Group #31819078 — subgroups

Subgroups — #31819078

Order 17792.

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

normal subgroups only

This group has 336 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×Dic₂₇₈	C₂	1	1	yes	maximal, characteristic
2	#22581303	C₂	1	3	yes	maximal
3	#22581248	C₂	1	3	yes	maximal
4	#22581160	C₂	1	3	yes	maximal
5	Q₈×C₁₃₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#14555775	C₂	1	3	yes	maximal
7	#557×C₁₃₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#14555775	C₂²	1	3	yes	—
9	#17984842	C₂²	1	3	yes	—
10	#17984841	—	2	3	no	—
11	Q₈×Dic₁₃₉	—	2	1	no	—
12	C₄×Dic₂₇₈	C₂²	1	3	yes	—
13	#17984775	C₂²	1	3	yes	—
14	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	3	yes	—
15	#17984766	C₂²	1	6	yes	—
16	#17984766	C₂²	1	3	yes	—
17	#17984765	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	C₂×Dic₂₇₈	C₂³	1	1	yes	characteristic
26	#14555775	—	2	6	no	—
27	C₂×C₁₃₉⋊C₈	C₂³	1	3	yes	—
28	C₂×C₁₃₉⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₃₉	—	2	3	no	—
30	C₄⋊Dic₁₃₉	C₂³	1	3	yes	—
31	Dic₁₃₉⋊C₄	—	2	3	no	—
32	C₄×Dic₁₃₉	—	2	3	no	—
33	C₂×Q₈×C₁₃₉	D₄	1	1	yes	characteristic
34	C₂×Q₈×C₁₃₉	C₂³	1	3	yes	—
35	C₄⋊C₄×C₁₃₉	—	2	3	no	—
36	C₄⋊C₄×C₁₃₉	—	2	3	no	—
37	C₄⋊C₄×C₁₃₉	—	2	3	no	—
38	C₄⋊C₄×C₁₃₉	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₃₉	C₂³	1	1	yes	characteristic
40	C₄²×C₁₃₉	—	2	3	no	—
41	C₄²×C₁₃₉	C₂³	1	3	yes	—
42	Dic₂₇₈	—	2	1	no	—
43	Dic₂₇₈	—	8	3	no	—
44	Dic₂₇₈	#41	1	2	yes	—
45	C₂×Dic₁₃₉	—	2	1	no	—
46	C₂×Dic₁₃₉	—	2	3	no	—
47	C₁₃₉⋊C₈	—	2	3	no	—
48	C₁₃₉⋊C₈	—	2	1	no	—
49	Q₈×C₁₃₉	—	2	3	no	—
50	Q₈×C₁₃₉	—	2	3	no	—
51	Q₈×C₁₃₉	Q₁₆	1	4	yes	complemented
52	C₂×C₄×C₁₃₉	—	2	1	no	—
53	C₂×C₄×C₁₃₉	—	2	3	no	—
54	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₁₃₉	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₃₉	C₂⁴	1	1	yes	characteristic
57	C₂²×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₃₉	—	2	3	no	—
59	C₄×C₁₃₉	—	4	3	no	—
60	C₄×C₁₃₉	—	4	1	no	—
61	C₄×C₁₃₉	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₃₉	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₃₉	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₃₉	—	2	2	no	—
65	Dic₁₃₉	—	4	3	no	—
66	C₂×C₁₃₉	#548	1	1	yes	characteristic
67	C₂×C₁₃₉	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₃₉	#577	1	1	yes	characteristic
69	C₁₃₉	#3363	1	1	yes	characteristic, complemented, Sylow
70	#3363	—	139	1	no	maximal, Sylow
71	Q₈²	—	139	1	no	—
72	#557	D₁₃₉	1	1	yes	characteristic
73	#487	—	139	3	no	—
74	#477	—	139	3	no	—
75	#462	—	139	3	no	—
76	C₈×Q₈	—	139	1	no	—
77	C₄×Q₁₆	—	139	3	no	—
78	C₂×Q₁₆	—	139	3	no	—
79	C₄⋊Q₈	—	278	3	no	—
80	C₄⋊Q₈	—	139	3	no	—
81	C₄⋊Q₈	C₂×D₁₃₉	1	3	yes	—
82	#125	—	2	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	278	1	no	—
85	C₄×Q₈	—	139	3	no	—
86	C₄×Q₈	C₂×D₁₃₉	1	3	yes	—
87	C₄×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
88	#107	—	139	3	no	—
89	C₄⋊C₈	—	139	3	no	—
90	#103	—	139	3	no	—
91	#103	—	139	6	no	—
92	#103	—	139	1	no	—
93	C₄×C₈	—	139	3	no	—
94	C₂×Q₈	—	278	3	no	—
95	C₂×Q₈	—	139	1	no	—
96	C₂×Q₈	C₂²⋊D₁₃₉	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₃₉	1	3	yes	—
98	Q₁₆	—	278	6	no	—
99	C₂×C₈	—	139	1	no	—
100	C₂×C₈	—	139	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	278	3	no	—
105	C₄⋊C₄	—	278	3	no	—
106	C₄⋊C₄	—	139	3	no	—
107	C₄⋊C₄	C₂²⋊D₁₃₉	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₃₉	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	278	3	no	—
111	C₄²	C₂²×D₁₃₉	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	278	1	no	—
115	Q₈	—	1112	3	no	—
116	Q₈	—	139	2	no	—
117	Q₈	#14555775	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	278	1	no	—
121	C₂×C₄	—	278	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	3	yes	—
124	C₂×C₄	C₂³×D₁₃₉	1	1	yes	characteristic
125	C₈	—	278	3	no	—
126	C₈	—	278	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₃₉	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	1	no	—
130	C₄	—	4	3	no	—
131	C₄	—	278	2	no	—
132	C₄	—	556	3	no	—
133	C₄	C₂×#14555775	1	6	yes	—
134	C₄	#41×D₁₃₉	1	1	yes	characteristic
135	C₄	#17984889	1	1	yes	characteristic
136	C₂	#22581610	1	1	yes	characteristic
137	C₂	C₂²×#14555775	1	1	yes	characteristic
138	C₂	#22581761	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₂₇₈

