Group #31807228 — subgroups

Subgroups — #31807228

Order 17792.

412 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 109 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×#17984839	C₂	1	1	yes	maximal, characteristic
2	C₂×#17984764	C₂	1	4	yes	maximal, complemented
3	#22580819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#97×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₅₅₆	—	2	2	no	—
6	C₂²×D₅₅₆	C₂²	1	2	yes	—
7	#17984839	—	2	2	no	—
8	#17984839	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₁₃₉	C₂²	1	2	yes	—
10	#17984764	—	2	8	no	—
11	C₁₃₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₃₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
14	#97×C₁₃₉	—	2	2	no	—
15	C₂³×D₁₃₉	—	2	2	no	—
16	C₂×D₅₅₆	—	2	8	no	—
17	C₂×D₅₅₆	—	2	4	no	—
18	C₂×D₅₅₆	—	2	2	no	—
19	C₂×D₅₅₆	—	2	4	no	—
20	C₂×D₅₅₆	—	4	8	no	—
21	C₂×D₅₅₆	C₂×C₄	1	4	yes	—
22	C₂²×Dic₁₃₉	—	2	2	no	—
23	C₄⋊Dic₁₃₉	—	2	4	no	—
24	C₂²×C₄×C₁₃₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₃₉	—	2	2	no	—
27	C₂×C₈×C₁₃₉	—	2	2	no	—
28	C₂×C₈×C₁₃₉	D₄	1	4	yes	complemented
29	C₄²×C₁₃₉	C₂×C₄	1	2	yes	—
30	C₄²×C₁₃₉	D₄	1	2	yes	—
31	C₂²×D₁₃₉	—	2	4	no	—
32	C₂²×D₁₃₉	—	4	2	no	—
33	C₂²×D₁₃₉	—	4	2	no	—
34	C₂²×D₁₃₉	—	4	8	no	—
35	D₅₅₆	—	2	8	no	—
36	D₅₅₆	—	4	16	no	—
37	D₅₅₆	—	4	4	no	—
38	C₂×Dic₁₃₉	—	4	2	no	—
39	C₂×Dic₁₃₉	—	4	4	no	—
40	C₂³×C₁₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₃₉	—	2	4	no	—
42	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₃₉	#41	1	2	yes	—
47	C₈×C₁₃₉	—	2	4	no	—
48	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₃₉	#124	1	1	yes	characteristic
50	C₂²×C₁₃₉	#127	1	1	yes	characteristic
51	C₂²×C₁₃₉	AGL(1,ℤ/8)	1	4	yes	—
52	C₂×D₁₃₉	—	4	4	no	—
53	C₂×D₁₃₉	—	4	8	no	—
54	C₂×D₁₃₉	—	8	4	no	—
55	C₂×D₁₃₉	—	8	4	no	—
56	C₄×C₁₃₉	—	2	4	no	—
57	C₄×C₁₃₉	C₄×D₄	1	2	yes	—
58	C₄×C₁₃₉	#121	1	2	yes	—
59	Dic₁₃₉	—	8	2	no	—
60	C₂×C₁₃₉	#390	1	1	yes	characteristic
61	C₂×C₁₃₉	#418	1	2	yes	—
62	C₂×C₁₃₉	#490	1	2	yes	—
63	C₂×C₁₃₉	#496	1	2	yes	—
