Finite Group Databasebeta

← #17002076

Subgroups — #17002076

Order 4352.

938 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 395 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537865C211yesmaximal, characteristic
2#14531705C211yesmaximal, characteristic
3#14527348C216yesmaximal
4#14527283C216yesmaximal
5C17×#3502C211yesmaximal, characteristic
6C2×Q8×Dic17—21no—
7Q8×#3622—24no—
8#11762862C2216yes—
9#11762708—23no—
10#11762675C2213yes—
11C4×C4⋊Dic17C2213yes—
12#11762546C2214yes—
13#11762546C22112yes—
14Q82×C17C414yescomplemented
15C17×#557—24no—
16C17×#557C414yescomplemented
17C17×#544—23no—
18C17×#533C2216yes—
19C17×#519—21no—
20C2×C4×Q8×C17C2211yescharacteristic
21C22×Q8×C17D411yescharacteristic
22C17×C4⋊Q8—212no—
23C17×C4⋊Q8C2×C416yes—
24C17×C4⋊Q8C2×C416yes—
25C17×#125—212no—
26C17×#125—26no—
27C17×C22⋊Q8—26no—
28C17×C22⋊Q8C2314yes—
29C4×Q8×C17—24no—
30C4×Q8×C17—24no—
31C4×Q8×C17—26no—
32C4×Q8×C17C2×C4112yes—
33C4×Q8×C17C2×C414yes—
34C4×Q8×C17D414yes—
35C17×#117—26no—
36C2×C4⋊C4×C17D413yes—
37C2×C42×C17C2313yes—
38Q8×Dic17—44no—
39C2×C4⋊Dic17—23no—
40C2×C4⋊Dic17—23no—
41C2×C4×Dic17—23no—
42#10587502—23no—
43#10587502C2311yescharacteristic
44#10587502C2313yes—
45C22⋊#3622C2314yes—
46C4⋊#3622—26no—
47C4×#3622—26no—
48C2×Q8×C17—26no—
49C2×Q8×C17C22⋊C414yes—
50C2×Q8×C17SD1614yes—
51C2×Q8×C17Q1614yes—
52C2×Q8×C17C22×C414yes—
53C22×C4×C17C2×D413yes—
54C22×C4×C17C2×D413yes—
55C22×C4×C17C2411yescharacteristic
56C22×Dic17—21no—
57C22×Dic17—23no—
58C4⋊C4×C17—26no—
59C4⋊C4×C17—212no—
60C4⋊C4×C17—26no—
61C4⋊C4×C17—26no—
62C4⋊C4×C17—24no—
63C4⋊C4×C17C22⋊C4112yes—
64C4⋊C4×C17C22×C414yes—
65C22⋊C4×C17—24no—
66C42×C17—212no—
67C42×C17C22×C416yes—
68C42×C17C2×D416yes—
69C4⋊Dic17—46no—
70C4⋊Dic17—46no—
71C4×Dic17—46no—
72C2×#3622—24no—
73#17619—171nomaximal, Sylow
74C23×C17C22×D411yescharacteristic
75Q8×C17—44no—
76Q8×C17—86no—
77Q8×C17#103116yes—
78C2×C4×C17—24no—
79C2×C4×C17—24no—
80C2×C4×C17—43no—
81C2×C4×C17—41no—
82C2×C4×C17C2×C22⋊C416yes—
83C2×C4×C17C2×C22⋊C416yes—
84C2×C4×C17C2×SD1616yes—
85C2×C4×C17C2×Q1616yes—
86C2×C4×C17C23×C411yescharacteristic
87C2×C4×C17C22×D411yescharacteristic
88C2×Dic17—22no—
89C2×Dic17—41no—
90C2×Dic17—41no—
91C2×Dic17—43no—
92C2×Dic17—43no—
93C2×Dic17—43no—
94#3622—44no—
95#3502D1711yescharacteristic
96#2982—171no—
97#2351—171no—
98#1518—176no—
99#1503—176no—
100C22×C17—21no—
101C22×C17—21no—
102C22×C17C22×C22⋊C411yescharacteristic
103C22×C17C22×SD1611yescharacteristic
104C22×C17C22×Q1611yescharacteristic
105C4×C17—21no—
106C4×C17—21no—
107C4×C17—26no—
108C4×C17—44no—
109C4×C17—44no—
110C4×C17C2×#103112yes—
111Dic17—42no—
112Dic17—83no—
113Q82Dic1714yescomplemented
114#557—24no—
115#557Dic1714yescomplemented
116#544—23no—
117#533C2×D1716yes—
118#519—21no—