C₂

yes

maximal, characteristic

#22581303

C₂

yes

maximal

#22581248

C₂

yes

maximal

#22581160

C₂

yes

maximal

Q₈×C₁₃₉⋊C₈

C₂

yes

maximal, characteristic

C₄×#14555775

C₂

yes

maximal

#557×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555775

C₂²

yes

—

#17984842

C₂²

yes

—

#17984841

—

Q₈×Dic₁₃₉

—

C₄×Dic₂₇₈

C₂²

yes

—

#17984775

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

#17984766

C₂²

yes

—

#17984766

C₂²

yes

—

#17984765

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

—

C₂×Dic₂₇₈

C₂³

yes

characteristic

#14555775

—

C₂×C₁₃₉⋊C₈

C₂³

yes

—

C₂×C₁₃₉⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

C₂³

yes

—

Dic₁₃₉⋊C₄

—

C₄×Dic₁₃₉

—

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

—

Dic₂₇₈

—

Dic₂₇₈

—

Dic₂₇₈

#41

yes

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

Q₁₆

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

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂×Q₁₆

yes

—

C₄×C₁₃₉

C₂×#41

yes

characteristic

C₄×C₁₃₉

2₋¹⁺⁴

yes

characteristic

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#548

yes

characteristic

C₂×C₁₃₉

C₂²×Q₁₆

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#3363

—

139

maximal, Sylow

Q₈²

—

139

—

#557

D₁₃₉

yes

characteristic

#487

—

139

—

#477

—

139

—

#462

—

139

—

C₈×Q₈

—

139

—

C₄×Q₁₆

—

139

—

C₂×Q₁₆

—

139

—

C₄⋊Q₈

—

278

—

C₄⋊Q₈

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

278

—

C₄×Q₈

—

139

—

C₄×Q₈

C₂×D₁₃₉

yes

—

C₄×Q₈

C₂×D₁₃₉

yes

characteristic

#107

—

139

—

C₄⋊C₈

—

139

—

#103

—

139

—

#103

—

139

—

#103

—

139

—

C₄×C₈

—

139

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂×Q₈

C₂²⋊D₁₃₉

yes

characteristic

C₂×Q₈

C₂²×D₁₃₉

yes

—

Q₁₆

—

278

—

C₂×C₈

—

139

—

100

C₂×C₈

—

139

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

278

—

105

C₄⋊C₄

—

278

—

106

C₄⋊C₄

—

139

—

107

C₄⋊C₄

C₂²⋊D₁₃₉

yes

—

108

C₄⋊C₄

C₂²×D₁₃₉

yes

characteristic

109

C₄²

—

110

C₄²

—

278

—

111

C₄²

C₂²×D₁₃₉

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

278

—

115

Q₈

—

1112

—

116

Q₈

—

139

—

117

Q₈

#14555775

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

278

—

121

C₂×C₄

—

278

—

122

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

—

124

C₂×C₄

C₂³×D₁₃₉

yes

characteristic

125

C₈

—

278

—

126

C₈

—

278

—

127

C₂²

C₂²×C₂²⋊D₁₃₉

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

278

—

132

C₄

—

556

—

133

C₄

C₂×#14555775

yes

—

134

C₄

#41×D₁₃₉

yes

characteristic

135

C₄

#17984889

yes

characteristic

136

C₂

#22581610

yes

characteristic

137

C₂

C₂²×#14555775

yes

characteristic

138

C₂

#22581761

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, prime radical	C₂²×C₁₃₉	C₂²×D₄
hypercenter	#557	D₁₃₉
hypocenter	C₁₃₉	#3363
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, prime radical

C₂²×C₁₃₉

C₂²×D₄

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
51. Q₈×C₁₃₉	Q₁₆	semidirect
69. C₁₃₉	#3363	semidirect
117. Q₈	#14555775	semidirect

H ≅ G/N

type

51. Q₈×C₁₃₉

Q₁₆

semidirect

69. C₁₃₉

#3363

semidirect

117. Q₈

#14555775

semidirect

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

subgroup	index
Q₈×Dic₂₇₈	2
#22581303	2
#22581248	2
#22581160	2
Q₈×C₁₃₉⋊C₈	2
C₄×#14555775	2
#557×C₁₃₉	2
#3363	139

subgroup

index

Q₈×Dic₂₇₈

Q₈×C₁₃₉⋊C₈

#557×C₁₃₉

139

Subgroups — #31819078

Conjugacy classes of subgroups

Complements

Maximal subgroups