Group #31819042 — subgroups

Subgroups — #31819042

Order 17792.

370 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×D₅₅₆	C₂	1	1	yes	maximal, characteristic
2	#22581303	C₂	1	3	yes	maximal, complemented
3	#22581238	C₂	1	3	yes	maximal, complemented
4	#22581155	C₂	1	3	yes	maximal
5	Q₈×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555772	C₂	1	3	yes	maximal
7	#557×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₁₃₉	—	2	1	no	—
9	C₂×#14555772	C₂²	1	3	yes	—
10	#17984842	C₂²	1	3	yes	—
11	#17984822	—	2	3	no	—
12	C₄×D₅₅₆	C₂²	1	3	yes	—
13	#17984773	C₂²	1	3	yes	—
14	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	3	yes	—
15	#17984766	C₂²	1	3	yes	—
16	#17984766	C₂²	1	6	yes	—
17	#17984762	C₂²	1	1	yes	characteristic
18	C₄×C₁₃₉⋊C₈	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₃₉	C₂²	1	3	yes	—
20	#125×C₁₃₉	—	2	3	no	—
21	C₄×Q₈×C₁₃₉	—	2	1	no	—
22	C₄×Q₈×C₁₃₉	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₇₈	—	2	3	no	—
25	Q₈×D₁₃₉	—	4	4	no	—
26	C₂×D₅₅₆	C₂³	1	1	yes	characteristic
27	C₂×C₄×D₁₃₉	—	2	3	no	—
28	#14555772	—	2	6	no	—
29	C₂×C₁₃₉⋊C₈	C₂³	1	3	yes	—
30	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₁₃₉	—	2	3	no	—
32	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
33	(C₂×D₁₃₉)⋊C₄	—	2	3	no	—
34	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₃₉	C₂³	1	3	yes	—
36	C₄⋊C₄×C₁₃₉	—	2	3	no	—
37	C₄⋊C₄×C₁₃₉	—	2	3	no	—
38	C₄⋊C₄×C₁₃₉	—	2	3	no	—
39	C₄⋊C₄×C₁₃₉	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
41	C₄²×C₁₃₉	—	2	3	no	—
42	C₄²×C₁₃₉	C₂³	1	3	yes	—
43	C₂²×D₁₃₉	—	2	1	no	—
44	Dic₂₇₈	—	4	6	no	—
45	D₅₅₆	—	2	1	no	—
46	D₅₅₆	#41	1	2	yes	—
47	C₂×Dic₁₃₉	—	2	3	no	—
48	C₄×D₁₃₉	—	4	6	no	—
49	C₁₃₉⋊C₈	—	2	3	no	—
50	C₁₃₉⋊C₈	—	2	1	no	—
51	Q₈×C₁₃₉	—	2	3	no	—
52	Q₈×C₁₃₉	—	2	3	no	—
53	Q₈×C₁₃₉	SD₁₆	1	4	yes	complemented
54	C₂×C₄×C₁₃₉	—	2	1	no	—
55	C₂×C₄×C₁₃₉	—	2	3	no	—
56	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
58	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
60	C₂×D₁₃₉	—	2	2	no	—
61	C₂×D₁₃₉	—	4	1	no	—
62	C₂×D₁₃₉	—	4	1	no	—
63	C₄×C₁₃₉	—	2	3	no	—
64	C₄×C₁₃₉	—	4	3	no	—
65	C₄×C₁₃₉	—	4	1	no	—
66	C₄×C₁₃₉	C₂×SD₁₆	1	6	yes	—
67	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
68	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₁₃₉	—	4	3	no	—
