Group #20295595 — subgroups

Subgroups — #20295595

Order 6976.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 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, complemented
2	C₂²×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#16812696	C₂	1	2	yes	maximal, complemented
4	#16812687	C₂	1	2	yes	maximal, complemented
5	#16812674	C₂	1	4	yes	maximal, complemented
6	#16812669	C₂	1	4	yes	maximal, complemented
7	#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₀₉	C₂²	1	2	yes	—
9	#13182029	—	2	4	no	—
10	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄₃₆	—	2	4	no	—
12	C₂×D₄₃₆	C₂²	1	2	yes	—
13	C₂×D₄₃₆	C₂²	1	2	yes	—
14	C₂×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₄₃₆	C₂²	1	8	yes	complemented
16	C₂×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
17	C₂×C₄×D₁₀₉	C₂²	1	2	yes	complemented
18	#13182006	C₂²	1	8	yes	complemented
19	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₄²×C₁₀₉	C₂²	1	2	yes	complemented
23	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
24	C₂²×D₁₀₉	—	2	4	no	—
25	C₂²×D₁₀₉	—	4	4	no	—
26	C₂²×D₁₀₉	C₂³	1	4	yes	—
27	C₂²×D₁₀₉	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₉	D₄	1	4	yes	complemented
29	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
30	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂²⋊D₁₀₉	—	2	4	no	—
33	C₂×Dic₁₀₉	C₂³	1	2	yes	—
34	D₄₃₆	—	2	4	no	—
35	D₄₃₆	—	2	4	no	—
36	D₄₃₆	—	2	2	no	—
37	D₄₃₆	—	4	8	no	—
38	C₄×D₁₀₉	—	2	4	no	—
39	Dic₂₁₈	—	2	2	no	—
40	C₂²×C₁₀₉	—	2	2	no	—
41	C₂²×C₁₀₉	C₂×D₄	1	2	yes	—
42	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₀₉	—	2	2	no	—
44	C₂×D₁₀₉	—	2	4	no	—
45	C₂×D₁₀₉	—	2	4	no	—
46	C₂×D₁₀₉	—	2	8	no	—
47	C₂×D₁₀₉	—	4	8	no	—
48	C₄×C₁₀₉	—	2	4	no	—
49	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
50	C₄×C₁₀₉	C₂×D₄	1	2	yes	—
51	Dic₁₀₉	—	2	2	no	—
52	C₂×C₁₀₉	—	2	2	no	—
53	C₂×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
55	D₁₀₉	—	4	2	no	—
56	D₁₀₉	—	4	4	no	—
57	C₁₀₉	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	109	1	no	maximal, Sylow
59	C₂×#41	—	109	1	no	—
60	C₂²×D₄	—	109	1	no	—
61	#127	—	109	2	no	—
62	#124	—	109	2	no	—
63	#121	—	109	4	no	—
64	#120	—	109	4	no	—
65	#117	D₁₀₉	1	1	yes	characteristic, complemented
66	C₂⁴	—	109	2	no	—
67	#41	—	218	4	no	—
68	C₂×Q₈	—	109	1	no	—
69	C₂×D₄	—	109	2	no	—
70	C₂×D₄	—	109	8	no	—
71	C₂×D₄	—	109	2	no	—
72	C₂×D₄	—	109	2	no	—
73	C₂×D₄	—	109	1	no	—
74	C₂×D₄	—	218	4	no	—
75	C₂²×C₄	—	109	2	no	—
76	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	complemented
78	C₂²⋊C₄	—	109	8	no	—
79	C₂²⋊C₄	C₂×D₁₀₉	1	2	yes	—
80	C₄²	C₂×D₁₀₉	1	2	yes	complemented
81	C₂³	—	109	4	no	—
82	C₂³	—	109	2	no	—
83	C₂³	—	436	4	no	—
84	C₂³	—	218	4	no	—
85	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
86	Q₈	—	218	2	no	—
87	D₄	—	436	8	no	—
88	D₄	—	218	4	no	—
89	D₄	—	218	4	no	—
90	D₄	—	218	4	no	—
91	D₄	—	218	2	no	—
92	C₂×C₄	—	109	2	no	—
93	C₂×C₄	—	218	4	no	—
94	C₂×C₄	D₄₃₆	1	4	yes	complemented
95	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
96	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	436	8	no	—
100	C₂²	—	218	2	no	—
101	C₂²	—	218	4	no	—
102	C₂²	—	218	4	no	—
103	C₂²	—	218	8	no	—
104	C₂²	C₂×D₄₃₆	1	2	yes	—
105	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	218	2	no	—
108	C₄	C₂×D₄₃₆	1	2	yes	—
109	C₄	C₂×D₄₃₆	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	436	4	no	—
112	C₂	—	436	2	no	—
113	C₂	C₂²×D₄₃₆	1	1	yes	characteristic
114	C₂	#16812745	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182029

