Group #31805065 — subgroups

Subgroups — #31805065

Order 17792.

246 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22581504	C₂	1	1	yes	maximal, characteristic
2	#22581005	C₂	1	2	yes	maximal, complemented
3	#22580702	C₂	1	1	yes	maximal, characteristic, complemented
4	#22580696	C₂	1	2	yes	maximal, complemented
5	#333×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14555786	C₄	1	2	yes	—
7	#17984839	C₂²	1	1	yes	characteristic
8	#17984831	C₄	1	2	yes	—
9	#17984817	—	2	2	no	—
10	#17984804	—	2	2	no	—
11	C₄²×D₁₃₉	C₂²	1	1	yes	characteristic
12	C₄⋊(C₁₃₉⋊C₈)	C₂²	1	2	yes	—
13	(C₂×D₁₃₉)⋊C₈	—	2	2	no	—
14	C₈×Dic₁₃₉	—	2	2	no	—
15	C₄⋊Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₉	C₂²	1	2	yes	—
17	#14555786	—	4	4	no	—
18	C₂×D₅₅₆	—	2	2	no	—
19	C₂×D₅₅₆	—	2	2	no	—
20	C₂×D₅₅₆	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₁₃₉	—	2	2	no	—
22	C₂×C₄×D₁₃₉	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₁₃₉	D₄	1	2	yes	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	(C₂×D₁₃₉)⋊C₄	—	2	4	no	—
26	C₄×Dic₁₃₉	C₂×C₄	1	1	yes	characteristic
27	C₄×Dic₁₃₉	D₄	1	2	yes	—
28	C₂×Q₈×C₁₃₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₃₉	—	2	2	no	—
30	C₄⋊C₄×C₁₃₉	—	2	2	no	—
31	C₄²×C₁₃₉	C₂³	1	1	yes	characteristic
32	C₂²×D₁₃₉	—	2	2	no	—
33	C₂²×D₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
34	D₅₅₆	—	4	2	no	—
35	D₅₅₆	—	4	4	no	—
36	D₅₅₆	—	4	2	no	—
37	D₅₅₆	—	4	2	no	—
38	C₂×Dic₁₃₉	C₂²⋊C₄	1	2	yes	—
39	C₂×Dic₁₃₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₄×D₁₃₉	—	2	2	no	—
41	C₄×D₁₃₉	—	2	2	no	—
42	C₄×D₁₃₉	—	2	2	no	—
43	C₄×D₁₃₉	—	4	4	no	—
44	C₁₃₉⋊C₈	—	4	2	no	—
45	Q₈×C₁₃₉	—	4	2	no	—
46	C₂×C₄×C₁₃₉	—	2	2	no	—
47	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
49	C₈×C₁₃₉	—	4	2	no	—
50	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂×D₁₃₉	—	2	1	no	—
52	C₂×D₁₃₉	—	2	2	no	—
53	C₂×D₁₃₉	—	4	2	no	—
54	C₂×D₁₃₉	—	4	4	no	—
55	C₄×C₁₃₉	—	2	1	no	—
56	C₄×C₁₃₉	—	4	2	no	—
57	C₄×C₁₃₉	#135	1	2	yes	—
