Group #31807152 — subgroups

Subgroups — #31807152

Order 17792.

208 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 93 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×#17984778	C₂	1	1	yes	maximal, characteristic
2	#22580991	C₂	1	1	yes	maximal, characteristic
3	#22580979	C₂	1	2	yes	maximal
4	#22580713	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	#17984801	C₂²	1	2	yes	—
8	#17984778	—	2	2	no	—
9	#17984776	C₄	1	2	yes	complemented
10	#17984774	C₄	1	2	yes	complemented
11	Dic₁₃₉⋊C₈	—	2	4	no	—
12	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₁₃₉	—	2	2	no	—
16	#14555769	—	2	2	no	—
17	#14555769	—	2	2	no	—
18	C₂×C₁₃₉⋊C₈	—	2	2	no	—
19	C₄⋊Dic₁₃₉	C₂×C₄	1	4	yes	—
20	C₂²⋊Dic₁₃₉	—	2	2	no	—
21	C₄×Dic₁₃₉	—	2	2	no	—
22	C₂²×C₄×C₁₃₉	—	2	1	no	—
23	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₃₉	—	2	1	no	—
25	C₂×C₈×C₁₃₉	—	2	1	no	—
26	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
29	C₂×C₈×C₁₃₉	D₄	1	2	yes	—
30	C₂×C₈×C₁₃₉	Q₈	1	2	yes	—
31	C₄²×C₁₃₉	—	2	1	no	—
32	C₄²×C₁₃₉	Q₈	1	2	yes	—
33	C₂×Dic₁₃₉	—	2	4	no	—
34	C₁₃₉⋊C₈	—	4	2	no	—
35	C₂³×C₁₃₉	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₃₉	—	2	4	no	—
37	C₂×C₄×C₁₃₉	—	2	2	no	—
38	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
44	C₈×C₁₃₉	—	2	2	no	—
45	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₃₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₃₉	—	2	2	no	—
48	C₂²×C₁₃₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₃₉	#125	1	1	yes	characteristic
51	C₄×C₁₃₉	—	2	4	no	—
52	C₄×C₁₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₃₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₃₉	#121	1	1	yes	characteristic
55	C₄×C₁₃₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₃₉	—	4	4	no	—
57	C₂×C₁₃₉	—	2	1	no	—
58	C₂×C₁₃₉	—	2	1	no	—
59	C₂×C₁₃₉	#389	1	1	yes	characteristic
60	C₂×C₁₃₉	#443	1	2	yes	—
61	C₁₃₉	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	139	1	no	maximal, Sylow
63	C₂×#108	—	139	1	no	—
64	#427	—	139	1	no	—
65	#424	—	139	2	no	—
66	C₂×C₄×C₈	D₁₃₉	1	1	yes	characteristic
67	#339	—	139	2	no	—
68	C₂×M₁₆	—	139	2	no	—
69	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₁₃₉	1	1	yes	characteristic
71	#117	—	139	2	no	—
72	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
73	#108	—	278	2	no	—
74	#107	—	139	2	no	—
75	#106	—	139	2	no	—
76	C₄⋊C₈	—	278	4	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
80	M₁₆	—	278	2	no	—
81	M₁₆	—	278	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	278	2	no	—
85	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
86	C₂×C₈	C₄×D₁₃₉	1	1	yes	characteristic
87	C₂×C₈	C₂²⋊D₁₃₉	1	2	yes	—
88	C₂×C₈	D₅₅₆	1	2	yes	—
89	C₂×C₈	Dic₂₇₈	1	2	yes	—
90	C₄⋊C₄	—	139	4	no	—
91	C₂²⋊C₄	—	278	2	no	—
92	C₄²	—	2	1	no	—
93	C₄²	—	278	2	no	—
94	C₄²	Dic₂₇₈	1	2	yes	—
95	C₂³	#14555785	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	278	4	no	—
99	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₅₅₆	1	1	yes	characteristic
102	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₂₇₈	1	1	yes	characteristic
104	C₂×C₄	#14555785	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	556	2	no	—
107	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
108	C₈	Dic₁₃₉⋊C₄	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₄×D₅₅₆	1	1	yes	characteristic
111	C₂²	#17984821	1	1	yes	characteristic
112	C₂²	#17984834	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	556	4	no	—
115	C₄	C₂×Dic₁₃₉⋊C₄	1	1	yes	characteristic
116	C₄	C₄×Dic₂₇₈	1	1	yes	characteristic
117	C₄	#17984816	1	1	yes	characteristic
118	C₄	#17984842	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#22580851	1	1	yes	characteristic
122	C₂	#22581071	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984778

