Group #31816918 — subgroups

Subgroups — #31816918

Order 17792.

356 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 173 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	#22581510	C₂	1	2	yes	maximal
2	#22581358	C₂	1	2	yes	maximal, complemented
3	#22581355	C₂	1	4	yes	maximal, complemented
4	#22581351	C₂	1	2	yes	maximal, complemented
5	#22580991	C₂	1	4	yes	maximal
6	C₂×C₄×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#17984842	C₂²	1	4	yes	—
8	#17984834	C₂²	1	4	yes	—
9	#17984821	—	2	4	no	—
10	C₄×Dic₂₇₈	—	2	4	no	—
11	#17984801	C₂²	1	4	yes	—
12	#17984776	C₂²	1	8	yes	—
13	#17984774	C₂²	1	8	yes	—
14	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₄×C₈×C₁₃₉	C₂²	1	2	yes	—
18	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₇₈	—	2	4	no	—
20	C₄⋊Dic₁₃₉	—	2	4	no	—
21	C₄⋊Dic₁₃₉	C₂³	1	8	yes	—
22	Dic₁₃₉⋊C₄	—	2	8	no	—
23	C₂²⋊Dic₁₃₉	—	2	4	no	—
24	C₄×Dic₁₃₉	—	2	4	no	—
25	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₃₉	Q₈	1	8	yes	complemented
28	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
29	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
30	C₄²×C₁₃₉	D₄	1	2	yes	—
31	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
33	Dic₂₇₈	—	4	8	no	—
34	C₂×Dic₁₃₉	—	2	8	no	—
35	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	—	2	4	no	—
37	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
42	C₈×C₁₃₉	C₂×Q₈	1	4	yes	—
43	C₈×C₁₃₉	C₂×Q₈	1	4	yes	—
44	C₂²×C₁₃₉	—	2	2	no	—
45	C₂²×C₁₃₉	C₄⋊Q₈	1	2	yes	—
46	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
47	C₄×C₁₃₉	—	2	2	no	—
48	C₄×C₁₃₉	C₄⋊Q₈	1	2	yes	—
49	C₄×C₁₃₉	#135	1	4	yes	—
50	C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
51	Dic₁₃₉	—	4	8	no	—
52	C₂×C₁₃₉	—	2	2	no	—
53	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
54	C₂×C₁₃₉	C₂×#135	1	2	yes	—
55	C₁₃₉	#3160	1	1	yes	characteristic, complemented, Sylow
56	#3160	—	139	1	no	maximal, Sylow
57	#533	—	139	2	no	—
58	#500	—	139	2	no	—
59	#499	—	139	4	no	—
60	#498	—	139	2	no	—
61	#427	—	139	4	no	—
62	C₂×C₄×C₈	D₁₃₉	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
64	C₄⋊Q₈	—	139	4	no	—
65	#125	—	139	4	no	—
66	C₂²⋊Q₈	—	278	4	no	—
67	C₄×Q₈	—	278	4	no	—
68	#117	—	139	4	no	—
69	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
70	#107	—	139	8	no	—
71	#106	—	139	8	no	—
72	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
73	C₄×C₈	C₂×D₁₃₉	1	2	yes	—
74	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
75	C₂×Q₈	—	278	4	no	—
76	C₂²×C₄	D₅₅₆	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
78	C₂×C₈	Dic₂₇₈	1	8	yes	complemented
79	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
80	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
81	C₄⋊C₄	—	278	8	no	—
82	C₄⋊C₄	—	278	4	no	—
83	C₄⋊C₄	—	139	8	no	—
84	C₂²⋊C₄	—	278	4	no	—
85	C₄²	—	278	4	no	—
86	C₄²	D₅₅₆	1	2	yes	—
87	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
88	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
89	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
90	Q₈	—	556	8	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	278	8	no	—
93	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
94	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
95	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
96	C₂×C₄	C₂×Dic₂₇₈	1	4	yes	—
97	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
98	C₈	C₂×Dic₂₇₈	1	4	yes	—
99	C₈	C₂×Dic₂₇₈	1	4	yes	—
100	C₂²	—	2	2	no	—
101	C₂²	#17984842	1	2	yes	—
102	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	556	8	no	—
105	C₄	#17984842	1	2	yes	—
106	C₄	#17984864	1	4	yes	—
107	C₄	C₂²×Dic₂₇₈	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂×#17984842	1	1	yes	characteristic
110	C₂	C₂×#17984864	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581510

