no	—
32	C₄×D₁₀₉	—	2	1	no	—
33	C₈×C₁₀₉	—	2	1	no	—
34	C₈×C₁₀₉	D₄	1	2	yes	complemented
35	#11724784	—	2	1	no	—
36	#11724784	—	2	1	no	—
37	C₂²×C₁₀₉	—	2	1	no	—
38	C₂²×C₁₀₉	—	2	1	no	—
39	C₂²×C₁₀₉	—	2	1	no	—
40	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	1	no	—
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	—	2	1	no	—
44	C₄×C₁₀₉	—	2	1	no	—
45	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
46	C₄×C₁₀₉	#41	1	1	yes	characteristic
47	Dic₁₀₉	—	2	1	no	—
48	Dic₁₀₉	—	2	1	no	—
49	Dic₁₀₉	M₁₆	1	2	yes	complemented
50	C₂×C₁₀₉	—	4	1	no	—
51	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
52	C₂×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₀₉	#131	1	1	yes	characteristic
54	D₁₀₉	—	4	1	no	—
55	C₁₀₉	#436	1	1	yes	characteristic, complemented, Sylow
56	#436	—	109	1	no	maximal, Sylow
57	C₂×M₁₆	—	109	1	no	—
58	C₂×M₁₆	D₁₀₉	1	1	yes	characteristic, complemented
59	C₄×D₄	—	109	1	no	—
60	C₄⋊C₈	—	109	1	no	—
61	C₂²⋊C₈	—	109	1	no	—
62	C₂²⋊C₈	—	109	1	no	—
63	C₄×C₈	—	109	1	no	—
64	C₂×D₄	—	109	1	no	—
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	M₁₆	—	218	2	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
72	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
73	C₄⋊C₄	—	109	1	no	—
74	C₂²⋊C₄	—	109	1	no	—
75	C₂²⋊C₄	—	109	1	no	—
76	C₄²	—	109	1	no	—
77	C₂³	—	109	1	no	—
78	C₂³	C₄×D₁₀₉	1	1	yes	characteristic
79	D₄	—	218	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	109	1	no	—
84	C₂×C₄	—	109	1	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	218	1	no	—
91	C₈	—	218	1	no	—
92	C₈	C₂²⋊D₁₀₉	1	2	yes	complemented
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	218	1	no	—
97	C₂²	—	218	1	no	—
98	C₂²	—	218	1	no	—
99	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	109	2	no	—
102	C₄	—	218	1	no	—
103	C₄	—	218	1	no	—
104	C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
105	C₄	#13182029	1	1	yes	characteristic
106	C₂	—	4	1	no	—
107	C₂	—	436	1	no	—
108	C₂	C₄×C₂²⋊D₁₀₉	1	1	yes	characteristic
109	C₂	M₁₆×D₁₀₉	1	1	yes	characteristic
110	C₂	#16812707	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182012

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₀₉

C₂

yes

maximal, characteristic

#16812628

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11724784

C₂

yes

maximal, characteristic, complemented

#16812611

C₂

yes

maximal, characteristic, complemented

C₈×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×M₁₆×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₀₉

C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182012

—

C₂×#11724784

C₂²

yes

characteristic

C₂×#11724784

C₂²

yes

characteristic

#13182008

C₄

yes

characteristic

C₂²⋊Dic₁₀₉

C₄

yes

characteristic

#13182006

C₄

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

characteristic

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

D₄

yes

complemented

#11724784

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

M₁₆

yes

complemented

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂×C₁₀₉

#131

yes

characteristic

D₁₀₉

—

C₁₀₉

#436

yes

characteristic, complemented, Sylow

#436

—

109

maximal, Sylow

C₂×M₁₆

—

109

—

C₂×M₁₆

D₁₀₉

yes

characteristic, complemented

C₄×D₄

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

M₁₆

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₂×C₈

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₂³

—

109

—

C₂³

C₄×D₁₀₉

yes

characteristic

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₈

—

218

—

C₈

C₂²⋊D₁₀₉

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

100

C₄

—

101

C₄

—

109

—

102

C₄

—

218

—

103

C₄

—

218

—

104

C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

105

C₄

#13182029

yes

characteristic

106

C₂

—

107

C₂

—

436

—

108

C₂

C₄×C₂²⋊D₁₀₉

yes

characteristic

109

C₂

M₁₆×D₁₀₉

yes

characteristic

110

C₂

#16812707

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×M₁₆×C₁₀₉	C₂
hypercenter	C₂×M₁₆	D₁₀₉
hypocenter	C₁₀₉	#436
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂×M₁₆×C₁₀₉

C₂

hypercenter

C₂×M₁₆

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#13182012	C₂	semidirect
3. #16812628	C₂	semidirect
4. C₂²⋊#11724784	C₂	semidirect
5. #16812611	C₂	semidirect
6. C₈×Dic₁₀₉	C₂	semidirect
7. C₂×M₁₆×C₁₀₉	C₂	semidirect
34. C₈×C₁₀₉	D₄	semidirect
49. Dic₁₀₉	M₁₆	semidirect
55. C₁₀₉	#436	semidirect
58. C₂×M₁₆	D₁₀₉	semidirect
92. C₈	C₂²⋊D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182012

C₂

semidirect

3. #16812628

C₂

semidirect

4. C₂²⋊#11724784

C₂

semidirect

5. #16812611

C₂

semidirect

6. C₈×Dic₁₀₉

C₂

semidirect

7. C₂×M₁₆×C₁₀₉

C₂

semidirect

34. C₈×C₁₀₉

D₄

semidirect

49. Dic₁₀₉

M₁₆

semidirect

55. C₁₀₉

#436

semidirect

58. C₂×M₁₆

D₁₀₉

semidirect

92. C₈

C₂²⋊D₁₀₉

semidirect

Subgroups — #20295102

Conjugacy classes of subgroups

Complements