Group #31814966 — subgroups

Subgroups — #31814966

Order 17792.

484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 213 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×M₁₆×D₁₃₉	C₂	1	2	yes	maximal
2	C₂×#17984799	C₂	1	1	yes	maximal, characteristic
3	#22580978	C₂	1	8	yes	maximal, complemented
4	#22580977	C₂	1	1	yes	maximal, characteristic, complemented
5	#22580976	C₂	1	2	yes	maximal, complemented
6	#423×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₁₃₉	C₂²	1	1	yes	characteristic
8	C₂×#14555769	C₂²	1	2	yes	—
9	C₂×#14555768	C₂²	1	4	yes	—
10	M₁₆×D₁₃₉	—	2	8	no	—
11	C₂×C₈×D₁₃₉	C₂²	1	4	yes	—
12	#17984799	C₄	1	4	yes	—
13	#17984799	C₂²	1	4	yes	—
14	C₂×Dic₁₃₉⋊C₄	C₄	1	2	yes	—
15	C₂×(C₂×D₁₃₉)⋊C₄	C₄	1	2	yes	—
16	C₂×C₄×Dic₁₃₉	C₂²	1	1	yes	characteristic
17	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	4	yes	complemented
18	Dic₁₃₉⋊C₈	C₂²	1	8	yes	complemented
19	C₂×M₁₆×C₁₃₉	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₉	C₂²	1	4	yes	complemented
22	C₂³×D₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₃₉	C₂×C₄	1	2	yes	—
24	C₂²×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₁₃₉	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₁₃₉	D₄	1	4	yes	—
27	C₂×C₄×D₁₃₉	Q₈	1	4	yes	—
28	C₂×C₄×D₁₃₉	C₂³	1	2	yes	—
29	#14555769	—	2	4	no	—
30	#14555768	—	2	8	no	—
31	C₂×C₁₃₉⋊C₈	C₂³	1	4	yes	—
32	C₈×D₁₃₉	—	2	8	no	—
33	Dic₁₃₉⋊C₄	C₂×C₄	1	8	yes	—
34	(C₂×D₁₃₉)⋊C₄	C₂×C₄	1	8	yes	—
35	C₄×Dic₁₃₉	C₂×C₄	1	2	yes	—
36	C₄×Dic₁₃₉	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
39	M₁₆×C₁₃₉	—	2	4	no	—
40	C₂×C₈×C₁₃₉	C₂³	1	4	yes	—
41	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
42	C₄²×C₁₃₉	C₂³	1	2	yes	—
43	C₂²×D₁₃₉	—	4	2	no	—
44	C₂²×D₁₃₉	C₄⋊C₄	1	4	yes	—
45	C₂²×D₁₃₉	C₂²×C₄	1	2	yes	—
46	C₂×Dic₁₃₉	—	4	2	no	—
47	C₂×Dic₁₃₉	C₄⋊C₄	1	4	yes	—
48	C₂×Dic₁₃₉	C₂²×C₄	1	4	yes	—
49	C₂×Dic₁₃₉	C₂²×C₄	1	2	yes	—
50	C₄×D₁₃₉	C₄⋊C₄	1	8	yes	—
51	C₄×D₁₃₉	C₂×D₄	1	4	yes	—
52	C₄×D₁₃₉	C₂×Q₈	1	4	yes	—
53	C₁₃₉⋊C₈	—	2	4	no	—
54	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₉	—	4	2	no	—
56	C₂×C₄×C₁₃₉	C₂²×C₄	1	4	yes	—
57	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
59	C₂×C₄×C₁₃₉	C₂×Q₈	1	2	yes	—
60	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₃₉	—	2	4	no	—
62	C₂²×C₁₃₉	—	2	2	no	—
63	C₂²×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₁₃₉	—	2	4	no	—
66	C₂×D₁₃₉	—	4	4	no	—
67	C₂×D₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₁₃₉	—	2	4	no	—
69	C₄×C₁₃₉	C₂×C₄⋊C₄	1	2	yes	—