C₂

yes

maximal, characteristic, complemented

C₂²×D₄₃₆

C₂

yes

maximal, characteristic, complemented

#16812696

C₂

yes

maximal, complemented

#16812687

C₂

yes

maximal, complemented

#16812674

C₂

yes

maximal, complemented

#16812669

C₂

yes

maximal, complemented

#117×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

—

#13182029

—

C₂×Dic₂₁₈

C₂²

yes

characteristic, complemented

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

—

C₂×D₄₃₆

C₂²

yes

characteristic, complemented

C₂×D₄₃₆

C₂²

yes

complemented

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

complemented

#13182006

C₂²

yes

complemented

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

complemented

C₂²⋊C₄×C₁₀₉

C₂²

yes

—

C₄²×C₁₀₉

C₂²

yes

complemented

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂³

yes

—

C₂²×D₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

D₄

yes

complemented

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

Dic₂₁₈

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×D₄

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×D₄

yes

—

C₄×C₁₀₉

C₂×D₄

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×C₁₀₉

H(5,𝔽₂)

yes

—

D₁₀₉

—

D₁₀₉

—

C₁₀₉

#535

yes

characteristic, complemented, Sylow

#535

—

109

maximal, Sylow

C₂×#41

—

109

—

C₂²×D₄

—

109

—

#127

—

109

—

#124

—

109

—

#121

—

109

—

#120

—

109

—

#117

D₁₀₉

yes

characteristic, complemented

C₂⁴

—

109

—

#41

—

218

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

C₂×D₁₀₉

yes

complemented

C₂²⋊C₄

—

109

—

C₂²⋊C₄

C₂×D₁₀₉

yes

—

C₄²

C₂×D₁₀₉

yes

complemented

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

436

—

C₂³

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

Q₈

—

218

—

D₄

—

436

—

D₄

—

218

—

D₄

—

218

—

D₄

—

218

—

D₄

—

218

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

D₄₃₆

yes

complemented

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²

—

C₂²

—

436

—

100

C₂²

—

218

—

101

C₂²

—

218

—

102

C₂²

—

218

—

103

C₂²

—

218

—

104

C₂²

C₂×D₄₃₆

yes

—

105

C₂²

C₂³×D₁₀₉

yes

characteristic

106

C₄

—

107

C₄

—

218

—

108

C₄

C₂×D₄₃₆

yes

—

109

C₄

C₂×D₄₃₆

yes

—

110

C₂

—

111

C₂

—

436

—

112

C₂

—

436

—

113

C₂

C₂²×D₄₃₆

yes

characteristic

114

C₂

#16812745

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂⁴
Fitting	#117×C₁₀₉	C₂
hypercenter	#117	D₁₀₉
hypocenter	C₁₀₉	#535
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂⁴

Fitting

#117×C₁₀₉

C₂

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#13182029	C₂	semidirect
2. C₂²×D₄₃₆	C₂	semidirect
3. #16812696	C₂	semidirect
4. #16812687	C₂	semidirect
5. #16812674	C₂	semidirect
6. #16812669	C₂	semidirect
7. #117×C₁₀₉	C₂	semidirect
10. C₂×Dic₂₁₈	C₂²	semidirect
14. C₂×D₄₃₆	C₂²	semidirect
15. C₂×D₄₃₆	C₂²	semidirect
17. C₂×C₄×D₁₀₉	C₂²	semidirect
18. #13182006	C₂²	semidirect
20. C₄⋊C₄×C₁₀₉	C₂²	semidirect
22. C₄²×C₁₀₉	C₂²	semidirect
28. C₂×C₄×C₁₀₉	D₄	semidirect
57. C₁₀₉	#535	semidirect
65. #117	D₁₀₉	semidirect
77. C₄⋊C₄	C₂×D₁₀₉	semidirect
80. C₄²	C₂×D₁₀₉	semidirect
94. C₂×C₄	D₄₃₆	semidirect

H ≅ G/N

type

1. C₂×#13182029

C₂

semidirect

2. C₂²×D₄₃₆

C₂

semidirect

3. #16812696

C₂

semidirect

4. #16812687

C₂

semidirect

5. #16812674

C₂

semidirect

6. #16812669

C₂

semidirect

7. #117×C₁₀₉

C₂

semidirect

10. C₂×Dic₂₁₈

C₂²

semidirect

14. C₂×D₄₃₆

C₂²

semidirect

15. C₂×D₄₃₆

C₂²

semidirect

17. C₂×C₄×D₁₀₉

C₂²

semidirect

18. #13182006

C₂²

semidirect

20. C₄⋊C₄×C₁₀₉

C₂²

semidirect

22. C₄²×C₁₀₉

C₂²

semidirect

28. C₂×C₄×C₁₀₉

D₄

semidirect

57. C₁₀₉

#535

semidirect

65. #117

D₁₀₉

semidirect

77. C₄⋊C₄

C₂×D₁₀₉

semidirect

80. C₄²

C₂×D₁₀₉

semidirect

94. C₂×C₄

D₄₃₆

semidirect

Subgroups — #20295595

Conjugacy classes of subgroups

Complements