64	D₁₃₉	—	8	4	no	—
65	C₁₃₉	#1961	1	1	yes	characteristic, complemented, Sylow
66	#1961	—	139	1	no	maximal, Sylow
67	C₂×#127	—	139	1	no	—
68	C₂×#102	—	139	4	no	—
69	C₂×#97	D₁₃₉	1	1	yes	characteristic, complemented
70	#382	—	139	1	no	—
71	C₂²×D₄	—	278	2	no	—
72	C₂²×D₄	—	139	2	no	—
73	C₂²×C₈	C₂×D₁₃₉	1	2	yes	—
74	#127	—	278	2	no	—
75	#127	—	139	4	no	—
76	C₂×C₄⋊C₄	—	139	2	no	—
77	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
78	#102	—	278	8	no	—
79	#97	—	2	2	no	—
80	#95	—	278	2	no	—
81	C₂⁴	—	278	2	no	—
82	C₂×D₄	—	278	8	no	—
83	C₂×D₄	—	278	2	no	—
84	C₂×D₄	—	278	4	no	—
85	C₂×D₄	—	278	4	no	—
86	C₂×D₄	—	556	8	no	—
87	C₂×D₄	—	139	4	no	—
88	C₂²×C₄	—	278	2	no	—
89	C₂²×C₄	D₅₅₆	1	2	yes	—
90	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	D₅₅₆	1	4	yes	complemented
94	C₄⋊C₄	—	278	4	no	—
95	C₄²	C₄×D₁₃₉	1	2	yes	—
96	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
97	C₂³	—	278	4	no	—
98	C₂³	—	556	2	no	—
99	C₂³	—	556	2	no	—
100	C₂³	—	556	8	no	—
101	C₂³	C₂×D₅₅₆	1	1	yes	characteristic
102	D₄	—	278	8	no	—
103	D₄	—	556	16	no	—
104	D₄	—	556	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	556	2	no	—
107	C₂×C₄	—	556	4	no	—
108	C₂×C₄	(C₂×D₁₃₉)⋊C₄	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₃₉	1	1	yes	characteristic
110	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
111	C₂×C₄	C₂×D₅₅₆	1	2	yes	—
112	C₂×C₄	#14555785	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₂²	—	556	4	no	—
115	C₂²	—	556	8	no	—
116	C₂²	—	1112	4	no	—
117	C₂²	—	1112	4	no	—
118	C₂²	C₂×(C₂×D₁₃₉)⋊C₄	1	1	yes	characteristic
119	C₂²	#17984830	1	1	yes	characteristic
120	C₂²	#17984839	1	1	yes	characteristic
121	C₂²	#17984870	1	4	yes	—
122	C₄	—	2	4	no	—
123	C₄	—	1112	2	no	—
124	C₄	C₄×D₅₅₆	1	2	yes	—
125	C₄	#17984816	1	2	yes	—
126	C₂	—	1112	4	no	—
127	C₂	#22580855	1	1	yes	characteristic
128	C₂	#22580959	1	2	yes	—
129	C₂	#22581318	1	2	yes	—
130	C₂	#22581342	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17984839

