Group #31816743 — subgroups

Subgroups — C₂×#22581321 #31816743

Order 17792.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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	1	yes	maximal, characteristic
2	#22581321	C₂	1	8	yes	maximal, direct product factor
3	C₂×#17984766	C₂	1	4	yes	maximal
4	C₂×#17984753	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃₉⋊C₈	C₂²	1	2	yes	—
7	#17984834	—	2	2	no	—
8	#17984834	C₂²	1	4	yes	—
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
10	C₂×Dic₁₃₉⋊C₄	—	2	2	no	—
11	#17984766	C₂²	1	16	yes	—
12	#17984753	C₂²	1	4	yes	—
13	C₂²×Q₈×C₁₃₉	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₃₉	—	2	2	no	—
15	C₄⋊Q₈×C₁₃₉	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₃₉	—	2	2	no	—
17	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₂²×Dic₁₃₉	—	2	2	no	—
19	C₂×C₁₃₉⋊C₈	—	2	4	no	—
20	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
21	C₄⋊Dic₁₃₉	—	2	2	no	—
22	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
23	Dic₁₃₉⋊C₄	—	2	8	no	—
24	C₂×Q₈×C₁₃₉	—	2	2	no	—
25	C₂×Q₈×C₁₃₉	—	2	8	no	—
26	C₂×Q₈×C₁₃₉	C₂³	1	4	yes	—
27	C₂²×C₄×C₁₃₉	—	2	2	no	—
28	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₃₉	—	2	8	no	—
31	C₄²×C₁₃₉	D₄	1	2	yes	—
32	C₄²×C₁₃₉	C₂³	1	2	yes	—
33	C₂×Dic₁₃₉	—	2	4	no	—
34	C₂×Dic₁₃₉	—	4	4	no	—
35	C₁₃₉⋊C₈	—	2	4	no	—
36	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₃₉	—	2	8	no	—
38	Q₈×C₁₃₉	—	4	4	no	—
39	C₂×C₄×C₁₃₉	—	2	4	no	—
40	C₂×C₄×C₁₃₉	—	2	4	no	—
41	C₂×C₄×C₁₃₉	—	4	4	no	—
42	C₂×C₄×C₁₃₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	#41	1	4	yes	—
45	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
46	C₂²×C₁₃₉	#124	1	2	yes	—
47	C₂²×C₁₃₉	#137	1	4	yes	—
48	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₃₉	—	2	4	no	—
50	C₄×C₁₃₉	—	4	4	no	—
51	C₄×C₁₃₉	#124	1	2	yes	—
52	C₄×C₁₃₉	C₂×#41	1	2	yes	—
53	Dic₁₃₉	—	4	4	no	—
54	C₂×C₁₃₉	#491	1	4	yes	complemented
55	C₂×C₁₃₉	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₃₉	C₂×#137	1	2	yes	—
57	C₁₃₉	C₂×#491	1	1	yes	characteristic, complemented, Sylow
58	C₂×#491	—	139	1	no	maximal, Sylow
59	C₂×C₄⋊Q₈	D₁₃₉	1	1	yes	characteristic
60	C₂×#125	—	139	1	no	—
61	#491	—	139	8	no	—
62	C₂×#103	—	139	4	no	—
63	C₂×#97	—	139	1	no	—
64	C₂²×Q₈	C₂×D₁₃₉	1	2	yes	—
65	C₂²×C₈	—	139	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	C₂×D₁₃₉	1	4	yes	—
68	#125	—	278	2	no	—
69	#125	—	139	4	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	278	2	no	—
72	C₂×C₄⋊C₄	—	139	2	no	—
73	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
74	#103	—	139	16	no	—
75	#97	—	139	4	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂²×D₁₃₉	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	278	2	no	—
81	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
82	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
83	C₂×C₈	—	278	4	no	—
84	C₂×C₈	—	139	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	278	8	no	—
87	C₄⋊C₄	—	278	2	no	—
88	C₄⋊C₄	—	139	4	no	—
89	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
90	C₄²	C₂²×D₁₃₉	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	Q₈	—	4	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	278	4	no	—
98	C₂×C₄	—	556	4	no	—
99	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
101	C₂×C₄	#14555784	1	4	yes	—
102	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
103	C₈	—	278	4	no	—
104	C₂²	#17984829	1	2	yes	—
105	C₂²	#17984873	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	556	4	no	—
110	C₄	#17984829	1	2	yes	—
111	C₄	C₂×#14555784	1	2	yes	—
112	C₂	#22581321	1	4	yes	direct product factor
113	C₂	C₂×#17984829	1	1	yes	characteristic
114	C₂	C₂×#17984873	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984834

C₂

yes

maximal, characteristic

#22581321

C₂

yes

maximal, direct product factor

C₂×#17984766

C₂

yes

maximal

C₂×#17984753

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×C₁₃₉⋊C₈

C₂²

yes

—

#17984834

—

#17984834

C₂²

yes

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

—

#17984766

C₂²

yes

—

#17984753

C₂²

yes

—

C₂²×Q₈×C₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₉

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂²×Dic₁₃₉

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₂²×C₁₃₉

#124

yes

—

C₂²×C₁₃₉

#137

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#124

yes

—

C₄×C₁₃₉

C₂×#41

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

—

C₁₃₉

C₂×#491

yes

characteristic, complemented, Sylow

C₂×#491

—

139

maximal, Sylow

C₂×C₄⋊Q₈

D₁₃₉

yes

characteristic

C₂×#125

—

139

—

#491

—

139

—

C₂×#103

—

139

—

C₂×#97

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#125

—

278

—

#125

—

139

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#103

—

139

—

#97

—

139

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₁₃₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

278

—

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

278

—

C₂×C₈

—

139

—

C₄⋊C₄

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

C₂²⋊D₁₃₉

yes

—

C₄²

C₂²×D₁₃₉

yes

—

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

556

—

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

100

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

101

C₂×C₄

#14555784

yes

—

102

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

103

C₈

—

278

—

104

C₂²

#17984829

yes

—

105

C₂²

#17984873

yes

—

106

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

556

—

110

C₄

#17984829

yes

—

111

C₄

C₂×#14555784

yes

—

112

C₂

#22581321

yes

direct product factor

113

C₂

C₂×#17984829

yes

characteristic

114

C₂

C₂×#17984873

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	C₂×C₄⋊Q₈×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₃₉
hypocenter	C₁₃₉	C₂×#491
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

C₂×C₄⋊Q₈×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₃₉

hypocenter

C₁₃₉

C₂×#491

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #22581321	C₂	direct (N × H)
54. C₂×C₁₃₉	#491	semidirect
57. C₁₃₉	C₂×#491	semidirect
112. C₂	#22581321	direct (N × H)

H ≅ G/N

type

2. #22581321

C₂

direct (N × H)

54. C₂×C₁₃₉

#491

semidirect

57. C₁₃₉

C₂×#491

semidirect

112. C₂

#22581321

direct (N × H)

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

subgroup	index
C₂×#17984834	2
#22581321	2
C₂×#17984766	2
C₂×#17984753	2
C₂×C₄⋊Q₈×C₁₃₉	2
C₂×#491	139

subgroup

index

C₂×C₄⋊Q₈×C₁₃₉

139

Subgroups — C₂×#22581321 #31816743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#22581321 #31816743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22581321 #31816743