Group #31816780 — subgroups

Subgroups — #31816780

Order 17792.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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×#17984839	C₂	1	1	yes	maximal, characteristic
2	#22581319	C₂	1	4	yes	maximal, complemented
3	#22581301	C₂	1	4	yes	maximal, complemented
4	#22580957	C₂	1	4	yes	maximal, complemented
5	C₄×#14555769	C₂	1	1	yes	maximal, characteristic, complemented
6	#533×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₅₅₆	—	2	2	no	—
8	C₂²×D₅₅₆	C₂²	1	2	yes	—
9	C₂×#14555769	C₂²	1	2	yes	—
10	#17984839	—	2	2	no	—
11	#17984839	C₂²	1	2	yes	—
12	#17984839	C₂²	1	2	yes	—
13	#17984762	C₂²	1	16	yes	complemented
14	#17984753	C₂²	1	2	yes	complemented
15	C₄×C₁₃₉⋊C₈	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₁₃₉	C₂²	1	2	yes	complemented
17	#125×C₁₃₉	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₁₃₉	—	2	2	no	—
19	C₄×Q₈×C₁₃₉	—	2	2	no	—
20	#117×C₁₃₉	C₂²	1	2	yes	—
21	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
22	C₂³×D₁₃₉	—	2	2	no	—
23	C₂×D₅₅₆	—	2	2	no	—
24	C₂×D₅₅₆	—	2	4	no	—
25	C₂×D₅₅₆	—	2	4	no	—
26	C₂×D₅₅₆	—	2	8	no	—
27	C₂×D₅₅₆	—	4	8	no	—
28	C₂×D₅₅₆	C₂³	1	4	yes	—
29	#14555769	—	2	4	no	—
30	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
31	C₂×Q₈×C₁₃₉	—	2	2	no	—
32	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
33	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₃₉	—	2	4	no	—
35	C₄⋊C₄×C₁₃₉	—	2	2	no	—
36	C₄⋊C₄×C₁₃₉	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₁₃₉	—	2	2	no	—
38	C₄²×C₁₃₉	—	2	2	no	—
39	C₄²×C₁₃₉	D₄	1	2	yes	—
40	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
41	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
42	C₂²×D₁₃₉	—	2	4	no	—
43	C₂²×D₁₃₉	—	4	8	no	—
44	C₂²×D₁₃₉	—	4	4	no	—
45	D₅₅₆	—	2	8	no	—
46	D₅₅₆	—	4	8	no	—
47	D₅₅₆	—	4	4	no	—
48	D₅₅₆	—	4	8	no	—
49	C₁₃₉⋊C₈	—	2	4	no	—
50	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
51	Q₈×C₁₃₉	—	4	4	no	—
52	C₂×C₄×C₁₃₉	—	2	4	no	—
53	C₂×C₄×C₁₃₉	—	2	4	no	—
54	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₃₉	#41	1	4	yes	—
58	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₃₉	—	2	2	no	—
60	C₂²×C₁₃₉	#124	1	2	yes	—
61	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₁₃₉	—	4	4	no	—
63	C₂×D₁₃₉	—	4	8	no	—
64	C₂×D₁₃₉	—	8	8	no	—
65	C₄×C₁₃₉	—	2	2	no	—
66	C₄×C₁₃₉	—	4	4	no	—
67	C₄×C₁₃₉	#124	1	2	yes	—
68	C₄×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₁₃₉	C₂×#41	1	2	yes	—
70	C₂×C₁₃₉	—	2	2	no	—
71	C₂×C₁₃₉	C₂×#124	1	1	yes	characteristic
72	C₂×C₁₃₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
73	D₁₃₉	—	8	4	no	—
74	C₁₃₉	#3138	1	1	yes	characteristic, complemented, Sylow
75	#3138	—	139	1	no	maximal, Sylow
76	#533	D₁₃₉	1	1	yes	characteristic, complemented
77	C₂×#127	—	139	1	no	—
78	#490	—	139	4	no	—
79	#486	—	139	4	no	—
80	#418	—	139	4	no	—
81	C₄×M₁₆	—	139	1	no	—
82	C₂²×D₄	—	278	2	no	—
83	C₂²×D₄	—	139	2	no	—
84	C₂×M₁₆	—	139	2	no	—
85	C₄⋊Q₈	C₂×D₁₃₉	1	2	yes	complemented
86	#127	—	278	2	no	—
87	#127	—	139	2	no	—
88	#127	—	139	2	no	—
89	#125	C₂×D₁₃₉	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₁₃₉	1	2	yes	—
93	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
94	#102	—	139	16	no	—
95	#97	—	139	2	no	—
96	C₄×C₈	—	139	2	no	—
97	C₂⁴	—	278	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	278	2	no	—
100	C₂×D₄	—	278	4	no	—
101	C₂×D₄	—	278	4	no	—
102	C₂×D₄	—	278	8	no	—
103	C₂×D₄	—	556	8	no	—
104	C₂×D₄	—	139	4	no	—
105	C₂²×C₄	C₂²⋊D₁₃₉	1	2	yes	—
106	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
107	M₁₆	—	278	4	no	—
108	C₂×C₈	—	139	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₁₃₉	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
115	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
116	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
117	C₂³	—	278	4	no	—
118	C₂³	—	556	8	no	—
119	C₂³	—	556	4	no	—
120	C₂³	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	278	8	no	—
123	D₄	—	556	8	no	—
124	D₄	—	556	4	no	—
125	D₄	—	556	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	2	yes	—
131	C₂×C₄	#14555786	1	4	yes	—
132	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
133	C₈	—	278	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	556	4	no	—
136	C₂²	—	556	8	no	—
137	C₂²	—	1112	8	no	—
138	C₂²	#17984831	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#17984831	1	2	yes	—
143	C₄	#17984867	1	4	yes	—
144	C₄	C₂×#14555786	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	1112	4	no	—
147	C₂	C₂×#17984831	1	1	yes	characteristic
148	C₂	C₂×#17984867	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984839

