Group #31808211 — subgroups

Subgroups — #31808211

Order 17792.

138 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₄⋊(C₁₃₉⋊C₁₆)	C₂	1	2	yes	maximal, complemented
2	C₂²⋊(C₁₃₉⋊C₁₆)	C₂	1	2	yes	maximal
3	#22580729	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₁₃₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#405×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃₉⋊C₁₆	C₂²	1	2	yes	—
7	C₂×C₁₃₉⋊C₁₆	C₂²	1	2	yes	—
8	C₂×M₁₆×C₁₃₉	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
10	#117×C₁₃₉	C₄	1	1	yes	characteristic
11	#97×C₁₃₉	C₄	1	1	yes	characteristic
12	#97×C₁₃₉	C₄	1	1	yes	characteristic
13	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₁₃₉⋊C₁₆	—	2	2	no	—
16	C₁₃₉⋊C₁₆	—	2	2	no	—
17	C₂²×C₄×C₁₃₉	C₂×C₄	1	1	yes	characteristic
18	M₁₆×C₁₃₉	—	2	2	no	—
19	C₂×C₈×C₁₃₉	—	2	2	no	—
20	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₁₃₉	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₃₉	C₈	1	2	yes	—
25	C₂²⋊C₄×C₁₃₉	C₈	1	2	yes	—
26	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₃₉	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃₉	—	2	1	no	—
30	C₂×C₄×C₁₃₉	—	2	1	no	—
31	C₂×C₄×C₁₃₉	C₂×C₈	1	2	yes	—
32	C₂×C₄×C₁₃₉	C₂×C₈	1	2	yes	—
33	C₂×C₄×C₁₃₉	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
35	C₈×C₁₃₉	—	2	1	no	—
36	C₈×C₁₃₉	—	2	1	no	—
37	C₈×C₁₃₉	M₁₆	1	2	yes	—
38	C₈×C₁₃₉	#41	1	2	yes	—
39	C₂²×C₁₃₉	—	2	1	no	—
40	C₂²×C₁₃₉	—	2	2	no	—
41	C₂²×C₁₃₉	C₂²×C₈	1	1	yes	characteristic
42	C₄×C₁₃₉	—	2	1	no	—
43	C₄×C₁₃₉	—	2	2	no	—
44	C₄×C₁₃₉	—	2	2	no	—
45	C₄×C₁₃₉	#117	1	1	yes	characteristic
46	C₄×C₁₃₉	C₂×M₁₆	1	1	yes	characteristic
47	C₂×C₁₃₉	—	4	1	no	—
48	C₂×C₁₃₉	#431	1	1	yes	characteristic
49	C₂×C₁₃₉	#504	1	2	yes	—
50	C₁₃₉	#2164	1	1	yes	characteristic, complemented, Sylow
51	#2164	—	139	1	no	maximal, Sylow
52	#405	D₁₃₉	1	1	yes	characteristic
53	C₄⋊C₁₆	—	139	2	no	—
54	C₂²⋊C₁₆	—	139	2	no	—
55	#346	—	139	1	no	—
56	C₄×C₁₆	—	139	1	no	—
57	C₂×M₁₆	Dic₁₃₉	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
59	#117	Dic₁₃₉	1	1	yes	characteristic
60	C₂×C₁₆	—	139	2	no	—
61	C₂×C₁₆	—	139	2	no	—
62	#97	Dic₁₃₉	1	1	yes	characteristic
63	#97	Dic₁₃₉	1	1	yes	characteristic
64	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
65	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₁₃₉	1	1	yes	characteristic
67	M₁₆	—	2	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
70	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
71	C₂×C₈	C₂×Dic₁₃₉	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₁₃₉	1	1	yes	characteristic
73	C₄⋊C₄	C₁₃₉⋊C₈	1	2	yes	—
74	C₂²⋊C₄	C₁₃₉⋊C₈	1	2	yes	—
75	C₄²	C₂×Dic₁₃₉	1	1	yes	characteristic
76	C₄²	C₂×Dic₁₃₉	1	1	yes	characteristic
77	C₁₆	—	278	2	no	—
78	C₁₆	—	278	2	no	—
79	C₂³	C₂×C₁₃₉⋊C₈	1	1	yes	characteristic
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	C₂×C₁₃₉⋊C₈	1	2	yes	—
83	C₂×C₄	C₂×C₁₃₉⋊C₈	1	2	yes	—
84	C₂×C₄	C₂×C₁₃₉⋊C₈	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₁₃₉	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	#14555769	1	2	yes	—
89	C₈	#14555785	1	2	yes	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₂²×C₁₃₉⋊C₈	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	#17984801	1	1	yes	characteristic
97	C₄	C₂×#14555769	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#22581006	1	1	yes	characteristic
100	C₂	#22581370	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₁₃₉⋊C₁₆)

