Group #20294875 — subgroups

Subgroups — C₄×#13182006 #20294875

Order 6976.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂²×C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#13182006	C₂	1	2	yes	maximal
3	C₂×C₄×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812606	C₂	1	2	yes	maximal, complemented
5	C₂×C₄²×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₀₉	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₀₉	C₂²	1	2	yes	—
9	C₂×C₄×D₁₀₉	—	2	2	no	—
10	C₂×C₄×D₁₀₉	—	2	2	no	—
11	C₂×C₄×D₁₀₉	C₄	1	4	yes	complemented
12	#13182006	C₄	1	8	yes	direct product factor
13	C₄×Dic₁₀₉	—	2	2	no	—
14	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
15	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
16	C₄²×C₁₀₉	—	2	2	no	—
17	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
18	C₂²×D₁₀₉	—	2	1	no	—
19	C₂²×D₁₀₉	—	2	1	no	—
20	C₂²×D₁₀₉	—	2	2	no	—
21	C₂²×D₁₀₉	C₂×C₄	1	4	yes	—
22	C₂²×D₁₀₉	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₁₀₉	—	2	2	no	—
24	C₂×C₄×C₁₀₉	—	2	2	no	—
25	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
27	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
28	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
29	C₂×Dic₁₀₉	—	2	2	no	—
30	C₂×Dic₁₀₉	—	2	1	no	—
31	C₂×Dic₁₀₉	—	2	1	no	—
32	C₂×Dic₁₀₉	—	2	2	no	—
33	C₂×Dic₁₀₉	C₂×C₄	1	2	yes	—
34	C₂×Dic₁₀₉	C₂×C₄	1	4	yes	complemented
35	C₄×D₁₀₉	—	2	8	no	—
36	C₂²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₁₀₉	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₀₉	#41	1	2	yes	—
41	C₂×D₁₀₉	—	2	2	no	—
42	C₂×D₁₀₉	—	2	2	no	—
43	C₂×D₁₀₉	—	2	4	no	—
44	C₂×D₁₀₉	—	2	4	no	—
45	C₂×D₁₀₉	C₄²	1	4	yes	complemented
46	C₄×C₁₀₉	—	2	4	no	—
47	C₄×C₁₀₉	C₂²⋊C₄	1	4	yes	complemented
48	Dic₁₀₉	—	2	2	no	—
49	Dic₁₀₉	—	2	4	no	—
50	C₂×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
51	C₂×C₁₀₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₁₀₉	#117	1	1	yes	characteristic
53	C₂×C₁₀₉	C₄×D₄	1	2	yes	—
54	C₂×C₁₀₉	C₄×D₄	1	2	yes	—
55	D₁₀₉	—	2	4	no	—
56	C₁₀₉	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
57	C₄×C₂²⋊C₄	—	109	1	no	maximal, Sylow
58	C₂³×C₄	—	109	1	no	—
59	C₂×C₂²⋊C₄	—	109	2	no	—
60	C₂×C₄²	—	109	1	no	—
61	C₂×C₄²	D₁₀₉	1	1	yes	characteristic, complemented
62	#95	—	109	2	no	—
63	C₂⁴	—	109	1	no	—
64	C₂²×C₄	—	109	1	no	—
65	C₂²×C₄	—	109	2	no	—
66	C₂²×C₄	—	109	4	no	—
67	C₂²×C₄	—	218	2	no	—
68	C₂²×C₄	—	218	2	no	—
69	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
70	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
71	C₂²⋊C₄	—	109	8	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	218	2	no	—
74	C₂³	—	109	4	no	—
75	C₂³	—	109	2	no	—
76	C₂³	—	218	1	no	—
77	C₂³	—	218	1	no	—
78	C₂³	—	218	2	no	—
79	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	109	4	no	—
83	C₂×C₄	—	109	2	no	—
84	C₂×C₄	—	218	1	no	—
85	C₂×C₄	—	218	2	no	—
86	C₂×C₄	—	218	1	no	—
87	C₂×C₄	—	218	2	no	—
88	C₂×C₄	—	218	8	no	—
89	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
90	C₂×C₄	C₄×D₁₀₉	1	4	yes	complemented
91	C₂×C₄	D₄₃₆	1	2	yes	—
92	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
93	C₂²	—	109	4	no	—
94	C₂²	—	218	2	no	—
95	C₂²	—	218	2	no	—
96	C₂²	—	218	4	no	—
97	C₂²	—	218	4	no	—
98	C₂²	C₂×C₄×D₁₀₉	1	1	yes	characteristic
99	C₂²	C₂×C₄×D₁₀₉	1	2	yes	—
100	C₂²	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
101	C₂²	C₂×D₄₃₆	1	1	yes	characteristic
102	C₂²	#13182029	1	2	yes	—
103	C₄	—	2	4	no	—
104	C₄	—	218	2	no	—
105	C₄	—	218	4	no	—
106	C₄	#13182006	1	4	yes	direct product factor
107	C₂	—	218	4	no	—
108	C₂	C₄²×D₁₀₉	1	1	yes	characteristic
109	C₂	C₂×#13182006	1	1	yes	characteristic
110	C₂	#16812656	1	1	yes	characteristic
111	C₂	C₄×C₂²⋊D₁₀₉	1	2	yes	—
112	C₂	C₄×D₄₃₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182006