C₂

yes

maximal, characteristic

#22581319

C₂

yes

maximal, complemented

#22581301

C₂

yes

maximal, complemented

#22580957

C₂

yes

maximal, complemented

C₄×#14555769

C₂

yes

maximal, characteristic, complemented

#533×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₅₆

—

C₂²×D₅₅₆

C₂²

yes

—

C₂×#14555769

C₂²

yes

—

#17984839

—

#17984839

C₂²

yes

—

#17984839

C₂²

yes

—

#17984762

C₂²

yes

complemented

#17984753

C₂²

yes

complemented

C₄×C₁₃₉⋊C₈

C₂²

yes

complemented

C₄⋊Q₈×C₁₃₉

C₂²

yes

complemented

#125×C₁₃₉

C₂²

yes

complemented

C₂²⋊Q₈×C₁₃₉

—

C₄×Q₈×C₁₃₉

—

#117×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂³×D₁₃₉

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂³

yes

—

#14555769

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₂×Q₈×C₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

C₂³

yes

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₂²×C₁₃₉

—

C₂²×C₁₃₉

#124

yes

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#124

yes

—

C₄×C₁₃₉

AGL(1,ℤ/8)

yes

—

C₄×C₁₃₉

C₂×#41

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂×#124

yes

characteristic

C₂×C₁₃₉

C₂×AGL(1,ℤ/8)

yes

—

D₁₃₉

—

C₁₃₉

#3138

yes

characteristic, complemented, Sylow

#3138

—

139

maximal, Sylow

#533

D₁₃₉

yes

characteristic, complemented

C₂×#127

—

139

—

#490

—

139

—

#486

—

139

—

#418

—

139

—

C₄×M₁₆

—

139

—

C₂²×D₄

—

278

—

C₂²×D₄

—

139

—

C₂×M₁₆

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

complemented

#127

—

278

—

#127

—

139

—

#127

—

139

—

#125

C₂×D₁₃₉

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#102

—

139

—

#97

—

139

—

C₄×C₈

—

139

—

C₂⁴

—

278

—

C₂×Q₈

—

C₂×D₄

—

278

—

100

C₂×D₄

—

278

—

101

C₂×D₄

—

278

—

102

C₂×D₄

—

278

—

103

C₂×D₄

—

556

—

104

C₂×D₄

—

139

—

105

C₂²×C₄

C₂²⋊D₁₃₉

yes

—

106

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

107

M₁₆

—

278

—

108

C₂×C₈

—

139

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₁₃₉

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₁₃₉

yes

—

115

C₄²

C₂²×D₁₃₉

yes

characteristic

116

C₄²

C₂²×D₁₃₉

yes

characteristic

117

C₂³

—

278

—

118

C₂³

—

556

—

119

C₂³

—

556

—

120

C₂³

C₂×C₂²⋊D₁₃₉

yes

characteristic

121

Q₈

—

122

D₄

—

278

—

123

D₄

—

556

—

124

D₄

—

556

—

125

D₄

—

556

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

129

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

131

C₂×C₄

#14555786

yes

—

132

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

133

C₈

—

278

—

134

C₂²

—

135

C₂²

—

556

—

136

C₂²

—

556

—

137

C₂²

—

1112

—

138

C₂²

#17984831

yes

—

139

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#17984831

yes

—

143

C₄

#17984867

yes

—

144

C₄

C₂×#14555786

yes

—

145

C₂

—

146

C₂

—

1112

—

147

C₂

C₂×#17984831

yes

characteristic

148

C₂

C₂×#17984867

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#533×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×#17984839	C₂
hypercenter	#533	D₁₃₉
hypocenter	C₁₃₉	#3138
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#533×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

C₂

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #22581319	C₂	semidirect
3. #22581301	C₂	semidirect
4. #22580957	C₂	semidirect
5. C₄×#14555769	C₂	semidirect
6. #533×C₁₃₉	C₂	semidirect
13. #17984762	C₂²	semidirect
14. #17984753	C₂²	semidirect
15. C₄×C₁₃₉⋊C₈	C₂²	semidirect
16. C₄⋊Q₈×C₁₃₉	C₂²	semidirect
17. #125×C₁₃₉	C₂²	semidirect
74. C₁₃₉	#3138	semidirect
76. #533	D₁₃₉	semidirect
85. C₄⋊Q₈	C₂×D₁₃₉	semidirect
89. #125	C₂×D₁₃₉	semidirect

H ≅ G/N

type

2. #22581319

C₂

semidirect

3. #22581301

C₂

semidirect

4. #22580957

C₂

semidirect

5. C₄×#14555769

C₂

semidirect

6. #533×C₁₃₉

C₂

semidirect

13. #17984762

C₂²

semidirect

14. #17984753

C₂²

semidirect

15. C₄×C₁₃₉⋊C₈

C₂²

semidirect

16. C₄⋊Q₈×C₁₃₉

C₂²

semidirect

17. #125×C₁₃₉

C₂²

semidirect

74. C₁₃₉

#3138

semidirect

76. #533

D₁₃₉

semidirect

85. C₄⋊Q₈

C₂×D₁₃₉

semidirect

89. #125

C₂×D₁₃₉

semidirect

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

subgroup	index
C₂×#17984839	2
#22581319	2
#22581301	2
#22580957	2
C₄×#14555769	2
#533×C₁₃₉	2
#3138	139

subgroup

index

#533×C₁₃₉

139

Subgroups — #31816780

Conjugacy classes of subgroups

Complements

Maximal subgroups