Group #20295507 — subgroups

Subgroups — #20295507

Order 6976.

294 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 151 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	#16812664	C₂	1	2	yes	maximal, complemented
3	#16812661	C₂	1	2	yes	maximal, complemented
4	C₄×C₂²⋊D₁₀₉	C₂	1	4	yes	maximal, complemented
5	#16812658	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812657	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₄×D₁₀₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₄⋊C₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#13182029	C₄	1	8	yes	complemented
10	C₂×Dic₂₁₈	C₂²	1	1	yes	characteristic
11	C₂×D₄₃₆	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄×D₁₀₉	C₂²	1	4	yes	—
14	C₂×C₄×D₁₀₉	C₂²	1	2	yes	—
15	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
17	#13182008	C₂²	1	4	yes	—
18	C₂²⋊Dic₁₀₉	C₂²	1	2	yes	—
19	#13182006	C₂²	1	4	yes	—
20	C₄×Dic₁₀₉	C₂²	1	4	yes	—
21	C₄×Dic₁₀₉	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₀₉	C₂²	1	2	yes	—
23	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
24	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
25	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
26	C₂³×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²×D₁₀₉	C₂³	1	2	yes	—
28	C₂×C₄×C₁₀₉	—	4	2	no	—
29	C₂×C₄×C₁₀₉	C₂×C₄	1	4	yes	complemented
30	C₂×C₄×C₁₀₉	C₂³	1	4	yes	—
31	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₀₉	C₂×C₄	1	8	yes	complemented
34	C₂×Dic₁₀₉	C₂³	1	2	yes	—
35	C₂×Dic₁₀₉	C₂³	1	2	yes	—
36	C₂×Dic₁₀₉	C₂³	1	2	yes	—
37	D₄₃₆	C₂×C₄	1	4	yes	complemented
38	C₄×D₁₀₉	—	4	4	no	—
39	C₄×D₁₀₉	C₂×C₄	1	8	yes	complemented
40	Dic₂₁₈	C₂×C₄	1	4	yes	complemented
41	C₂²×C₁₀₉	—	2	1	no	—
42	C₂²×C₁₀₉	—	2	1	no	—
43	C₂²×C₁₀₉	C₂²×C₄	1	2	yes	—
44	C₂²×C₁₀₉	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₀₉	—	2	2	no	—
46	C₂×D₁₀₉	—	2	2	no	—
47	C₂×D₁₀₉	C₂²×C₄	1	4	yes	—
48	C₄×C₁₀₉	—	2	4	no	—
49	C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
50	C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
51	Dic₁₀₉	—	2	2	no	—
52	Dic₁₀₉	—	2	2	no	—
53	Dic₁₀₉	C₂²×C₄	1	4	yes	—
54	C₂×C₁₀₉	—	2	2	no	—
55	C₂×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
56	C₂×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
58	D₁₀₉	—	2	4	no	—
59	C₁₀₉	#520	1	1	yes	characteristic, complemented, Sylow
60	#520	—	109	1	no	maximal, Sylow
61	C₂×#41	—	109	1	no	—
62	C₄×Q₈	—	109	2	no	—
63	C₄×D₄	—	109	4	no	—
64	C₄×D₄	—	109	2	no	—
65	#117	—	109	1	no	—
66	#117	—	109	2	no	—
67	C₂×C₄⋊C₄	—	109	2	no	—
68	C₂×C₄⋊C₄	D₁₀₉	1	1	yes	characteristic, complemented
69	#41	—	109	8	no	—
70	C₂×Q₈	—	109	1	no	—
71	C₂×D₄	—	109	2	no	—
72	C₂×D₄	—	109	1	no	—
73	C₂²×C₄	—	109	2	no	—
74	C₂²×C₄	—	109	4	no	—
75	C₂²×C₄	C₂×D₁₀₉	1	2	yes	—
76	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
77	C₄⋊C₄	—	109	4	no	—
78	C₄⋊C₄	—	109	1	no	—
79	C₄⋊C₄	—	109	1	no	—
80	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
81	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
82	C₂²⋊C₄	—	109	2	no	—
83	C₂²⋊C₄	—	109	4	no	—
84	C₄²	—	109	4	no	—
85	C₄²	—	109	2	no	—
86	C₂³	—	109	2	no	—
87	C₂³	C₂²×D₁₀₉	1	1	yes	characteristic
88	Q₈	—	109	4	no	—
89	D₄	—	109	8	no	—
90	D₄	—	109	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	109	2	no	—
93	C₂×C₄	—	109	8	no	—
94	C₂×C₄	—	109	2	no	—
95	C₂×C₄	—	109	2	no	—
96	C₂×C₄	—	436	4	no	—
97	C₂×C₄	C₄×D₁₀₉	1	4	yes	complemented
98	C₂×C₄	C₂²×D₁₀₉	1	4	yes	—
99	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	109	4	no	—
104	C₂²	—	218	2	no	—
105	C₂²	—	218	2	no	—
106	C₂²	C₂×C₄×D₁₀₉	1	2	yes	—
107	C₂²	C₂³×D₁₀₉	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	109	4	no	—
110	C₄	—	218	2	no	—
111	C₄	—	218	2	no	—
112	C₄	C₂×C₄×D₁₀₉	1	2	yes	—
113	C₄	C₂×C₄×D₁₀₉	1	2	yes	—
114	C₂	—	2	2	no	—
115	C₂	—	218	4	no	—
116	C₂	C₂²×C₄×D₁₀₉	1	1	yes	characteristic
117	C₂	#16812744	1	1	yes	characteristic
118	C₂	#16812747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13182029

