Group #28336608 — subgroups

Subgroups — #28336608

Order 13696.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
2	#20292397	C₂	1	2	yes	maximal
3	#20292386	C₂	1	2	yes	maximal
4	C₁₀₇⋊#338	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133802	C₂²	1	2	yes	—
7	C₂×#13133802	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
9	#16812019	C₄	1	2	yes	complemented
10	C₈×Dic₁₀₇	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₁₀₇	C₂²	1	1	yes	characteristic
12	#117×C₁₀₇	C₂²	1	2	yes	—
13	C₂²×Dic₁₀₇	—	2	1	no	—
14	#13133802	—	2	2	no	—
15	#13133802	—	2	2	no	—
16	#13133802	—	2	2	no	—
17	C₂×C₁₀₇⋊C₈	—	2	2	no	—
18	C₂×C₁₀₇⋊C₈	C₂×C₄	1	2	yes	—
19	C₄×Dic₁₀₇	—	2	1	no	—
20	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₁₀₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₀₇	—	2	2	no	—
24	C₂×C₈×C₁₀₇	C₂×C₄	1	2	yes	—
25	C₄⋊C₄×C₁₀₇	—	2	2	no	—
26	C₂²⋊C₄×C₁₀₇	—	2	2	no	—
27	C₄²×C₁₀₇	—	2	2	no	—
28	C₂³×C₁₀₇	#41	1	1	yes	characteristic
29	C₂×C₄×C₁₀₇	—	4	2	no	—
30	C₂×C₄×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₀₇	#41	1	2	yes	—
32	C₂×C₄×C₁₀₇	#41	1	2	yes	—
33	C₂×C₄×C₁₀₇	#41	1	1	yes	characteristic
34	C₂×Dic₁₀₇	—	2	1	no	—
35	C₂×Dic₁₀₇	—	2	1	no	—
36	C₂×Dic₁₀₇	—	4	2	no	—
37	C₈×C₁₀₇	—	2	2	no	—
38	C₁₀₇⋊C₈	—	2	2	no	—
39	C₁₀₇⋊C₈	—	4	2	no	—
40	C₂²×C₁₀₇	—	4	1	no	—
41	C₂²×C₁₀₇	#117	1	1	yes	characteristic
42	C₂²×C₁₀₇	#126	1	2	yes	—
43	C₄×C₁₀₇	—	8	2	no	—
44	C₄×C₁₀₇	#117	1	1	yes	characteristic
45	C₄×C₁₀₇	#117	1	1	yes	characteristic
46	C₄×C₁₀₇	#126	1	2	yes	—
47	Dic₁₀₇	—	2	2	no	—
48	Dic₁₀₇	—	4	1	no	—
49	C₂×C₁₀₇	—	2	1	no	—
50	C₂×C₁₀₇	—	2	2	no	—
51	C₂×C₁₀₇	#383	1	1	yes	characteristic
52	#1841	—	107	1	no	maximal, Sylow
53	C₁₀₇	#1841	1	1	yes	characteristic, complemented, Sylow
54	C₄×M₁₆	—	107	1	no	—
55	#344	—	107	2	no	—
56	#339	—	107	2	no	—
57	#338	—	107	1	no	—
58	H(3,ℤ/4)	D₁₀₇	1	1	yes	characteristic
59	C₂×M₁₆	—	107	1	no	—
60	C₂×M₁₆	—	107	2	no	—
61	C₂×M₁₆	C₂×D₁₀₇	1	1	yes	characteristic
62	#117	C₂×D₁₀₇	1	2	yes	—
63	C₂×C₄²	—	107	1	no	—
64	#97	—	107	2	no	—
65	C₄×C₈	—	107	2	no	—
66	C₂²×C₄	—	214	1	no	—
67	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	214	2	no	—
70	M₁₆	—	214	2	no	—
71	M₁₆	—	214	2	no	—
72	C₂×C₈	—	107	2	no	—
73	C₂×C₈	—	214	2	no	—
74	C₂×C₈	C₄×D₁₀₇	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	107	1	no	—
79	C₄²	—	107	1	no	—
80	C₄²	—	214	1	no	—
81	C₂³	#13133817	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	428	2	no	—
84	C₂×C₄	—	214	1	no	—
85	C₂×C₄	—	214	1	no	—
86	C₂×C₄	C₂×C₄×D₁₀₇	1	1	yes	characteristic
87	C₂×C₄	#13133817	1	2	yes	—
88	C₂×C₄	#13133818	1	2	yes	—
89	C₂×C₄	#13133819	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	428	2	no	—
92	C₈	—	214	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#16812067	1	1	yes	characteristic
95	C₂²	#16812102	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	428	1	no	—
98	C₄	—	214	2	no	—
99	C₄	#16812065	1	1	yes	characteristic
100	C₄	#16812066	1	1	yes	characteristic
101	C₄	#16812104	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#20292494	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₁₀₇

