Group #31806070 — subgroups

Subgroups — #31806070

Order 17792.

242 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 69 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, complemented
2	#22581330	C₂	1	2	yes	maximal, complemented
3	#22581326	C₂	1	1	yes	maximal, characteristic, complemented
4	#22581300	C₂	1	2	yes	maximal, complemented
5	#322×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555774	—	2	2	no	—
7	C₂×D₁₁₁₂	—	2	4	no	—
8	C₂×D₁₁₁₂	—	2	2	no	—
9	#17984842	C₂²	1	1	yes	characteristic
10	#17984839	C₂²	1	2	yes	—
11	#17984839	C₂²	1	1	yes	characteristic
12	#17984764	—	2	4	no	—
13	C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
14	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	2	1	no	—
16	C₂×D₅₅₆	—	2	2	no	—
17	C₂×D₅₅₆	—	2	2	no	—
18	C₂×D₅₅₆	—	2	2	no	—
19	C₂×D₅₅₆	—	2	1	no	—
20	C₂×D₅₅₆	—	2	2	no	—
21	#14555774	—	4	4	no	—
22	D₁₁₁₂	—	4	8	no	—
23	D₁₁₁₂	—	4	4	no	—
24	C₄⋊Dic₁₃₉	—	2	2	no	—
25	C₂×C₈×C₁₃₉	D₄	1	4	yes	—
26	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
27	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₂²×D₁₃₉	—	4	1	no	—
29	C₂²×D₁₃₉	—	4	2	no	—
30	Dic₂₇₈	—	4	2	no	—
31	D₅₅₆	—	4	4	no	—
32	D₅₅₆	—	4	2	no	—
33	D₅₅₆	—	4	4	no	—
34	D₅₅₆	—	4	4	no	—
35	D₅₅₆	—	4	4	no	—
36	C₂×Dic₁₃₉	—	4	1	no	—
37	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
39	C₈×C₁₃₉	—	2	4	no	—
40	C₈×C₁₃₉	D₈	1	4	yes	complemented
41	C₂²×C₁₃₉	#127	1	1	yes	characteristic
42	C₂×D₁₃₉	—	8	2	no	—
43	C₂×D₁₃₉	—	8	1	no	—
44	C₂×D₁₃₉	—	8	2	no	—
45	C₂×D₁₃₉	—	8	2	no	—
46	C₂×D₁₃₉	—	8	2	no	—
47	C₄×C₁₃₉	C₂×D₈	1	2	yes	—
48	C₄×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
49	Dic₁₃₉	—	8	1	no	—
50	C₂×C₁₃₉	#493	1	1	yes	characteristic
51	C₂×C₁₃₉	#496	1	2	yes	—
52	D₁₃₉	—	16	2	no	—
53	D₁₃₉	—	16	1	no	—
54	C₁₃₉	#1721	1	1	yes	characteristic, complemented, Sylow
55	#1721	—	139	1	no	maximal, Sylow
56	#493	—	139	2	no	—
57	#493	—	139	1	no	—
58	#492	—	139	1	no	—
59	#486	—	139	2	no	—
60	#322	D₁₃₉	1	1	yes	characteristic, complemented
61	C₂×SD₁₆	—	278	2	no	—
62	C₂×D₈	—	278	4	no	—
63	C₂×D₈	—	278	2	no	—
64	C₄⋊Q₈	—	139	1	no	—
65	#127	—	139	2	no	—
66	#127	—	139	1	no	—
67	#102	—	278	4	no	—
68	C₄×C₈	C₂×D₁₃₉	1	2	yes	—
69	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
70	C₂×Q₈	—	278	1	no	—
71	C₂×D₄	—	278	2	no	—
72	C₂×D₄	—	278	2	no	—
73	C₂×D₄	—	278	2	no	—
74	C₂×D₄	—	278	2	no	—
75	C₂×D₄	—	278	1	no	—
76	SD₁₆	—	556	4	no	—
77	D₈	—	556	8	no	—
78	D₈	—	556	4	no	—
79	C₂×C₈	D₅₅₆	1	4	yes	—
80	C₂×C₈	D₅₅₆	1	2	yes	—
81	C₄⋊C₄	—	278	2	no	—
82	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
83	C₂³	—	556	2	no	—
84	C₂³	—	556	1	no	—
85	Q₈	—	556	2	no	—
86	D₄	—	556	4	no	—
87	D₄	—	556	2	no	—
88	D₄	—	556	4	no	—
89	D₄	—	556	4	no	—
90	D₄	—	556	4	no	—
91	C₂×C₄	—	556	1	no	—
92	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
93	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	D₁₁₁₂	1	4	yes	complemented
96	C₂²	—	1112	2	no	—
97	C₂²	—	1112	1	no	—
98	C₂²	—	1112	2	no	—
99	C₂²	—	1112	2	no	—
100	C₂²	—	1112	2	no	—
101	C₂²	#17984839	1	1	yes	characteristic
102	C₄	—	1112	1	no	—
103	C₄	C₂×D₁₁₁₂	1	2	yes	—
104	C₄	#17984870	1	4	yes	—
105	C₂	—	2224	2	no	—
106	C₂	—	2224	1	no	—
107	C₂	#22581330	1	1	yes	characteristic
108	C₂	#22581342	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581330

