Group #31774467 — subgroups

Subgroups — #31774467

Order 17344.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#16864653	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#22576160	C₂	1	1	yes	maximal, characteristic, complemented
4	#22576158	C₂	1	1	yes	maximal, characteristic, complemented
5	#22576147	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576142	C₂	1	4	yes	maximal, complemented
7	#22576137	C₂	1	4	yes	maximal, complemented
8	#22576117	C₂	1	1	yes	maximal, characteristic, complemented
9	#124×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864653	—	2	4	no	—
11	C₂×Dic₅₄₂	C₂²	1	2	yes	—
12	C₂×Dic₅₄₂	C₂²	1	1	yes	characteristic
13	C₂×Dic₅₄₂	C₂²	1	1	yes	characteristic
14	Q₈×D₂₇₁	—	2	4	no	—
15	C₂×C₂²⋊D₂₇₁	C₂²	1	2	yes	—
16	C₂²×Dic₂₇₁	C₂²	1	2	yes	—
17	C₂×C₄×D₂₇₁	C₂²	1	2	yes	—
18	C₂×C₄×D₂₇₁	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₂₇₁	C₂²	1	4	yes	—
20	Dic₂₇₁⋊C₄	C₂²	1	2	yes	—
21	Dic₂₇₁⋊C₄	C₂²	1	4	yes	—
22	C₂²⋊Dic₂₇₁	C₂²	1	4	yes	—
23	(C₂×D₂₇₁)⋊C₄	C₂²	1	2	yes	—
24	C₄×Dic₂₇₁	C₂²	1	1	yes	characteristic
25	C₂×Q₈×C₂₇₁	C₂²	1	1	yes	characteristic
26	C₂×D₄×C₂₇₁	C₂²	1	1	yes	characteristic
27	C₂²⋊C₄×C₂₇₁	C₂²	1	4	yes	—
28	C₄²×C₂₇₁	C₂²	1	1	yes	characteristic
29	C₂²×D₂₇₁	C₂³	1	1	yes	characteristic
30	Dic₅₄₂	—	2	2	no	—
31	Dic₅₄₂	—	2	2	no	—
32	Dic₅₄₂	—	2	4	no	—
33	C₂²⋊D₂₇₁	—	2	4	no	—
34	C₂×Dic₂₇₁	—	2	4	no	—
35	C₂×Dic₂₇₁	C₂³	1	2	yes	—
36	C₂×Dic₂₇₁	C₂³	1	2	yes	—
37	C₂×Dic₂₇₁	C₂³	1	2	yes	—
38	C₂×Dic₂₇₁	C₂³	1	1	yes	characteristic
39	C₄×D₂₇₁	—	2	4	no	—
40	C₄×D₂₇₁	D₄	1	4	yes	—
41	C₂³×C₂₇₁	C₂³	1	2	yes	—
42	Q₈×C₂₇₁	—	2	2	no	—
43	D₄×C₂₇₁	—	2	2	no	—
44	C₂×C₄×C₂₇₁	C₂³	1	2	yes	—
45	C₂×C₄×C₂₇₁	C₂³	1	2	yes	—
46	C₂×C₄×C₂₇₁	C₂³	1	1	yes	characteristic
47	C₂²×C₂₇₁	—	2	2	no	—
48	C₂²×C₂₇₁	—	2	4	no	—
49	C₂²×C₂₇₁	C₂⁴	1	1	yes	characteristic
50	C₂×D₂₇₁	—	2	2	no	—
51	C₂×D₂₇₁	C₂×D₄	1	2	yes	—
52	C₄×C₂₇₁	—	2	2	no	—
53	C₄×C₂₇₁	—	2	2	no	—
54	C₄×C₂₇₁	C₂×D₄	1	2	yes	—
55	Dic₂₇₁	—	2	2	no	—
56	Dic₂₇₁	—	2	2	no	—
57	Dic₂₇₁	—	2	2	no	—
58	Dic₂₇₁	C₂×D₄	1	2	yes	—
59	C₂×C₂₇₁	—	4	2	no	—
60	C₂×C₂₇₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×C₂₇₁	2₋¹⁺⁴	1	2	yes	—
62	D₂₇₁	—	2	2	no	—
63	C₂₇₁	#536	1	1	yes	characteristic, complemented, Sylow
64	#536	—	271	1	no	maximal, Sylow
65	C₂×#41	—	271	1	no	—
66	C₂²×Q₈	—	271	1	no	—
67	C₄⋊Q₈	—	271	1	no	—
68	C₄⋊Q₈	—	271	1	no	—
69	#124	—	271	1	no	—
70	#124	D₂₇₁	1	1	yes	characteristic, complemented
71	#123	—	271	4	no	—
72	C₂²⋊Q₈	—	271	4	no	—
73	#117	—	271	1	no	—
74	#41	—	542	4	no	—
75	C₂×Q₈	—	271	2	no	—
76	C₂×Q₈	—	271	1	no	—
77	C₂×Q₈	—	271	1	no	—
78	C₂×Q₈	—	542	4	no	—
79	C₂×Q₈	C₂×D₂₇₁	1	1	yes	characteristic
80	C₂×D₄	—	271	2	no	—
81	C₂×D₄	C₂×D₂₇₁	1	1	yes	characteristic
82	C₂²×C₄	—	271	1	no	—
83	C₂²×C₄	—	271	2	no	—
84	C₂²×C₄	—	271	2	no	—
85	C₄⋊C₄	—	271	2	no	—
86	C₄⋊C₄	—	271	4	no	—
87	C₄⋊C₄	—	271	4	no	—
88	C₂²⋊C₄	—	271	4	no	—
89	C₂²⋊C₄	—	271	2	no	—
90	C₂²⋊C₄	C₂×D₂₇₁	1	4	yes	—
91	C₄²	—	271	1	no	—
92	C₄²	C₂×D₂₇₁	1	1	yes	characteristic
93	C₂³	—	271	1	no	—
94	C₂³	C₂²×D₂₇₁	1	2	yes	—
95	Q₈	—	2	2	no	—
96	Q₈	—	542	4	no	—
97	Q₈	—	542	2	no	—
98	Q₈	—	542	2	no	—
99	D₄	—	2	2	no	—
100	D₄	—	542	4	no	—
101	C₂×C₄	—	271	1	no	—
102	C₂×C₄	—	271	4	no	—
103	C₂×C₄	—	271	2	no	—
104	C₂×C₄	—	271	2	no	—
105	C₂×C₄	—	271	2	no	—
106	C₂×C₄	—	542	4	no	—
107	C₂×C₄	—	542	4	no	—
108	C₂×C₄	C₂²×D₂₇₁	1	2	yes	—
109	C₂×C₄	C₂²×D₂₇₁	1	2	yes	—
110	C₂×C₄	C₂²×D₂₇₁	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	2	4	no	—
113	C₂²	—	271	2	no	—
114	C₂²	—	542	2	no	—
115	C₂²	C₂³×D₂₇₁	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	271	2	no	—
119	C₄	—	542	2	no	—
120	C₄	—	542	2	no	—
121	C₄	—	542	2	no	—
122	C₄	D₄×D₂₇₁	1	2	yes	—
123	C₂	—	4	2	no	—
124	C₂	—	542	2	no	—
125	C₂	C₂×D₄×D₂₇₁	1	1	yes	characteristic
126	C₂	#22576207	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16864653

