Group #31912186 — subgroups

Subgroups — #31912186

Order 18176.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22590595	C₂	1	2	yes	maximal
2	#22590592	C₂	1	2	yes	maximal, complemented
3	#22587958	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586257	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#1709	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556074	—	2	4	no	—
7	C₂×#14556072	—	2	4	no	—
8	#17986404	—	2	2	no	—
9	#17986379	C₂²	1	2	yes	—
10	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
11	#494×C₇₁	C₂²	1	1	yes	characteristic
12	#493×C₇₁	C₂²	1	1	yes	characteristic
13	#486×C₇₁	—	2	2	no	—
14	C₈²×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₈₄	—	2	2	no	—
16	C₂×Dic₂₈₄	—	2	2	no	—
17	#14556128	—	2	2	no	—
18	#14556074	—	4	8	no	—
19	#14556072	—	4	8	no	—
20	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
21	#14556062	—	4	4	no	—
22	C₂×Q₁₆×C₇₁	—	2	2	no	—
23	C₂×D₈×C₇₁	—	2	2	no	—
24	C₄⋊Q₈×C₇₁	—	2	1	no	—
25	#127×C₇₁	—	2	1	no	—
26	#102×C₇₁	—	4	4	no	—
27	C₄×C₈×C₇₁	D₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₂	—	4	2	no	—
30	Dic₂₈₄	—	4	4	no	—
31	Dic₂₈₄	—	4	4	no	—
32	C₄⋊Dic₇₁	—	4	4	no	—
33	C₇₁⋊C₁₆	D₈	1	8	yes	—
34	C₂×Q₈×C₇₁	—	4	1	no	—
35	C₂×D₄×C₇₁	—	4	1	no	—
36	C₂×D₄×C₇₁	—	4	2	no	—
37	Q₁₆×C₇₁	—	4	4	no	—
38	D₈×C₇₁	—	4	4	no	—
39	C₂×C₈×C₇₁	D₈	1	4	yes	—
40	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
41	C₄⋊C₄×C₇₁	—	4	2	no	—
42	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₂³×C₇₁	—	8	1	no	—
44	Q₈×C₇₁	—	8	2	no	—
45	D₄×C₇₁	—	8	2	no	—
46	D₄×C₇₁	—	8	4	no	—
47	C₂×C₄×C₇₁	—	8	1	no	—
48	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
49	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
50	C₂×Dic₇₁	—	8	2	no	—
51	Dic₁₄₂	—	8	2	no	—
52	Dic₁₄₂	—	8	2	no	—
53	C₈×C₇₁	—	2	4	no	—
54	C₈×C₇₁	C₂×D₈	1	4	yes	—
55	C₂²×C₇₁	—	16	1	no	—
56	C₂²×C₇₁	—	16	2	no	—
57	C₂²×C₇₁	#493	1	1	yes	characteristic
58	C₄×C₇₁	—	16	1	no	—
59	C₄×C₇₁	#493	1	2	yes	—
60	C₄×C₇₁	#510	1	4	yes	—
61	Dic₇₁	—	16	2	no	—
62	#12520	—	71	1	no	maximal, Sylow
63	C₂×C₇₁	—	32	1	no	—
64	C₂×C₇₁	#1712	1	1	yes	characteristic
65	C₂×C₇₁	#2251	1	2	yes	—
66	#2248	—	71	2	no	—
67	#2247	—	71	2	no	—
68	#1713	—	71	1	no	—
69	#1709	D₇₁	1	1	yes	characteristic
70	#1311	—	71	1	no	—
71	C₇₁	#12520	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	142	4	no	—
73	C₂×SD₃₂	—	142	4	no	—
74	#500	—	142	2	no	—
75	#494	—	71	2	no	—
76	#494	C₂×D₇₁	1	1	yes	characteristic
77	#493	C₂×D₇₁	1	1	yes	characteristic
78	#486	—	2	2	no	—
79	C₄×C₁₆	—	71	2	no	—
80	C₈²	C₂×D₇₁	1	1	yes	characteristic
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	142	2	no	—
83	C₂×Q₁₆	—	142	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	—	142	2	no	—
87	#127	—	2	1	no	—
88	Q₃₂	—	284	8	no	—
89	SD₃₂	—	284	8	no	—
90	C₂×C₁₆	—	71	4	no	—
91	#107	—	284	4	no	—
92	#102	—	4	4	no	—
93	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
94	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	284	2	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂×D₄	—	4	2	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	284	4	no	—
101	Q₁₆	—	284	4	no	—
102	D₈	—	4	4	no	—
103	C₂×C₈	#11765850	1	4	yes	—
104	C₂×C₈	D₄×D₇₁	1	2	yes	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	284	4	no	—
107	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
108	C₁₆	—	71	8	no	—
109	C₂³	—	8	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	568	2	no	—
112	Q₈	—	568	2	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	4	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	568	2	no	—
117	C₂×C₄	#14556124	1	1	yes	characteristic
118	C₂×C₄	C₂×#11765850	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	D₈×D₇₁	1	4	yes	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#17986375	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	1136	2	no	—
126	C₄	#17986374	1	2	yes	—
127	C₄	#17986439	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#22587954	1	1	yes	characteristic
130	C₂	#22590612	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590595

