Group #20294988 — subgroups

Subgroups — #20294988

Order 6976.

190 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 61 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	C₂²×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182012	C₂	1	2	yes	maximal, complemented
3	C₂²⋊#11724784	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812611	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₀₉	C₄	1	1	yes	characteristic
7	C₂²×Dic₁₀₉	C₄	1	1	yes	characteristic
8	C₂×C₄×D₁₀₉	—	2	4	no	—
9	C₂×C₄×D₁₀₉	C₄	1	1	yes	characteristic
10	C₂×C₄×D₁₀₉	C₄	1	1	yes	characteristic
11	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
13	#13182012	—	2	4	no	—
14	C₂×#11724784	C₂²	1	2	yes	complemented
15	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₈×C₁₀₉	C₂²	1	2	yes	complemented
17	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₂²×D₁₀₉	—	2	2	no	—
19	C₂²×D₁₀₉	—	2	2	no	—
20	C₂²×D₁₀₉	—	2	2	no	—
21	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₀₉	—	2	2	no	—
27	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₄×D₁₀₉	—	2	2	no	—
30	C₄×D₁₀₉	—	2	2	no	—
31	C₄×D₁₀₉	—	2	2	no	—
32	C₄×D₁₀₉	D₄	1	4	yes	—
33	C₈×C₁₀₉	—	2	2	no	—
34	#11724784	—	2	2	no	—
35	C₂²×C₁₀₉	—	2	1	no	—
36	C₂²×C₁₀₉	—	2	1	no	—
37	C₂²×C₁₀₉	M₁₆	1	2	yes	complemented
38	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₀₉	—	2	8	no	—
40	C₂×D₁₀₉	—	2	1	no	—
41	C₂×D₁₀₉	—	2	1	no	—
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	—	2	1	no	—
44	C₂×D₁₀₉	C₂²⋊C₄	1	2	yes	—
45	C₂×D₁₀₉	M₁₆	1	2	yes	complemented
46	C₄×C₁₀₉	—	2	1	no	—
47	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
48	Dic₁₀₉	—	2	1	no	—
49	Dic₁₀₉	C₂²⋊C₄	1	2	yes	—
50	C₂×C₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
54	D₁₀₉	—	2	2	no	—
55	D₁₀₉	—	2	2	no	—
56	C₁₀₉	#407	1	1	yes	characteristic, complemented, Sylow
57	#407	—	109	1	no	maximal, Sylow
58	C₂³×C₄	—	109	1	no	—
59	C₂×M₁₆	—	109	2	no	—
60	C₂²⋊C₈	—	109	1	no	—
61	C₂²⋊C₈	—	109	2	no	—
62	C₂²⋊C₈	D₁₀₉	1	1	yes	characteristic, complemented
63	C₂⁴	—	109	1	no	—
64	C₂²×C₄	—	109	1	no	—
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	—	109	1	no	—
67	C₂²×C₄	—	109	1	no	—
68	C₂²×C₄	—	109	1	no	—
69	C₂²×C₄	—	218	4	no	—
70	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
71	M₁₆	—	218	4	no	—
72	C₂×C₈	—	109	2	no	—
73	C₂×C₈	C₂×D₁₀₉	1	2	yes	complemented
74	C₂³	—	109	1	no	—
75	C₂³	—	109	1	no	—
76	C₂³	—	218	2	no	—
77	C₂³	—	218	2	no	—
78	C₂³	—	218	2	no	—
79	C₂³	C₄×D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	109	4	no	—
83	C₂×C₄	—	109	1	no	—
84	C₂×C₄	—	218	2	no	—
85	C₂×C₄	—	218	2	no	—
86	C₂×C₄	—	218	2	no	—
87	C₂×C₄	—	218	2	no	—
88	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	218	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	109	2	no	—
95	C₂²	—	109	2	no	—
96	C₂²	—	218	8	no	—
97	C₂²	—	218	1	no	—
98	C₂²	—	218	1	no	—
99	C₂²	—	218	1	no	—
100	C₂²	—	218	1	no	—
101	C₂²	#13182012	1	2	yes	complemented
102	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	109	2	no	—
105	C₄	—	218	1	no	—
106	C₄	D₄×D₁₀₉	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	—	218	2	no	—
109	C₂	—	218	2	no	—
110	C₂	C₂²⋊C₄×D₁₀₉	1	1	yes	characteristic
111	C₂	M₁₆×D₁₀₉	1	1	yes	characteristic
112	C₂	C₂×#13182012	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182012

C₂

yes

maximal, complemented

C₂²⋊#11724784

C₂

yes

maximal, characteristic, complemented

#16812611

C₂

yes

maximal, complemented

C₂²⋊C₈×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₄

yes

characteristic

C₂²×Dic₁₀₉

C₄

yes

characteristic

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₄

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182012

—

C₂×#11724784

C₂²

yes

complemented

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₂×C₈×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

D₄

yes

—

C₈×C₁₀₉

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

M₁₆

yes

complemented

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₄

yes

—

C₂×D₁₀₉

M₁₆

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

C₂×M₁₆

yes

characteristic

C₂×C₁₀₉

C₂×M₁₆

yes

characteristic

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#407

yes

characteristic, complemented, Sylow

#407

—

109

maximal, Sylow

C₂³×C₄

—

109

—

C₂×M₁₆

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

D₁₀₉

yes

characteristic, complemented

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

C₂×D₁₀₉

yes

complemented

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₄×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

109

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

100

C₂²

—

218

—

101

C₂²

#13182012

yes

complemented

102

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

103

C₄

—

104

C₄

—

109

—

105

C₄

—

218

—

106

C₄

D₄×D₁₀₉

yes

—

107

C₂

—

108

C₂

—

218

—

109

C₂

—

218

—

110

C₂

C₂²⋊C₄×D₁₀₉

yes

characteristic

111

C₂

M₁₆×D₁₀₉

yes

characteristic

112

C₂

C₂×#13182012

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	#407
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
2. C₂×#13182012	C₂	semidirect
3. C₂²⋊#11724784	C₂	semidirect
4. #16812611	C₂	semidirect
5. C₂²⋊C₈×C₁₀₉	C₂	semidirect
14. C₂×#11724784	C₂²	semidirect
16. C₂×C₈×C₁₀₉	C₂²	semidirect
37. C₂²×C₁₀₉	M₁₆	semidirect
45. C₂×D₁₀₉	M₁₆	semidirect
56. C₁₀₉	#407	semidirect
62. C₂²⋊C₈	D₁₀₉	semidirect
73. C₂×C₈	C₂×D₁₀₉	semidirect
101. C₂²	#13182012	semidirect

H ≅ G/N

type

2. C₂×#13182012

C₂

semidirect

3. C₂²⋊#11724784

C₂

semidirect

4. #16812611

C₂

semidirect

5. C₂²⋊C₈×C₁₀₉

C₂

semidirect

14. C₂×#11724784

C₂²

semidirect

16. C₂×C₈×C₁₀₉

C₂²

semidirect

37. C₂²×C₁₀₉

M₁₆

semidirect

45. C₂×D₁₀₉

M₁₆

semidirect

56. C₁₀₉

#407

semidirect

62. C₂²⋊C₈

D₁₀₉

semidirect

73. C₂×C₈

C₂×D₁₀₉

semidirect

101. C₂²

#13182012

semidirect

Subgroups — #20294988

Conjugacy classes of subgroups

Complements