Group #14530052 — subgroups

Subgroups — #14530052

Order 2176.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂²×Dic₆₈	C₂	1	1	yes	maximal, characteristic
2	#11762817	C₂	1	2	yes	maximal
3	C₂×#10587495	C₂	1	2	yes	maximal
4	#11762574	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#424	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₄	C₂²	1	2	yes	—
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	C₁₇×#117	C₂²	1	1	yes	characteristic
10	C₂×#63223	C₂²	1	2	yes	—
11	C₂×Dic₆₈	—	2	4	no	—
12	C₂×Dic₆₈	—	2	2	no	—
13	C₁₇×C₄⋊C₈	—	2	2	no	—
14	#10587495	—	2	4	no	—
15	#10587481	—	2	4	no	—
16	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₁₇	—	2	2	no	—
18	C₂×Dic₃₄	—	2	2	no	—
19	C₂×Dic₃₄	—	2	2	no	—
20	C₂×Dic₃₄	—	2	2	no	—
21	C₂×Dic₃₄	—	2	2	no	—
22	C₂×Dic₃₄	—	2	2	no	—
23	C₂×Dic₃₄	D₄	1	4	yes	—
24	M₁₆×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	2	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	D₄	1	2	yes	—
28	C₄⋊C₄×C₁₇	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₇	—	2	1	no	—
30	C₄²×C₁₇	—	2	1	no	—
31	#63223	—	2	2	no	—
32	#63223	—	2	2	no	—
33	C₂×#3622	—	2	2	no	—
34	Dic₆₈	—	2	8	no	—
35	C₂³×C₁₇	#41	1	1	yes	characteristic
36	C₂×C₄×C₁₇	—	2	2	no	—
37	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	#41	1	2	yes	—
41	C₂×Dic₁₇	—	2	4	no	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₂×Dic₁₇	—	4	2	no	—
44	Dic₃₄	—	2	4	no	—
45	Dic₃₄	—	2	4	no	—
46	Dic₃₄	—	4	2	no	—
47	Dic₃₄	—	4	2	no	—
48	C₈×C₁₇	—	2	2	no	—
49	C₈×C₁₇	—	4	1	no	—
50	#3622	—	4	2	no	—
51	#2047	—	17	1	no	maximal, Sylow
52	C₂²×C₁₇	—	2	2	no	—
53	C₂²×C₁₇	#121	1	1	yes	characteristic
54	C₂²×C₁₇	#123	1	2	yes	—
55	C₄×C₁₇	—	4	2	no	—
56	C₄×C₁₇	#120	1	1	yes	characteristic
57	C₄×C₁₇	#121	1	2	yes	—
58	C₄×C₁₇	#124	1	1	yes	characteristic
59	Dic₁₇	—	4	4	no	—
60	C₂²×Q₁₆	—	17	1	no	—
61	#424	D₁₇	1	1	yes	characteristic
62	#419	—	17	2	no	—
63	C₂×#101	—	17	2	no	—
64	#341	—	17	1	no	—
65	C₂×C₁₇	—	2	2	no	—
66	C₂×C₁₇	#394	1	1	yes	characteristic
67	C₂×C₁₇	#473	1	2	yes	—
68	C₂²×Q₈	—	17	2	no	—
69	C₂×Q₁₆	—	34	4	no	—
70	C₂×Q₁₆	—	34	2	no	—
71	C₂×M₁₆	—	17	2	no	—
72	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
73	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
74	#117	C₂×D₁₇	1	1	yes	characteristic
75	C₄⋊C₈	—	2	2	no	—
76	#103	—	34	4	no	—
77	#101	—	34	4	no	—
78	C₁₇	#2047	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	17	4	no	—
80	C₂×Q₈	—	34	2	no	—
81	C₂×Q₈	—	34	2	no	—
82	C₂×Q₈	—	34	2	no	—
83	C₂×Q₈	—	34	2	no	—
84	C₂×Q₈	—	34	2	no	—
85	C₂²×C₄	—	34	2	no	—
86	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
87	Q₁₆	—	34	8	no	—
88	M₁₆	—	2	2	no	—
89	M₁₆	—	34	2	no	—
90	M₁₆	—	34	2	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	34	2	no	—
94	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
95	C₄⋊C₄	D₆₈	1	2	yes	—
96	C₂²⋊C₄	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	#63252	1	1	yes	characteristic
99	Q₈	—	34	4	no	—
100	Q₈	—	34	4	no	—
101	Q₈	—	68	2	no	—
102	Q₈	—	68	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	34	4	no	—
105	C₂×C₄	—	68	2	no	—
106	C₂×C₄	—	68	2	no	—
107	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
108	C₂×C₄	D₄×D₁₇	1	2	yes	—
109	C₂×C₄	#63256	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
111	C₈	—	2	2	no	—
112	C₈	—	4	1	no	—
113	C₈	—	68	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#10587580	1	2	yes	—
116	C₂²	#10587616	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	68	4	no	—
119	C₄	#10587581	1	2	yes	—
120	C₄	#10587623	1	1	yes	characteristic
121	C₄	#10587633	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#11762726	1	1	yes	characteristic
124	C₂	#11763084	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₈

C₂

yes

maximal, characteristic

#11762817

C₂

yes

maximal

C₂×#10587495

C₂

yes

maximal

#11762574

C₂

yes

maximal, characteristic

C₁₇×#424

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₁₇×#117

C₂²

yes

characteristic

C₂×#63223

C₂²

yes

—

C₂×Dic₆₈

—

C₂×Dic₆₈

—

C₁₇×C₄⋊C₈

—

#10587495

—

#10587481

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

D₄

yes

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

D₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

#63223

—

#63223

—

C₂×#3622

—

Dic₆₈

—

C₂³×C₁₇

#41

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

#3622

—

#2047

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

#121

yes

characteristic

C₂²×C₁₇

#123

yes

—

C₄×C₁₇

—

C₄×C₁₇

#120

yes

characteristic

C₄×C₁₇

#121

yes

—

C₄×C₁₇

#124

yes

characteristic

Dic₁₇

—

C₂²×Q₁₆

—

#424

D₁₇

yes

—

—

—

C₂×C₁₇

—

C₂×C₁₇

#394

yes

characteristic

C₂×C₁₇

#473

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

characteristic

#117

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

#103

—

#101

—

C₁₇

#2047

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

D₆₈

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#63252

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×D₆₈

yes

characteristic

108

C₂×C₄

D₄×D₁₇

yes

—

109

C₂×C₄

#63256

yes

—

110

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

114

C₂²

—

115

C₂²

#10587580

yes

—

116

C₂²

#10587616

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

#10587581

yes

—

120

C₄

#10587623

yes

characteristic

121

C₄

#10587633

yes

characteristic

122

C₂

—

123

C₂

#11762726

yes

characteristic

124

C₂

#11763084

yes

—

	subgroup	G/H
center	C₂²	#10587616
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#424	C₂
socle	C₂²×C₁₇	#121
prime radical	C₂³×C₁₇	#41
hypercenter	#424	D₁₇
hypocenter	C₁₇	#2047
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#10587616

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#424

C₂

C₂²×C₁₇

C₂³×C₁₇

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
78. C₁₇	#2047	semidirect

H ≅ G/N

type

78. C₁₇

#2047

semidirect

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

subgroup	index
C₂²×Dic₆₈	2
#11762817	2
C₂×#10587495	2
#11762574	2
C₁₇×#424	2
#2047	17

subgroup

index

C₂²×Dic₆₈

C₁₇×#424

Subgroups — #14530052

Conjugacy classes of subgroups

Complements

Maximal subgroups