58	C₄×C₁₃₉	#135	1	2	yes	—
59	Dic₁₃₉	—	2	2	no	—
60	Dic₁₃₉	#100	1	2	yes	—
61	C₂×C₁₃₉	C₂×#100	1	1	yes	characteristic
62	C₂×C₁₃₉	#416	1	2	yes	—
63	D₁₃₉	—	4	1	no	—
64	D₁₃₉	—	8	2	no	—
65	C₁₃₉	#1543	1	1	yes	characteristic, complemented, Sylow
66	#1543	—	139	1	no	maximal, Sylow
67	#532	—	139	1	no	—
68	#431	—	139	2	no	—
69	#333	—	139	1	no	—
70	#333	D₁₃₉	1	1	yes	characteristic, complemented
71	#331	—	139	2	no	—
72	C₂×#41	—	139	2	no	—
73	C₄⋊Q₈	C₂×D₁₃₉	1	1	yes	characteristic
74	#127	—	139	1	no	—
75	#124	—	139	2	no	—
76	#121	—	278	2	no	—
77	C₄×D₄	—	278	2	no	—
78	C₂×C₄²	—	139	1	no	—
79	C₄⋊C₈	—	139	2	no	—
80	C₄⋊C₈	C₂×D₁₃₉	1	2	yes	—
81	C₂²⋊C₈	—	278	2	no	—
82	C₄×C₈	—	278	2	no	—
83	#41	—	556	4	no	—
84	C₂×Q₈	C₄×D₁₃₉	1	2	yes	—
85	C₂×D₄	—	278	2	no	—
86	C₂×D₄	—	278	2	no	—
87	C₂×D₄	—	139	2	no	—
88	C₂²×C₄	—	278	2	no	—
89	C₂²×C₄	—	139	1	no	—
90	C₂²×C₄	—	139	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	278	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₂²⋊C₄	—	278	4	no	—
95	C₄²	—	139	1	no	—
96	C₄²	—	139	2	no	—
97	C₄²	C₂²×D₁₃₉	1	1	yes	characteristic
98	C₂³	—	278	2	no	—
99	C₂³	—	139	1	no	—
100	Q₈	—	4	2	no	—
101	D₄	—	556	2	no	—
102	D₄	—	556	4	no	—
103	D₄	—	556	2	no	—
104	D₄	—	556	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	278	2	no	—
107	C₂×C₄	—	278	2	no	—
108	C₂×C₄	—	278	2	no	—
109	C₂×C₄	—	556	4	no	—
110	C₂×C₄	—	139	2	no	—
111	C₂×C₄	—	139	1	no	—
112	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
113	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
114	C₈	—	4	2	no	—
115	C₈	—	556	2	no	—
116	C₂²	—	278	1	no	—
117	C₂²	—	278	2	no	—
118	C₂²	—	556	2	no	—
119	C₂²	—	556	4	no	—
120	C₂²	C₂²⋊C₄×D₁₃₉	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	4	2	no	—
123	C₄	—	278	2	no	—
124	C₄	—	139	2	no	—
125	C₄	#17984863	1	2	yes	—
126	C₄	#17984865	1	2	yes	—
127	C₂	—	556	1	no	—
128	C₂	—	1112	2	no	—
129	C₂	#100×D₁₃₉	1	1	yes	characteristic
130	C₂	#22580948	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22581504