C₂

yes

maximal, characteristic

#16812664

C₂

yes

maximal, complemented

#16812661

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₀₉

C₂

yes

maximal, complemented

#16812658

C₂

yes

maximal, characteristic, complemented

#16812657

C₂

yes

maximal, complemented

C₄⋊C₄×D₁₀₉

C₂

yes

maximal, complemented

C₂×C₄⋊C₄×C₁₀₉

C₂

yes

maximal, characteristic, complemented

#13182029

C₄

yes

complemented

C₂×Dic₂₁₈

C₂²

yes

characteristic

C₂×D₄₃₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₂×C₄×D₁₀₉

C₂²

yes

—

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

C₄⋊Dic₁₀₉

C₂²

yes

characteristic

#13182008

C₂²

yes

—

C₂²⋊Dic₁₀₉

C₂²

yes

—

#13182006

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₄×Dic₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

—

C₂²×C₄×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂³×C₁₀₉

C₂³

yes

characteristic

C₂²×D₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₀₉

C₂³

yes

—

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²⋊D₁₀₉

C₂×C₄

yes

complemented

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

C₂×Dic₁₀₉

C₂³

yes

—

D₄₃₆

C₂×C₄

yes

complemented

C₄×D₁₀₉

—

C₄×D₁₀₉

C₂×C₄

yes

complemented

Dic₂₁₈

C₂×C₄

yes

complemented

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×C₄

yes

—

C₂²×C₁₀₉

C₂⁴

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

C₂²×C₄

yes

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂²×C₄

yes

—

C₄×C₁₀₉

C₂²×C₄

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂²×C₄

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₂×C₁₀₉

2₋¹⁺⁴

yes

characteristic

D₁₀₉

—

C₁₀₉

#520

yes

characteristic, complemented, Sylow

#520

—

109

maximal, Sylow

C₂×#41

—

109

—

C₄×Q₈

—

109

—

C₄×D₄

—

109

—

C₄×D₄

—

109

—

#117

—

109

—

#117

—

109

—

C₂×C₄⋊C₄

—

109

—

C₂×C₄⋊C₄

D₁₀₉

yes

characteristic, complemented

#41

—

109

—

C₂×Q₈

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

C₂×D₁₀₉

yes

—

C₂²×C₄

C₂×D₁₀₉

yes

characteristic

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂²⋊C₄

—

109

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₂³

—

109

—

C₂³

C₂²×D₁₀₉

yes