119C2×C4×Q8—341no—
120C2×C4×Q8C2×D1711yescharacteristic
121C8×Q8—344no—
122#431—176no—
123#389—343no—
124#382—173no—
125C4×C4⋊C4—173no—
126#328—174no—
127#328—1712no—
128C2×C17—21no—
129C2×C17—21no—
130C2×C17#288411yescharacteristic
131C2×C17C22×#10311yescharacteristic
132C2×C17#290111yescharacteristic
133C22×Q8C22⋊D1711yescharacteristic
134C4⋊Q8—212no—
135C4⋊Q8C2×Dic1716yes—
136C4⋊Q8C2×Dic1716yes—
137#125—212no—
138#125—26no—
139C22⋊Q8—26no—
140C22⋊Q8C22×D1714yes—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—26no—
144C4×Q8—684no—
145C4×Q8C2×Dic17112yes—
146C4×Q8C2×Dic1714yes—
147C4×Q8C22⋊D1714yes—
148#117—26no—
149C2×C4⋊C4—343no—
150C2×C4⋊C4—343no—
151C2×C4⋊C4C22⋊D1713yes—
152C2×C42—343no—
153C2×C42C22×D1713yes—
154C4⋊C8—346no—
155C22⋊C8—174no—
156C4×C8—346no—
157#95—171no—
158#95—173no—
159#95—343no—
160C17#1761911yescharacteristic, complemented, Sylow
161C2×Q8—26no—
162C2×Q8#6323014yes—
163C2×Q8#6323114yes—
164C2×Q8C22⋊Dic1714yes—
165C2×Q8C22×Dic1714yes—
166C22×C4—341no—
167C22×C4—343no—
168C22×C4C2×C22⋊D1713yes—
169C22×C4C2×C22⋊D1713yes—
170C22×C4C23×D1711yescharacteristic
171C2×C8—344no—
172C4⋊C4—24no—
173C4⋊C4—26no—
174C4⋊C4—26no—
175C4⋊C4—26no—
176C4⋊C4—212no—
177C4⋊C4—686no—
178C4⋊C4—686no—
179C4⋊C4C22⋊Dic17112yes—
180C4⋊C4C22×Dic1714yes—
181C22⋊C4—24no—
182C42—212no—
183C42—686no—
184C42C22×Dic1716yes—
185C42C2×C22⋊D1716yes—
186C23C22×C22⋊D1711yescharacteristic
187Q8—44no—
188Q8—86no—
189Q8#10587506116yes—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—43no—
193C2×C4—41no—
194C2×C4—342no—
195C2×C4—681no—
196C2×C4—681no—
197C2×C4—683no—
198C2×C4—683no—
199C2×C4—683no—
200C2×C4C2×#6323016yes—
201C2×C4C2×#6323116yes—
202C2×C4C2×C22⋊Dic1716yes—
203C2×C4C2×C22⋊Dic1716yes—
204C2×C4C23×Dic1711yescharacteristic
205C2×C4C22×C22⋊D1711yescharacteristic
206C8—684no—
207C22—21no—
208C22—21no—
209C22C22×C22⋊Dic1711yescharacteristic
210C22C22×#6323011yescharacteristic
211C22C22×#6323111yescharacteristic
212C4—21no—
213C4—21no—
214C4—26no—
215C4—44no—
216C4—44no—
217C4—682no—
218C4—1363no—
219C4C2×#10587506112yes—
220C2—21no—
221C2—21no—
222C2#1453724811yescharacteristic
223C2C22×#1058750611yescharacteristic
224C2#1453733111yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC17×#3502C2
socleC22×C17C22×C22⋊C4
prime radicalC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenter#3502D17
hypocenterC17#17619
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #3502

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
14. Q82×C17C4semidirect
16. C17×#557C4semidirect
113. Q82Dic17semidirect
115. #557Dic17semidirect
160. C17#17619semidirect

Maximal subgroups

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

subgroupindex
#145378652
#145317052
#145273482
#145272832
C17×#35022
#1761917