C₂

yes

maximal, characteristic

#22580991

C₂

yes

maximal, characteristic

#22580979

C₂

yes

maximal

#22580713

C₂

yes

maximal

C₂×C₄×C₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

#17984801

C₂²

yes

—

#17984778

—

#17984776

C₄

yes

complemented

#17984774

C₄

yes

complemented

Dic₁₃₉⋊C₈

—

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂²×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄×C₈×C₁₃₉

—

#14555769

—

#14555769

—

C₂×C₁₃₉⋊C₈

—

C₄⋊Dic₁₃₉

C₂×C₄

yes

—

C₂²⋊Dic₁₃₉

—

C₄×Dic₁₃₉

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

D₄

yes

—

C₂×C₈×C₁₃₉

Q₈

yes

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

Q₈

yes

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

#41

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

characteristic

C₈×C₁₃₉

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₈×C₁₃₉

C₄⋊C₄

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₄×D₄

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₃₉

#125

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₃₉

C₄×Q₈

yes

characteristic

C₄×C₁₃₉

#121

yes

characteristic

C₄×C₁₃₉

C₄⋊Q₈

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#389

yes

characteristic

C₂×C₁₃₉

#443

yes

—

C₁₃₉

#1945

yes

characteristic, complemented, Sylow

#1945

—

139

maximal, Sylow

C₂×#108

—

139

—

#427

—

139

—

#424

—

139

—

C₂×C₄×C₈

D₁₃₉

yes

characteristic

#339

—

139

—

C₂×M₁₆

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

C₂²×C₈

C₂×D₁₃₉

yes

characteristic

#117

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#108

—

278

—

#107

—

139

—

#106

—

139

—

C₄⋊C₈

—

278

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

M₁₆

—

278

—

M₁₆

—

278

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

278

—

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₄×D₁₃₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₃₉

yes

—

C₂×C₈

D₅₅₆

yes

—

C₂×C₈

Dic₂₇₈

yes

—

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

278

—

C₄²

—

C₄²

—

278

—

C₄²

Dic₂₇₈

yes

—

C₂³

#14555785

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

101

C₂×C₄

C₂×D₅₅₆

yes

characteristic

102

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

103

C₂×C₄

C₂×Dic₂₇₈

yes

characteristic

104

C₂×C₄

#14555785

yes

characteristic

105

C₈

—

106

C₈

—

556

—

107

C₈

Dic₁₃₉⋊C₄

yes

—

108

C₈

Dic₁₃₉⋊C₄

yes

—

109

C₂²

—

110

C₂²

C₄×D₅₅₆

yes

characteristic

111

C₂²

#17984821

yes

characteristic

112

C₂²

#17984834

yes

characteristic

113

C₄

—

114

C₄

—

556

—

115

C₄

C₂×Dic₁₃₉⋊C₄

yes

characteristic

116

C₄

C₄×Dic₂₇₈

yes

characteristic

117

C₄

#17984816

yes

characteristic

118

C₄

#17984842

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#22580851

yes

characteristic

122

C₂

#22581071

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₅₅₆
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×C₄×C₈×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₄×D₄
prime radical	C₂³×C₁₃₉	#41
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₉
hypocenter	C₁₃₉	#1945
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₅₅₆

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×C₄×C₈×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₄×D₄

prime radical

C₂³×C₁₃₉

#41

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₂×C₄×C₈

D₁₃₉

hypocenter

C₁₃₉

#1945

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. #17984776	C₄	semidirect
10. #17984774	C₄	semidirect
61. C₁₃₉	#1945	semidirect

H ≅ G/N

type

9. #17984776

C₄

semidirect

10. #17984774

C₄

semidirect

61. C₁₃₉

#1945

semidirect

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

subgroup	index
C₂×#17984778	2
#22580991	2
#22580979	2
#22580713	2
C₂×C₄×C₈×C₁₃₉	2
#1945	139

subgroup

index

C₂×C₄×C₈×C₁₃₉

139

Subgroups — #31807152

Conjugacy classes of subgroups

Complements

Maximal subgroups