C₂

yes

maximal, characteristic

C₂×Q₈×D₂₇₁

C₂

yes

maximal, characteristic, complemented

#22576160

C₂

yes

maximal, characteristic, complemented

#22576158

C₂

yes

maximal, characteristic, complemented

#22576147

C₂

yes

maximal, characteristic, complemented

#22576142

C₂

yes

maximal, complemented

#22576137

C₂

yes

maximal, complemented

#22576117

C₂

yes

maximal, characteristic, complemented

#124×C₂₇₁

C₂

yes

maximal, characteristic, complemented

#16864653

—

C₂×Dic₅₄₂

C₂²

yes

—

C₂×Dic₅₄₂

C₂²

yes

characteristic

C₂×Dic₅₄₂

C₂²

yes

characteristic

Q₈×D₂₇₁

—

C₂×C₂²⋊D₂₇₁

C₂²

yes

—

C₂²×Dic₂₇₁

C₂²

yes

—

C₂×C₄×D₂₇₁

C₂²

yes

—

C₂×C₄×D₂₇₁

C₂²

yes

characteristic

C₄⋊Dic₂₇₁

C₂²

yes

—

Dic₂₇₁⋊C₄

C₂²

yes

—

Dic₂₇₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₂₇₁

C₂²

yes

—

(C₂×D₂₇₁)⋊C₄

C₂²

yes

—

C₄×Dic₂₇₁

C₂²

yes

characteristic

C₂×Q₈×C₂₇₁

C₂²

yes

characteristic

C₂×D₄×C₂₇₁

C₂²

yes

characteristic

C₂²⋊C₄×C₂₇₁

C₂²

yes

—

C₄²×C₂₇₁

C₂²

yes

characteristic

C₂²×D₂₇₁

C₂³

yes

characteristic

Dic₅₄₂

—

Dic₅₄₂

—

Dic₅₄₂

—

C₂²⋊D₂₇₁

—

C₂×Dic₂₇₁

—

C₂×Dic₂₇₁

C₂³

yes

—

C₂×Dic₂₇₁

C₂³

yes

—

C₂×Dic₂₇₁

C₂³

yes

—

C₂×Dic₂₇₁

C₂³

yes

characteristic

C₄×D₂₇₁

—

C₄×D₂₇₁

D₄

yes

—

C₂³×C₂₇₁

C₂³

yes

—

Q₈×C₂₇₁

—

D₄×C₂₇₁

—

C₂×C₄×C₂₇₁

C₂³

yes

—

C₂×C₄×C₂₇₁

C₂³

yes

—

C₂×C₄×C₂₇₁

C₂³

yes

characteristic

C₂²×C₂₇₁

—

C₂²×C₂₇₁

—

C₂²×C₂₇₁

C₂⁴

yes

characteristic

C₂×D₂₇₁

—

C₂×D₂₇₁

C₂×D₄

yes

—

C₄×C₂₇₁

—

C₄×C₂₇₁

—

C₄×C₂₇₁

C₂×D₄

yes

—

Dic₂₇₁

—

Dic₂₇₁

—

Dic₂₇₁

—

Dic₂₇₁

C₂×D₄

yes

—

C₂×C₂₇₁

—

C₂×C₂₇₁

C₂²×D₄

yes

characteristic

C₂×C₂₇₁

2₋¹⁺⁴

yes

—

D₂₇₁

—

C₂₇₁

#536

yes

characteristic, complemented, Sylow

#536

—

271

maximal, Sylow

C₂×#41

—

271

—

C₂²×Q₈

—

271

—

C₄⋊Q₈

—

271

—

C₄⋊Q₈

—

271

—

#124

—

271

—

#124

D₂₇₁

yes

characteristic, complemented

#123

—

271

—

C₂²⋊Q₈

—

271

—

#117

—

271

—

#41

—

542

—

C₂×Q₈

—

271

—

C₂×Q₈

—

271

—

C₂×Q₈

—

271

—

C₂×Q₈

—

542

—

C₂×Q₈

C₂×D₂₇₁

yes

characteristic

C₂×D₄

—

271

—

C₂×D₄

C₂×D₂₇₁

yes

characteristic

C₂²×C₄

—

271

—

C₂²×C₄

—

271

—

C₂²×C₄

—

271

