Group #31816913 — subgroups

Subgroups — #31816913

Order 17792.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#22581508	C₂	1	2	yes	maximal
2	#22581356	C₂	1	2	yes	maximal, complemented
3	#22581353	C₂	1	4	yes	maximal, complemented
4	#22581351	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580988	C₂	1	4	yes	maximal
6	C₄×C₈×D₁₃₉	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₃₉	C₂²	1	2	yes	—
9	#17984842	C₂²	1	2	yes	—
10	#17984833	C₂²	1	4	yes	—
11	#17984822	—	2	2	no	—
12	#17984818	—	2	2	no	—
13	Q₈×Dic₁₃₉	—	2	2	no	—
14	#17984807	—	2	2	no	—
15	#17984800	C₂²	1	4	yes	—
16	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
17	#17984775	C₂²	1	8	yes	—
18	#17984774	C₂²	1	4	yes	—
19	C₄×C₁₃₉⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₃₉	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	—
22	#106×C₁₃₉	C₂²	1	4	yes	—
23	C₄×C₈×C₁₃₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₇₈	—	2	2	no	—
25	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
26	C₂×C₄×D₁₃₉	C₂³	1	1	yes	characteristic
27	C₂×C₁₃₉⋊C₈	C₂³	1	2	yes	—
28	C₈×D₁₃₉	Q₈	1	8	yes	complemented
29	C₄⋊Dic₁₃₉	—	2	4	no	—
30	C₄⋊Dic₁₃₉	—	2	2	no	—
31	C₄⋊Dic₁₃₉	C₂³	1	4	yes	—
32	Dic₁₃₉⋊C₄	—	2	4	no	—
33	(C₂×D₁₃₉)⋊C₄	—	2	4	no	—
34	C₄×Dic₁₃₉	—	2	4	no	—
35	C₄×Dic₁₃₉	D₄	1	2	yes	—
36	C₄×Dic₁₃₉	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₃₉	—	2	2	no	—
38	C₂×C₈×C₁₃₉	C₂³	1	2	yes	—
39	C₄⋊C₄×C₁₃₉	—	2	2	no	—
40	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
41	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
42	C₂²×D₁₃₉	C₂×D₄	1	1	yes	characteristic
43	Dic₂₇₈	—	4	4	no	—
44	C₂×Dic₁₃₉	—	2	4	no	—
45	C₂×Dic₁₃₉	C₂×D₄	1	2	yes	—
46	C₂×Dic₁₃₉	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₃₉	—	2	4	no	—
48	C₄×D₁₃₉	C₂×Q₈	1	4	yes	—
49	C₁₃₉⋊C₈	C₂×Q₈	1	4	yes	—
50	Q₈×C₁₃₉	—	4	4	no	—
51	C₂×C₄×C₁₃₉	—	2	4	no	—
52	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₃₉	C₂×Q₈	1	4	yes	—
55	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₁₃₉	—	2	2	no	—
57	C₂×D₁₃₉	C₄⋊Q₈	1	2	yes	—
58	C₄×C₁₃₉	—	4	4	no	—
59	C₄×C₁₃₉	#135	1	4	yes	—
60	C₄×C₁₃₉	C₂²×Q₈	1	2	yes	—
