Group #31807556 — subgroups

Subgroups — #31807556

Order 17792.

258 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 83 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×#17984834	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984767	C₂	1	2	yes	maximal
3	#22580859	C₂	1	1	yes	maximal, characteristic
4	#22580716	C₂	1	2	yes	maximal
5	#336×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₇₈	C₂²	1	1	yes	characteristic
7	#17984834	—	2	4	no	—
8	C₂×C₄⋊Dic₁₃₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
10	C₂×Dic₁₃₉⋊C₄	—	2	2	no	—
11	#17984767	—	2	4	no	—
12	C₁₃₉⋊#95	—	2	2	no	—
13	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₇₈	—	2	2	no	—
16	C₂×Dic₂₇₈	—	2	1	no	—
17	C₂×Dic₂₇₈	—	2	2	no	—
18	C₂×Dic₂₇₈	Q₈	1	2	yes	—
19	C₂²×Dic₁₃₉	—	2	2	no	—
20	C₂²×Dic₁₃₉	—	2	1	no	—
21	C₂²×Dic₁₃₉	—	2	1	no	—
22	C₄⋊Dic₁₃₉	—	2	2	no	—
23	C₄⋊Dic₁₃₉	—	2	2	no	—
24	C₄⋊Dic₁₃₉	D₄	1	4	yes	—
25	Dic₁₃₉⋊C₄	—	2	8	no	—
26	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
27	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₃₉	—	2	2	no	—
29	C₂×C₈×C₁₃₉	—	2	2	no	—
30	C₂×C₈×C₁₃₉	—	2	2	no	—
31	C₄²×C₁₃₉	—	2	2	no	—
32	Dic₂₇₈	—	2	4	no	—
33	Dic₂₇₈	—	4	2	no	—
34	C₂×Dic₁₃₉	—	2	4	no	—
35	C₂×Dic₁₃₉	—	2	2	no	—
36	C₂×Dic₁₃₉	—	4	1	no	—
37	C₂×Dic₁₃₉	—	4	1	no	—
38	C₂×Dic₁₃₉	—	4	2	no	—
39	C₂×Dic₁₃₉	—	4	2	no	—
40	C₂×Dic₁₃₉	—	4	1	no	—
41	C₂×Dic₁₃₉	—	4	2	no	—
42	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₉	—	2	2	no	—
44	C₂×C₄×C₁₃₉	—	2	2	no	—
45	C₂×C₄×C₁₃₉	—	2	2	no	—
46	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₃₉	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₃₉	#41	1	1	yes	characteristic
49	C₂×C₄×C₁₃₉	#41	1	2	yes	—
50	C₈×C₁₃₉	—	4	2	no	—
51	C₂²×C₁₃₉	#120	1	1	yes	characteristic
52	C₂²×C₁₃₉	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₃₉	#124	1	1	yes	characteristic
54	C₂²×C₁₃₉	#135	1	2	yes	—
55	C₂²×C₁₃₉	#137	1	2	yes	—
56	C₄×C₁₃₉	—	4	2	no	—
57	C₄×C₁₃₉	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₃₉	#124	1	2	yes	—
59	Dic₁₃₉	—	4	2	no	—
60	Dic₁₃₉	—	4	4	no	—
61	Dic₁₃₉	—	8	1	no	—
62	C₂×C₁₃₉	#393	1	1	yes	characteristic
63	C₂×C₁₃₉	#452	1	2	yes	—
64	C₂×C₁₃₉	#479	1	2	yes	—
65	C₂×C₁₃₉	#488	1	1	yes	characteristic
66	C₂×C₁₃₉	#491	1	1	yes	characteristic
67	C₁₃₉	#2026	1	1	yes	characteristic, complemented, Sylow
68	#2026	—	139	1	no	maximal, Sylow
69	C₂×#125	—	139	1	no	—
70	C₂×#103	—	139	2	no	—
71	#391	—	139	1	no	—
72	#340	—	139	2	no	—
73	#336	D₁₃₉	1	1	yes	characteristic
74	C₂²×Q₈	—	139	1	no	—
75	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
76	#125	—	278	4	no	—
77	C₂×C₄⋊C₄	—	278	2	no	—
78	C₂×C₄⋊C₄	—	139	1	no	—
79	C₂×C₄⋊C₄	—	139	2	no	—
80	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
81	#103	—	278	4	no	—
82	#95	—	278	2	no	—
83	C₂×Q₈	—	278	1	no	—
84	C₂×Q₈	—	278	2	no	—
85	C₂×Q₈	—	278	2	no	—
86	C₂×Q₈	—	139	2	no	—
87	C₂²×C₄	—	278	1	no	—
88	C₂²×C₄	—	278	2	no	—
89	C₂²×C₄	—	278	1	no	—
90	C₂²×C₄	D₅₅₆	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₄⋊C₄	—	278	8	no	—
96	C₄⋊C₄	—	278	2	no	—
97	C₄⋊C₄	—	278	2	no	—
98	C₄⋊C₄	—	139	4	no	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
101	Q₈	—	278	4	no	—
102	Q₈	—	556	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	278	4	no	—
107	C₂×C₄	—	278	2	no	—
108	C₂×C₄	—	556	1	no	—
109	C₂×C₄	—	556	1	no	—
110	C₂×C₄	—	556	2	no	—
111	C₂×C₄	—	556	2	no	—
112	C₂×C₄	—	556	1	no	—
113	C₂×C₄	—	556	2	no	—
114	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
115	C₂×C₄	Q₈×D₁₃₉	1	1	yes	characteristic
116	C₂×C₄	#14555784	1	1	yes	characteristic
117	C₂×C₄	#14555785	1	2	yes	—
118	C₈	—	4	2	no	—
119	C₂²	#17984812	1	1	yes	characteristic
120	C₂²	#17984822	1	1	yes	characteristic
121	C₂²	#17984830	1	1	yes	characteristic
122	C₂²	#17984864	1	2	yes	—
123	C₂²	#17984875	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	556	4	no	—
126	C₄	—	556	2	no	—
127	C₄	—	1112	1	no	—
128	C₄	#17984820	1	2	yes	—
129	C₄	#17984828	1	2	yes	—
130	C₂	#22580866	1	1	yes	characteristic
131	C₂	#22581117	1	2	yes	—
132	C₂	#22581261	1	2	yes	—
133	C₂	#22581308	1	1	yes	characteristic
134	C₂	#22581322	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984834

