Group #31912122 — subgroups

Subgroups — #31912122

Order 18176.

302 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22590590	C₂	1	1	yes	maximal, characteristic
3	#22590564	C₂	1	2	yes	maximal, complemented
4	#22587932	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₃₆	—	2	2	no	—
7	#17986404	C₂²	1	2	yes	—
8	#17986376	C₂²	1	2	yes	—
9	#17986372	—	2	2	no	—
10	#17986346	—	2	2	no	—
11	#17985827	—	2	2	no	—
12	#17985801	—	2	4	no	—
13	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₈²×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×#11765854	—	4	4	no	—
17	C₂×D₅₆₈	—	2	2	no	—
18	C₂×D₅₆₈	—	2	2	no	—
19	#14556128	—	2	2	no	—
20	#14556125	—	2	2	no	—
21	D₁₁₃₆	—	4	4	no	—
22	#14556062	—	2	2	no	—
23	#14556062	—	2	2	no	—
24	#14556050	—	4	4	no	—
25	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
27	C₄×C₈×C₇₁	D₄	1	2	yes	—
28	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₄₂	—	4	2	no	—
30	C₂×D₂₈₄	—	4	2	no	—
31	C₂×D₂₈₄	—	4	2	no	—
32	C₂×D₂₈₄	—	4	2	no	—
33	#11765854	—	8	8	no	—
34	D₅₆₈	—	4	4	no	—
35	D₅₆₈	—	4	4	no	—
36	C₄⋊Dic₇₁	—	4	2	no	—
37	C₄⋊Dic₇₁	—	4	2	no	—
38	C₂×C₈×C₇₁	D₈	1	4	yes	—
39	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₇₁	—	2	2	no	—
43	C₁₆×C₇₁	SD₁₆	1	4	yes	complemented
44	C₂²×D₇₁	—	8	2	no	—
45	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
47	C₂×Dic₇₁	—	8	2	no	—
48	D₂₈₄	—	8	2	no	—
49	D₂₈₄	—	8	2	no	—
50	D₂₈₄	—	8	4	no	—
51	D₂₈₄	—	8	4	no	—
52	Dic₁₄₂	—	8	4	no	—
53	C₈×C₇₁	—	2	4	no	—
54	C₈×C₇₁	C₂×SD₁₆	1	2	yes	—
55	C₈×C₇₁	#135	1	2	yes	—
56	C₂²×C₇₁	#493	1	1	yes	characteristic
57	C₂×D₇₁	—	16	2	no	—
58	C₂×D₇₁	—	16	2	no	—
59	C₂×D₇₁	—	16	2	no	—
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	#1706	1	1	yes	characteristic
66	C₂×C₇₁	#2250	1	2	yes	—
67	D₇₁	—	32	2	no	—
68	#2254	—	71	1	no	—
69	#2246	—	71	1	no	—
70	#2240	—	71	2	no	—
71	#1706	—	71	2	no	—
72	#1311	D₇₁	1	1	yes	characteristic, complemented
73	C₇₁	#12506	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₁₆	—	142	2	no	—
75	#500	—	71	2	no	—
76	#493	—	71	2	no	—
77	#492	—	142	2	no	—
78	#486	—	142	2	no	—
79	#367	—	142	2	no	—
80	#357	—	142	4	no	—
81	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
82	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
83	C₈²	C₂×D₇₁	1	1	yes	characteristic
84	C₂×SD₁₆	—	284	4	no	—
85	C₂×D₈	—	142	2	no	—
86	C₂×D₈	—	142	2	no	—
87	C₄⋊Q₈	—	142	2	no	—
88	#127	—	142	2	no	—
89	D₁₆	—	284	4	no	—
90	C₂×C₁₆	D₂₈₄	1	2	yes	—
91	C₂×C₁₆	D₂₈₄	1	2	yes	—
92	#107	—	142	2	no	—
93	#107	—	142	2	no	—
94	#102	—	284	4	no	—
95	C₄×C₈	D₂₈₄	1	2	yes	—
96	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
97	C₂×Q₈	—	284	2	no	—
98	C₂×D₄	—	284	2	no	—
99	C₂×D₄	—	284	2	no	—
100	C₂×D₄	—	284	2	no	—
101	SD₁₆	—	568	8	no	—
102	D₈	—	284	4	no	—
103	D₈	—	284	4	no	—
104	C₂×C₈	D₅₆₈	1	4	yes	—
105	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
106	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
107	C₄⋊C₄	—	284	2	no	—
108	C₄⋊C₄	—	284	2	no	—
109	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
110	C₁₆	—	2	2	no	—
111	C₁₆	#11765854	1	4	yes	complemented
112	C₂³	—	568	2	no	—
113	Q₈	—	568	4	no	—
114	D₄	—	568	2	no	—
115	D₄	—	568	2	no	—
116	D₄	—	568	4	no	—
117	D₄	—	568	4	no	—
118	C₂×C₄	—	568	2	no	—
119	C₂×C₄	#14556125	1	1	yes	characteristic
120	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	C₂×#11765854	1	2	yes	—
123	C₈	#14556150	1	2	yes	—
124	C₂²	—	1136	2	no	—
125	C₂²	—	1136	2	no	—
126	C₂²	—	1136	2	no	—
127	C₂²	#17986376	1	1	yes	characteristic
128	C₄	—	1136	2	no	—
129	C₄	#17986372	1	1	yes	characteristic
130	C₄	#17986383	1	1	yes	characteristic
131	C₄	#17986437	1	4	yes	—
132	C₂	—	2272	2	no	—
133	C₂	#22587932	1	1	yes	characteristic
134	C₂	#22590606	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22590622