61	Dic₁₃₉	—	2	2	no	—
62	Dic₁₃₉	—	4	4	no	—
63	Dic₁₃₉	C₄⋊Q₈	1	2	yes	—
64	C₂×C₁₃₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₃₉	C₂×#135	1	2	yes	—
66	D₁₃₉	—	2	2	no	—
67	C₁₃₉	#3160	1	1	yes	characteristic, complemented, Sylow
68	#3160	—	139	1	no	maximal, Sylow
69	#533	—	139	2	no	—
70	#500	—	139	2	no	—
71	#499	—	139	4	no	—
72	#498	—	139	1	no	—
73	#498	D₁₃₉	1	1	yes	characteristic, complemented
74	#427	—	139	4	no	—
75	C₂×C₄×C₈	—	139	1	no	—
76	C₂²×C₈	—	139	2	no	—
77	C₄⋊Q₈	—	139	2	no	—
78	C₄⋊Q₈	C₂×D₁₃₉	1	2	yes	—
79	#125	—	139	4	no	—
80	C₂²⋊Q₈	—	278	2	no	—
81	C₂²⋊Q₈	—	278	2	no	—
82	C₄×Q₈	—	278	2	no	—
83	C₄×Q₈	—	278	2	no	—
84	#117	—	139	4	no	—
85	C₂×C₄²	—	139	1	no	—
86	#107	—	139	8	no	—
87	#106	—	139	4	no	—
88	#106	C₂×D₁₃₉	1	4	yes	—
89	C₄×C₈	—	139	2	no	—
90	C₄×C₈	—	139	1	no	—
91	C₄×C₈	C₂×D₁₃₉	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	278	2	no	—
94	C₂²×C₄	—	139	2	no	—
95	C₂²×C₄	—	139	1	no	—
96	C₂×C₈	—	139	2	no	—
97	C₂×C₈	—	139	8	no	—
98	C₂×C₈	C₂²×D₁₃₉	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	278	4	no	—
101	C₄⋊C₄	—	278	4	no	—
102	C₄⋊C₄	—	278	2	no	—
103	C₄⋊C₄	—	139	4	no	—
104	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
105	C₂²⋊C₄	—	278	4	no	—
106	C₄²	—	278	4	no	—
107	C₄²	—	139	2	no	—
108	C₄²	—	139	1	no	—
109	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
110	C₂³	—	139	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	556	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	278	4	no	—
115	C₂×C₄	—	278	4	no	—
116	C₂×C₄	—	139	2	no	—
117	C₂×C₄	—	139	1	no	—
118	C₂×C₄	—	139	4	no	—
119	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
120	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
121	C₈	—	139	4	no	—
122	C₈	Q₈×D₁₃₉	1	4	yes	—
123	C₂²	—	278	2	no	—
124	C₂²	—	139	2	no	—
125	C₂²	C₂×D₄×D₁₃₉	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	278	2	no	—
128	C₄	—	556	4	no	—
129	C₄	—	139	2	no	—
130	C₄	#17984863	1	4	yes	—
131	C₄	C₂×Q₈×D₁₃₉	1	2	yes	—
132	C₂	—	278	2	no	—
133	C₂	C₄⋊Q₈×D₁₃₉	1	1	yes	characteristic
134	C₂	C₂×#17984863	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581508