C₂

yes

maximal, characteristic

C₂×#17984767

C₂

yes

maximal

#22580859

C₂

yes

maximal, characteristic

#22580716

C₂

yes

maximal

#336×C₁₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₂₇₈

C₂²

yes

characteristic

#17984834

—

C₂×C₄⋊Dic₁₃₉

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

C₂×Dic₁₃₉⋊C₄

—

#17984767

—

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

—

C₂×Dic₂₇₈

Q₈

yes

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₄⋊Dic₁₃₉

D₄

yes

—

Dic₁₃₉⋊C₄

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₄²×C₁₃₉

—

Dic₂₇₈

—

Dic₂₇₈

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×Q₈

yes

C₂×C₄×C₁₃₉

yes

C₂×C₄×C₁₃₉

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

#120

yes

characteristic

C₂²×C₁₃₉

C₂²⋊Q₈

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

—

C₂²×C₁₃₉

#137

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²⋊Q₈

yes

—

C₄×C₁₃₉

#124

yes

—

Dic₁₃₉

—

Dic₁₃₉

—

Dic₁₃₉

—

C₂×C₁₃₉

#393

yes

characteristic

C₂×C₁₃₉

#452

yes

—

C₂×C₁₃₉

#479

yes

—

C₂×C₁₃₉

yes

C₂×C₁₃₉

yes

C₁₃₉

yes

characteristic, complemented, Sylow

#2026

—

139

maximal, Sylow

C₂×#125

—

139

—

C₂×#103

—

139

—

#391

—

139

—

#340

—

139

—

#336

D₁₃₉

yes

characteristic

C₂²×Q₈

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

—

#125

—

278

—

C₂×C₄⋊C₄

—

278

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#103

—

278

—

#95

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

278

—

C₂×Q₈

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

—

278

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

278

—

C₄⋊C₄

—

139

—

C₄²

—

100

C₂³

C₂×D₅₅₆

yes

characteristic

101

Q₈

—

278

—

102

Q₈

—

556

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

278

—

107

C₂×C₄

—

278

—

108

C₂×C₄

—

556

—

109

C₂×C₄

—

556

—

110

C₂×C₄

—

556

—

111

C₂×C₄

—

556

—

112

C₂×C₄

—

556

—

113

C₂×C₄

—

556

—

114

C₂×C₄

D₄×D₁₃₉

yes

—

115

C₂×C₄

Q₈×D₁₃₉

yes

characteristic

116

C₂×C₄

#14555784

yes

characteristic

117

C₂×C₄

#14555785

yes

—

118

C₈

—

119

C₂²

#17984812

yes

characteristic

120

C₂²

#17984822

yes

characteristic

121

C₂²

#17984830

yes

characteristic

122

C₂²

#17984864

yes

—

123

C₂²

#17984875

yes

—

124

C₄

—

125

C₄

—

556

—

126

C₄

—

556

—

127

C₄

—

1112

—

128

C₄

#17984820

yes

—

129

C₄

#17984828

yes

—

130

C₂

#22580866

yes

characteristic

131

C₂

#22581117

yes

—

132

C₂

#22581261

yes

—

133

C₂

#22581308

yes

characteristic

134

C₂

#22581322

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₅₅₆
commutator, cosocle	C₂²×C₄×C₁₃₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	#336×C₁₃₉	C₂
socle, prime radical	C₂³×C₁₃₉	C₂×D₄
hypercenter	#336	D₁₃₉
hypocenter	C₁₃₉	#2026
LCS	G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

subgroup

G/H

center

C₂³

C₂×D₅₅₆

commutator, cosocle

C₂²×C₄×C₁₃₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

#336×C₁₃₉

C₂

socle, prime radical

C₂³×C₁₃₉

C₂×D₄

D₁₃₉

C₁₃₉

G ▷ C₂²×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #336

N	H ≅ G/N	type
67. C₁₃₉	#2026	semidirect

H ≅ G/N

type

67. C₁₃₉

#2026

semidirect

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

subgroup	index
C₂×#17984834	2
C₂×#17984767	2
#22580859	2
#22580716	2
#336×C₁₃₉	2
#2026	139

subgroup

index

#336×C₁₃₉

139

Subgroups — #31807556

Conjugacy classes of subgroups

Complements

Maximal subgroups