Group #31807223 — subgroups

Subgroups — #31807223

Order 17792.

308 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 105 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×#17984842	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984765	C₂	1	4	yes	maximal
3	C₂×#17984753	C₂	1	1	yes	maximal, characteristic
4	#382×C₁₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₇₈	C₂²	1	2	yes	—
6	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
7	#17984842	—	2	2	no	—
8	#17984842	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₃₉	—	2	2	no	—
10	#17984765	—	2	8	no	—
11	#17984753	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
14	#95×C₁₃₉	—	2	2	no	—
15	C₂×Dic₂₇₈	—	2	2	no	—
16	C₂×Dic₂₇₈	—	2	4	no	—
17	C₂×Dic₂₇₈	—	2	4	no	—
18	C₂×Dic₂₇₈	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₃₉	—	2	2	no	—
20	C₂×C₁₃₉⋊C₈	—	2	2	no	—
21	C₂×C₁₃₉⋊C₈	—	2	2	no	—
22	C₂×C₁₃₉⋊C₈	D₄	1	4	yes	—
23	C₄⋊Dic₁₃₉	—	2	8	no	—
24	C₂²×C₄×C₁₃₉	—	2	2	no	—
25	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
26	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₃₉	—	2	4	no	—
28	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
29	C₄²×C₁₃₉	D₄	1	2	yes	—
30	Dic₂₇₈	—	2	8	no	—
31	Dic₂₇₈	—	4	4	no	—
32	C₂×Dic₁₃₉	—	2	4	no	—
33	C₂×Dic₁₃₉	—	4	2	no	—
34	C₂×Dic₁₃₉	—	4	2	no	—
35	C₁₃₉⋊C₈	—	2	4	no	—
36	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₃₉	—	2	4	no	—
38	C₂×C₄×C₁₃₉	—	4	2	no	—
39	C₂×C₄×C₁₃₉	—	4	4	no	—
40	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	#41	1	2	yes	—
45	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₁₃₉	#124	1	1	yes	characteristic
47	C₂²×C₁₃₉	#127	1	1	yes	characteristic
48	C₂²×C₁₃₉	#137	1	4	yes	—
49	C₄×C₁₃₉	—	2	4	no	—
50	C₄×C₁₃₉	—	8	2	no	—
51	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
52	C₄×C₁₃₉	#121	1	2	yes	—
53	Dic₁₃₉	—	4	4	no	—
54	C₂×C₁₃₉	#390	1	1	yes	characteristic
55	C₂×C₁₃₉	#419	1	2	yes	—
56	C₂×C₁₃₉	#491	1	2	yes	—
57	C₂×C₁₃₉	#497	1	2	yes	—
58	C₁₃₉	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	139	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	—	139	1	no	—
61	C₂×#103	—	139	4	no	—
62	C₂×#97	—	139	1	no	—
63	#382	D₁₃₉	1	1	yes	characteristic
64	C₂²×Q₈	—	139	2	no	—
65	C₂²×C₈	—	139	2	no	—
66	C₄⋊Q₈	—	278	2	no	—
67	C₄⋊Q₈	—	139	4	no	—
68	C₂×C₄⋊C₄	—	278	2	no	—
69	C₂×C₄⋊C₄	C₂×D₁₃₉	1	2	yes	—
70	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
71	#103	—	278	8	no	—
72	#97	—	278	2	no	—
73	#95	—	2	2	no	—
74	C₂×Q₈	—	278	2	no	—
75	C₂×Q₈	—	278	4	no	—
76	C₂×Q₈	—	278	4	no	—
77	C₂×Q₈	—	139	4	no	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	278	2	no	—
80	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
82	C₂×C₈	—	278	2	no	—
83	C₂×C₈	—	278	2	no	—
84	C₂×C₈	—	139	4	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄⋊C₄	—	278	8	no	—
87	C₄²	C₄×D₁₃₉	1	2	yes	—
88	C₄²	D₅₅₆	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
90	Q₈	—	278	8	no	—
91	Q₈	—	556	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	4	no	—
95	C₂×C₄	—	278	4	no	—
96	C₂×C₄	—	556	2	no	—
97	C₂×C₄	—	556	2	no	—
98	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
101	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
102	C₂×C₄	#14555786	1	2	yes	—
103	C₈	—	278	4	no	—
104	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
105	C₂²	#17984831	1	1	yes	characteristic
106	C₂²	#17984838	1	1	yes	characteristic
107	C₂²	#17984872	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	8	2	no	—
110	C₄	—	556	4	no	—
111	C₄	#17984804	1	2	yes	—
112	C₄	#17984817	1	2	yes	—
113	C₂	#22580854	1	1	yes	characteristic
114	C₂	#22580961	1	2	yes	—
115	C₂	#22581323	1	2	yes	—
116	C₂	#22581348	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984842

C₂

yes

maximal, characteristic

C₂×#17984765

C₂

yes

maximal

C₂×#17984753

C₂

yes

maximal, characteristic

#382×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

—

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

#17984842

—

#17984842

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

—

#17984765

—

#17984753

—

C₂×C₄⋊C₄×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#95×C₁₃₉

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

C₂×C₄

yes

—

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

D₄

yes

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

D₄

yes

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

#121

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

#390

yes

characteristic

C₂×C₁₃₉

#419

yes

—

C₂×C₁₃₉

#491

yes

—

C₂×C₁₃₉

#497

yes

—

C₁₃₉

#1960

yes

characteristic, complemented, Sylow

#1960

—

139

maximal, Sylow

C₂×C₄⋊Q₈

—

139

—

C₂×#103

—

139

—

C₂×#97

—

139

—

#382

D₁₃₉

yes

characteristic

C₂²×Q₈

—

139

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#103

—

278

—

#97

—

278

—

#95

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂²×C₄

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄²

C₄×D₁₃₉

yes

—

C₄²

D₅₅₆

yes

—

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

Q₈

—

278

—

Q₈

—

556

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

—

556

—

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

100

C₂×C₄

D₄×D₁₃₉

yes

—

101

C₂×C₄

C₂×D₅₅₆

yes

characteristic

102

C₂×C₄

#14555786

yes

—

103

C₈

—

278

—

104

C₂²

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

105

C₂²

#17984831

yes

characteristic

106

C₂²

#17984838

yes

characteristic

107

C₂²

#17984872

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

556

—

111

C₄

#17984804

yes

—

112

C₄

#17984817

yes

—

113

C₂

#22580854

yes

characteristic

114

C₂

#22580961

yes

—

115

C₂

#22581323

yes

—

116

C₂

#22581348

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#382×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	#382	D₁₃₉
hypocenter	C₁₃₉	#1960
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#382×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #17984842	C₄	semidirect
58. C₁₃₉	#1960	semidirect

H ≅ G/N

type

8. #17984842

C₄

semidirect

58. C₁₃₉

#1960

semidirect

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

subgroup	index
C₂×#17984842	2
C₂×#17984765	2
C₂×#17984753	2
#382×C₁₃₉	2
#1960	139

subgroup

index

#382×C₁₃₉

139

Subgroups — #31807223

Conjugacy classes of subgroups

Complements

Maximal subgroups