C₂

yes

maximal, characteristic

#22581005

C₂

yes

maximal, complemented

#22580702

C₂

yes

maximal, characteristic, complemented

#22580696

C₂

yes

maximal, complemented

#333×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×#14555786

C₄

yes

—

#17984839

C₂²

yes

characteristic

#17984831

C₄

yes

—

#17984817

—

#17984804

—

C₄²×D₁₃₉

C₂²

yes

characteristic

C₄⋊(C₁₃₉⋊C₈)

C₂²

yes

—

(C₂×D₁₃₉)⋊C₈

—

C₈×Dic₁₃₉

—

C₄⋊Q₈×C₁₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₉

C₂²

yes

—

#14555786

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂×C₄

yes

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

C₂×C₄

yes

characteristic

C₂×C₄×D₁₃₉

D₄

yes

—

C₂×C₁₃₉⋊C₈

—

(C₂×D₁₃₉)⋊C₄

—

C₄×Dic₁₃₉

C₂×C₄

yes

characteristic

C₄×Dic₁₃₉

D₄

yes

—

C₂×Q₈×C₁₃₉

C₂×C₄

yes

—

C₂×C₈×C₁₃₉

—

C₄⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

C₂³

yes

characteristic

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₂²⋊C₄

yes

characteristic

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

—

C₂×Dic₁₃₉

C₂²⋊C₄

yes

characteristic

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₄×D₁₃₉

—

C₁₃₉⋊C₈

—

Q₈×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#135

yes

—

C₄×C₁₃₉

#135

yes

—

Dic₁₃₉

—

Dic₁₃₉

#100

yes

—

C₂×C₁₃₉

C₂×#100

yes

characteristic

C₂×C₁₃₉

#416

yes

—

D₁₃₉

—

D₁₃₉

—

C₁₃₉

#1543

yes

characteristic, complemented, Sylow

#1543

—

139

maximal, Sylow

#532

—

139

—

#431

—

139

—

#333

—

139

—

#333

D₁₃₉

yes

characteristic, complemented

#331

—

139

—

C₂×#41

—

139

—

C₄⋊Q₈

C₂×D₁₃₉

yes

characteristic

#127

—

139

—

#124

—

139

—

#121

—

278

—

C₄×D₄

—

278

—

C₂×C₄²

—

139

—

C₄⋊C₈

—

139

—

C₄⋊C₈

C₂×D₁₃₉

yes

—

C₂²⋊C₈

—

278

—

C₄×C₈

—

278

—

#41

—

556

—

C₂×Q₈

C₄×D₁₃₉

yes

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂×C₈

—

C₂×C₈

—

278

—

C₄⋊C₄

—

C₂²⋊C₄

—

278

—

C₄²

—

139

—

C₄²

—

139

—

C₄²

C₂²×D₁₃₉

yes

characteristic

C₂³

—

278

—

C₂³

—

139

—

100

Q₈

—

101

D₄

—

556

—

102

D₄

—

556

—

103

D₄

—

556

—

104

D₄

—

556

—

105

C₂×C₄

—

106

C₂×C₄

—

278

—

107

C₂×C₄

—

278

—

108

C₂×C₄

—

278

—

109

C₂×C₄

—

556

—

110

C₂×C₄

—

139

—

111

C₂×C₄

—

139

—

112

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

113

C₂×C₄

D₄×D₁₃₉

yes

—

114

C₈

—

115

C₈

—

556

—

116

C₂²

—

278

—

117

C₂²

—

278

—

118

C₂²

—

556

—

119

C₂²

—

556

—

120

C₂²

C₂²⋊C₄×D₁₃₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

278

—

124

C₄

—

139

—

125

C₄

#17984863

yes

—

126

C₄

#17984865

yes

—

127

C₂

—

556

—

128

C₂

—

1112

—

129

C₂

#100×D₁₃₉

yes

characteristic

130

C₂

#22580948

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₃₉
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₄²	C₂²×D₁₃₉
Fitting	#333×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₂²⋊C₄
prime radical	#22581504	C₂
cosocle	C₄²×C₁₃₉	C₂³
hypercenter	#333	D₁₃₉
hypocenter	C₁₃₉	#1543
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₃₉

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₄²

C₂²×D₁₃₉

Fitting

#333×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₂²⋊C₄

prime radical

#22581504

C₂

cosocle

C₄²×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂×C₄×C₁₃₉ ▷ C₂²×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #22581005	C₂	semidirect
3. #22580702	C₂	semidirect
4. #22580696	C₂	semidirect
5. #333×C₁₃₉	C₂	semidirect
65. C₁₃₉	#1543	semidirect
70. #333	D₁₃₉	semidirect

H ≅ G/N

type

2. #22581005

C₂

semidirect

3. #22580702

C₂

semidirect

4. #22580696

C₂

semidirect

5. #333×C₁₃₉

C₂

semidirect

65. C₁₃₉

#1543

semidirect

70. #333

D₁₃₉

semidirect

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

subgroup	index
#22581504	2
#22581005	2
#22580702	2
#22580696	2
#333×C₁₃₉	2
#1543	139

subgroup

index

#333×C₁₃₉

139

Subgroups — #31805065

Conjugacy classes of subgroups

Complements

Maximal subgroups