Group #31819035 — subgroups

Subgroups — #31819035

Order 17792.

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

normal subgroups only

This group has 426 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, complemented
2	#22581299	C₂	1	3	yes	maximal, complemented
3	#22581233	C₂	1	3	yes	maximal, complemented
4	#22581145	C₂	1	3	yes	maximal, complemented
5	Q₈×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14555770	C₂	1	3	yes	maximal, complemented
7	#550×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₁₃₉	—	2	1	no	—
9	C₂×#14555770	C₂²	1	3	yes	—
10	#17984842	C₂²	1	3	yes	—
11	#17984822	—	2	3	no	—
12	C₄×D₅₅₆	C₂²	1	3	yes	—
13	#17984775	C₂²	1	3	yes	—
14	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	3	yes	—
15	#17984763	C₂²	1	3	yes	—
16	#17984763	C₂²	1	6	yes	—
17	#17984762	C₂²	1	1	yes	characteristic
18	C₄×C₁₃₉⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₁₃₉	—	2	1	no	—
20	#127×C₁₃₉	C₂²	1	3	yes	—
21	#121×C₁₃₉	—	2	3	no	—
22	C₄×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₃₉	C₂²	1	3	yes	—
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	#14555770	—	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	#41×C₁₃₉	—	4	4	no	—
35	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₃₉	—	2	3	no	—
37	C₂×D₄×C₁₃₉	—	2	3	no	—
38	C₂×D₄×C₁₃₉	C₂³	1	3	yes	—
39	C₂²×C₄×C₁₃₉	—	2	3	no	—
40	C₄⋊C₄×C₁₃₉	D₄	1	3	yes	—
41	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
42	C₂²⋊C₄×C₁₃₉	—	2	3	no	—
43	C₄²×C₁₃₉	C₂³	1	3	yes	—
44	C₂²×D₁₃₉	—	2	1	no	—
45	Dic₂₇₈	—	4	6	no	—
46	D₅₅₆	—	2	1	no	—
47	D₅₅₆	#41	1	2	yes	complemented
48	C₂×Dic₁₃₉	—	2	3	no	—
49	C₄×D₁₃₉	—	4	6	no	—
50	C₁₃₉⋊C₈	—	2	3	no	—
51	C₁₃₉⋊C₈	—	2	1	no	—
52	C₂³×C₁₃₉	—	2	3	no	—
53	Q₈×C₁₃₉	D₈	1	4	yes	complemented
54	D₄×C₁₃₉	—	2	3	no	—
55	D₄×C₁₃₉	—	2	3	no	—
56	D₄×C₁₃₉	—	4	6	no	—
57	D₄×C₁₃₉	—	8	3	no	—
58	C₂×C₄×C₁₃₉	—	2	1	no	—
59	C₂×C₄×C₁₃₉	—	4	6	no	—
60	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
61	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
62	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
63	C₂²×C₁₃₉	—	4	3	no	—
64	C₂²×C₁₃₉	—	4	3	no	—
65	C₂²×C₁₃₉	—	4	3	no	—
66	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
67	C₂×D₁₃₉	—	2	2	no	—
68	C₂×D₁₃₉	—	4	1	no	—
69	C₂×D₁₃₉	—	4	1	no	—
70	C₄×C₁₃₉	—	2	3	no	—
71	C₄×C₁₃₉	—	4	1	no	—
72	C₄×C₁₃₉	C₂×D₈	1	6	yes	—
73	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
74	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
75	Dic₁₃₉	—	4	3	no	—
76	C₂×C₁₃₉	—	8	3	no	—
77	C₂×C₁₃₉	#548	1	1	yes	characteristic
78	C₂×C₁₃₉	C₂²×D₈	1	1	yes	characteristic
79	C₂×C₁₃₉	#577	1	1	yes	characteristic
80	D₁₃₉	—	4	2	no	—
81	C₁₃₉	#3358	1	1	yes	characteristic, complemented, Sylow
82	#3358	—	139	1	no	maximal, Sylow
83	#550	D₁₃₉	1	1	yes	characteristic, complemented
84	D₄×Q₈	—	139	1	no	—
85	#486	—	139	3	no	—
86	#474	—	139	3	no	—
87	#459	—	139	3	no	—
88	C₈×Q₈	—	139	1	no	—
89	C₄×D₈	—	139	3	no	—
90	C₂×#41	—	2	1	no	—
91	C₂²×Q₈	—	278	1	no	—
92	C₂×D₈	—	139	3	no	—
93	C₄⋊Q₈	—	139	3	no	—
94	#127	C₂×D₁₃₉	1	3	yes	—
95	C₂²⋊Q₈	—	278	3	no	—
96	#121	—	2	3	no	—
97	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
98	C₄×D₄	—	139	3	no	—
99	C₄×D₄	C₂×D₁₃₉	1	3	yes	—
100	#107	—	139	3	no	—
101	C₄⋊C₈	—	139	3	no	—
102	#102	—	139	3	no	—
103	#102	—	139	6	no	—
104	#102	—	139	1	no	—
105	C₄×C₈	—	139	3	no	—
106	#41	—	4	4	no	—
107	C₂×Q₈	—	278	3	no	—
108	C₂×Q₈	—	556	4	no	—
109	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	2	3	no	—
112	C₂×D₄	—	139	1	no	—
113	C₂×D₄	C₂²×D₁₃₉	1	3	yes	—
114	C₂²×C₄	—	2	3	no	—
115	C₂²×C₄	—	278	3	no	—
116	D₈	—	278	6	no	—
117	C₂×C₈	—	139	1	no	—
118	C₂×C₈	—	139	3	no	—
119	C₄⋊C₄	—	278	3	no	—
120	C₄⋊C₄	—	139	3	no	—
121	C₄⋊C₄	C₂²⋊D₁₃₉	1	3	yes	—
122	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
123	C₂²⋊C₄	—	2	3	no	—
124	C₂²⋊C₄	—	278	3	no	—
125	C₄²	C₂²×D₁₃₉	1	3	yes	—
126	C₂³	—	2	3	no	—
127	C₂³	—	278	1	no	—
128	Q₈	—	556	6	no	—
129	Q₈	#14555770	1	4	yes	complemented
130	D₄	—	2	3	no	—
131	D₄	—	2	3	no	—
132	D₄	—	4	6	no	—
133	D₄	—	8	3	no	—
134	D₄	—	278	1	no	—
135	D₄	—	139	2	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	—	278	3	no	—
139	C₂×C₄	—	556	6	no	—
140	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
141	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
142	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
143	C₈	—	278	3	no	—
144	C₈	—	278	1	no	—
145	C₂²	—	4	3	no	—
146	C₂²	—	4	3	no	—
147	C₂²	—	4	3	no	—
148	C₂²	—	278	2	no	—
149	C₂²	—	556	1	no	—
150	C₂²	—	556	1	no	—
151	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
152	C₄	—	2	3	no	—
153	C₄	—	4	1	no	—
154	C₄	—	556	3	no	—
155	C₄	C₂×#14555770	1	6	yes	—
156	C₄	#41×D₁₃₉	1	1	yes	characteristic
157	C₄	#17984889	1	1	yes	characteristic
158	C₂	—	8	3	no	—
159	C₂	—	556	2	no	—
160	C₂	#22581610	1	1	yes	characteristic
161	C₂	C₂²×#14555770	1	1	yes	characteristic
162	C₂	#22581764	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₅₅₆