C₂

yes

maximal, complemented

C₂²⋊(C₁₃₉⋊C₁₆)

C₂

yes

maximal

#22580729

C₂

yes

maximal, characteristic, complemented

C₄×C₁₃₉⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#405×C₁₃₉

C₂

yes

maximal, characteristic

C₂×C₁₃₉⋊C₁₆

C₂²

yes

—

C₂×C₁₃₉⋊C₁₆

C₂²

yes

—

C₂×M₁₆×C₁₃₉

C₄

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

#117×C₁₃₉

C₄

yes

characteristic

#97×C₁₃₉

C₄

yes

characteristic

#97×C₁₃₉

C₄

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₁₃₉⋊C₁₆

—

C₁₃₉⋊C₁₆

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

C₈

yes

—

C₂²⋊C₄×C₁₃₉

C₈

yes

—

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

C₄²×C₁₃₉

C₂×C₄

yes

characteristic

C₂³×C₁₃₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×C₈

yes

—

C₂×C₄×C₁₃₉

C₂×C₈

yes

—

C₂×C₄×C₁₃₉

C₂×C₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

—

C₈×C₁₃₉

M₁₆

yes

—

C₈×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×C₈

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#117

yes

characteristic

C₄×C₁₃₉

C₂×M₁₆

yes

characteristic

C₂×C₁₃₉

—

C₂×C₁₃₉

#431

yes

characteristic

C₂×C₁₃₉

#504

yes

—

C₁₃₉

#2164

yes

characteristic, complemented, Sylow

#2164

—

139

maximal, Sylow

#405

D₁₃₉

yes

characteristic

C₄⋊C₁₆

—

139

—

C₂²⋊C₁₆

—

139

—

#346

—

139

—

C₄×C₁₆

—

139

—

C₂×M₁₆

Dic₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

Dic₁₃₉

yes

characteristic

C₂×C₁₆

—

139

—

C₂×C₁₆

—

139

—

#97

Dic₁₃₉

yes

characteristic

#97

Dic₁₃₉

yes

characteristic

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₂×C₈

C₂×Dic₁₃₉

yes

characteristic

C₂×C₈

C₂²×D₁₃₉

yes

characteristic

C₄⋊C₄

C₁₃₉⋊C₈

yes

—

C₂²⋊C₄

C₁₃₉⋊C₈

yes

—

C₄²

C₂×Dic₁₃₉

yes

characteristic

C₄²

C₂×Dic₁₃₉

yes

characteristic

C₁₆

—

278

—

C₁₆

—

278

—

C₂³

C₂×C₁₃₉⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₁₃₉⋊C₈

yes

—

C₂×C₄

C₂×C₁₃₉⋊C₈

yes

—

C₂×C₄

C₂×C₁₃₉⋊C₈

yes

characteristic

C₂×C₄

C₂²×Dic₁₃₉

yes

characteristic

C₈

—

C₈

—

C₈

#14555769

yes

—

C₈

#14555785

yes

—

C₂²

—

C₂²

—

C₂²

C₂²×C₁₃₉⋊C₈

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#22581006

yes

characteristic

100

C₂

#22581370

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂²×C₈
Fitting	#405×C₁₃₉	C₂
prime radical	C₂³×C₁₃₉	C₂×C₈
cosocle	C₂×C₈×C₁₃₉	C₂³
hypercenter	#405	D₁₃₉
hypocenter	C₁₃₉	#2164
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₈ ◁ #405

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂²×C₈

Fitting

#405×C₁₃₉

C₂

prime radical

C₂³×C₁₃₉

C₂×C₈

cosocle

C₂×C₈×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂×C₈ ◁ #405

N	H ≅ G/N	type
1. C₄⋊(C₁₃₉⋊C₁₆)	C₂	semidirect
3. #22580729	C₂	semidirect
4. C₄×C₁₃₉⋊C₁₆	C₂	semidirect
50. C₁₃₉	#2164	semidirect

H ≅ G/N

type

1. C₄⋊(C₁₃₉⋊C₁₆)

C₂

semidirect

3. #22580729

C₂

semidirect

4. C₄×C₁₃₉⋊C₁₆

C₂

semidirect

50. C₁₃₉

#2164

semidirect

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

subgroup	index
C₄⋊(C₁₃₉⋊C₁₆)	2
C₂²⋊(C₁₃₉⋊C₁₆)	2
#22580729	2
C₄×C₁₃₉⋊C₁₆	2
#405×C₁₃₉	2
#2164	139

subgroup

index

C₄⋊(C₁₃₉⋊C₁₆)

C₂²⋊(C₁₃₉⋊C₁₆)

#22580729

C₄×C₁₃₉⋊C₁₆

#405×C₁₃₉

#2164

139

Subgroups — #31808211

Conjugacy classes of subgroups

Complements

Maximal subgroups