Group #14530202 — subgroups

Subgroups — #14530202

Order 2176.

222 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 81 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×#10587506	C₂	1	2	yes	maximal
2	#11762675	C₂	1	1	yes	maximal, characteristic
3	#11762571	C₂	1	2	yes	maximal
4	#11762564	C₂	1	1	yes	maximal, characteristic
5	C₁₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
10	C₂²×#3622	C₂²	1	2	yes	—
11	C₁₇×#95	—	2	2	no	—
12	#10587506	—	2	4	no	—
13	#10587502	—	2	2	no	—
14	C₂×Q₈×C₁₇	—	2	1	no	—
15	C₂×Q₈×C₁₇	—	2	2	no	—
16	C₂×Q₈×C₁₇	—	2	2	no	—
17	C₂×Q₈×C₁₇	D₄	1	2	yes	—
18	C₂²×C₄×C₁₇	—	2	1	no	—
19	C₂²×C₄×C₁₇	—	2	1	no	—
20	C₂²×C₄×C₁₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	2	no	—
23	C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₄²×C₁₇	—	2	2	no	—
25	C₄⋊Dic₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₂×#3622	—	2	2	no	—
28	C₂×#3622	—	2	2	no	—
29	C₂×#3622	—	2	2	no	—
30	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₁₇	—	2	4	no	—
32	Q₈×C₁₇	—	4	2	no	—
33	C₂×C₄×C₁₇	—	2	2	no	—
34	C₂×C₄×C₁₇	—	2	2	no	—
35	C₂×C₄×C₁₇	—	2	2	no	—
36	C₂×C₄×C₁₇	—	2	2	no	—
37	C₂×C₄×C₁₇	—	4	1	no	—
38	C₂×C₄×C₁₇	—	4	1	no	—
39	C₂×C₄×C₁₇	—	4	1	no	—
40	C₂×C₄×C₁₇	—	4	2	no	—
41	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
43	C₂×C₄×C₁₇	#41	1	2	yes	—
44	C₂×C₄×C₁₇	#41	1	2	yes	—
45	C₂×Dic₁₇	—	4	2	no	—
46	C₂×Dic₁₇	—	4	2	no	—
47	C₂×Dic₁₇	—	4	2	no	—
48	#3622	—	4	2	no	—
49	#2074	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	#121	1	1	yes	characteristic
51	C₂²×C₁₇	#123	1	1	yes	characteristic
52	C₂²×C₁₇	#124	1	1	yes	characteristic
53	C₂²×C₁₇	#135	1	2	yes	—
54	C₂²×C₁₇	#137	1	2	yes	—
55	C₄×C₁₇	—	4	2	no	—
56	C₄×C₁₇	—	4	2	no	—
57	C₄×C₁₇	—	8	1	no	—
58	C₄×C₁₇	#123	1	2	yes	—
59	C₄×C₁₇	#126	1	2	yes	—
60	Dic₁₇	—	8	2	no	—
61	C₂×#103	—	17	2	no	—
62	#391	D₁₇	1	1	yes	characteristic
63	#382	—	17	1	no	—
64	#340	—	17	2	no	—
65	#336	—	17	1	no	—
66	C₂×C₁₇	#397	1	1	yes	characteristic
67	C₂×C₁₇	#464	1	2	yes	—
68	C₂×C₁₇	#485	1	2	yes	—
69	C₂×C₁₇	#488	1	1	yes	characteristic
70	C₂×C₁₇	#491	1	1	yes	characteristic
71	C₂²×Q₈	C₂×D₁₇	1	1	yes	characteristic
72	C₂²×C₈	—	17	2	no	—
73	C₂×C₄⋊C₄	—	17	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
75	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
76	#103	—	34	4	no	—
77	#95	—	2	2	no	—
78	#95	—	34	2	no	—
79	C₁₇	#2074	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	D₆₈	1	2	yes	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	34	2	no	—
87	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
89	C₂×C₈	—	34	2	no	—
90	C₂×C₈	—	34	2	no	—
91	C₂×C₈	—	34	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	34	2	no	—
94	C₄⋊C₄	—	34	2	no	—
95	C₄²	—	2	2	no	—
96	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
97	Q₈	—	2	4	no	—
98	Q₈	—	4	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	68	2	no	—
108	C₂×C₄	—	68	2	no	—
109	C₂×C₄	—	68	2	no	—
110	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
111	C₂×C₄	#63252	1	1	yes	characteristic
112	C₂×C₄	#63254	1	2	yes	—
113	C₂×C₄	#63254	1	2	yes	—
114	C₈	—	68	2	no	—
115	C₂²	#10587616	1	1	yes	characteristic
116	C₂²	#10587621	1	1	yes	characteristic
117	C₂²	#10587622	1	1	yes	characteristic
118	C₂²	#10587628	1	2	yes	—
119	C₂²	#10587636	1	2	yes	—
120	C₄	—	4	2	no	—
121	C₄	—	4	2	no	—
122	C₄	—	8	1	no	—
123	C₄	—	136	2	no	—
124	C₄	#10587565	1	2	yes	—
125	C₄	#10587572	1	2	yes	—
126	C₂	#11762734	1	1	yes	characteristic
127	C₂	#11763031	1	2	yes	—
128	C₂	#11763147	1	2	yes	—
129	C₂	#11763163	1	1	yes	characteristic
130	C₂	#11763177	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587506

C₂

yes

maximal

#11762675

C₂

yes

maximal, characteristic

#11762571

C₂

yes

maximal

#11762564

C₂

yes

maximal, characteristic

C₁₇×#391

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂²×#3622

C₂²

yes

—

C₁₇×#95

—

#10587506

—

#10587502

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

C₂×C₄×C₁₇

yes

C₂×C₄×C₁₇

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

—

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₂²×C₁₇

#137

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#123

yes

—

C₄×C₁₇

#126

yes

—

Dic₁₇

—

C₂×#103

—

#391

D₁₇

yes

—

—

—

C₂×C₁₇

#397

yes

characteristic

C₂×C₁₇

#464

yes

—

C₂×C₁₇

#485

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂²×Q₈

C₂×D₁₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂×C₄²

C₂×D₁₇

yes

—

—

—

C₁₇

#2074

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₆₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂×D₆₈

yes

characteristic

111

C₂×C₄

#63252

yes

characteristic

112

C₂×C₄

#63254

yes

—

113

C₂×C₄

#63254

yes

—

114

C₈

—

115

C₂²

#10587616

yes

characteristic

116

C₂²

#10587621

yes

characteristic

117

C₂²

#10587622

yes

characteristic

118

C₂²

#10587628

yes

—

119

C₂²

#10587636

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

136

—

124

C₄

#10587565

yes

—

125

C₄

#10587572

yes

—

126

C₂

#11762734

yes

characteristic

127

C₂

#11763031

yes

—

128

C₂

#11763147

yes

—

129

C₂

#11763163

yes

characteristic

130

C₂

#11763177

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator, cosocle	C₂²×C₄×C₁₇	C₂³
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#391	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
hypercenter	#391	D₁₇
hypocenter	C₁₇	#2074
LCS	G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator, cosocle

C₂²×C₄×C₁₇

C₂³

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#391

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

D₁₇

C₁₇

G ▷ C₂²×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
79. C₁₇	#2074	semidirect

H ≅ G/N

type

79. C₁₇

#2074

semidirect

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

subgroup	index
C₂×#10587506	2
#11762675	2
#11762571	2
#11762564	2
C₁₇×#391	2
#2074	17

subgroup

index

C₁₇×#391

Subgroups — #14530202

Conjugacy classes of subgroups

Complements

Maximal subgroups