C₂

yes

maximal

C₂×C₄×Dic₁₀₉

C₂

yes

maximal, characteristic, complemented

#16812606

C₂

yes

maximal, complemented

C₂×C₄²×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

characteristic

C₂²×Dic₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

—

C₂×C₄×D₁₀₉

C₄

yes

complemented

#13182006

C₄

yes

direct product factor

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄²×C₁₀₉

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

—

C₂²×D₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×C₄×C₁₀₉

D₄

yes

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

—

C₂×Dic₁₀₉

C₂×C₄

yes

—

C₂×Dic₁₀₉

C₂×C₄

yes

complemented

C₄×D₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

characteristic

C₂²×C₁₀₉

C₂²×C₄

yes

—

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×C₁₀₉

#41

yes

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₄²

yes

complemented

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²⋊C₄

yes

complemented

Dic₁₀₉

—

Dic₁₀₉

—

C₂×C₁₀₉

C₂×C₄²

yes

characteristic

C₂×C₁₀₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₀₉

#117

yes

characteristic

C₂×C₁₀₉

C₄×D₄

yes

—

C₂×C₁₀₉

C₄×D₄

yes

—

D₁₀₉

—

C₁₀₉

C₄×C₂²⋊C₄

yes

characteristic, complemented, Sylow

C₄×C₂²⋊C₄

—

109

maximal, Sylow

C₂³×C₄

—

109

—

C₂×C₂²⋊C₄

—

109

—

C₂×C₄²

—

109

—

C₂×C₄²

D₁₀₉

yes

characteristic, complemented

#95

—

109

—

C₂⁴

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₂²⋊C₄

—

109

—

C₄²

—

C₄²

—

218

—

C₂³

—

109

—

C₂³

—

109

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

—

218

—

C₂³

C₂²×D₁₀₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

—

218

—

C₂×C₄

C₄×D₁₀₉

yes

—

C₂×C₄

C₄×D₁₀₉

yes

complemented

C₂×C₄

D₄₃₆

yes

—

C₂×C₄

C₂²⋊D₁₀₉

yes

—

C₂²

—

109

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

—

218

—

C₂²

C₂×C₄×D₁₀₉

yes

characteristic

C₂²

C₂×C₄×D₁₀₉

yes

—

100

C₂²

C₂×C₂²⋊D₁₀₉

yes

characteristic

101

C₂²

C₂×D₄₃₆

yes

characteristic

102

C₂²

#13182029

yes

—

103

C₄

—

104

C₄

—

218

—

105

C₄

—

218

—

106

C₄

#13182006

yes

direct product factor

107

C₂

—

218

—

108

C₂

C₄²×D₁₀₉

yes

characteristic

109

C₂

C₂×#13182006

yes

characteristic

110

C₂

#16812656

yes

characteristic

111

C₂

C₄×C₂²⋊D₁₀₉

yes

—

112

C₂

C₄×D₄₃₆

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₀₉
commutator	C₂×C₁₀₉	C₂×C₄²
Frattini	C₂³	C₂²×D₁₀₉
Fitting	C₂×C₄²×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂³
hypercenter	C₂×C₄²	D₁₀₉
hypocenter	C₁₀₉	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²×C₄

C₂×D₁₀₉

commutator

C₂×C₁₀₉

C₂×C₄²

Frattini

C₂³

C₂²×D₁₀₉

Fitting

C₂×C₄²×C₁₀₉

C₂

socle

C₂³×C₁₀₉

C₂³

hypercenter

C₂×C₄²

D₁₀₉

hypocenter

C₁₀₉

C₄×C₂²⋊C₄

LCS

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
3. C₂×C₄×Dic₁₀₉	C₂	semidirect
4. #16812606	C₂	semidirect
5. C₂×C₄²×C₁₀₉	C₂	semidirect
11. C₂×C₄×D₁₀₉	C₄	semidirect
12. #13182006	C₄	direct (N × H)
26. C₂×C₄×C₁₀₉	C₂×C₄	semidirect
34. C₂×Dic₁₀₉	C₂×C₄	semidirect
45. C₂×D₁₀₉	C₄²	semidirect
47. C₄×C₁₀₉	C₂²⋊C₄	semidirect
56. C₁₀₉	C₄×C₂²⋊C₄	semidirect
61. C₂×C₄²	D₁₀₉	semidirect
90. C₂×C₄	C₄×D₁₀₉	semidirect
106. C₄	#13182006	direct (N × H)

H ≅ G/N

type

3. C₂×C₄×Dic₁₀₉

C₂

semidirect

4. #16812606

C₂

semidirect

5. C₂×C₄²×C₁₀₉

C₂

semidirect

11. C₂×C₄×D₁₀₉

C₄

semidirect

12. #13182006

C₄

direct (N × H)

26. C₂×C₄×C₁₀₉

C₂×C₄

semidirect

34. C₂×Dic₁₀₉

C₂×C₄

semidirect

45. C₂×D₁₀₉

C₄²

semidirect

47. C₄×C₁₀₉

C₂²⋊C₄

semidirect

56. C₁₀₉

C₄×C₂²⋊C₄

semidirect

61. C₂×C₄²

D₁₀₉

semidirect

90. C₂×C₄

C₄×D₁₀₉

semidirect

106. C₄

#13182006

direct (N × H)

Subgroups — C₄×#13182006 #20294875

Conjugacy classes of subgroups

Complements

Subgroups — C4×#13182006 #20294875

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#13182006 #20294875