C₂

yes

maximal, characteristic, complemented

#22581330

C₂

yes

maximal, complemented

#22581326

C₂

yes

maximal, characteristic, complemented

#22581300

C₂

yes

maximal, complemented

#322×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555774

—

C₂×D₁₁₁₂

—

C₂×D₁₁₁₂

—

#17984842

C₂²

yes

characteristic

#17984839

C₂²

yes

—

#17984839

C₂²

yes

characteristic

#17984764

—

C₄×C₈×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

#14555774

—

D₁₁₁₂

—

D₁₁₁₂

—

C₄⋊Dic₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₈×C₁₃₉

—

C₈×C₁₃₉

D₈

yes

complemented

C₂²×C₁₃₉

#127

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

C₂×D₈

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

#493

yes

characteristic

C₂×C₁₃₉

#496

yes

—

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#1721

yes

characteristic, complemented, Sylow

#1721

—

139

maximal, Sylow

#493

—

139

—

#493

—

139

—

#492

—

139

—

#486

—

139

—

#322

D₁₃₉

yes

characteristic, complemented

C₂×SD₁₆

—

278

—

C₂×D₈

—

278

—

C₂×D₈

—

278

—

C₄⋊Q₈

—

139

—

#127

—

139

—

#127

—

139

—

#102

—

278

—

C₄×C₈

C₂×D₁₃₉

yes

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

SD₁₆

—

556

—

D₈

—

556

—

D₈

—

556

—

C₂×C₈

D₅₅₆

yes

—

C₂×C₈

D₅₅₆

yes

—

C₄⋊C₄

—

278

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

556

—

C₂³

—

556

—

Q₈

—

556

—

D₄

—

556

—

D₄

—

556

—

D₄

—

556

—

D₄

—

556

—

D₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

C₂×D₅₅₆

yes

characteristic

C₂×C₄

C₂×D₅₅₆

yes

—

C₈

—

C₈

D₁₁₁₂

yes

complemented

C₂²

—

1112

—

C₂²

—

1112

—

C₂²

—

1112

—

C₂²

—

1112

—

100

C₂²

—

1112

—

101

C₂²

#17984839

yes

characteristic

102

C₄

—

1112

—

103

C₄

C₂×D₁₁₁₂

yes

—

104

C₄

#17984870

yes

—

105

C₂

—

2224

—

106

C₂

—

2224

—

107

C₂

#22581330

yes

characteristic

108

C₂

#22581342

yes

—

	subgroup	G/H
center	C₂²	#17984839
commutator, cosocle	C₄²×C₁₃₉	C₂³
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#322×C₁₃₉	C₂
socle	C₂²×C₁₃₉	#127
hypercenter	#322	D₁₃₉
hypocenter	C₁₃₉	#1721
LCS	G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#17984839

commutator, cosocle

C₄²×C₁₃₉

C₂³

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#322×C₁₃₉

C₂

C₂²×C₁₃₉

D₁₃₉

C₁₃₉

G ▷ C₄²×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #22581330	C₂	semidirect
2. #22581330	C₂	semidirect
3. #22581326	C₂	semidirect
4. #22581300	C₂	semidirect
5. #322×C₁₃₉	C₂	semidirect
40. C₈×C₁₃₉	D₈	semidirect
54. C₁₃₉	#1721	semidirect
60. #322	D₁₃₉	semidirect
95. C₈	D₁₁₁₂	semidirect

H ≅ G/N

type

1. #22581330

C₂

semidirect

2. #22581330

C₂

semidirect

3. #22581326

C₂

semidirect

4. #22581300

C₂

semidirect

5. #322×C₁₃₉

C₂

semidirect

40. C₈×C₁₃₉

D₈

semidirect

54. C₁₃₉

#1721

semidirect

60. #322

D₁₃₉

semidirect

95. C₈

D₁₁₁₂

semidirect

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

subgroup	index
#22581330	2
#22581326	2
#22581300	2
#322×C₁₃₉	2
#1721	139

subgroup

index

#322×C₁₃₉

139

Subgroups — #31806070

Conjugacy classes of subgroups

Complements

Maximal subgroups