C₂

yes

maximal, characteristic

#20292397

C₂

yes

maximal

#20292386

C₂

yes

maximal

C₁₀₇⋊#338

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₀₇

C₂

yes

maximal, characteristic

C₂×#13133802

C₂²

yes

—

C₂×#13133802

C₂²

yes

characteristic

C₂×C₄×Dic₁₀₇

C₂²

yes

characteristic

#16812019

C₄

yes

complemented

C₈×Dic₁₀₇

C₄

yes

complemented

C₂×M₁₆×C₁₀₇

C₂²

yes

characteristic

#117×C₁₀₇

C₂²

yes

—

C₂²×Dic₁₀₇

—

#13133802

—

#13133802

—

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₂×C₁₀₇⋊C₈

C₂×C₄

yes

—

C₄×Dic₁₀₇

—

C₄×Dic₁₀₇

C₂×C₄

yes

characteristic

C₄×Dic₁₀₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₇

—

C₂²⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

—

C₂³×C₁₀₇

#41

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

—

C₂×C₄×C₁₀₇

#41

yes

characteristic

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

#117

yes

characteristic

C₂²×C₁₀₇

#126

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#383

yes

characteristic

#1841

—

107

maximal, Sylow

C₁₀₇

#1841

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

107

—

#344

—

107

—

#339

—

107

—

#338

—

107

—

H(3,ℤ/4)

D₁₀₇

yes

characteristic

C₂×M₁₆

—

107

—

C₂×M₁₆

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

characteristic

#117

C₂×D₁₀₇

yes

—

C₂×C₄²

—

107

—

#97

—

107

—

C₄×C₈

—

107

—

C₂²×C₄

—

214

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

M₁₆

—

214

—

M₁₆

—

214

—

M₁₆

—

214

—

C₂×C₈

—

107

—

C₂×C₈

—

214

—

C₂×C₈

C₄×D₁₀₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

107

—

C₄²

—

107

—

C₄²

—

214

—

C₂³

#13133817

yes

characteristic

C₂×C₄

—

C₂×C₄

—

428

—

C₂×C₄

—

214

—

C₂×C₄

—

214

—

C₂×C₄

C₂×C₄×D₁₀₇

yes

characteristic

C₂×C₄

#13133817

yes

—

C₂×C₄

#13133818

yes

—

C₂×C₄

#13133819

yes

characteristic

C₈

—

C₈

—

428

—

C₈

—

214

—

C₂²

—

C₂²

#16812067

yes

characteristic

C₂²

#16812102

yes

—

C₄

—

C₄

—

428

—

C₄

—

214

—

C₄

#16812065

yes

characteristic

100

C₄

#16812066

yes

characteristic

101

C₄

#16812104

yes

—

102

C₂

—

103

C₂

—

104

C₂

#20292494

yes

characteristic

	subgroup	G/H
center	C₄	#16812066
commutator	C₂×C₄×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	H(3,ℤ/4)×C₁₀₇	C₂
socle	C₂×C₁₀₇	#383
prime radical	C₂³×C₁₀₇	#41
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	H(3,ℤ/4)	D₁₀₇
hypocenter	C₁₀₇	#1841
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16812066

commutator

C₂×C₄×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

H(3,ℤ/4)×C₁₀₇

C₂

C₂×C₁₀₇

C₂³×C₁₀₇

C₂²×C₄×C₁₀₇

C₂³

D₁₀₇

C₁₀₇

G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #16812019	C₄	semidirect
10. C₈×Dic₁₀₇	C₄	semidirect
53. C₁₀₇	#1841	semidirect

H ≅ G/N

type

9. #16812019

C₄

semidirect

10. C₈×Dic₁₀₇

C₄

semidirect

53. C₁₀₇

#1841

semidirect

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

subgroup	index
M₁₆×Dic₁₀₇	2
#20292397	2
#20292386	2
C₁₀₇⋊#338	2
H(3,ℤ/4)×C₁₀₇	2
#1841	107

subgroup

index

M₁₆×Dic₁₀₇

#20292397

#20292386

C₁₀₇⋊#338

H(3,ℤ/4)×C₁₀₇

#1841

107

Subgroups — #28336608

Conjugacy classes of subgroups

Complements

Maximal subgroups