70	C₄×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₁₃₉	C₂²×Q₈	1	1	yes	characteristic
72	Dic₁₃₉	—	2	4	no	—
73	Dic₁₃₉	C₂×C₄⋊C₄	1	4	yes	—
74	C₂×C₁₃₉	—	2	2	no	—
75	C₂×C₁₃₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₁₃₉	#568	1	2	yes	—
77	D₁₃₉	—	2	4	no	—
78	C₁₃₉	#2905	1	1	yes	characteristic, complemented, Sylow
79	#2905	—	139	1	no	maximal, Sylow
80	C₂²×M₁₆	—	139	2	no	—
81	C₂×#117	—	139	1	no	—
82	#424	—	139	8	no	—
83	#423	—	139	2	no	—
84	#423	—	139	1	no	—
85	#423	D₁₃₉	1	1	yes	characteristic, complemented
86	C₂³×C₄	—	139	1	no	—
87	C₂×M₁₆	—	278	8	no	—
88	C₂×M₁₆	—	139	2	no	—
89	C₂×M₁₆	—	139	4	no	—
90	C₂×M₁₆	C₂×D₁₃₉	1	2	yes	—
91	C₂²×C₈	—	139	4	no	—
92	#117	—	139	4	no	—
93	#117	—	139	4	no	—
94	C₂×C₄⋊C₄	—	139	2	no	—
95	C₂×C₂²⋊C₄	—	139	2	no	—
96	C₂×C₄²	—	139	1	no	—
97	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
98	C₄⋊C₈	—	139	8	no	—
99	C₄⋊C₈	—	139	4	no	—
100	C₄⋊C₈	C₂×D₁₃₉	1	4	yes	complemented
101	C₂⁴	—	139	1	no	—
102	C₂²×C₄	—	139	2	no	—
103	C₂²×C₄	—	139	1	no	—
104	C₂²×C₄	—	139	2	no	—
105	C₂²×C₄	—	139	4	no	—
106	C₂²×C₄	—	139	4	no	—
107	C₂²×C₄	—	139	2	no	—
108	C₂²×C₄	C₄×D₁₃₉	1	2	yes	—
109	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	278	8	no	—
112	M₁₆	—	278	4	no	—
113	C₂×C₈	—	278	8	no	—
114	C₂×C₈	—	139	4	no	—
115	C₂×C₈	C₂²×D₁₃₉	1	4	yes	—
116	C₄⋊C₄	—	139	8	no	—
117	C₂²⋊C₄	—	139	8	no	—
118	C₄²	—	139	2	no	—
119	C₄²	—	139	2	no	—
120	C₄²	C₄×D₁₃₉	1	2	yes	—
121	C₄²	C₂²×D₁₃₉	1	2	yes	—
122	C₂³	—	556	2	no	—
123	C₂³	—	139	2	no	—
124	C₂³	—	139	4	no	—
125	C₂³	C₂×C₄×D₁₃₉	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	556	2	no	—
128	C₂×C₄	—	139	4	no	—
129	C₂×C₄	—	139	4	no	—
130	C₂×C₄	—	139	2	no	—
131	C₂×C₄	—	139	8	no	—
132	C₂×C₄	—	139	4	no	—
133	C₂×C₄	—	139	4	no	—
134	C₂×C₄	C₂×C₄×D₁₃₉	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
136	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
137	C₂×C₄	Q₈×D₁₃₉	1	2	yes	—
138	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	278	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	278	4	no	—
143	C₂²	—	556	4	no	—
144	C₂²	—	139	4	no	—
145	C₂²	C₄⋊C₄×D₁₃₉	1	2	yes	—
146	C₂²	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	278	4	no	—
149	C₄	—	139	4	no	—
150	C₄	C₄⋊C₄×D₁₃₉	1	2	yes	—
151	C₄	C₂×D₄×D₁₃₉	1	1	yes	characteristic
152	C₄	C₂×Q₈×D₁₃₉	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	278	4	no	—
155	C₂	C₂×C₄⋊C₄×D₁₃₉	1	1	yes	characteristic
156	C₂	#22581719	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₁₃₉