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

271

—

C₄⋊C₄

—

271

—

C₂²⋊C₄

—

271

—

C₂²⋊C₄

—

271

—

C₂²⋊C₄

C₂×D₂₇₁

yes

—

C₄²

—

271

—

C₄²

C₂×D₂₇₁

yes

characteristic

C₂³

—

271

—

C₂³

C₂²×D₂₇₁

yes

—

Q₈

—

Q₈

—

542

—

Q₈

—

542

—

Q₈

—

542

—

D₄

—

100

D₄

—

542

—

101

C₂×C₄

—

271

—

102

C₂×C₄

—

271

—

103

C₂×C₄

—

271

—

104

C₂×C₄

—

271

—

105

C₂×C₄

—

271

—

106

C₂×C₄

—

542

—

107

C₂×C₄

—

542

—

108

C₂×C₄

C₂²×D₂₇₁

yes

—

109

C₂×C₄

C₂²×D₂₇₁

yes

—

110

C₂×C₄

C₂²×D₂₇₁

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

271

—

114

C₂²

—

542

—

115

C₂²

C₂³×D₂₇₁

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

271

—

119

C₄

—

542

—

120

C₄

—

542

—

121

C₄

—

542

—

122

C₄

D₄×D₂₇₁

yes

—

123

C₂

—

124

C₂

—

542

—

125

C₂

C₂×D₄×D₂₇₁

yes

characteristic

126

C₂

#22576207

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₇₁
commutator, socle, cosocle	C₂²×C₂₇₁	C₂⁴
Fitting	#124×C₂₇₁	C₂
prime radical	C₂×#16864653	C₂
hypercenter	#124	D₂₇₁
hypocenter	C₂₇₁	#536
LCS	G ▷ C₂²×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₇₁

commutator, socle, cosocle

C₂²×C₂₇₁

C₂⁴

Fitting

#124×C₂₇₁

C₂

C₂

D₂₇₁

C₂₇₁

G ▷ C₂²×C₂₇₁ ▷ C₂₇₁

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
2. C₂×Q₈×D₂₇₁	C₂	semidirect
3. #22576160	C₂	semidirect
4. #22576158	C₂	semidirect
5. #22576147	C₂	semidirect
6. #22576142	C₂	semidirect
7. #22576137	C₂	semidirect
8. #22576117	C₂	semidirect
9. #124×C₂₇₁	C₂	semidirect
63. C₂₇₁	#536	semidirect
70. #124	D₂₇₁	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₂₇₁

C₂

semidirect

3. #22576160

C₂

semidirect

4. #22576158

C₂

semidirect

5. #22576147

C₂

semidirect

6. #22576142

C₂

semidirect

7. #22576137

C₂

semidirect

8. #22576117

C₂

semidirect

9. #124×C₂₇₁

C₂

semidirect

63. C₂₇₁

#536

semidirect

70. #124

D₂₇₁

semidirect

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

subgroup	index
C₂×#16864653	2
C₂×Q₈×D₂₇₁	2
#22576160	2
#22576158	2
#22576147	2
#22576142	2
#22576137	2
#22576117	2
#124×C₂₇₁	2
#536	271

subgroup

index

C₂×Q₈×D₂₇₁

#124×C₂₇₁

271

Subgroups — #31774467

Conjugacy classes of subgroups

Complements

Maximal subgroups