70	C₂×C₁₃₉	#548	1	1	yes	characteristic
71	C₂×C₁₃₉	C₂²×SD₁₆	1	1	yes	characteristic
72	C₂×C₁₃₉	#578	1	1	yes	characteristic
73	D₁₃₉	—	4	2	no	—
74	C₁₃₉	#3359	1	1	yes	characteristic, complemented, Sylow
75	#3359	—	139	1	no	maximal, Sylow
76	#557	D₁₃₉	1	1	yes	characteristic, complemented
77	D₄×Q₈	—	139	1	no	—
78	#487	—	139	3	no	—
79	#475	—	139	3	no	—
80	#461	—	139	3	no	—
81	C₈×Q₈	—	139	1	no	—
82	C₄×SD₁₆	—	139	3	no	—
83	C₂²×Q₈	—	278	1	no	—
84	C₂×SD₁₆	—	139	3	no	—
85	C₄⋊Q₈	—	139	3	no	—
86	C₄⋊Q₈	C₂×D₁₃₉	1	3	yes	—
87	#125	—	2	3	no	—
88	C₂²⋊Q₈	—	278	3	no	—
89	C₄×Q₈	—	2	1	no	—
90	C₄×Q₈	C₂×D₁₃₉	1	3	yes	—
91	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
92	C₄×D₄	—	139	3	no	—
93	#106	—	139	3	no	—
94	C₄⋊C₈	—	139	3	no	—
95	#103	—	139	3	no	—
96	#103	—	139	6	no	—
97	#102	—	139	1	no	—
98	C₄×C₈	—	139	3	no	—
99	C₂×Q₈	—	278	3	no	—
100	C₂×Q₈	—	556	4	no	—
101	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
102	C₂×Q₈	C₂²×D₁₃₉	1	3	yes	—
103	C₂×D₄	—	139	1	no	—
104	C₂²×C₄	—	278	3	no	—
105	SD₁₆	—	278	6	no	—
106	C₂×C₈	—	139	1	no	—
107	C₂×C₈	—	139	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	2	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	278	3	no	—
112	C₄⋊C₄	—	139	3	no	—
113	C₄⋊C₄	C₂²⋊D₁₃₉	1	3	yes	—
114	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
115	C₂²⋊C₄	—	278	3	no	—
116	C₄²	—	2	3	no	—
117	C₄²	C₂²×D₁₃₉	1	3	yes	—
118	C₂³	—	278	1	no	—
119	Q₈	—	2	3	no	—
120	Q₈	—	2	3	no	—
121	Q₈	—	556	6	no	—
122	Q₈	#14555772	1	4	yes	complemented
123	D₄	—	278	1	no	—
124	D₄	—	139	2	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	278	3	no	—
128	C₂×C₄	—	556	6	no	—
129	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
131	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
132	C₈	—	278	3	no	—
133	C₈	—	278	1	no	—
134	C₂²	—	278	2	no	—
135	C₂²	—	556	1	no	—
136	C₂²	—	556	1	no	—
137	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	1	no	—
140	C₄	—	4	3	no	—
141	C₄	—	556	3	no	—
142	C₄	C₂×#14555772	1	6	yes	—
143	C₄	#41×D₁₃₉	1	1	yes	characteristic
144	C₄	#17984889	1	1	yes	characteristic
145	C₂	—	556	2	no	—
146	C₂	#22581610	1	1	yes	characteristic
147	C₂	C₂²×#14555772	1	1	yes	characteristic
148	C₂	#22581768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₅₅₆

