Group #28336215 — subgroups

Subgroups — #28336215

Order 13696.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20292585	C₂	1	1	yes	maximal, characteristic
2	#20292583	C₂	1	2	yes	maximal
3	C₁₀₇⋊#338	C₂	1	1	yes	maximal, characteristic
4	#20292383	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133802	C₂²	1	2	yes	—
7	C₂²×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
8	#16812068	C₂²	1	1	yes	characteristic
9	#16812020	C₄	1	4	yes	complemented
10	#16812019	C₄	1	2	yes	complemented
11	C₄×C₁₀₇⋊C₈	C₄	1	4	yes	complemented
12	C₈×Dic₁₀₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
14	#117×C₁₀₇	C₂²	1	2	yes	—
15	#13133802	—	2	2	no	—
16	#13133802	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₁₀₇⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₂×C₁₀₇⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₁₀₇	—	2	1	no	—
23	C₂²⋊Dic₁₀₇	—	2	1	no	—
24	C₄×Dic₁₀₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₀₇	—	2	2	no	—
27	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₁₀₇	—	2	2	no	—
29	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
30	C₄²×C₁₀₇	C₂×C₄	1	4	yes	—
31	C₂³×C₁₀₇	#41	1	1	yes	characteristic
32	C₂×C₄×C₁₀₇	—	2	4	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	2	yes	—
36	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
37	C₂×Dic₁₀₇	—	2	2	no	—
38	C₈×C₁₀₇	—	2	2	no	—
39	C₁₀₇⋊C₈	—	2	4	no	—
40	C₁₀₇⋊C₈	C₄²	1	1	yes	characteristic
41	C₁₀₇⋊C₈	C₄²	1	1	yes	characteristic
42	C₁₀₇⋊C₈	C₄²	1	2	yes	—
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	4	no	—
47	C₄×C₁₀₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₁₀₇	#117	1	1	yes	characteristic
49	C₄×C₁₀₇	#117	1	2	yes	—
50	Dic₁₀₇	—	4	2	no	—
51	C₂×C₁₀₇	—	2	1	no	—
52	C₂×C₁₀₇	—	2	2	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	1	no	—
57	#405	—	107	2	no	—
58	#338	—	107	1	no	—
59	H(3,ℤ/4)	—	107	2	no	—
60	H(3,ℤ/4)	D₁₀₇	1	1	yes	characteristic
61	C₂×M₁₆	—	107	2	no	—
62	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
63	C₂²×C₈	—	107	1	no	—
64	#117	—	107	1	no	—
65	#117	C₂×D₁₀₇	1	2	yes	—
66	#97	—	107	2	no	—
67	#97	—	107	4	no	—
68	C₄×C₈	—	107	2	no	—
69	C₄×C₈	—	107	4	no	—
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	1	no	—
75	C₂×C₈	—	107	2	no	—
76	C₂×C₈	—	107	1	no	—
77	C₂×C₈	—	107	2	no	—
78	C₂×C₈	—	107	4	no	—
79	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	214	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	214	1	no	—
84	C₄²	—	107	2	no	—
85	C₄²	C₄×D₁₀₇	1	4	yes	—
86	C₂³	#13133817	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	214	2	no	—
89	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₁₀₇	1	2	yes	—
91	C₂×C₄	#13133817	1	2	yes	—
92	C₂×C₄	#13133819	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	107	1	no	—
95	C₈	—	107	1	no	—
96	C₈	—	107	2	no	—
97	C₈	—	214	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#16812065	1	2	yes	—
100	C₂²	#16812067	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	428	2	no	—
103	C₄	C₄²×D₁₀₇	1	1	yes	characteristic
104	C₄	#16812065	1	1	yes	characteristic
105	C₄	#16812067	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#20292472	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20292585