C₂

yes

maximal

C₂×#17984799

C₂

yes

maximal, characteristic

#22580978

C₂

yes

maximal, complemented

#22580977

C₂

yes

maximal, characteristic, complemented

#22580976

C₂

yes

maximal, complemented

#423×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₁₃₉

C₂²

yes

characteristic

C₂×#14555769

C₂²

yes

—

C₂×#14555768

C₂²

yes

—

M₁₆×D₁₃₉

—

C₂×C₈×D₁₃₉

C₂²

yes

—

#17984799

C₄

yes

—

#17984799

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

C₄

yes

—

C₂×(C₂×D₁₃₉)⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₃₉

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

complemented

Dic₁₃₉⋊C₈

C₂²

yes

complemented

C₂×M₁₆×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

complemented

C₂³×D₁₃₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₃₉

C₂×C₄

yes

—

C₂²×Dic₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₉

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₄×D₁₃₉

Q₈

yes

—

C₂×C₄×D₁₃₉

C₂³

yes

—

#14555769

—

#14555768

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₈×D₁₃₉

—

Dic₁₃₉⋊C₄

C₂×C₄

yes

—

(C₂×D₁₃₉)⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₃₉

C₂×C₄

yes

—

C₄×Dic₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

M₁₆×C₁₃₉

—

C₂×C₈×C₁₃₉

C₂³

yes

—

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₄⋊C₄

yes

—

C₂²×D₁₃₉

C₂²×C₄

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₄⋊C₄

yes

—

C₂×Dic₁₃₉

C₂²×C₄

yes

—

C₂×Dic₁₃₉

C₂²×C₄

yes

—

C₄×D₁₃₉

C₄⋊C₄

yes

—

C₄×D₁₃₉

C₂×D₄

yes

—

C₄×D₁₃₉

C₂×Q₈

yes

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

—

C₂×C₄×C₁₃₉

C₂⁴

yes

characteristic

C₈×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₃₉

C₂³×C₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₃₉

C₂²×D₄

yes

characteristic

C₄×C₁₃₉

C₂²×Q₈

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

C₂×C₄⋊C₄

yes

—

C₂×C₁₃₉

—

C₂×C₁₃₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₃₉

#568

yes

—

D₁₃₉

—

C₁₃₉

#2905

yes

characteristic, complemented, Sylow

#2905

—

139

maximal, Sylow

C₂²×M₁₆

—

139

—

C₂×#117

—

139

—

#424

—

139

—

#423

—

139

—

#423

—

139

—

#423

D₁₃₉

yes

characteristic, complemented

C₂³×C₄

—

139

—

C₂×M₁₆

—

278

—

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

C₂×M₁₆

C₂×D₁₃₉

yes

—

C₂²×C₈

—

139

—

#117

—

139

—

#117

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

139

—

C₄⋊C₈

—

139

—

100

C₄⋊C₈

C₂×D₁₃₉

yes

complemented

101

C₂⁴

—

139

—

102

C₂²×C₄

—

139

—

103

C₂²×C₄

—

139

—

104

C₂²×C₄

—

139

—

105

C₂²×C₄

—

139

—

106

C₂²×C₄

—

139

—

107

C₂²×C₄

—

139

—

108

