Group #20295044 — subgroups

Subgroups — #20295044

Order 6976.

154 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 59 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×#13182029	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182013	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812627	C₂	1	4	yes	maximal, complemented
4	#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#13182029	—	2	1	no	—
6	#13182029	—	2	1	no	—
7	#13182029	C₄	1	2	yes	complemented
8	#13182029	C₂²	1	2	yes	—
9	C₂×Dic₂₁₈	C₄	1	1	yes	characteristic, complemented
10	C₂×D₄₃₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
12	C₂×C₄×D₁₀₉	—	2	1	no	—
13	#13182013	—	2	1	no	—
14	#13182013	C₂²	1	2	yes	complemented
15	C₂×#11724784	—	2	1	no	—
16	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₁₀₉	—	2	1	no	—
18	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
19	C₄²×C₁₀₉	C₂²	1	2	yes	complemented
20	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
21	C₂²×D₁₀₉	—	2	1	no	—
22	C₂×C₄×C₁₀₉	—	2	2	no	—
23	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
29	C₂²⋊D₁₀₉	—	2	2	no	—
30	C₂²⋊D₁₀₉	—	2	2	no	—
31	C₂×Dic₁₀₉	—	2	1	no	—
32	D₄₃₆	—	2	1	no	—
33	D₄₃₆	C₂×C₄	1	2	yes	—
34	C₄×D₁₀₉	—	2	2	no	—
35	C₄×D₁₀₉	—	2	2	no	—
36	Dic₂₁₈	—	2	1	no	—
37	Dic₂₁₈	C₂×C₄	1	2	yes	—
38	#11724784	—	2	2	no	—
39	C₂²×C₁₀₉	—	4	1	no	—
40	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	C₂×D₁₀₉	—	2	2	no	—
44	C₂×D₁₀₉	—	4	2	no	—
45	C₄×C₁₀₉	—	4	2	no	—
46	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
48	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
49	C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
50	Dic₁₀₉	—	2	2	no	—
51	C₂×C₁₀₉	—	2	1	no	—
52	C₂×C₁₀₉	—	2	1	no	—
53	C₂×C₁₀₉	—	2	1	no	—
54	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	D₁₀₉	—	4	2	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	D₁₀₉	1	1	yes	characteristic, complemented
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	—	109	1	no	—
68	C₂×D₄	—	218	1	no	—
69	C₂²×C₄	—	218	1	no	—
70	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
71	M₁₆	—	109	2	no	—
72	M₁₆	—	218	1	no	—
73	C₂×C₈	—	218	1	no	—
74	C₄⋊C₄	—	2	1	no	—
75	C₂²⋊C₄	—	2	1	no	—
76	C₄²	C₂×D₁₀₉	1	2	yes	complemented
77	C₂³	—	218	1	no	—
78	C₂³	C₂²⋊D₁₀₉	1	1	yes	characteristic
79	Q₈	—	109	2	no	—
80	Q₈	—	218	1	no	—
81	D₄	—	109	2	no	—
82	D₄	—	218	2	no	—
83	D₄	—	218	2	no	—
84	D₄	—	218	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	—	218	2	no	—
88	C₂×C₄	—	218	2	no	—
89	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
90	C₂×C₄	C₄×D₁₀₉	1	1	yes	characteristic
91	C₂×C₄	D₄₃₆	1	1	yes	characteristic
92	C₂×C₄	D₄₃₆	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
94	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
95	C₈	—	218	2	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	436	2	no	—
98	C₂²	—	218	2	no	—
99	C₂²	#13182006	1	1	yes	characteristic
100	C₂²	#13182006	1	1	yes	characteristic
101	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
102	C₄	—	4	2	no	—
103	C₄	—	218	2	no	—
104	C₄	#13182006	1	1	yes	characteristic
105	C₄	#13182006	1	1	yes	characteristic
106	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
107	C₄	C₂×D₄₃₆	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	—	436	2	no	—
112	C₂	C₂×#13182006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182029

