Group #20296053 — subgroups

Subgroups — #20296053

Order 6976.

134 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 56 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×D₁₀₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812766	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812756	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#11724786	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11724786	C₂	1	1	yes	maximal, characteristic, complemented
7	#16812659	C₂	1	1	yes	maximal, characteristic
8	D₁₀₉⋊C₈	—	2	2	no	—
9	C₂×#11724786	—	2	2	no	—
10	C₂×#11724786	C₂²	1	1	yes	characteristic
11	C₂×#11724786	C₂²	1	1	yes	characteristic
12	C₂×#11724786	C₂²	1	1	yes	characteristic
13	C₂×#11724786	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₁₀₉	C₄	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	#13182008	C₄	1	1	yes	characteristic
18	#13182006	C₄	1	1	yes	characteristic
19	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₁₀₉	C₄	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	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊D₁₀₉	C₈	1	4	yes	complemented
26	C₂×Dic₁₀₉	—	2	1	no	—
27	C₂×Dic₁₀₉	—	2	1	no	—
28	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
29	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
31	C₄×D₁₀₉	—	2	1	no	—
32	C₄×D₁₀₉	—	2	1	no	—
33	#11724786	—	2	1	no	—
34	#11724786	—	2	1	no	—
35	#11724786	—	2	1	no	—
36	#11724786	D₄	1	2	yes	complemented
37	C₂²×C₁₀₉	—	2	1	no	—
38	C₂²×C₁₀₉	—	2	1	no	—
39	C₂²×C₁₀₉	C₂×C₈	1	2	yes	complemented
40	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×D₁₀₉	—	2	1	no	—
42	C₂×D₁₀₉	—	2	1	no	—
43	C₂×D₁₀₉	C₂×C₈	1	2	yes	complemented
44	C₄×C₁₀₉	—	2	1	no	—
45	C₄×C₁₀₉	—	2	1	no	—
46	Dic₁₀₉	—	2	1	no	—
47	Dic₁₀₉	C₂×C₈	1	2	yes	complemented
48	Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
49	Dic₁₀₉	#41	1	1	yes	characteristic
50	C₂×C₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
52	C₂×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₁₀₉	#131	1	1	yes	characteristic
54	D₁₀₉	—	2	2	no	—
55	C₁₀₉	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
56	C₈×D₄	—	109	1	no	maximal, Sylow
57	C₂²×C₈	—	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	—	109	1	no	—
63	C₄×C₈	—	109	1	no	—
64	C₂×D₄	—	109	1	no	—
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	—	109	1	no	—
67	C₂×C₈	—	109	1	no	—
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	—	218	2	no	—
72	C₂×C₈	—	218	2	no	—
73	C₄⋊C₄	—	109	1	no	—
74	C₂²⋊C₄	—	109	1	no	—
75	C₂²⋊C₄	#89137	1	1	yes	characteristic
76	C₄²	—	109	1	no	—
77	C₂³	—	109	1	no	—
78	C₂³	C₂×#89137	1	1	yes	characteristic
79	D₄	—	109	4	no	—
80	C₂×C₄	—	109	1	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	218	1	no	—
84	C₂×C₄	—	218	1	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	C₂×#89137	1	1	yes	characteristic
88	C₂×C₄	C₂×#89137	1	1	yes	characteristic
89	C₈	—	109	2	no	—
90	C₈	—	218	1	no	—
91	C₈	—	218	1	no	—
92	C₈	—	218	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	109	2	no	—
96	C₂²	—	218	1	no	—
97	C₂²	—	218	1	no	—
98	C₂²	D₁₀₉⋊C₈	1	2	yes	complemented
99	C₂²	C₂²×#89137	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	109	1	no	—
103	C₄	—	109	1	no	—
104	C₄	—	109	2	no	—
105	C₄	—	218	1	no	—
106	C₂	—	2	2	no	—
107	C₂	—	218	2	no	—
108	C₂	D₄×#89137	1	1	yes	characteristic
109	C₂	C₂×D₁₀₉⋊C₈	1	1	yes	characteristic
110	C₂	#16812786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×D₁₀₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂²×#11724786

