Group #31912119 — subgroups

Subgroups — #31912119

Order 18176.

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

normal subgroups only

This group has 302 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	#22590622	C₂	1	1	yes	maximal, characteristic, complemented
2	#22590563	C₂	1	2	yes	maximal, complemented
3	#22587929	C₂	1	2	yes	maximal, complemented
4	#22586256	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#2246	C₂	1	1	yes	maximal, characteristic
6	#17986404	C₂²	1	2	yes	—
7	#17986371	—	2	2	no	—
8	#17986345	—	2	2	no	—
9	#17985827	—	2	2	no	—
10	#17985800	—	2	4	no	—
11	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×D₁₆×C₇₁	—	2	2	no	—
14	#493×C₇₁	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#11765853	—	4	4	no	—
17	#14556128	—	2	2	no	—
18	C₂×C₇₁⋊C₁₆	D₄	1	2	yes	—
19	#14556062	—	2	2	no	—
20	#14556062	—	2	2	no	—
21	#14556049	—	4	4	no	—
22	C₄×C₇₁⋊C₈	D₄	1	2	yes	—
23	C₂×D₈×C₇₁	—	2	2	no	—
24	C₂×D₈×C₇₁	—	2	2	no	—
25	#127×C₇₁	—	2	2	no	—
26	D₁₆×C₇₁	—	4	4	no	—
27	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₂	—	4	2	no	—
30	#11765853	—	8	8	no	—
31	C₂×C₇₁⋊C₈	D₈	1	4	yes	—
32	C₄⋊Dic₇₁	—	4	2	no	—
33	C₄⋊Dic₇₁	—	4	2	no	—
34	C₇₁⋊C₁₆	—	2	2	no	—
35	C₂×D₄×C₇₁	—	4	2	no	—
36	C₂×D₄×C₇₁	—	4	2	no	—
37	C₂×D₄×C₇₁	—	4	2	no	—
38	D₈×C₇₁	—	4	4	no	—
39	D₈×C₇₁	—	4	4	no	—
40	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₁	SD₁₆	1	4	yes	complemented
44	C₂³×C₇₁	—	8	2	no	—
45	D₄×C₇₁	—	8	2	no	—
46	D₄×C₇₁	—	8	2	no	—
47	D₄×C₇₁	—	8	4	no	—
48	D₄×C₇₁	—	8	4	no	—
49	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
50	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
51	C₂×Dic₇₁	—	8	2	no	—
52	Dic₁₄₂	—	8	4	no	—
53	C₈×C₇₁	C₂×SD₁₆	1	2	yes	—
54	C₈×C₇₁	#135	1	2	yes	—
55	C₇₁⋊C₈	—	2	4	no	—
56	C₂²×C₇₁	—	16	2	no	—
57	C₂²×C₇₁	—	16	2	no	—
58	C₂²×C₇₁	—	16	2	no	—
59	C₂²×C₇₁	#493	1	1	yes	characteristic
60	C₄×C₇₁	#492	1	1	yes	characteristic
61	C₄×C₇₁	#495	1	1	yes	characteristic
62	C₄×C₇₁	#509	1	4	yes	—
63	Dic₇₁	—	16	2	no	—
64	#12506	—	71	1	no	maximal, Sylow
65	C₂×C₇₁	—	32	2	no	—
66	C₂×C₇₁	#1706	1	1	yes	characteristic
67	C₂×C₇₁	#2250	1	2	yes	—
68	#2254	—	71	1	no	—
69	#2246	D₇₁	1	1	yes	characteristic
70	#2240	—	71	2	no	—
71	#1706	—	71	2	no	—
72	#1311	—	71	1	no	—
73	C₇₁	#12506	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₁₆	—	2	2	no	—
75	#500	—	71	2	no	—
76	#493	C₂×D₇₁	1	2	yes	—
77	#492	—	142	2	no	—
78	#486	—	142	2	no	—
79	#367	—	142	2	no	—
80	#357	—	142	4	no	—
81	C₄×C₁₆	—	71	1	no	—
82	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
83	C₈²	—	71	1	no	—
84	C₂×SD₁₆	—	284	4	no	—
85	C₂×D₈	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₄⋊Q₈	—	142	2	no	—
88	#127	—	2	2	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	71	2	no	—
91	C₂×C₁₆	C₂²⋊D₇₁	1	2	yes	—
92	#107	—	142	2	no	—
93	#107	—	142	2	no	—
94	#102	—	284	4	no	—
95	C₄×C₈	—	71	2	no	—
96	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
97	C₂×Q₈	—	284	2	no	—
98	C₂×D₄	—	4	2	no	—
99	C₂×D₄	—	4	2	no	—
100	C₂×D₄	—	4	2	no	—
101	SD₁₆	—	568	8	no	—
102	D₈	—	4	4	no	—
103	D₈	—	4	4	no	—
104	C₂×C₈	—	71	4	no	—
105	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
106	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
107	C₄⋊C₄	—	284	2	no	—
108	C₄⋊C₄	—	284	2	no	—
109	C₄²	D₄×D₇₁	1	1	yes	characteristic
110	C₁₆	—	142	2	no	—
111	C₁₆	#11765853	1	4	yes	complemented
112	C₂³	—	8	2	no	—
113	Q₈	—	568	4	no	—
114	D₄	—	8	2	no	—
115	D₄	—	8	2	no	—
116	D₄	—	8	4	no	—
117	D₄	—	8	4	no	—
118	C₂×C₄	—	568	2	no	—
119	C₂×C₄	#14556124	1	1	yes	characteristic
120	C₂×C₄	D₈×D₇₁	1	2	yes	—
121	C₈	—	142	4	no	—
122	C₈	C₂×#11765853	1	2	yes	—
123	C₈	#14556147	1	2	yes	—
124	C₂²	—	16	2	no	—
125	C₂²	—	16	2	no	—
126	C₂²	—	16	2	no	—
127	C₂²	#17986374	1	1	yes	characteristic
128	C₄	—	1136	2	no	—
129	C₄	#17986371	1	1	yes	characteristic
130	C₄	#17986382	1	1	yes	characteristic
131	C₄	#17986433	1	4	yes	—
132	C₂	—	32	2	no	—
133	C₂	#22587929	1	1	yes	characteristic
134	C₂	#22590603	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590622