C₂

yes

maximal

#22581356

C₂

yes

maximal, complemented

#22581353

C₂

yes

maximal, complemented

#22581351

C₂

yes

maximal, characteristic, complemented

#22580988

C₂

yes

maximal

C₄×C₈×D₁₃₉

C₂

yes

maximal, characteristic

#498×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₃₉

C₂²

yes

—

#17984842

C₂²

yes

—

#17984833

C₂²

yes

—

#17984822

—

#17984818

—

Q₈×Dic₁₃₉

—

#17984807

—

#17984800

C₂²

yes

—

C₄²×D₁₃₉

C₂²

yes

characteristic

#17984775

C₂²

yes

—

#17984774

C₂²

yes

—

C₄×C₁₃₉⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₃₉

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

#106×C₁₃₉

C₂²

yes

—

C₄×C₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₄×D₁₃₉

C₂³

yes

characteristic

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₈×D₁₃₉

Q₈

yes

complemented

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

—

C₄×Dic₁₃₉

D₄

yes

—

C₄×Dic₁₃₉

C₂³

yes

characteristic

C₂×Q₈×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

C₂×D₄

yes

characteristic

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂×D₄

yes

—

C₂×Dic₁₃₉

C₂×D₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

C₂×Q₈

yes

—

C₁₃₉⋊C₈

C₂×Q₈

yes

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

C₂×Q₈

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

C₄⋊Q₈

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

C₂²×Q₈

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

C₄⋊Q₈

yes

—

C₂×C₁₃₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₃₉

C₂×#135

yes

—

D₁₃₉

—

C₁₃₉

#3160

yes

characteristic, complemented, Sylow

#3160

—

139

maximal, Sylow

#533

—

139

—

#500

—

139

—

#499

—

139

—

#498

—

139

—

#498

D₁₃₉

yes

characteristic, complemented

#427

—

139

—

C₂×C₄×C₈

—

139

—

C₂²×C₈

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#125

—

139

—

C₂²⋊Q₈

—

278

—

C₂²⋊Q₈

—

278

—

C₄×Q₈

—

278

—

C₄×Q₈

—

278

—

#117

—

139

—

C₂×C₄²

—

139

—

#107

—

139

—

#106

—

139

—

#106

C₂×D₁₃₉

yes

—

C₄×C₈

—

139

—

C₄×C₈

—

139

—

C₄×C₈

C₂×D₁₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂×C₈

—

139

—

C₂×C₈

—

139

—

C₂×C₈

C₂²×D₁₃₉

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

278

—

101

C₄⋊C₄

—

278

—

102

C₄⋊C₄

—

278

—

103

C₄⋊C₄

—

139

—

104

C₄⋊C₄

C₂²×D₁₃₉

yes

—

105

C₂²⋊C₄

—

278

—

106

C₄²

—

278

—

107

C₄²

—

139

—

108

C₄²

—

139

—

109

C₄²

C₂²×D₁₃₉

yes

characteristic

110

C₂³

—

139

—

111

Q₈

—

112

Q₈

—

556

—

113

C₂×C₄

—

114

C₂×C₄

—

278

—

115

C₂×C₄

—

278

—

116

C₂×C₄

—

139

—

117

C₂×C₄

—

139

—

118

C₂×C₄

—

139

—

119

C₂×C₄

D₄×D₁₃₉

yes

—

120

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

121

C₈

—

139

—

122

C₈

Q₈×D₁₃₉

yes

—

123

C₂²

—

278

—

124

C₂²

—

139

—

125

C₂²

C₂×D₄×D₁₃₉

yes

characteristic

126

C₄

—

127

C₄

—

278

—

128

C₄

—

556

—

129

C₄

—

139

—

130

C₄

#17984863

yes

—

131

C₄

C₂×Q₈×D₁₃₉

yes

—

132

C₂

—

278

—

133

C₂

C₄⋊Q₈×D₁₃₉

yes

characteristic

134

C₂

C₂×#17984863

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#498×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂²×D₁₃₉	C₂×D₄
hypercenter	#498	D₁₃₉
hypocenter	C₁₃₉	#3160
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#498×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

prime radical

C₂²×D₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #22581356	C₂	semidirect
3. #22581353	C₂	semidirect
4. #22581351	C₂	semidirect
7. #498×C₁₃₉	C₂	semidirect
28. C₈×D₁₃₉	Q₈	semidirect
67. C₁₃₉	#3160	semidirect
73. #498	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581356

C₂

semidirect

3. #22581353

C₂

semidirect

4. #22581351

C₂

semidirect

7. #498×C₁₃₉

C₂

semidirect

28. C₈×D₁₃₉

Q₈

semidirect

67. C₁₃₉

#3160

semidirect

73. #498

D₁₃₉

semidirect

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

subgroup	index
#22581508	2
#22581356	2
#22581353	2
#22581351	2
#22580988	2
C₄×C₈×D₁₃₉	2
#498×C₁₃₉	2
#3160	139

subgroup

index

C₄×C₈×D₁₃₉

#498×C₁₃₉

#3160

139

Subgroups — #31816913

Conjugacy classes of subgroups

Complements

Maximal subgroups