Group #31814767 — subgroups

Subgroups — #31814767

Order 17792.

472 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂×#17984834	C₂	1	2	yes	maximal
2	#22580851	C₂	1	8	yes	maximal
3	#22580823	C₂	1	2	yes	maximal
4	#22580819	C₂	1	2	yes	maximal
5	C₄³×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	#17984834	—	2	8	no	—
7	C₂×C₄⋊Dic₁₃₉	C₂²	1	4	yes	—
8	C₂×C₄⋊Dic₁₃₉	C₂²	1	8	yes	—
9	C₂×Dic₁₃₉⋊C₄	C₂²	1	8	yes	—
10	C₁₃₉⋊#95	C₂²	1	8	yes	—
11	C₂×C₄²×C₁₃₉	C₂²	1	4	yes	—
12	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₃₉	C₂³	1	8	yes	—
16	C₄⋊Dic₁₃₉	—	2	8	no	—
17	C₄⋊Dic₁₃₉	—	2	16	no	—
18	Dic₁₃₉⋊C₄	—	2	16	no	—
19	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
20	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
21	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₉	C₂³	1	2	yes	—
23	C₄²×C₁₃₉	—	2	8	no	—
24	C₄²×C₁₃₉	D₄	1	4	yes	—
25	C₄²×C₁₃₉	Q₈	1	4	yes	—
26	C₄²×C₁₃₉	Q₈	1	4	yes	—
27	C₂×Dic₁₃₉	—	2	8	no	—
28	C₂×Dic₁₃₉	—	2	16	no	—
29	C₂³×C₁₃₉	C₂⁴	1	1	yes	characteristic
30	C₂×C₄×C₁₃₉	—	2	4	no	—
31	C₂×C₄×C₁₃₉	—	2	4	no	—
32	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
35	C₂×C₄×C₁₃₉	C₂×Q₈	1	4	yes	—
36	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
37	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₃₉	#41	1	8	yes	—
39	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
40	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
41	C₂²×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
42	C₂²×C₁₃₉	C₂×#41	1	4	yes	—
43	C₄×C₁₃₉	—	2	4	no	—
44	C₄×C₁₃₉	—	2	4	no	—
45	C₄×C₁₃₉	#125	1	8	yes	complemented
46	C₄×C₁₃₉	C₄⋊Q₈	1	4	yes	—
47	Dic₁₃₉	—	4	8	no	—
48	C₂×C₁₃₉	C₂×#125	1	2	yes	—
49	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₁₃₉	#532	1	2	yes	—
51	C₂×C₁₃₉	#533	1	2	yes	—
52	C₁₃₉	#2859	1	1	yes	characteristic, complemented, Sylow
53	#2859	—	139	1	no	maximal, Sylow
54	C₂×#125	—	139	2	no	—
55	#389	—	139	8	no	—
56	#384	—	139	2	no	—
57	#382	—	139	2	no	—
58	C₄³	D₁₃₉	1	1	yes	characteristic
59	#125	—	278	8	no	—
60	C₂×C₄⋊C₄	—	139	8	no	—
61	C₂×C₄⋊C₄	—	139	4	no	—
62	C₂×C₄⋊C₄	—	139	8	no	—
63	C₂×C₄²	C₂×D₁₃₉	1	4	yes	—
64	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
67	#95	—	139	8	no	—
68	C₂²×C₄	—	139	8	no	—
69	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
70	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
71	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₃₉	1	2	yes	—
73	C₄⋊C₄	—	278	16	no	—
74	C₄⋊C₄	—	278	8	no	—
75	C₄⋊C₄	—	278	16	no	—
76	C₄²	—	2	8	no	—
77	C₄²	D₅₅₆	1	4	yes	—
78	C₄²	Dic₂₇₈	1	4	yes	—
79	C₄²	Dic₂₇₈	1	4	yes	—
80	C₂³	C₂³×D₁₃₉	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	278	8	no	—
84	C₂×C₄	—	278	16	no	—
85	C₂×C₄	C₂×D₅₅₆	1	4	yes	—
86	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
87	C₂×C₄	C₂×Dic₂₇₈	1	4	yes	—
88	C₂×C₄	C₂×Dic₂₇₈	1	4	yes	—
89	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
90	C₂×C₄	C₂×Dic₂₇₈	1	2	yes	—
91	C₂×C₄	#14555785	1	8	yes	—
92	C₂²	C₂²×D₅₅₆	1	1	yes	characteristic
93	C₂²	C₂²×Dic₂₇₈	1	1	yes	characteristic
94	C₂²	C₂²×Dic₂₇₈	1	1	yes	characteristic
95	C₂²	C₂×#14555785	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	2	4	no	—
98	C₄	—	556	8	no	—
99	C₄	#17984834	1	8	yes	complemented
100	C₄	#17984842	1	4	yes	—
101	C₂	C₂×#17984834	1	2	yes	—
102	C₂	C₂×#17984842	1	1	yes	characteristic
103	C₂	#22581503	1	2	yes	—
104	C₂	#22581510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984834