C₂

yes

maximal, characteristic, complemented

#22590563

C₂

yes

maximal, complemented

#22587929

C₂

yes

maximal, complemented

#22586256

C₂

yes

maximal, characteristic, complemented

C₇₁×#2246

C₂

yes

maximal, characteristic

#17986404

C₂²

yes

—

—

—

—

—

C₄×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₈×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×D₁₆×C₇₁

—

#493×C₇₁

C₂²

yes

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×#11765853

—

#14556128

—

C₂×C₇₁⋊C₁₆

D₄

yes

—

#14556062

—

#14556062

—

#14556049

—

C₄×C₇₁⋊C₈

D₄

yes

—

C₂×D₈×C₇₁

—

C₂×D₈×C₇₁

—

#127×C₇₁

—

D₁₆×C₇₁

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

#11765853

—

C₂×C₇₁⋊C₈

D₈

yes

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

D₈×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

SD₁₆

yes

complemented

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

Dic₁₄₂

—

C₈×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

#135

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

#12506

—

maximal, Sylow

C₂×C₇₁

—

C₂×C₇₁

#1706

yes

characteristic

C₂×C₇₁

#2250

yes

—

#2254

—

#2246

D₇₁

yes

—

—

—

C₇₁

#12506

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

#500

—

#493

C₂×D₇₁

yes

—

#492

—

142

—

#486

—

142

—

#367

—

142

—

#357

—

142

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

—

C₂×SD₁₆

—

284

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

142

—

#127

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₁

yes

—

#107

—

142

—

#107

—

142

—

#102

—

284

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

SD₁₆

—

568

—

102

D₈

—

103

D₈

—

104

C₂×C₈

—

105

C₂×C₈

D₄×D₇₁

yes

characteristic

106

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

107

C₄⋊C₄

—

284

—

108

C₄⋊C₄

—

284

—

109

C₄²

D₄×D₇₁

yes

characteristic

110

C₁₆

—

142

—

111

C₁₆

#11765853

yes

complemented

112

C₂³

—

113

Q₈

—

568

—

114

D₄

—

115

D₄

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

568

—

119

C₂×C₄

#14556124

yes

characteristic

120

C₂×C₄

D₈×D₇₁

yes

—

121

C₈

—

142

—

122

C₈

C₂×#11765853

yes

—

123

C₈

#14556147

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

#17986374

yes

characteristic

128

C₄

—

1136

—

129

C₄

#17986371

yes

characteristic

130

C₄

#17986382

yes

characteristic

131

C₄

#17986433

yes

—

132

C₂

—

133

C₂

#22587929

yes

characteristic

134

C₂

#22590603

yes

—

	subgroup	G/H
center	C₂²	#17986374
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting, prime radical	C₇₁×#2246	C₂
socle	C₂²×C₇₁	#493
hypercenter	#2246	D₇₁
hypocenter	C₇₁	#12506
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#17986374

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting, prime radical

C₇₁×#2246

C₂

C₂²×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #22590622	C₂	semidirect
2. #22590563	C₂	semidirect
3. #22587929	C₂	semidirect
4. #22586256	C₂	semidirect
43. C₁₆×C₇₁	SD₁₆	semidirect
73. C₇₁	#12506	semidirect
111. C₁₆	#11765853	semidirect

H ≅ G/N

type

1. #22590622

C₂

semidirect

2. #22590563

C₂

semidirect

3. #22587929

C₂

semidirect

4. #22586256

C₂

semidirect

43. C₁₆×C₇₁

SD₁₆

semidirect

73. C₇₁

#12506

semidirect

111. C₁₆

#11765853

semidirect

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

subgroup	index
#22590622	2
#22590563	2
#22587929	2
#22586256	2
C₇₁×#2246	2
#12506	71

subgroup

index

C₇₁×#2246

Subgroups — #31912119

Conjugacy classes of subgroups

Complements

Maximal subgroups