Finite Group Databasebeta

← #16931998

Subgroups — #16931998

Order 4352.

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

normal subgroups only

This group has 424 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.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14533971C211yesmaximal, characteristic
2#14527410C212yesmaximal
3#14527128C211yesmaximal, characteristic
4#14526639C212yesmaximal
5C17×#2983C211yesmaximal, characteristic
6C2×#10587556C2211yescharacteristic
7#11762866C2212yes—
8#11762742—21no—
9#11762742—21no—
10#11762716C411yescharacteristic
11#11762716C411yescharacteristic
12#11762693—21no—
13#11762693—21no—
14#11762678C411yescharacteristic
15#11762674C411yescharacteristic
16C4×C4⋊Dic17C2211yescharacteristic
17#11762550—24no—
18#11762535—21no—
19#11762535—21no—
20C2×C4×Q8×C17C2211yescharacteristic
21C17×#446—24no—
22C17×#432—21no—
23C17×#432C2212yes—
24C17×#431—21no—
25C22×Q8×C17C2×C411yescharacteristic
26C4×Q8×C17—22no—
27C4×Q8×C17—22no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C42×C17D411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C42×C17C2311yescharacteristic
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17C2×C411yescharacteristic
43#10587556—22no—
44#10587556—22no—
45C17×C4⋊C8—22no—
46C17×C4⋊C8—22no—
47C17×C4⋊C8—22no—
48C17×C22⋊C8—22no—
49C17×#97—22no—
50C17×#97D414yes—
51C4×C8×C17—22no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59C22⋊#3622—22no—
60C4⋊#3622—22no—
61#10587474—22no—
62C2×Q8×C17—22no—
63C2×Q8×C17—22no—
64C2×Q8×C17—41no—
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C2×C8×C17—24no—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17—22no—
80C42×C17—21no—
81C42×C17—21no—
82C42×C17—21no—
83C42×C17—21no—
84C42×C17C2×D412yes—
85C42×C17#4112yes—
86C4⋊Dic17—22no—
87C4⋊Dic17—22no—
88C4⋊Dic17—42no—
89#63225—42no—
90#63225—42no—
91#63225—41no—
92#63225—42no—
93#63225—41no—
94C4×Dic17—41no—
95C4×Dic17—41no—
96C2×#3622—42no—
97#9450—171nomaximal, Sylow
98C23×C17C2×C22⋊C411yescharacteristic
99Q8×C17—44no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17#10112yes—
114C2×C4×C17C4×D412yes—
115C2×C4×C17#12011yescharacteristic
116C2×C4×C17#12112yes—
117C2×C4×C17#12311yescharacteristic
118C2×Dic17—22no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—42no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C8×C17—44no—
130#3622—82no—
131#2983D1711yescharacteristic
132#2565—171no—
133#1534—172no—
134#1467—171no—
135#1285—172no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17#38611yescharacteristic
139C22×C17C2×#10111yescharacteristic
140C22×C17#41311yescharacteristic
141C4×C17—22no—
142C4×C17—22no—
143C4×C17—22no—
144C4×C17—21no—
145C4×C17—21no—
146C4×C17—41no—
147C4×C17—41no—
148C4×C17—41no—
149Dic17—42no—
150Dic17—81no—
151Dic17—81no—
152Dic17—81no—
153C2×#125—171no—
154C2×C4×Q8C2×D1711yescharacteristic
155#446—24no—
156#432—21no—
157#432—172no—
158#432C2×D1712yes—
159#431—21no—
160#398—341no—
161#398—341no—
162#391—171no—
163#391—171no—
164#387—341no—
165#387—341no—
166#384—171no—
167#382—171no—
168C4×C4⋊C4—171no—
169#330—344no—
170#324—341no—
171#324—341no—
172C2×C17—21no—
173C2×C17—21no—
174C2×C17#188411yescharacteristic
175C2×C17#190111yescharacteristic
176C2×C17#191211yescharacteristic
177C22×Q8C4×D1711yescharacteristic
178#125—342no—
179#125—342no—
180C4×Q8—22no—
181C4×Q8—22no—
182C2×C4⋊C4—171no—
183C2×C4⋊C4—171no—
184C2×C4⋊C4—341no—
185C2×C4⋊C4—341no—
186C2×C4⋊C4—341no—
187C2×C4⋊C4—341no—
188C2×C4⋊C4—341no—
189C2×C4⋊C4C4×D1711yescharacteristic
190C2×C4⋊C4D6811yescharacteristic
191C2×C4⋊C4D6811yescharacteristic
192C2×C42—171no—
193C2×C42—171no—
194C2×C42C22⋊D1711yescharacteristic
195C2×C42C22⋊D1711yescharacteristic
196C2×C42C22×D1711yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—342no—
201C22⋊C8—22no—
202C22⋊C8—342no—
203#97—22no—
204#97—342no—
205#97C22⋊D1714yes—
206C4×C8—22no—
207#95—341no—
208#95—341no—
209#95—341no—
210#95—341no—
211#95—341no—
212#95—341no—
213#95—341no—
214C17#945011yescharacteristic, complemented, Sylow
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—41no—
218C22×C4—341no—
219C22×C4—341no—
220C22×C4—341no—
221C22×C4—341no—
222C22×C4#6322711yescharacteristic
223C22×C4#6322711yescharacteristic
224C22×C4#6322711yescharacteristic
225C22×C4#6322711yescharacteristic
226C22×C4C2×C4×D1711yescharacteristic
227C22×C4C2×D6811yescharacteristic
228C22×C4C2×C22⋊D1711yescharacteristic
229C2×C8—24no—
230C2×C8—682no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—342no—
235C4⋊C4—342no—
236C4⋊C4—682no—
237C4⋊C4—682no—
238C4⋊C4—682no—
239C4⋊C4—681no—
240C4⋊C4—682no—
241C4⋊C4—681no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—21no—
246C42—681no—
247C42—681no—
248C42#6325212yes—
249C42C2×C22⋊D1712yes—
250C23C2×#6322711yescharacteristic
251Q8—44no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—342no—
266C2×C4—681no—
267C2×C4—681no—
268C2×C4—681no—
269C2×C4—681no—
270C2×C4—681no—
271C2×C4—681no—
272C2×C4—682no—
273C2×C4—681no—
274C2×C4—681no—
275C2×C4—681no—
276C2×C4#1058749512yes—
277C2×C4C4×C22⋊D1712yes—
278C2×C4#1058761511yescharacteristic
279C2×C4#1058761612yes—
280C2×C4#1058763911yescharacteristic
281C8—44no—
282C8—1362no—
283C22—21no—
284C22—21no—
285C22#1176268811yescharacteristic
286C22C2×#1058749511yescharacteristic
287C22#1176279111yescharacteristic
288C4—21no—
289C4—21no—
290C4—22no—
291C4—22no—
292C4—22no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—682no—
297C4—1361no—
298C4—1361no—
299C4—1361no—
300C2—21no—
301C2—21no—
302C2#1452916711yescharacteristic
303C2#1452925211yescharacteristic
304C2#1452932111yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2983C2
socleC22×C17#386
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2983D17
hypocenterC17#9450
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
214. C17#9450semidirect

Maximal subgroups

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

subgroupindex
#145339712
#145274102
#145271282
#145266392
C17×#29832
#945017