Group #20296057 — subgroups

Subgroups — #20296057

Order 6976.

124 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 48 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×#13182038	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	C₂²⋊#11724786	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#11724786	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812754	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
8	#13182038	—	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	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₉	C₄	1	1	yes	characteristic
17	C₂²⋊Dic₁₀₉	C₄	1	1	yes	characteristic
18	C₂²⋊Dic₁₀₉	C₄	1	1	yes	characteristic
19	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₁₀₉	C₄	1	1	yes	characteristic
21	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	D₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₀₉	—	2	1	no	—
26	C₂×Dic₁₀₉	—	2	1	no	—
27	C₂×Dic₁₀₉	—	2	1	no	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
30	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
31	C₂×Dic₁₀₉	C₂×C₄	1	1	yes	characteristic
32	C₂×Dic₁₀₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
40	C₂²×C₁₀₉	—	2	1	no	—
41	C₂²×C₁₀₉	—	2	1	no	—
42	C₂²×C₁₀₉	M₁₆	1	2	yes	complemented
43	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
44	C₄×C₁₀₉	—	2	1	no	—
45	Dic₁₀₉	—	2	1	no	—
46	Dic₁₀₉	—	2	1	no	—
47	Dic₁₀₉	—	2	1	no	—
48	Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
49	Dic₁₀₉	#41	1	1	yes	characteristic
50	C₂×C₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	—	4	1	no	—
52	C₂×C₁₀₉	C₄×D₄	1	1	yes	characteristic
53	C₂×C₁₀₉	C₂×M₁₆	1	1	yes	characteristic
54	C₂×C₁₀₉	#131	1	1	yes	characteristic
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	—	109	1	no	—
63	#97	—	109	1	no	—
64	C₂×D₄	#89137	1	1	yes	characteristic
65	C₂²×C₄	—	109	1	no	—
66	C₂²×C₄	—	109	1	no	—
67	M₁₆	—	218	2	no	—
68	C₂×C₈	—	109	1	no	—
69	C₂×C₈	—	109	1	no	—
70	C₂×C₈	—	109	1	no	—
71	C₂×C₈	—	109	1	no	—
72	C₂×C₈	—	218	2	no	—
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	C₂×#89137	1	1	yes	characteristic
78	C₂³	C₂×#89137	1	1	yes	characteristic
79	D₄	—	2	2	no	—
80	C₂×C₄	—	109	1	no	—
81	C₂×C₄	—	109	1	no	—
82	C₂×C₄	—	109	1	no	—
83	C₂×C₄	—	109	1	no	—
84	C₂×C₄	—	218	1	no	—
85	C₂×C₄	—	218	1	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	—	218	1	no	—
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	—	2	1	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#13182038	1	2	yes	complemented
99	C₂²	C₂²×#89137	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	109	1	no	—
102	C₄	—	109	1	no	—
103	C₄	—	218	1	no	—
104	C₄	—	218	1	no	—
105	C₄	—	218	1	no	—
106	C₂	—	2	2	no	—
107	C₂	—	4	1	no	—
108	C₂	D₄×#89137	1	1	yes	characteristic
109	C₂	C₂×#13182038	1	1	yes	characteristic
110	C₂	#16812786	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182038

C₂

yes

maximal, characteristic, complemented

C₂²×#11724786

C₂

yes

maximal, characteristic, complemented

#16812766

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11724786

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11724786

C₂

yes

maximal, characteristic, complemented

#16812754

C₂

yes

maximal, characteristic, complemented

D₄×Dic₁₀₉

C₂

yes

maximal, characteristic

#13182038

—

C₂×#11724786

—

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

characteristic

C₂×#11724786

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₄

yes

characteristic

C₂²⋊Dic₁₀₉

C₄

yes

characteristic

C₂²⋊Dic₁₀₉

C₄

yes

characteristic

C₄×Dic₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

C₂³×C₁₀₉

C₂×C₄

yes

characteristic

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂³

yes

—

—

—

D₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

M₁₆

yes

complemented

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

#41

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×D₄

yes

characteristic

C₂×C₁₀₉

C₂×M₁₆

yes

C₂×C₁₀₉

yes

C₁₀₉

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

—

109

—

#97

—

109

—

C₂×D₄

#89137

yes

characteristic

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₄⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₂³

yes

C₂³

yes

D₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₂×#89137

yes

characteristic

C₈

—

109

—

C₈

—

218

—

C₈

—

218

—

C₈

—

218

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#13182038

yes

complemented

C₂²

C₂²×#89137

yes

characteristic

100

C₄

—

101

C₄

—

109

—

102

C₄

—

109

—

103

C₄

—

218

—

104

C₄

—

218

—

105

C₄

—

218

—

106

C₂

—

107

C₂

—

108

C₂

D₄×#89137

yes

characteristic

109

C₂

C₂×#13182038

yes

characteristic

110

C₂

#16812786

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#89137
commutator, socle	C₂²×C₁₀₉	C₂²×C₄
Fitting	C₂×D₄×C₁₀₉	C₄
hypercenter	C₂×D₄	#89137
hypocenter	C₁₀₉	#435
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#89137

commutator, socle

C₂²×C₁₀₉

C₂²×C₄

Fitting

C₂×D₄×C₁₀₉

C₄

C₂×D₄

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. C₂×#13182038	C₂	semidirect
2. C₂²×#11724786	C₂	semidirect
3. #16812766	C₂	semidirect
4. C₂²⋊#11724786	C₂	semidirect
5. C₂²⋊#11724786	C₂	semidirect
6. #16812754	C₂	semidirect
36. #11724786	D₄	semidirect
42. C₂²×C₁₀₉	M₁₆	semidirect
55. C₁₀₉	#435	semidirect
98. C₂²	#13182038	semidirect

H ≅ G/N

type

1. C₂×#13182038

C₂

semidirect

2. C₂²×#11724786

C₂

semidirect

3. #16812766

C₂

semidirect

4. C₂²⋊#11724786

C₂

semidirect

5. C₂²⋊#11724786

C₂

semidirect

6. #16812754

C₂

semidirect

36. #11724786

D₄

semidirect

42. C₂²×C₁₀₉

M₁₆

semidirect

55. C₁₀₉

#435

semidirect

98. C₂²

#13182038

semidirect

Subgroups — #20296057

Conjugacy classes of subgroups

Complements