C₂

yes

maximal, characteristic

C₂×#17984764

C₂

yes

maximal, complemented

#22580819

C₂

yes

maximal, characteristic, complemented

C₂×#97×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₅₆

—

C₂²×D₅₅₆

C₂²

yes

—

#17984839

—

#17984839

C₄

yes

complemented

C₂×C₄⋊Dic₁₃₉

C₂²

yes

—

#17984764

—

C₁₃₉⋊#95

—

C₂²×C₈×C₁₃₉

C₂²

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

characteristic

#97×C₁₃₉

—

C₂³×D₁₃₉

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

—

C₂×D₅₅₆

C₂×C₄

yes

—

C₂²×Dic₁₃₉

—

C₄⋊Dic₁₃₉

—

C₂²×C₄×C₁₃₉

D₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

—

C₂×C₈×C₁₃₉

D₄

yes

complemented

C₄²×C₁₃₉

C₂×C₄

yes

—

C₄²×C₁₃₉

D₄

yes

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

D₅₅₆

—

D₅₅₆

—

D₅₅₆

—

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

—

C₂³×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

#41

yes

—

C₈×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

C₂²×C₁₃₉

yes

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

#121

yes

—

Dic₁₃₉

—

C₂×C₁₃₉

#390

yes

characteristic

C₂×C₁₃₉

#418

yes

—

C₂×C₁₃₉

#490

yes

—

C₂×C₁₃₉

#496

yes

—

D₁₃₉

—

C₁₃₉

#1961

yes

characteristic, complemented, Sylow

#1961

—

139

maximal, Sylow

C₂×#127

—

139

—

C₂×#102

—

139

—

C₂×#97

D₁₃₉

yes

characteristic, complemented

#382

—

139

—

C₂²×D₄

—

278

—

C₂²×D₄

—

139

—

C₂²×C₈

C₂×D₁₃₉

yes

—

#127

—

278

—

#127

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

#102

—

278

—

#97

—

#95

—

278

—

C₂⁴

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

278

—

C₂×D₄

—

556

—

C₂×D₄

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

D₅₅₆

yes

—

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₅₅₆

yes

complemented

C₄⋊C₄

—

278

—

C₄²

C₄×D₁₃₉

yes

—

C₄²

C₂²⋊D₁₃₉

yes

—

C₂³

—

278

—

C₂³

—

556

—

C₂³

—

556

—

100

C₂³

—

556

—

101

C₂³

C₂×D₅₅₆

yes

characteristic

102

D₄

—

278

—

103

D₄

—

556

—

104

D₄

—

556

—

105

C₂×C₄

—

106

C₂×C₄

—

556

—

107

C₂×C₄

—

556

—

108

C₂×C₄

(C₂×D₁₃₉)⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₁₃₉

yes

characteristic

110

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

111

C₂×C₄

C₂×D₅₅₆

yes

—

112

C₂×C₄

#14555785

yes

—

113

C₈

—

114

C₂²

—

556

—

115

C₂²

—

556

—

116

C₂²

—

1112

—

117

C₂²

—

1112

—

118

C₂²

C₂×(C₂×D₁₃₉)⋊C₄

yes

characteristic

119

C₂²

#17984830

yes

characteristic

120

C₂²

#17984839

yes

characteristic

121

C₂²

#17984870

yes

—

122

C₄

—

123

C₄

—

1112

—

124

C₄

C₄×D₅₅₆

yes

—

125

C₄

#17984816

yes

—

126

C₂

—

1112

—

127

C₂

#22580855

yes

characteristic

128

C₂

#22580959

yes

—

129

C₂

#22581318

yes

—

130

C₂

#22581342

yes

—

	subgroup	G/H
center	C₂³	C₂×D₅₅₆
commutator	C₂×C₄×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#97×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂×D₄
prime radical	C₂×#17984839	C₂
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#97	D₁₃₉
hypocenter	C₁₃₉	#1961
LCS	G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₅₅₆

commutator

C₂×C₄×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#97×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂×D₄

prime radical

C₂×#17984839

C₂

cosocle

C₂²×C₄×C₁₃₉

C₂³

hypercenter

C₂×#97

D₁₃₉

hypocenter

C₁₃₉

#1961

LCS

G ▷ C₂×C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
2. C₂×#17984764	C₂	semidirect
3. #22580819	C₂	semidirect
4. C₂×#97×C₁₃₉	C₂	semidirect
8. #17984839	C₄	semidirect
28. C₂×C₈×C₁₃₉	D₄	semidirect
65. C₁₃₉	#1961	semidirect
69. C₂×#97	D₁₃₉	semidirect
93. C₂×C₈	D₅₅₆	semidirect

H ≅ G/N

type

2. C₂×#17984764

C₂

semidirect

3. #22580819

C₂

semidirect

4. C₂×#97×C₁₃₉

C₂

semidirect

8. #17984839

C₄

semidirect

28. C₂×C₈×C₁₃₉

D₄

semidirect

65. C₁₃₉

#1961

semidirect

69. C₂×#97

D₁₃₉

semidirect

93. C₂×C₈

D₅₅₆

semidirect

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

subgroup	index
C₂×#17984839	2
C₂×#17984764	2
#22580819	2
C₂×#97×C₁₃₉	2
#1961	139

subgroup

index

C₂×#97×C₁₃₉

139

Subgroups — #31807228

Conjugacy classes of subgroups

Complements

Maximal subgroups