C₂

yes

maximal

#22580851

C₂

yes

maximal

#22580823

C₂

yes

maximal

#22580819

C₂

yes

maximal

C₄³×C₁₃₉

C₂

yes

maximal, characteristic

#17984834

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

C₂²

yes

—

C₁₃₉⋊#95

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

Dic₁₃₉⋊C₄

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²×C₄×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

Q₈

yes

—

C₄²×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂⁴

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂²×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂²×Q₈

yes

characteristic

C₂²×C₁₃₉

C₂×#41

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#125

yes

complemented

C₄×C₁₃₉

C₄⋊Q₈

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

C₂×#125

yes

—

C₂×C₁₃₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₉

#532

yes

—

C₂×C₁₃₉

#533

yes

—

C₁₃₉

#2859

yes

characteristic, complemented, Sylow

#2859

—

139

maximal, Sylow

C₂×#125

—

139

—

#389

—

139

—

#384

—

139

—

#382

—

139

—

C₄³

D₁₃₉

yes

characteristic

#125

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#95

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂²×C₄

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄²

—

C₄²

D₅₅₆

yes

—

C₄²

Dic₂₇₈

yes

—

C₄²

Dic₂₇₈

yes

—

C₂³

C₂³×D₁₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

C₂×D₅₅₆

yes

—

C₂×C₄

C₂×D₅₅₆

yes

—

C₂×C₄

C₂×Dic₂₇₈

yes

—

C₂×C₄

C₂×Dic₂₇₈

yes

—

C₂×C₄

C₂×Dic₂₇₈

yes

—

C₂×C₄

C₂×Dic₂₇₈

yes

—

C₂×C₄

#14555785

yes

—

C₂²

C₂²×D₅₅₆

yes

characteristic

C₂²

C₂²×Dic₂₇₈

yes

characteristic

C₂²

C₂²×Dic₂₇₈

yes

characteristic

C₂²

C₂×#14555785

yes

—

C₄

—

C₄

—

C₄

—

556

—

C₄

#17984834

yes

complemented

100

C₄

#17984842

yes

—

101

C₂

C₂×#17984834

yes

—

102

C₂

C₂×#17984842

yes

characteristic

103

C₂

#22581503

yes

—

104

C₂

#22581510

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₃₉
commutator, socle, prime radical, cosocle	C₂³×C₁₃₉	C₂⁴
Fitting	C₄³×C₁₃₉	C₂
hypercenter	C₄³	D₁₃₉
hypocenter	C₁₃₉	#2859
LCS	G ▷ C₂³×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₄³

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₃₉

commutator, socle, prime radical, cosocle

C₂³×C₁₃₉

C₂⁴

Fitting

C₄³×C₁₃₉

C₂

hypercenter

C₄³

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂³ ◁ C₄³

N	H ≅ G/N	type
45. C₄×C₁₃₉	#125	semidirect
52. C₁₃₉	#2859	semidirect
99. C₄	#17984834	semidirect

H ≅ G/N

type

45. C₄×C₁₃₉

#125

semidirect

52. C₁₃₉

#2859

semidirect

99. C₄

#17984834

semidirect

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

subgroup	index
C₂×#17984834	2
#22580851	2
#22580823	2
#22580819	2
C₄³×C₁₃₉	2
#2859	139

subgroup

index

C₄³×C₁₃₉

139

Subgroups — #31814767

Conjugacy classes of subgroups

Complements

Maximal subgroups