Group #20296024 — subgroups

Subgroups — #20296024

Order 6976.

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

normal subgroups only

This group has 154 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	D₁₀₉⋊M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812778	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812763	C₂	1	4	yes	maximal, complemented
4	C₂×#13182028	C₂	1	1	yes	maximal, characteristic
5	#13182039	C₂²	1	2	yes	complemented
6	#13182038	—	2	1	no	—
7	D₁₀₉⋊C₈	—	2	1	no	—
8	C₄⋊#89137	—	2	1	no	—
9	C₂²⋊#89137	—	2	1	no	—
10	C₄×#89137	C₂²	1	2	yes	complemented
11	#13182028	—	2	1	no	—
12	#13182028	—	2	1	no	—
13	#13182028	C₄	1	2	yes	complemented
14	#13182028	C₂²	1	2	yes	—
15	C₂×Dic₂₁₈	C₄	1	1	yes	characteristic, complemented
16	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
17	C₂²×Dic₁₀₉	—	2	1	no	—
18	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₁₀₉	C₄	1	1	yes	characteristic, complemented
20	C₂³×C₁₀₉	—	2	1	no	—
21	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
22	C₂×#89137	—	2	2	no	—
23	D₄×C₁₀₉	—	2	1	no	—
24	D₄×C₁₀₉	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
26	C₂²⋊D₁₀₉	—	2	2	no	—
27	C₂²⋊D₁₀₉	—	2	2	no	—
28	C₂×Dic₁₀₉	—	2	1	no	—
29	C₂×Dic₁₀₉	—	2	2	no	—
30	C₂×Dic₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
32	C₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₄×D₁₀₉	D₄	1	1	yes	characteristic
34	C₄×D₁₀₉	D₄	1	1	yes	characteristic
35	C₄×D₁₀₉	C₂³	1	1	yes	characteristic
36	Dic₂₁₈	—	2	1	no	—
37	Dic₂₁₈	C₂×C₄	1	2	yes	—
38	#11724786	—	2	2	no	—
39	C₂²×C₁₀₉	—	2	2	no	—
40	C₂²×C₁₀₉	—	4	2	no	—
41	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×D₁₀₉	—	4	1	no	—
43	C₂×D₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×D₁₀₉	C₂×D₄	1	1	yes	characteristic
45	#89137	—	4	2	no	—
46	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
48	Dic₁₀₉	—	2	2	no	—
49	Dic₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
50	Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₁₀₉	—	2	1	no	—
52	C₂×C₁₀₉	—	4	2	no	—
53	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	D₁₀₉	—	2	1	no	—
55	D₁₀₉	—	2	1	no	—
56	C₁₀₉	#421	1	1	yes	characteristic, complemented, Sylow
57	#421	—	109	1	no	maximal, Sylow
58	C₂×#41	—	109	1	no	—
59	C₂×M₁₆	—	109	1	no	—
60	#117	—	109	1	no	—
61	#104	—	109	4	no	—
62	#41	—	109	2	no	—
63	#41	—	109	2	no	—
64	#41	—	218	1	no	—
65	#41	—	218	1	no	—
66	C₂×Q₈	—	109	1	no	—
67	C₂×D₄	—	218	1	no	—
68	C₂×D₄	#89137	1	1	yes	characteristic, complemented
69	C₂²×C₄	—	109	1	no	—
70	C₂²×C₄	—	218	1	no	—
71	M₁₆	—	109	2	no	—
72	M₁₆	—	218	1	no	—
73	C₂×C₈	—	218	1	no	—
74	C₄⋊C₄	—	218	1	no	—
75	C₂²⋊C₄	—	218	1	no	—
76	C₄²	—	109	2	no	—
77	C₂³	—	2	1	no	—
78	C₂³	—	109	1	no	—
79	Q₈	—	109	2	no	—
80	Q₈	—	218	1	no	—
81	D₄	—	2	1	no	—
82	D₄	—	218	2	no	—
83	D₄	—	218	2	no	—
84	D₄	C₂×#89137	1	2	yes	—
85	C₂×C₄	—	109	1	no	—
86	C₂×C₄	—	109	1	no	—
87	C₂×C₄	—	109	1	no	—
88	C₂×C₄	—	109	1	no	—
89	C₂×C₄	—	109	1	no	—
90	C₂×C₄	—	218	2	no	—
91	C₂×C₄	—	218	2	no	—
92	C₂×C₄	—	218	2	no	—
93	C₂×C₄	—	218	1	no	—
94	C₂×C₄	C₂×#89137	1	1	yes	characteristic
95	C₈	—	218	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	4	2	no	—
98	C₂²	—	109	1	no	—
99	C₂²	—	109	1	no	—
100	C₂²	—	436	1	no	—
101	C₂²	C₂²⋊#89137	1	1	yes	characteristic
102	C₄	—	109	1	no	—
103	C₄	—	109	1	no	—
104	C₄	—	436	2	no	—
105	C₄	—	218	2	no	—
106	C₄	C₂²⋊#89137	1	1	yes	characteristic
107	C₄	C₂²×#89137	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	4	2	no	—
110	C₂	—	218	1	no	—
111	C₂	—	218	1	no	—
112	C₂	C₂×C₂²⋊#89137	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₁₀₉⋊M₁₆

