Group #20295100 — subgroups

Subgroups — #20295100

Order 6976.

124 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 51 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×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₂²⋊D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812628	C₂	1	1	yes	maximal, characteristic, complemented
5	#16812611	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812609	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₈×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	#13182013	—	2	2	no	—
11	C₂×#11724784	C₂²	1	1	yes	characteristic
12	C₂×#11724784	C₂²	1	1	yes	characteristic
13	C₈×D₁₀₉	—	2	2	no	—
14	#13182008	C₄	1	1	yes	characteristic
15	C₂²⋊Dic₁₀₉	C₄	1	1	yes	characteristic
16	#13182006	C₄	1	1	yes	characteristic
17	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
35	#11724784	—	2	1	no	—
36	#11724784	D₄	1	2	yes	complemented
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	M₁₆	1	2	yes	complemented
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	—	2	1	no	—
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	—	2	2	no	—
55	C₁₀₉	#435	1	1	yes	characteristic, complemented, Sylow
56	#435	—	109	1	no	maximal, Sylow
57	C₂×M₁₆	—	109	1	no	—
58	C₂²×C₈	—	109	1	no	—
59	C₄×D₄	—	109	1	no	—
60	C₄⋊C₈	—	109	1	no	—
61	C₂²⋊C₈	—	109	1	no	—
62	C₂²⋊C₈	D₁₀₉	1	1	yes	characteristic, complemented
63	#97	—	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	—	218	2	no	—
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	218	2	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	—	2	1	no	—
91	C₈	—	109	2	no	—
92	C₈	—	218	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	2	1	no	—
96	C₂²	—	109	2	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	—	218	1	no	—
102	C₄	—	218	1	no	—
103	C₄	—	218	1	no	—
104	C₄	D₄×D₁₀₉	1	1	yes	characteristic
105	C₄	#13182028	1	1	yes	characteristic
106	C₂	—	4	1	no	—
107	C₂	—	218	2	no	—
108	C₂	#16812659	1	1	yes	characteristic
109	C₂	M₁₆×D₁₀₉	1	1	yes	characteristic
110	C₂	#16812705	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182013

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₀₉

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₁₀₉

C₂

yes

maximal, characteristic

#16812628

C₂

yes

maximal, characteristic, complemented

#16812611

C₂

yes

maximal, characteristic, complemented

#16812609

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊D₁₀₉

C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182013

—

C₂×#11724784

C₂²

yes

characteristic

C₂×#11724784

C₂²

yes

characteristic

C₈×D₁₀₉

—

#13182008

C₄

yes

characteristic

C₂²⋊Dic₁₀₉

C₄

yes

characteristic

#13182006

C₄

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

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

—

#11724784

—

#11724784

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

M₁₆

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₄×C₁₀₉

#41

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂×C₁₀₉

#131

yes

characteristic

D₁₀₉

—

C₁₀₉

#435

yes

characteristic, complemented, Sylow

#435

—

109

maximal, Sylow

C₂×M₁₆

—

109

—

C₂²×C₈

—

109

—

C₄×D₄

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

D₁₀₉

yes

characteristic, complemented

#97

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

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

—

C₈

—

109

—

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

100

C₄

—

101

C₄

—

218

—

102

C₄

—

218

—

103

C₄

—

218

—

104

C₄

D₄×D₁₀₉

yes

characteristic

105

C₄

#13182028

yes

characteristic

106

C₂

—

107

C₂

—

218

—

108

C₂

#16812659

yes

characteristic

109

C₂

M₁₆×D₁₀₉

yes

characteristic

110

C₂

#16812705

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂²⋊C₈×C₁₀₉	C₂
hypercenter	C₂²⋊C₈	D₁₀₉
hypocenter	C₁₀₉	#435
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂²⋊C₈×C₁₀₉

C₂

hypercenter

C₂²⋊C₈

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂²⋊C₈

N	H ≅ G/N	type
1. C₂×#13182013	C₂	semidirect
2. C₂×C₈×D₁₀₉	C₂	semidirect
4. #16812628	C₂	semidirect
5. #16812611	C₂	semidirect
6. #16812609	C₂	semidirect
7. C₂²⋊C₈×C₁₀₉	C₂	semidirect
36. #11724784	D₄	semidirect
43. C₂×D₁₀₉	M₁₆	semidirect
55. C₁₀₉	#435	semidirect
62. C₂²⋊C₈	D₁₀₉	semidirect

H ≅ G/N

type

1. C₂×#13182013

C₂

semidirect

2. C₂×C₈×D₁₀₉

C₂

semidirect

4. #16812628

C₂

semidirect

5. #16812611

C₂

semidirect

6. #16812609

C₂

semidirect

7. C₂²⋊C₈×C₁₀₉

C₂

semidirect

36. #11724784

D₄

semidirect

43. C₂×D₁₀₉

M₁₆

semidirect

55. C₁₀₉

#435

semidirect

62. C₂²⋊C₈

D₁₀₉

semidirect

Subgroups — #20295100

Conjugacy classes of subgroups

Complements