C₂

yes

maximal, characteristic, complemented

#22590590

C₂

yes

maximal, characteristic

#22590564

C₂

yes

maximal, complemented

#22587932

C₂

yes

maximal, complemented

C₇₁×#1311

C₂

yes

maximal, characteristic, complemented

C₂×D₁₁₃₆

—

#17986404

C₂²

yes

—

#17986376

C₂²

yes

—

—

—

—

—

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₁₆×C₇₁

C₂²

yes

characteristic

C₈²×C₇₁

C₂²

yes

characteristic

C₂×#11765854

—

C₂×D₅₆₈

—

C₂×D₅₆₈

—

#14556128

—

#14556125

—

D₁₁₃₆

—

#14556062

—

#14556062

—

#14556050

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₂×C₁₆×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

D₄

yes

—

C₄×C₈×C₇₁

C₂³

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

#11765854

—

D₅₆₈

—

D₅₆₈

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂×C₈×C₇₁

D₈

yes

—

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

characteristic

C₄²×C₇₁

C₂×D₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

SD₁₆

yes

complemented

C₂²×D₇₁

—

C₂×C₄×C₇₁

#127

yes

characteristic

C₂×C₄×C₇₁

C₂×D₈

yes

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×SD₁₆

yes

—

C₈×C₇₁

#135

yes

—

C₂²×C₇₁

#493

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

D₇₁

—

—

—

—

—

D₇₁

yes

characteristic, complemented

C₇₁

#12506

yes

characteristic, complemented, Sylow

C₂×D₁₆

—

142

—

#500

—

#493

—

#492

—

142

—

#486

—

142

—

#367

—

142

—

#357

—

142

—

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₄×C₁₆

C₂×D₇₁

yes

characteristic

C₈²

C₂×D₇₁

yes

characteristic

C₂×SD₁₆

—

284

—

C₂×D₈

—

142

—

C₂×D₈

—

142

—

C₄⋊Q₈

—

142

—

#127

—

142

—

D₁₆

—

284

—

C₂×C₁₆

D₂₈₄

yes

—

C₂×C₁₆

D₂₈₄

yes

—

#107

—

142

—

#107

—

142

—

#102

—

284

—

C₄×C₈

D₂₈₄

yes

—

C₄×C₈

C₂²×D₇₁

yes

characteristic

C₂×Q₈

—

284

—

C₂×D₄

—

284

—

C₂×D₄

—

284

—

100

C₂×D₄

—

284

—

101

SD₁₆

—

568

—

102

D₈

—

284

—

103

D₈

—

284

—

104

C₂×C₈

D₅₆₈

yes

—

105

C₂×C₈

C₂×D₂₈₄

yes

characteristic

106

C₂×C₈

C₂×D₂₈₄

yes

characteristic

107

C₄⋊C₄

—

284

—

108

C₄⋊C₄

—

284

—

109

C₄²

C₂×D₂₈₄

yes

characteristic

110

C₁₆

—

111

C₁₆

#11765854

yes

complemented

112

C₂³

—

568

—

113

Q₈

—

568

—

114

D₄

—

568

—

115

D₄

—

568

—

116

D₄

—

568

—

117

D₄

—

568

—

118

C₂×C₄

—

568

—

119

C₂×C₄

#14556125

yes

characteristic

120

C₂×C₄

C₂×D₅₆₈

yes

—

121

C₈

—

122

C₈

C₂×#11765854

yes

—

123

C₈

#14556150

yes

—

124

C₂²

—

1136

—

125

C₂²

—

1136

—

126

C₂²

—

1136

—

127

C₂²

#17986376

yes

characteristic

128

C₄

—

1136

—

129

C₄

#17986372

yes

characteristic

130

C₄

#17986383

yes

characteristic

131

C₄

#17986437

yes

—

132

C₂

—

2272

—

133

C₂

#22587932

yes

characteristic

134

C₂

#22590606

yes

—

	subgroup	G/H
center	C₂²	#17986376
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1311	C₂
socle	C₂²×C₇₁	#493
prime radical	#22590590	C₂
hypercenter	#1311	D₇₁
hypocenter	C₇₁	#12506
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#17986376

commutator, cosocle

C₄×C₈×C₇₁

C₂³

Frattini

C₄×C₈

C₂²×D₇₁

Fitting

C₇₁×#1311

C₂

C₂²×C₇₁

C₂

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #22590622	C₂	semidirect
3. #22590564	C₂	semidirect
4. #22587932	C₂	semidirect
5. C₇₁×#1311	C₂	semidirect
43. C₁₆×C₇₁	SD₁₆	semidirect
72. #1311	D₇₁	semidirect
73. C₇₁	#12506	semidirect
111. C₁₆	#11765854	semidirect

H ≅ G/N

type

1. #22590622

C₂

semidirect

3. #22590564

C₂

semidirect

4. #22587932

C₂

semidirect

5. C₇₁×#1311

C₂

semidirect

43. C₁₆×C₇₁

SD₁₆

semidirect

72. #1311

D₇₁

semidirect

73. C₇₁

#12506

semidirect

111. C₁₆

#11765854

semidirect

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

subgroup	index
#22590622	2
#22590590	2
#22590564	2
#22587932	2
C₇₁×#1311	2
#12506	71

subgroup

index

C₇₁×#1311

Subgroups — #31912122

Conjugacy classes of subgroups

Complements

Maximal subgroups