C₂

yes

maximal

#22581358

C₂

yes

maximal, complemented

#22581355

C₂

yes

maximal, complemented

#22581351

C₂

yes

maximal, complemented

#22580991

C₂

yes

maximal

C₂×C₄×C₈×C₁₃₉

C₂

yes

maximal, characteristic

#17984842

C₂²

yes

—

#17984834

C₂²

yes

—

#17984821

—

C₄×Dic₂₇₈

—

#17984801

C₂²

yes

—

#17984776

C₂²

yes

—

#17984774

C₂²

yes

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

Q₈

yes

complemented

C₂×C₈×C₁₃₉

C₂³

yes

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×Q₈

yes

—

C₈×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄⋊Q₈

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄⋊Q₈

yes

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

C₂²×Q₈

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₉

C₂×#135

yes

—

C₁₃₉

#3160

yes

characteristic, complemented, Sylow

#3160

—

139

maximal, Sylow

#533

—

139

—

#500

—

139

—

#499

—

139

—

#498

—

139

—

#427

—

139

—

C₂×C₄×C₈

D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

—

C₄⋊Q₈

—

139

—

#125

—

139

—

C₂²⋊Q₈

—

278

—

C₄×Q₈

—

278

—

#117

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#107

—

139

—

#106

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₄×C₈

C₂×D₁₃₉

yes

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

278

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

Dic₂₇₈

yes

complemented

C₂×C₈

C₂²×D₁₃₉

yes

—

C₂×C₈

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

C₄²

—

278

—

C₄²

D₅₅₆

yes

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

C₂×D₅₅₆

yes

characteristic

Q₈

—

556

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

C₂×D₅₅₆

yes

—

C₂×C₄

C₂×D₅₅₆

yes

—

C₂×C₄

C₂×D₅₅₆

yes

characteristic

C₂×C₄

C₂×Dic₂₇₈

yes

—

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

C₈

C₂×Dic₂₇₈

yes

—

C₈

C₂×Dic₂₇₈

yes

—

100

C₂²

—

101

C₂²

#17984842

yes

—

102

C₂²

C₂²×D₅₅₆

yes

characteristic

103

C₄

—

104

C₄

—

556

—

105

C₄

#17984842

yes

—

106

C₄

#17984864

yes

—

107

C₄

C₂²×Dic₂₇₈

yes

—

108

C₂

—

109

C₂

C₂×#17984842

yes

characteristic

110

C₂

C₂×#17984864

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₅₆
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×C₄×C₈×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	C₂×C₄×C₈	D₁₃₉
hypocenter	C₁₃₉	#3160
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₅₅₆

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

C₂×C₄×C₈×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

prime radical

C₂³×C₁₃₉

C₂×D₄

hypercenter

C₂×C₄×C₈

D₁₃₉

hypocenter

C₁₃₉

#3160

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #22581358	C₂	semidirect
3. #22581355	C₂	semidirect
4. #22581351	C₂	semidirect
27. C₂×C₈×C₁₃₉	Q₈	semidirect
55. C₁₃₉	#3160	semidirect
78. C₂×C₈	Dic₂₇₈	semidirect

H ≅ G/N

type

2. #22581358

C₂

semidirect

3. #22581355

C₂

semidirect

4. #22581351

C₂

semidirect

27. C₂×C₈×C₁₃₉

Q₈

semidirect

55. C₁₃₉

#3160

semidirect

78. C₂×C₈

Dic₂₇₈

semidirect

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

subgroup	index
#22581510	2
#22581358	2
#22581355	2
#22581351	2
#22580991	2
C₂×C₄×C₈×C₁₃₉	2
#3160	139

subgroup

index

C₂×C₄×C₈×C₁₃₉

139

Subgroups — #31816918

Conjugacy classes of subgroups

Complements

Maximal subgroups