C₂²×C₄

C₄×D₁₃₉

yes

—

109

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

110

M₁₆

—

111

M₁₆

—

278

—

112

M₁₆

—

278

—

113

C₂×C₈

—

278

—

114

C₂×C₈

—

139

—

115

C₂×C₈

C₂²×D₁₃₉

yes

—

116

C₄⋊C₄

—

139

—

117

C₂²⋊C₄

—

139

—

118

C₄²

—

139

—

119

C₄²

—

139

—

120

C₄²

C₄×D₁₃₉

yes

—

121

C₄²

C₂²×D₁₃₉

yes

—

122

C₂³

—

556

—

123

C₂³

—

139

—

124

C₂³

—

139

—

125

C₂³

C₂×C₄×D₁₃₉

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

556

—

128

C₂×C₄

—

139

—

129

C₂×C₄

—

139

—

130

C₂×C₄

—

139

—

131

C₂×C₄

—

139

—

132

C₂×C₄

—

139

—

133

C₂×C₄

—

139

—

134

C₂×C₄

C₂×C₄×D₁₃₉

yes

—

135

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

136

C₂×C₄

D₄×D₁₃₉

yes

—

137

C₂×C₄

Q₈×D₁₃₉

yes

—

138

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

139

C₈

—

140

C₈

—

278

—

141

C₂²

—

142

C₂²

—

278

—

143

C₂²

—

556

—

144

C₂²

—

139

—

145

C₂²

C₄⋊C₄×D₁₃₉

yes

—

146

C₂²

C₂²×C₄×D₁₃₉

yes

characteristic

147

C₄

—

148

C₄

—

278

—

149

C₄

—

139

—

150

C₄

C₄⋊C₄×D₁₃₉

yes

—

151

C₄

C₂×D₄×D₁₃₉

yes

characteristic

152

C₄

C₂×Q₈×D₁₃₉

yes

characteristic

153

C₂

—

154

C₂

—

278

—

155

C₂

C₂×C₄⋊C₄×D₁₃₉

yes

characteristic

156

C₂

#22581719

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₉
commutator, socle	C₂²×C₁₃₉	C₂³×C₄
Fitting	#423×C₁₃₉	C₂
prime radical	C₂³×D₁₃₉	C₂×C₄
cosocle	C₂×C₄×C₁₃₉	C₂⁴
hypercenter	#423	D₁₃₉
hypocenter	C₁₃₉	#2905
LCS	G ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₉

commutator, socle

C₂²×C₁₃₉

C₂³×C₄

Fitting

#423×C₁₃₉

C₂

prime radical

C₂³×D₁₃₉

C₂×C₄

cosocle

C₂×C₄×C₁₃₉

C₂⁴

D₁₃₉

C₁₃₉

G ▷ C₂²×C₁₃₉ ▷ C₁₃₉

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
3. #22580978	C₂	semidirect
4. #22580977	C₂	semidirect
5. #22580976	C₂	semidirect
6. #423×C₁₃₉	C₂	semidirect
17. C₄⋊(C₁₃₉⋊C₈)	C₂²	semidirect
18. Dic₁₃₉⋊C₈	C₂²	semidirect
21. C₄⋊C₈×C₁₃₉	C₂²	semidirect
78. C₁₃₉	#2905	semidirect
85. #423	D₁₃₉	semidirect
100. C₄⋊C₈	C₂×D₁₃₉	semidirect

H ≅ G/N

type

3. #22580978

C₂

semidirect

4. #22580977

C₂

semidirect

5. #22580976

C₂

semidirect

6. #423×C₁₃₉

C₂

semidirect

17. C₄⋊(C₁₃₉⋊C₈)

C₂²

semidirect

18. Dic₁₃₉⋊C₈

C₂²

semidirect

21. C₄⋊C₈×C₁₃₉

C₂²

semidirect

78. C₁₃₉

#2905

semidirect

85. #423

D₁₃₉

semidirect

100. C₄⋊C₈

C₂×D₁₃₉

semidirect

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

subgroup	index
C₂×M₁₆×D₁₃₉	2
C₂×#17984799	2
#22580978	2
#22580977	2
#22580976	2
#423×C₁₃₉	2
#2905	139

subgroup

index

C₂×M₁₆×D₁₃₉

#423×C₁₃₉

139

Subgroups — #31814966

Conjugacy classes of subgroups

Complements

Maximal subgroups