Group #14529003 — subgroups

Subgroups — #14529003

Order 2176.

252 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 85 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	#11763576	C₂	1	1	yes	maximal, characteristic
2	#11762857	C₂	1	2	yes	maximal, complemented
3	#11762793	C₂	1	1	yes	maximal, characteristic
4	#11762574	C₂	1	2	yes	maximal, complemented
5	C₁₇×#408	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
9	#10587634	—	2	4	no	—
10	C₂×#63223	—	2	2	no	—
11	C₂×#63223	C₂²	1	2	yes	—
12	C₂×#63223	C₂²	1	2	yes	—
13	C₁₇×C₂²⋊C₈	—	2	2	no	—
14	#10587507	C₄	1	2	yes	—
15	#10587504	C₄	1	2	yes	—
16	#10587493	—	2	4	no	—
17	#41×C₁₇	—	2	2	no	—
18	#41×C₁₇	—	2	2	no	—
19	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₁₇	Q₈	1	2	yes	—
22	C₂²×C₄×C₁₇	D₄	1	2	yes	—
23	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₁₇	—	2	2	no	—
25	C₂×C₈×C₁₇	—	2	1	no	—
26	C₂×C₈×C₁₇	—	2	1	no	—
27	C₂×C₈×C₁₇	—	2	2	no	—
28	#63223	—	2	4	no	—
29	#63223	—	2	2	no	—
30	#63223	—	2	2	no	—
31	#63223	C₂×C₄	1	4	yes	—
32	#63223	D₄	1	4	yes	—
33	C₂×#3622	—	2	4	no	—
34	C₂×#3622	—	2	2	no	—
35	C₂×#3622	—	2	2	no	—
36	C₂³×C₁₇	C₄⋊C₄	1	2	yes	—
37	C₂³×C₁₇	C₂×Q₈	1	1	yes	characteristic
38	Q₈×C₁₇	—	2	2	no	—
39	D₄×C₁₇	—	2	4	no	—
40	D₄×C₁₇	—	2	2	no	—
41	C₂×C₄×C₁₇	—	2	2	no	—
42	C₂×C₄×C₁₇	—	2	2	no	—
43	C₂×C₄×C₁₇	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇	#41	1	2	yes	—
48	C₈×C₁₇	—	4	1	no	—
49	C₈×C₁₇	—	4	1	no	—
50	#3622	—	2	4	no	—
51	#3622	—	2	4	no	—
52	#1858	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	—	2	2	no	—
54	C₂²×C₁₇	—	4	1	no	—
55	C₂²×C₁₇	—	4	2	no	—
56	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₁₇	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₇	—	2	2	no	—
59	C₄×C₁₇	C₄×D₄	1	2	yes	—
60	C₄×C₁₇	#120	1	1	yes	characteristic
61	C₄×C₁₇	#123	1	1	yes	characteristic
62	#568	—	17	1	no	—
63	#430	—	17	2	no	—
64	#413	—	17	1	no	—
65	#408	D₁₇	1	1	yes	characteristic
66	#341	—	17	2	no	—
67	C₂×C₁₇	—	2	1	no	—
68	C₂×C₁₇	—	2	2	no	—
69	C₂×C₁₇	—	4	2	no	—
70	C₂×C₁₇	#385	1	1	yes	characteristic
71	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
72	#131	—	34	4	no	—
73	C₂×M₁₆	—	17	2	no	—
74	C₂×M₁₆	—	17	2	no	—
75	C₂×M₁₆	—	34	2	no	—
76	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
77	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
78	#108	—	34	4	no	—
79	#101	—	17	2	no	—
80	#100	—	17	2	no	—
81	C₂²⋊C₈	—	2	2	no	—
82	C₁₇	#1858	1	1	yes	characteristic, complemented, Sylow
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	C₄×D₁₇	1	1	yes	characteristic
86	C₂×D₄	Dic₃₄	1	2	yes	—
87	C₂×D₄	C₄×D₁₇	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	17	4	no	—
92	M₁₆	—	17	4	no	—
93	M₁₆	—	34	4	no	—
94	M₁₆	—	34	2	no	—
95	M₁₆	—	34	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	34	4	no	—
100	C₂×C₈	—	34	2	no	—
101	C₂×C₈	—	34	2	no	—
102	C₂³	#63225	1	2	yes	—
103	C₂³	C₂×Dic₃₄	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	#63225	1	2	yes	—
110	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₇	1	2	yes	—
112	C₂×C₄	#63254	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
114	C₈	—	4	1	no	—
115	C₈	—	4	1	no	—
116	C₈	—	34	4	no	—
117	C₈	—	34	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	4	1	no	—
120	C₂²	—	4	2	no	—
121	C₂²	#10587564	1	2	yes	—
122	C₂²	C₂×#63225	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	#10587567	1	2	yes	—
125	C₄	#10587621	1	1	yes	characteristic
126	C₄	#10587623	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	#11762681	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763576

C₂

yes

maximal, characteristic

#11762857

C₂

yes

maximal, complemented

#11762793

C₂

yes

maximal, characteristic

#11762574

C₂

yes

maximal, complemented

C₁₇×#408

C₂

yes

maximal, characteristic

C₂×#41×C₁₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₇

C₂²

yes

characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

—

C₂²

yes

—

C₂×#63223

C₂²

yes

—

C₁₇×C₂²⋊C₈

—

#10587507

C₄

yes

—

#10587504

C₄

yes

—

#10587493

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₇

C₂×C₄

yes

characteristic

C₂×D₄×C₁₇

Q₈

yes

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

—

—

—

C₂×C₄

yes

—

#63223

D₄

yes

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

C₄⋊C₄

yes

—

C₂³×C₁₇

C₂×Q₈

yes

characteristic

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₄×D₄

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

—

—

—

D₁₇

yes

characteristic

#341

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#385

yes

characteristic

C₂×#41

C₂×D₁₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

characteristic

C₂²×C₈

C₂×D₁₇

yes

—

—

—

C₂²⋊C₈

—

C₁₇

#1858

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

C₄×D₁₇

yes

characteristic

C₂×D₄

Dic₃₄

yes

—

C₂×D₄

C₄×D₁₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂³

#63225

yes

—

103

C₂³

C₂×Dic₃₄

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#63225

yes

—

110

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

111

C₂×C₄

D₄×D₁₇

yes

—

112

C₂×C₄

#63254

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#10587564

yes

—

122

C₂²

C₂×#63225

yes

characteristic

123

C₄

—

124

C₄

#10587567

yes

—

125

C₄

#10587621

yes

characteristic

126

C₄

#10587623

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

#11762681

yes

characteristic

	subgroup	G/H
center	C₄	#10587623
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#408	C₂
socle	C₂×C₁₇	#385
prime radical	C₂×#41×C₁₇	C₂²
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#408	D₁₇
hypocenter	C₁₇	#1858
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₄

#10587623

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#408

C₂

socle

C₂×C₁₇

#385

prime radical

C₂×#41×C₁₇

C₂²

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. #11762857	C₂	semidirect
4. #11762574	C₂	semidirect
82. C₁₇	#1858	semidirect

H ≅ G/N

type

2. #11762857

C₂

semidirect

4. #11762574

C₂

semidirect

82. C₁₇

#1858

semidirect

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

subgroup	index
#11763576	2
#11762857	2
#11762793	2
#11762574	2
C₁₇×#408	2
#1858	17

subgroup

index

C₁₇×#408

Subgroups — #14529003

Conjugacy classes of subgroups

Complements

Maximal subgroups