Group #28336216 — subgroups

Subgroups — #28336216

Order 13696.

200 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 105 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	#20292582	C₂	1	2	yes	maximal
2	C₁₀₇⋊#338	C₂	1	1	yes	maximal, characteristic
3	#20292384	C₂	1	1	yes	maximal, characteristic
4	#20292383	C₂	1	2	yes	maximal
5	#405×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133802	C₂²	1	2	yes	—
7	#16812068	C₂²	1	2	yes	—
8	#16812019	C₄	1	4	yes	complemented
9	C₈×Dic₁₀₇	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
12	#117×C₁₀₇	C₂²	1	1	yes	characteristic
13	#97×C₁₀₇	C₄	1	2	yes	complemented
14	C₄×C₈×C₁₀₇	C₄	1	2	yes	complemented
15	#13133802	—	2	2	no	—
16	#13133802	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₀₇	—	2	2	no	—
19	C₂²⋊Dic₁₀₇	—	2	2	no	—
20	C₄×Dic₁₀₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₁₀₇	—	2	2	no	—
23	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₀₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₁₀₇	—	2	1	no	—
29	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
30	C₄²×C₁₀₇	C₂×C₄	1	2	yes	—
31	C₂³×C₁₀₇	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	—	2	2	no	—
33	C₂×C₄×C₁₀₇	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₀₇	#41	1	2	yes	—
37	C₂×Dic₁₀₇	—	2	4	no	—
38	C₈×C₁₀₇	—	2	2	no	—
39	C₈×C₁₀₇	C₄²	1	2	yes	—
40	C₈×C₁₀₇	C₄²	1	1	yes	characteristic
41	C₈×C₁₀₇	C₄²	1	1	yes	characteristic
42	C₁₀₇⋊C₈	—	2	4	no	—
43	C₂²×C₁₀₇	—	4	1	no	—
44	C₂²×C₁₀₇	#117	1	2	yes	—
45	C₂²×C₁₀₇	#117	1	1	yes	characteristic
46	C₄×C₁₀₇	—	4	2	no	—
47	C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₇	#117	1	2	yes	—
49	C₄×C₁₀₇	#117	1	1	yes	characteristic
50	Dic₁₀₇	—	4	4	no	—
51	C₂×C₁₀₇	—	2	2	no	—
52	C₂×C₁₀₇	—	2	1	no	—
53	C₂×C₁₀₇	#376	1	1	yes	characteristic
54	#1747	—	107	1	no	maximal, Sylow
55	C₁₀₇	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	107	2	no	—
57	#405	D₁₀₇	1	1	yes	characteristic
58	#338	—	107	1	no	—
59	H(3,ℤ/4)	—	107	1	no	—
60	H(3,ℤ/4)	—	107	2	no	—
61	C₂×M₁₆	—	107	2	no	—
62	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
64	#117	—	107	2	no	—
65	#117	C₂×D₁₀₇	1	1	yes	characteristic
66	#97	—	107	4	no	—
67	#97	Dic₁₀₇	1	2	yes	complemented
68	C₄×C₈	—	107	4	no	—
69	C₄×C₈	Dic₁₀₇	1	2	yes	complemented
70	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	214	2	no	—
73	M₁₆	—	214	2	no	—
74	C₂×C₈	—	107	4	no	—
75	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
76	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
77	C₂×C₈	C₂×Dic₁₀₇	1	2	yes	—
78	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₁₀₇	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	214	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	214	2	no	—
84	C₄²	—	107	4	no	—
85	C₄²	C₂×Dic₁₀₇	1	2	yes	—
86	C₂³	#13133818	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	214	4	no	—
89	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
90	C₂×C₄	C₂²×Dic₁₀₇	1	1	yes	characteristic
91	C₂×C₄	#13133818	1	1	yes	characteristic
92	C₂×C₄	#13133818	1	2	yes	—
93	C₈	—	2	2	no	—
94	C₈	—	214	4	no	—
95	C₈	C₄×Dic₁₀₇	1	1	yes	characteristic
96	C₈	C₄×Dic₁₀₇	1	2	yes	—
97	C₈	C₄×Dic₁₀₇	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#16812066	1	2	yes	—
100	C₂²	#16812068	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	428	4	no	—
103	C₄	C₂×C₄×Dic₁₀₇	1	1	yes	characteristic
104	C₄	#16812066	1	2	yes	—
105	C₄	#16812068	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#20292473	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292582