C₂

yes

maximal, characteristic, complemented

#22581299

C₂

yes

maximal, complemented

#22581233

C₂

yes

maximal, complemented

#22581145

C₂

yes

maximal, complemented

Q₈×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#14555770

C₂

yes

maximal, complemented

#550×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₃₉

—

C₂×#14555770

C₂²

yes

—

#17984842

C₂²

yes

—

#17984822

—

C₄×D₅₅₆

C₂²

yes

—

#17984775

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984763

C₂²

yes

—

#17984763

C₂²

yes

—

#17984762

C₂²

yes

characteristic

C₄×C₁₃₉⋊C₈

C₂²

yes

—

C₂×#41×C₁₃₉

—

#127×C₁₃₉

C₂²

yes

—

#121×C₁₃₉

—

C₄×Q₈×C₁₃₉

C₂²

yes

characteristic

C₄×D₄×C₁₃₉

C₂²

yes

—

C₂×Dic₂₇₈

—

Q₈×D₁₃₉

—

C₂×D₅₅₆

C₂³

yes

characteristic

C₂×C₄×D₁₃₉

—

#14555770

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₂×C₁₃₉⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

(C₂×D₁₃₉)⋊C₄

—

#41×C₁₃₉

—

C₂×Q₈×C₁₃₉

D₄

yes

characteristic

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

C₂³

yes

—

C₂²×C₄×C₁₃₉

—

C₄⋊C₄×C₁₃₉

D₄

yes

—

C₄⋊C₄×C₁₃₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

—

C₂²×D₁₃₉

—

Dic₂₇₈

—

D₅₅₆

—

D₅₅₆

#41

yes

complemented

C₂×Dic₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

Q₈×C₁₃₉

D₈

yes

complemented

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

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×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×D₈

