Group #31806033 — subgroups

Subgroups — #31806033

Order 17792.

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

normal subgroups only

This group has 220 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	#22581330	C₂	1	1	yes	maximal, characteristic
2	#22581327	C₂	1	2	yes	maximal, complemented
3	#22581301	C₂	1	2	yes	maximal, complemented
4	#22580674	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555772	—	2	4	no	—
7	C₂×D₁₁₁₂	—	2	2	no	—
8	#17984839	C₂²	1	1	yes	characteristic
9	#17984839	C₂²	1	1	yes	characteristic
10	#17984762	—	2	4	no	—
11	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
13	#106×C₁₃₉	—	2	2	no	—
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×D₅₅₆	—	2	1	no	—
16	C₂×D₅₅₆	—	2	1	no	—
17	C₂×D₅₅₆	—	2	2	no	—
18	C₂×D₅₅₆	—	2	2	no	—
19	#14555772	—	4	8	no	—
20	D₁₁₁₂	—	4	4	no	—
21	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
22	C₂×Q₈×C₁₃₉	—	2	2	no	—
23	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
24	C₄⋊C₄×C₁₃₉	—	2	2	no	—
25	C₄⋊C₄×C₁₃₉	—	2	2	no	—
26	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₂²×D₁₃₉	—	4	1	no	—
28	C₂²×D₁₃₉	—	4	1	no	—
29	D₅₅₆	—	4	2	no	—
30	D₅₅₆	—	4	2	no	—
31	D₅₅₆	—	4	4	no	—
32	D₅₅₆	—	4	4	no	—
33	C₁₃₉⋊C₈	—	2	4	no	—
34	Q₈×C₁₃₉	—	4	4	no	—
35	C₂×C₄×C₁₃₉	—	4	2	no	—
36	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₃₉	SD₁₆	1	4	yes	complemented
39	C₂²×C₁₃₉	#127	1	1	yes	characteristic
40	C₂×D₁₃₉	—	8	1	no	—
41	C₂×D₁₃₉	—	8	1	no	—
42	C₂×D₁₃₉	—	8	2	no	—
43	C₂×D₁₃₉	—	8	2	no	—
44	C₄×C₁₃₉	—	8	2	no	—
45	C₄×C₁₃₉	C₂×SD₁₆	1	2	yes	—
46	C₄×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₁₃₉	#492	1	1	yes	characteristic
48	C₂×C₁₃₉	#496	1	2	yes	—
49	D₁₃₉	—	16	1	no	—
50	D₁₃₉	—	16	1	no	—
51	C₁₃₉	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	139	1	no	maximal, Sylow
53	#498	D₁₃₉	1	1	yes	characteristic, complemented
54	#493	—	139	1	no	—
55	#492	—	139	2	no	—
56	#486	—	139	2	no	—
57	#322	—	139	1	no	—
58	C₂×SD₁₆	—	278	4	no	—
59	C₂×D₈	—	278	2	no	—
60	C₄⋊Q₈	C₂×D₁₃₉	1	2	yes	—
61	#127	—	139	1	no	—
62	#127	—	139	1	no	—
63	#106	—	2	2	no	—
64	#102	—	278	4	no	—
65	C₄×C₈	—	139	2	no	—
66	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	278	2	no	—
69	C₂×D₄	—	278	2	no	—
70	C₂×D₄	—	278	1	no	—
71	C₂×D₄	—	278	1	no	—
72	SD₁₆	—	556	8	no	—
73	D₈	—	556	4	no	—
74	C₂×C₈	—	139	4	no	—
75	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
79	C₂³	—	556	1	no	—
80	C₂³	—	556	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	556	2	no	—
83	D₄	—	556	2	no	—
84	D₄	—	556	4	no	—
85	D₄	—	556	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
89	C₈	—	278	4	no	—
90	C₈	#14555772	1	4	yes	complemented
91	C₂²	—	1112	1	no	—
92	C₂²	—	1112	1	no	—
93	C₂²	—	1112	2	no	—
94	C₂²	—	1112	2	no	—
95	C₂²	#17984838	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#14555772	1	2	yes	—
98	C₄	#17984869	1	4	yes	—
99	C₂	—	2224	1	no	—
100	C₂	—	2224	1	no	—
101	C₂	#22581327	1	1	yes	characteristic
102	C₂	#22581339	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581330

C₂

yes

maximal, characteristic

#22581327

C₂

yes

maximal, complemented

#22581301

C₂

yes

maximal, complemented

#22580674

C₂

yes

maximal, characteristic, complemented

#498×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555772

—

C₂×D₁₁₁₂

—

#17984839

C₂²

yes

characteristic

#17984839

C₂²

yes

characteristic

#17984762

—

C₄×C₁₃₉⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

#106×C₁₃₉

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

#14555772

—

D₁₁₁₂

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₂×Q₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₈×C₁₃₉

SD₁₆

yes

complemented

C₂²×C₁₃₉

#127

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×SD₁₆

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₂×C₁₃₉

#492

yes

characteristic

C₂×C₁₃₉

#496

yes

—

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#1715

yes

characteristic, complemented, Sylow

#1715

—

139

maximal, Sylow

#498

D₁₃₉

yes

characteristic, complemented

#493

—

139

—

#492

—

139

—

#486

—

139

—

#322

—

139

—

C₂×SD₁₆

—

278

—

C₂×D₈

—

278

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#127

—

139

—

#127

—

139

—

#106

—

#102

—

278

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

SD₁₆

—

556

—

D₈

—

556

—

C₂×C₈

—

139

—

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

556

—

C₂³

—

556

—

Q₈

—

D₄

—

556

—

D₄

—

556

—

D₄

—

556

—

D₄

—

556

—

C₂×C₄

—

C₂×C₄

D₄×D₁₃₉

yes

—

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

C₈

—

278

—

C₈

#14555772

yes

complemented

C₂²

—

1112

—

C₂²

—

1112

—

C₂²

—

1112

—

C₂²

—

1112

—

C₂²

#17984838

yes

characteristic

C₄

—

C₄

C₂×#14555772

yes

—

C₄

#17984869

yes

—

C₂

—

2224

—

100

C₂

—

2224

—

101

C₂

#22581327

yes

characteristic

102

C₂

#22581339

yes

—

	subgroup	G/H
center	C₂²	#17984838
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#498×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#127
prime radical	#22581330	C₂
hypercenter	#498	D₁₃₉
hypocenter	C₁₃₉	#1715
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#17984838

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#498×C₁₃₉

C₂

C₂²×C₁₃₉

C₂

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #22581327	C₂	semidirect
3. #22581301	C₂	semidirect
4. #22580674	C₂	semidirect
5. #498×C₁₃₉	C₂	semidirect
38. C₈×C₁₃₉	SD₁₆	semidirect
51. C₁₃₉	#1715	semidirect
53. #498	D₁₃₉	semidirect
90. C₈	#14555772	semidirect

H ≅ G/N

type

2. #22581327

C₂

semidirect

3. #22581301

C₂

semidirect

4. #22580674

C₂

semidirect

5. #498×C₁₃₉

C₂

semidirect

38. C₈×C₁₃₉

SD₁₆

semidirect

51. C₁₃₉

#1715

semidirect

53. #498

D₁₃₉

semidirect

90. C₈

#14555772

semidirect

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

subgroup	index
#22581330	2
#22581327	2
#22581301	2
#22580674	2
#498×C₁₃₉	2
#1715	139

subgroup

index

#498×C₁₃₉

139

Subgroups — #31806033

Conjugacy classes of subgroups

Complements

Maximal subgroups