C₂

yes

maximal, characteristic, complemented

#16812766

C₂

yes

maximal, characteristic, complemented

#16812756

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11724786

C₂

yes

maximal, characteristic, complemented

C₄×#11724786

C₂

yes

maximal, characteristic, complemented

#16812659

C₂

yes

maximal, characteristic

D₁₀₉⋊C₈

—

C₂×#11724786

—

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₄

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

#13182008

C₄

yes

characteristic

#13182006

C₄

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₀₉

C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×D₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²⋊D₁₀₉

C₈

yes

complemented

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

characteristic

C₄×D₁₀₉

—

C₄×D₁₀₉

—

—

—

—

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₈

yes

complemented

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂×C₈

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×C₈

yes

complemented

Dic₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

#41

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₂²×C₈

yes

characteristic

C₂×C₁₀₉

#131

yes

characteristic

D₁₀₉

—

C₁₀₉

C₈×D₄

yes

characteristic, complemented, Sylow

C₈×D₄

—

109

maximal, Sylow

C₂²×C₈

—

109

—

C₂²×C₈

—

109

—

C₄×D₄

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

C₄×C₈

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

#89137

yes

characteristic

C₄²

—

109

—

C₂³

—

109

—

C₂³

C₂×#89137

yes

characteristic

D₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

109

—

C₈

—

218

—

C₈

—

218

—

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

D₁₀₉⋊C₈

yes

complemented

C₂²

C₂²×#89137

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

109

—

103

C₄

—

109

—

104

C₄

—

109

—

105

C₄

—

218

—

106

C₂

—

107

C₂

—

218

—

108

C₂

D₄×#89137

yes

characteristic

109

C₂

C₂×D₁₀₉⋊C₈

yes

characteristic

110

C₂

#16812786

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#89137
commutator	C₂×C₁₀₉	C₂²×C₈
Fitting	C₂²⋊C₄×C₁₀₉	C₄
socle	C₂²×C₁₀₉	C₂²×C₄
hypercenter	C₂²⋊C₄	#89137
hypocenter	C₁₀₉	C₈×D₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#89137

commutator

C₂×C₁₀₉

C₂²×C₈

Fitting

C₂²⋊C₄×C₁₀₉

C₄

socle

C₂²×C₁₀₉

C₂²×C₄

hypercenter

C₂²⋊C₄

#89137

hypocenter

C₁₀₉

C₈×D₄

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂²⋊C₄

N	H ≅ G/N	type
1. C₂×D₁₀₉⋊C₈	C₂	semidirect
2. C₂²×#11724786	C₂	semidirect
3. #16812766	C₂	semidirect
4. #16812756	C₂	semidirect
5. C₂²⋊#11724786	C₂	semidirect
6. C₄×#11724786	C₂	semidirect
25. C₂²⋊D₁₀₉	C₈	semidirect
36. #11724786	D₄	semidirect
39. C₂²×C₁₀₉	C₂×C₈	semidirect
43. C₂×D₁₀₉	C₂×C₈	semidirect
47. Dic₁₀₉	C₂×C₈	semidirect
55. C₁₀₉	C₈×D₄	semidirect
98. C₂²	D₁₀₉⋊C₈	semidirect

H ≅ G/N

type

1. C₂×D₁₀₉⋊C₈

C₂

semidirect

2. C₂²×#11724786

C₂

semidirect

3. #16812766

C₂

semidirect

4. #16812756

C₂

semidirect

5. C₂²⋊#11724786

C₂

semidirect

6. C₄×#11724786

C₂

semidirect

25. C₂²⋊D₁₀₉

C₈

semidirect

36. #11724786

D₄

semidirect

39. C₂²×C₁₀₉

C₂×C₈

semidirect

43. C₂×D₁₀₉

C₂×C₈

semidirect

47. Dic₁₀₉

C₂×C₈

semidirect

55. C₁₀₉

C₈×D₄

semidirect

98. C₂²

D₁₀₉⋊C₈

semidirect

Subgroups — #20296053

Conjugacy classes of subgroups

Complements