yes

—

C₄×C₁₃₉

C₂×#41

yes

characteristic

C₄×C₁₃₉

2₋¹⁺⁴

yes

characteristic

Dic₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#548

yes

characteristic

C₂×C₁₃₉

C₂²×D₈

yes

characteristic

C₂×C₁₃₉

#577

yes

characteristic

D₁₃₉

—

C₁₃₉

#3358

yes

characteristic, complemented, Sylow

#3358

—

139

maximal, Sylow

#550

D₁₃₉

yes

characteristic, complemented

D₄×Q₈

—

139

—

#486

—

139

—

#474

—

139

—

#459

—

139

—

C₈×Q₈

—

139

—

C₄×D₈

—

139

—

C₂×#41

—

C₂²×Q₈

—

278

—

C₂×D₈

—

139

—

C₄⋊Q₈

—

139

—

#127

C₂×D₁₃₉

yes

—

C₂²⋊Q₈

—

278

—

#121

—

C₄×Q₈

C₂×D₁₃₉

yes

characteristic

C₄×D₄

—

139

—

C₄×D₄

C₂×D₁₃₉

yes

—

100

#107

—

139

—

101

C₄⋊C₈

—

139

—

102

#102

—

139

—

103

#102

—

139

—

104

#102

—

139

—

105

C₄×C₈

—

139

—

106

#41

—

107

C₂×Q₈

—

278

—

108

C₂×Q₈

—

556

—

109

C₂×Q₈

C₂²⋊D₁₃₉

yes

characteristic

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

139

—

113

C₂×D₄

C₂²×D₁₃₉

yes

—

114

C₂²×C₄

—

115

C₂²×C₄

—

278

—

116

D₈

—

278

—

117

C₂×C₈

—

139

—

118

C₂×C₈

—

139

—

119

C₄⋊C₄

—

278

—

120

C₄⋊C₄

—

139

—

121

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

122

C₄⋊C₄

C₂²×D₁₃₉

yes

characteristic

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

278

—

125

C₄²

C₂²×D₁₃₉

yes

—

126

C₂³

—

127

C₂³

—

278

—

128

Q₈

—

556

—

129

Q₈

#14555770

yes

complemented

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

278

—

135

D₄

—

139

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

278

—

139

C₂×C₄

—

556

—

140

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

141

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

142

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

143

C₈

—

278

—

144

C₈

—

278

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

278

—

149

C₂²

—

556

—

150

C₂²

—

556

—

151

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

556

—

155

C₄

C₂×#14555770

yes

—

156

C₄

#41×D₁₃₉

yes

characteristic

157

C₄

#17984889

yes

characteristic

158

C₂

—

159

C₂

—

556

—

160

C₂

#22581610

yes

characteristic

161

C₂

C₂²×#14555770

yes

characteristic

162

C₂

#22581764

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃₉
commutator, cosocle	C₂×C₄×C₁₃₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃₉
Fitting	#550×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#550	D₁₃₉
hypocenter	C₁₃₉	#3358
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₃₉

commutator, cosocle

C₂×C₄×C₁₃₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₃₉

Fitting

#550×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂²×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×D₅₅₆	C₂	semidirect
2. #22581299	C₂	semidirect
3. #22581233	C₂	semidirect
4. #22581145	C₂	semidirect
5. Q₈×C₁₃₉⋊C₈	C₂	semidirect
6. C₄×#14555770	C₂	semidirect
7. #550×C₁₃₉	C₂	semidirect
47. D₅₅₆	#41	semidirect
53. Q₈×C₁₃₉	D₈	semidirect
81. C₁₃₉	#3358	semidirect
83. #550	D₁₃₉	semidirect
129. Q₈	#14555770	semidirect

H ≅ G/N

type

1. Q₈×D₅₅₆

C₂

semidirect

2. #22581299

C₂

semidirect

3. #22581233

C₂

semidirect

4. #22581145

C₂

semidirect

5. Q₈×C₁₃₉⋊C₈

C₂

semidirect

6. C₄×#14555770

C₂

semidirect

7. #550×C₁₃₉

C₂

semidirect

47. D₅₅₆

#41

semidirect

53. Q₈×C₁₃₉

D₈

semidirect

81. C₁₃₉

#3358

semidirect

83. #550

D₁₃₉

semidirect

129. Q₈

#14555770

semidirect

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

subgroup	index
Q₈×D₅₅₆	2
#22581299	2
#22581233	2
#22581145	2
Q₈×C₁₃₉⋊C₈	2
C₄×#14555770	2
#550×C₁₃₉	2
#3358	139

subgroup

index

Q₈×D₅₅₆

Q₈×C₁₃₉⋊C₈

#550×C₁₃₉

139

Subgroups — #31819035

Conjugacy classes of subgroups

Complements

Maximal subgroups