C₂

yes

maximal, characteristic

#22581303

C₂

yes

maximal, complemented

#22581238

C₂

yes

maximal, complemented

#22581155

C₂

yes

maximal

Q₈×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#14555772

C₂

yes

maximal

#557×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₃₉

—

C₂×#14555772

C₂²

yes

—

#17984842

C₂²

yes

—

#17984822

—

C₄×D₅₅₆

C₂²

yes

—

#17984773

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984766

C₂²

yes

—

#17984766

C₂²

yes

—

#17984762

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

—

#125×C₁₃₉

—

C₄×Q₈×C₁₃₉

—

C₄×Q₈×C₁₃₉

C₂²

yes

—

C₄×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

Q₈×D₁₃₉

—

C₂×D₅₅₆

C₂³

yes

characteristic

C₂×C₄×D₁₃₉

—

#14555772

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₂×C₁₃₉⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

(C₂×D₁₃₉)⋊C₄

—

C₂×Q₈×C₁₃₉

D₄

yes

characteristic

C₂×Q₈×C₁₃₉

C₂³

yes

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

characteristic

C₄²×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

D₅₅₆

—

D₅₅₆

#41

yes

—

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

SD₁₆

yes

complemented

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×SD₁₆

yes

—

C₄×C₁₃₉

C₂×#41

yes

characteristic

C₄×C₁₃₉

2₋¹⁺⁴

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

#548

yes

characteristic

C₂×C₁₃₉

C₂²×SD₁₆

yes

characteristic

C₂×C₁₃₉

#578

yes

characteristic

D₁₃₉

—

C₁₃₉

#3359

yes

characteristic, complemented, Sylow

#3359

—

139

maximal, Sylow

#557

D₁₃₉

yes

characteristic, complemented

D₄×Q₈

—

139

—

#487

—

139

—

#475

—

139

—

#461

—

139

—

C₈×Q₈

—

139

—

C₄×SD₁₆

—

139

—

C₂²×Q₈

—

278

—

C₂×SD₁₆

—

139

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#125

—

C₂²⋊Q₈

—

278

—

C₄×Q₈

—

C₄×Q₈

C₂×D₁₃₉

yes

—

C₄×Q₈

C₂×D₁₃₉

yes

characteristic

C₄×D₄

—

139

—

#106

—

139

—

C₄⋊C₈

—

139

—

#103

—

139

—

#103

—

139

—

#102

—

139

—

C₄×C₈

—

139

—

C₂×Q₈

—

278

—

100

C₂×Q₈

—

556

—

101

C₂×Q₈

C₂²⋊D₁₃₉

yes

characteristic

102

C₂×Q₈

C₂²×D₁₃₉

yes

—

103

C₂×D₄

—

139

—

104

C₂²×C₄

—

278

—

105

SD₁₆

—

278

—

106

C₂×C₈

—

139

—

107

C₂×C₈

—

139

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

278

—

112

C₄⋊C₄

—

139

—

113

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

114

C₄⋊C₄

C₂²×D₁₃₉

yes

characteristic

115

C₂²⋊C₄

—

278

—

116

C₄²

—

117

C₄²

C₂²×D₁₃₉

yes

—

118

C₂³

—

278

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

556

—

122

Q₈

#14555772

yes

complemented

123

D₄

—

278

—

124

D₄

—

139

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

278

—

128

C₂×C₄

—

556

—

129

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

131

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

132

C₈

—

278

—

133

C₈

—

278

—

134

C₂²

—

278

—

135

C₂²

—

556

—

136

C₂²

—

556

—

137

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

556

—

142

C₄

C₂×#14555772

yes

—

143

C₄

#41×D₁₃₉

yes

characteristic

144

C₄

#17984889

yes

characteristic

145

C₂

—

556

—

146

C₂

#22581610

yes

characteristic

147

C₂

C₂²×#14555772

yes

characteristic

148

C₂

#22581768

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#557×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
prime radical	C₂×D₅₅₆	C₂³
hypercenter	#557	D₁₃₉
hypocenter	C₁₃₉	#3359
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#557×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

prime radical

C₂×D₅₅₆

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
2. #22581303	C₂	semidirect
3. #22581238	C₂	semidirect
5. Q₈×C₁₃₉⋊C₈	C₂	semidirect
7. #557×C₁₃₉	C₂	semidirect
53. Q₈×C₁₃₉	SD₁₆	semidirect
74. C₁₃₉	#3359	semidirect
76. #557	D₁₃₉	semidirect
122. Q₈	#14555772	semidirect

H ≅ G/N

type

2. #22581303

C₂

semidirect

3. #22581238

C₂

semidirect

5. Q₈×C₁₃₉⋊C₈

C₂

semidirect

7. #557×C₁₃₉

C₂

semidirect

53. Q₈×C₁₃₉

SD₁₆

semidirect

74. C₁₃₉

#3359

semidirect

76. #557

D₁₃₉

semidirect

122. Q₈

#14555772

semidirect

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

subgroup	index
Q₈×D₅₅₆	2
#22581303	2
#22581238	2
#22581155	2
Q₈×C₁₃₉⋊C₈	2
C₄×#14555772	2
#557×C₁₃₉	2
#3359	139

subgroup

index

Q₈×D₅₅₆

Q₈×C₁₃₉⋊C₈

#557×C₁₃₉

139

Subgroups — #31819042

Conjugacy classes of subgroups

Complements

Maximal subgroups