C₂

yes

maximal

#22590592

C₂

yes

maximal, complemented

#22587958

C₂

yes

maximal, characteristic, complemented

#22586257

C₂

yes

maximal, characteristic, complemented

C₇₁×#1709

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×C₇₁⋊C₁₆

C₂²

yes

—

#494×C₇₁

C₂²

yes

characteristic

#493×C₇₁

C₂²

yes

characteristic

#486×C₇₁

—

C₈²×C₇₁

C₂²

yes

characteristic

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

#14556128

—

#14556074

—

#14556072

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556062

—

C₂×Q₁₆×C₇₁

—

C₂×D₈×C₇₁

—

C₄⋊Q₈×C₇₁

—

#127×C₇₁

—

#102×C₇₁

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

D₈

yes

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

Q₁₆×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×D₈

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#493

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

#493

yes

—

C₄×C₇₁

#510

yes

—

Dic₇₁

—

#12520

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1712

yes

characteristic

C₂×C₇₁

#2251

yes

—

—

—

—

D₇₁

yes

characteristic

#1311

—

C₇₁

#12520

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

142

—

C₂×SD₃₂

—

142

—

#500

—

142

—

#494

—

#494

C₂×D₇₁

yes

characteristic

#493

C₂×D₇₁

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₇₁

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

142

—

C₂×Q₁₆

—

142

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

#127

—

Q₃₂

—

284

—

SD₃₂

—

284

—

C₂×C₁₆

—

#107

—

284

—

#102

—

C₄×C₈

C₂²⋊D₇₁

yes

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

284

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

284

—

101

Q₁₆

—

284

—

102

D₈

—

103

C₂×C₈

#11765850

yes

—

104

C₂×C₈

D₄×D₇₁

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

284

—

107

C₄²

C₂×C₂²⋊D₇₁

yes

characteristic

108

C₁₆

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

568

—

112

Q₈

—

568

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

568

—

117

C₂×C₄

#14556124

yes

characteristic

118

C₂×C₄

C₂×#11765850

yes

—

119

C₈

—

120

C₈

D₈×D₇₁

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#17986375

yes

characteristic

124

C₄

—

125

C₄

—

1136

—

126

C₄

#17986374

yes

—

127

C₄

#17986439

yes

—

128

C₂

—

129

C₂

#22587954

yes

characteristic

130

C₂

#22590612

yes

—

	subgroup	G/H
center	C₂²	#17986375
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1709	C₂
socle	C₂²×C₇₁	#493
prime radical	#493×C₇₁	C₂²
hypercenter	#1709	D₇₁
hypocenter	C₇₁	#12520
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#17986375

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1709

C₂

socle

C₂²×C₇₁

#493

prime radical

#493×C₇₁

C₂²

D₇₁

C₇₁

G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #22590592	C₂	semidirect
3. #22587958	C₂	semidirect
4. #22586257	C₂	semidirect
71. C₇₁	#12520	semidirect

H ≅ G/N

type

2. #22590592

C₂

semidirect

3. #22587958

C₂

semidirect

4. #22586257

C₂

semidirect

71. C₇₁

#12520

semidirect

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

subgroup	index
#22590595	2
#22590592	2
#22587958	2
#22586257	2
C₇₁×#1709	2
#12520	71

subgroup

index

C₇₁×#1709

Subgroups — #31912186

Conjugacy classes of subgroups

Complements

Maximal subgroups