C₂

yes

maximal, characteristic

#20292583

C₂

yes

maximal

C₁₀₇⋊#338

C₂

yes

maximal, characteristic

#20292383

C₂

yes

maximal

H(3,ℤ/4)×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₂²×C₁₀₇⋊C₈

C₂²

yes

characteristic

#16812068

C₂²

yes

characteristic

#16812020

C₄

yes

complemented

#16812019

C₄

yes

complemented

C₄×C₁₀₇⋊C₈

C₄

yes

complemented

C₈×Dic₁₀₇

C₄

yes

complemented

C₂×M₁₆×C₁₀₇

C₂²

yes

characteristic

#117×C₁₀₇

C₂²

yes

—

#13133802

—

#13133802

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

characteristic

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

—

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

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

characteristic

C₂×Dic₁₀₇

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

C₄²

yes

characteristic

C₁₀₇⋊C₈

C₄²

yes

characteristic

C₁₀₇⋊C₈

C₄²

yes

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#117

yes

—

C₂²×C₁₀₇

#117

yes

characteristic

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×C₄²

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#376

yes

characteristic

#1747

—

107

maximal, Sylow

C₁₀₇

#1747

yes

characteristic, complemented, Sylow

#405

—

107

—

#405

—

107

—

#338

—

107

—

H(3,ℤ/4)

—

107

—

H(3,ℤ/4)

D₁₀₇

yes

characteristic

C₂×M₁₆

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

characteristic

C₂²×C₈

—

107

—

#117

—

107

—

#117

C₂×D₁₀₇

yes

—

#97

—

107

—

#97

—

107

—

C₄×C₈

—

107

—

C₄×C₈

—

107

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

M₁₆

—

214

—

M₁₆

—

214

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

—

107

—

C₂×C₈

C₄×D₁₀₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

214

—

C₂²⋊C₄

—

C₂²⋊C₄

—

214

—

C₄²

—

107

—

C₄²

C₄×D₁₀₇

yes

—

C₂³

#13133817

yes

characteristic

C₂×C₄

—

C₂×C₄

—

214

—

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

C₂×C₄×D₁₀₇

yes

—

C₂×C₄

#13133817

yes

—

C₂×C₄

#13133819

yes

characteristic

C₈

—

C₈

—

107

—

C₈

—

107

—

C₈

—

107

—

C₈

—

214

—

C₂²

—

C₂²

#16812065

yes

—

100

C₂²

#16812067

yes

characteristic

101

C₄

—

102

C₄

—

428

—

103

C₄

C₄²×D₁₀₇

yes

characteristic

104

C₄

#16812065

yes

characteristic

105

C₄

#16812067

yes

—

106

C₂

—

107

C₂

—

108

C₂

#20292472

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₁₀₇
commutator	C₄×C₁₀₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	H(3,ℤ/4)×C₁₀₇	C₂
socle	C₂×C₁₀₇	#376
prime radical	C₂³×C₁₀₇	#41
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	H(3,ℤ/4)	D₁₀₇
hypocenter	C₁₀₇	#1747
LCS	G ▷ C₄×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₁₀₇

commutator

C₄×C₁₀₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

H(3,ℤ/4)×C₁₀₇

C₂

C₂×C₁₀₇

C₂³×C₁₀₇

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₄×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #16812020	C₄	semidirect
10. #16812019	C₄	semidirect
11. C₄×C₁₀₇⋊C₈	C₄	semidirect
12. C₈×Dic₁₀₇	C₄	semidirect
55. C₁₀₇	#1747	semidirect

H ≅ G/N

type

9. #16812020

C₄

semidirect

10. #16812019

C₄

semidirect

11. C₄×C₁₀₇⋊C₈

C₄

semidirect

12. C₈×Dic₁₀₇

C₄

semidirect

55. C₁₀₇

#1747

semidirect

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

subgroup	index
#20292585	2
#20292583	2
C₁₀₇⋊#338	2
#20292383	2
H(3,ℤ/4)×C₁₀₇	2
#1747	107

subgroup

index

C₁₀₇⋊#338

H(3,ℤ/4)×C₁₀₇

107

Subgroups — #28336215

Conjugacy classes of subgroups

Complements

Maximal subgroups