characteristic

Q₈

—

109

—

D₄

—

109

—

D₄

—

109

—

C₂×C₄

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

109

—

C₂×C₄

—

436

—

C₂×C₄

C₄×D₁₀₉

yes

complemented

C₂×C₄

C₂²×D₁₀₉

yes

—

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

100

C₂×C₄

C₂²×D₁₀₉

yes

characteristic

101

C₂²

—

102

C₂²

—

103

C₂²

—

109

—

104

C₂²

—

218

—

105

C₂²

—

218

—

106

C₂²

C₂×C₄×D₁₀₉

yes

—

107

C₂²

C₂³×D₁₀₉

yes

characteristic

108

C₄

—

109

C₄

—

109

—

110

C₄

—

218

—

111

C₄

—

218

—

112

C₄

C₂×C₄×D₁₀₉

yes

—

113

C₄

C₂×C₄×D₁₀₉

yes

—

114

C₂

—

115

C₂

—

218

—

116

C₂

C₂²×C₄×D₁₀₉

yes

characteristic

117

C₂

#16812744

yes

characteristic

118

C₂

#16812747

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₉
commutator	C₂×C₁₀₉	C₂³×C₄
Fitting	C₂×C₄⋊C₄×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂⁴
hypercenter	C₂×C₄⋊C₄	D₁₀₉
hypocenter	C₁₀₉	#520
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₉

commutator

C₂×C₁₀₉

C₂³×C₄

Fitting

C₂×C₄⋊C₄×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂⁴

hypercenter

C₂×C₄⋊C₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
2. #16812664	C₂	semidirect
3. #16812661	C₂	semidirect
4. C₄×C₂²⋊D₁₀₉	C₂	semidirect
5. #16812658	C₂	semidirect
6. #16812657	C₂	semidirect
7. C₄⋊C₄×D₁₀₉	C₂	semidirect
8. C₂×C₄⋊C₄×C₁₀₉	C₂	semidirect
9. #13182029	C₄	semidirect
29. C₂×C₄×C₁₀₉	C₂×C₄	semidirect
33. C₂²⋊D₁₀₉	C₂×C₄	semidirect
37. D₄₃₆	C₂×C₄	semidirect
39. C₄×D₁₀₉	C₂×C₄	semidirect
40. Dic₂₁₈	C₂×C₄	semidirect
59. C₁₀₉	#520	semidirect
68. C₂×C₄⋊C₄	D₁₀₉	semidirect
97. C₂×C₄	C₄×D₁₀₉	semidirect

H ≅ G/N

type

2. #16812664

C₂

semidirect

3. #16812661

C₂

semidirect

4. C₄×C₂²⋊D₁₀₉

C₂

semidirect

5. #16812658

C₂

semidirect

6. #16812657

C₂

semidirect

7. C₄⋊C₄×D₁₀₉

C₂

semidirect

8. C₂×C₄⋊C₄×C₁₀₉

C₂

semidirect

9. #13182029

C₄

semidirect

29. C₂×C₄×C₁₀₉

C₂×C₄

semidirect

33. C₂²⋊D₁₀₉

C₂×C₄

semidirect

37. D₄₃₆

C₂×C₄

semidirect

39. C₄×D₁₀₉

C₂×C₄

semidirect

40. Dic₂₁₈

C₂×C₄

semidirect

59. C₁₀₉

#520

semidirect

68. C₂×C₄⋊C₄

D₁₀₉

semidirect

97. C₂×C₄

C₄×D₁₀₉

semidirect

Subgroups — #20295507

Conjugacy classes of subgroups

Complements