C₂

yes

maximal

C₁₀₇⋊#338

C₂

yes

maximal, characteristic

#20292384

C₂

yes

maximal, characteristic

#20292383

C₂

yes

maximal

#405×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

#16812068

C₂²

yes

—

#16812019

C₄

yes

complemented

C₈×Dic₁₀₇

C₄

yes

complemented

C₂×M₁₆×C₁₀₇

C₂²

yes

characteristic

C₂²×C₈×C₁₀₇

C₂²

yes

characteristic

#117×C₁₀₇

C₂²

yes

characteristic

#97×C₁₀₇

C₄

yes

complemented

C₄×C₈×C₁₀₇

C₄

yes

complemented

#13133802

—

#13133802

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₀₇

—

C₂²⋊Dic₁₀₇

—

C₄×Dic₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₇

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂×C₄

yes

—

C₂³×C₁₀₇

#41

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

C₂×C₄×C₁₀₇

yes

C₂×C₄×C₁₀₇

yes

—

C₂×Dic₁₀₇

—

C₈×C₁₀₇

—

C₈×C₁₀₇

C₄²

yes

—

C₈×C₁₀₇

C₄²

yes

characteristic

C₈×C₁₀₇

C₄²

yes

characteristic

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#117

yes

—

C₂²×C₁₀₇

#117

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₄²

yes

characteristic

C₄×C₁₀₇

#117

yes

—

C₄×C₁₀₇

#117

yes

characteristic

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#376

yes

characteristic

#1747

—

107

maximal, Sylow

C₁₀₇

#1747

yes

characteristic, complemented, Sylow

#405

—

107

—

#405

D₁₀₇

yes

characteristic

#338

—

107

—

H(3,ℤ/4)

—

107

—

H(3,ℤ/4)

—

107

—

C₂×M₁₆

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

characteristic

C₂²×C₈

C₂×D₁₀₇

yes

characteristic

#117

—

107

—

#117

C₂×D₁₀₇

yes

characteristic

#97

—

107

—

#97

Dic₁₀₇

yes

complemented

C₄×C₈

—

107

—

C₄×C₈

Dic₁₀₇

yes

complemented

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

M₁₆

—

214

—

M₁₆

—

214

—

C₂×C₈

—

107

—

C₂×C₈

C₄×D₁₀₇

yes

—

C₂×C₈

C₄×D₁₀₇

yes

—

C₂×C₈

C₂×Dic₁₀₇

yes

—

C₂×C₈

C₂×Dic₁₀₇

yes

characteristic

C₂×C₈

C₂×Dic₁₀₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₂²⋊C₄

—

C₂²⋊C₄

—

214

—

C₄²

—

107

—

C₄²

C₂×Dic₁₀₇

yes

—

C₂³

#13133818

yes

characteristic

C₂×C₄

—

C₂×C₄

—

214

—

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

C₂×C₄

C₂²×Dic₁₀₇

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

214

—

C₈

C₄×Dic₁₀₇

yes

characteristic

C₈

C₄×Dic₁₀₇

yes

—

C₈

C₄×Dic₁₀₇

yes

characteristic

C₂²

—

C₂²

#16812066

yes

—

100

C₂²

#16812068

yes

characteristic

101

C₄

—

102

C₄

—

428

—

103

C₄

C₂×C₄×Dic₁₀₇

yes

characteristic

104

C₄

#16812066

yes

—

105

C₄

#16812068

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#20292473

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₁₀₇
commutator	C₄×C₁₀₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	#405×C₁₀₇	C₂
socle	C₂×C₁₀₇	#376
prime radical	C₂³×C₁₀₇	#41
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	#405	D₁₀₇
hypocenter	C₁₀₇	#1747
LCS	G ▷ C₄×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₁₀₇

commutator

C₄×C₁₀₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

#405×C₁₀₇

C₂

C₂×C₁₀₇

C₂³×C₁₀₇

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₄×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #16812019	C₄	semidirect
9. C₈×Dic₁₀₇	C₄	semidirect
13. #97×C₁₀₇	C₄	semidirect
14. C₄×C₈×C₁₀₇	C₄	semidirect
55. C₁₀₇	#1747	semidirect
67. #97	Dic₁₀₇	semidirect
69. C₄×C₈	Dic₁₀₇	semidirect

H ≅ G/N

type

8. #16812019

C₄

semidirect

9. C₈×Dic₁₀₇

C₄

semidirect

13. #97×C₁₀₇

C₄

semidirect

14. C₄×C₈×C₁₀₇

C₄

semidirect

55. C₁₀₇

#1747

semidirect

67. #97

Dic₁₀₇

semidirect

69. C₄×C₈

Dic₁₀₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20292582	2
C₁₀₇⋊#338	2
#20292384	2
#20292383	2
#405×C₁₀₇	2
#1747	107

subgroup

index

C₁₀₇⋊#338

#405×C₁₀₇

107

Subgroups — #28336216

Conjugacy classes of subgroups

Complements

Maximal subgroups