C₂

yes

maximal, characteristic

C₂×#13182013

C₂

yes

maximal, characteristic, complemented

#16812627

C₂

yes

maximal, complemented

#117×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

—

#13182029

—

#13182029

C₄

yes

complemented

#13182029

C₂²

yes

—

C₂×Dic₂₁₈

C₄

yes

characteristic, complemented

C₂×D₄₃₆

C₄

yes

characteristic, complemented

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

—

#13182013

—

#13182013

C₂²

yes

complemented

C₂×#11724784

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

C₂×C₄

yes

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

Dic₂₁₈

—

Dic₂₁₈

C₂×C₄

yes

—

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂²⋊C₄

yes

characteristic

C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₄×C₁₀₉

C₂×D₄

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

D₁₀₉

—

C₁₀₉

#421

yes

characteristic, complemented, Sylow

#421

—

109

maximal, Sylow

C₂×#41

—

109

—

C₂×M₁₆

—

109

—

#117

D₁₀₉

yes

characteristic, complemented

#104

—

109

—

#41

—

109

—

#41

—

109

—

#41

—

218

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

M₁₆

—

109

—

M₁₆

—

218

—

C₂×C₈

—

218

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₁₀₉

yes

complemented

C₂³

—

218

—

C₂³

C₂²⋊D₁₀₉

yes

characteristic

Q₈

—

109

—

Q₈

—

218

—

D₄

—

109

—

D₄

—

218

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

C₄×D₁₀₉

yes

characteristic

C₂×C₄

D₄₃₆

yes

characteristic

C₂×C₄

D₄₃₆

yes

characteristic

C₂×C₄

C₂²⋊D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₈

—

218

—

C₂²

—

C₂²

—

436

—

C₂²

—

218

—

C₂²

#13182006

yes

characteristic

100

C₂²

#13182006

yes

characteristic

101

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

102

C₄

—

103

C₄

—

218

—

104

C₄

#13182006

yes

characteristic

105

C₄

#13182006

yes

characteristic

106

C₄

C₂×C₄×D₁₀₉

yes

characteristic

107

C₄

C₂×D₄₃₆

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

—

111

C₂

—

436

—

112

C₂

C₂×#13182006

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₄₃₆
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#117×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×C₂²⋊C₄
hypercenter	#117	D₁₀₉
hypocenter	C₁₀₉	#421
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

subgroup

G/H

center

C₄

C₂×D₄₃₆

commutator

C₄×C₁₀₉

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₁₀₉

Fitting

#117×C₁₀₉

C₂

socle

C₂×C₁₀₉

C₂×C₂²⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #117

N	H ≅ G/N	type
2. C₂×#13182013	C₂	semidirect
3. #16812627	C₂	semidirect
4. #117×C₁₀₉	C₂	semidirect
7. #13182029	C₄	semidirect
9. C₂×Dic₂₁₈	C₄	semidirect
10. C₂×D₄₃₆	C₄	semidirect
14. #13182013	C₂²	semidirect
19. C₄²×C₁₀₉	C₂²	semidirect
56. C₁₀₉	#421	semidirect
60. #117	D₁₀₉	semidirect
76. C₄²	C₂×D₁₀₉	semidirect

H ≅ G/N

type

2. C₂×#13182013

C₂

semidirect

3. #16812627

C₂

semidirect

4. #117×C₁₀₉

C₂

semidirect

7. #13182029

C₄

semidirect

9. C₂×Dic₂₁₈

C₄

semidirect

10. C₂×D₄₃₆

C₄

semidirect

14. #13182013

C₂²

semidirect

19. C₄²×C₁₀₉

C₂²

semidirect

56. C₁₀₉

#421

semidirect

60. #117

D₁₀₉

semidirect

76. C₄²

C₂×D₁₀₉

semidirect

Subgroups — #20295044

Conjugacy classes of subgroups

Complements