C₂

yes

maximal, characteristic, complemented

#16812778

C₂

yes

maximal, characteristic, complemented

#16812763

C₂

yes

maximal, complemented

C₂×#13182028

C₂

yes

maximal, characteristic

#13182039

C₂²

yes

complemented

#13182038

—

D₁₀₉⋊C₈

—

C₄⋊#89137

—

C₂²⋊#89137

—

C₄×#89137

C₂²

yes

—

—

C₄

yes

complemented

#13182028

C₂²

yes

—

C₂×Dic₂₁₈

C₄

yes

characteristic, complemented

C₂×C₂²⋊D₁₀₉

—

C₂²×Dic₁₀₉

—

C₂×C₄×D₁₀₉

C₂²

yes

characteristic

C₂×D₄×C₁₀₉

C₄

yes

characteristic, complemented

C₂³×C₁₀₉

—

C₂²×D₁₀₉

D₄

yes

characteristic

C₂×#89137

—

D₄×C₁₀₉

—

D₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

C₂×C₄

yes

characteristic

C₄×D₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

D₄

yes

characteristic

C₄×D₁₀₉

C₂³

yes

characteristic

Dic₂₁₈

—

Dic₂₁₈

C₂×C₄

yes

—

#11724786

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²⋊C₄

yes

characteristic

C₂×D₁₀₉

C₂×D₄

yes

characteristic

#89137

—

C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

Dic₁₀₉

—

Dic₁₀₉

C₂²⋊C₄

yes

characteristic

Dic₁₀₉

C₂×D₄

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#421

yes

characteristic, complemented, Sylow

#421

—

109

maximal, Sylow

C₂×#41

—

109

—

C₂×M₁₆

—

109

—

#117

—

109

—

#104

—

109

—

#41

—

109

—

#41

—

109

—

#41

—

218

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×D₄

—

218

—

C₂×D₄

#89137

yes

characteristic, complemented

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

218

—

C₂²⋊C₄

—

218

—

C₄²

—

109

—

C₂³

—

C₂³

—

109

—

Q₈

—

109

—

Q₈

—

218

—

D₄

—

D₄

—

218

—

D₄

—

218

—

D₄

C₂×#89137

yes

—

C₂×C₄

—

109

—

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

—

218

—

C₂²

—

C₂²

—

C₂²

—

109

—

C₂²

—

109

—

100

C₂²

—

436

—

101

C₂²

C₂²⋊#89137

yes

characteristic

102

C₄

—

109

—

103

C₄

—

109

—

104

C₄

—

436

—

105

C₄

—

218

—

106

C₄

C₂²⋊#89137

yes

characteristic

107

C₄

C₂²×#89137

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

—

218

—

111

C₂

—

218

—

112

C₂

C₂×C₂²⋊#89137

yes

characteristic

	subgroup	G/H
center	C₂	C₂×C₂²⋊#89137
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₄	C₂²×#89137
Fitting	C₂×D₄×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₂²⋊C₄
hypercenter	C₂×D₄	#89137
hypocenter	C₁₀₉	#421
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

subgroup

G/H

center

C₂

C₂×C₂²⋊#89137

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₄

C₂²×#89137

Fitting

C₂×D₄×C₁₀₉

C₄

socle

C₂×C₁₀₉

C₂×C₂²⋊C₄

C₂×D₄

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

N	H ≅ G/N	type
1. D₁₀₉⋊M₁₆	C₂	semidirect
2. #16812778	C₂	semidirect
3. #16812763	C₂	semidirect
5. #13182039	C₂²	semidirect
10. C₄×#89137	C₂²	semidirect
13. #13182028	C₄	semidirect
15. C₂×Dic₂₁₈	C₄	semidirect
19. C₂×D₄×C₁₀₉	C₄	semidirect
56. C₁₀₉	#421	semidirect
68. C₂×D₄	#89137	semidirect

H ≅ G/N

type

1. D₁₀₉⋊M₁₆

C₂

semidirect

2. #16812778

C₂

semidirect

3. #16812763

C₂

semidirect

5. #13182039

C₂²

semidirect

10. C₄×#89137

C₂²

semidirect

13. #13182028

C₄

semidirect

15. C₂×Dic₂₁₈

C₄

semidirect

19. C₂×D₄×C₁₀₉

C₄

semidirect

56. C₁₀₉

#421

semidirect

68. C₂×D₄

#89137

semidirect

Subgroups — #20296024

Conjugacy classes of subgroups

Complements