Finite Group Databasebeta

← #16950591

Subgroups — #16950591

Order 4352.

642 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 99 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#14531275C211yesmaximal, characteristic
2#14528712C211yesmaximal, characteristic
3#14526910C214yesmaximal, complemented
4C17×#1785C211yesmaximal, characteristic
5C2×#10587612C411yescharacteristic
6C22×C4×Dic17C411yescharacteristic
7C2×#10587507C2212yes—
8C2×#10587504—21no—
9#11762773—21no—
10#11762716C2212yes—
11#11762712C412yes—
12#11762689—21no—
13#11762689—21no—
14C4×C22⋊Dic17—21no—
15C4×C22⋊Dic17—21no—
16C22×#41×C17C2211yescharacteristic
17C2×C17×#101C2212yes—
18C2×C17×#100—21no—
19C17×#407—21no—
20C2×#41×C17—24no—
21C2×#41×C17—42no—
22C2×#41×C17—42no—
23C22×Q8×C17C2311yescharacteristic
24C22×D4×C17C2×C411yescharacteristic
25C22×D4×C17D411yescharacteristic
26C22×D4×C17D411yescharacteristic
27C23×C4×C17C2×C411yescharacteristic
28C23×C4×C17D411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C23×Dic17—21no—
31C2×M16×C17—22no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—22no—
34C2×C22⋊Dic17—22no—
35#10587612—42no—
36C2×C4⋊Dic17C2×C412yes—
37C2×C4×Dic17—21no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17—42no—
40C2×C4×Dic17C2×C412yes—
41C2×#63223—22no—
42C17×#101—24no—
43C17×#100—42no—
44C17×C22⋊C8—42no—
45#10587507—24no—
46#10587504—42no—
47#10587502—22no—
48#10587502—22no—
49C22⋊#3622—42no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C24×C17C22⋊C411yescharacteristic
53#41×C17—416no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17#4114yes—
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17—22no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C2×D4×C17—41no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—22no—
72C22×C4×C17—22no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×Dic17—22no—
80C22×Dic17—22no—
81C22×Dic17—44no—
82C22×Dic17—42no—
83M16×C17—42no—
84M16×C17—42no—
85C2×C8×C17—42no—
86C22⋊Dic17—44no—
87C22⋊Dic17—44no—
88C22⋊Dic17—42no—
89C4⋊Dic17—42no—
90C4×Dic17—42no—
91C4×Dic17—42no—
92C4×Dic17—42no—
93#63223—42no—
94#63223—42no—
95C2×#3622—42no—
96#12861—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—42no—
106C23×C17—42no—
107C23×C17—42no—
108C23×C17#9912yes—
109C23×C17C2×C22⋊C411yescharacteristic
110Q8×C17—44no—
111D4×C17—44no—
112D4×C17—44no—
113D4×C17—44no—
114C2×C4×C17—28no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—44no—
124C2×C4×C17—44no—
125C2×C4×C17#9912yes—
126C2×C4×C17#11712yes—
127C2×C4×C17#12312yes—
128C2×C4×C17#12312yes—
129C2×Dic17—24no—
130C2×Dic17—42no—
131C2×Dic17—48no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C8×C17—82no—
137#3622—82no—
138#2297—171no—
139#1785—171no—
140#1785D1711yescharacteristic
141#1394—174no—
142C22×C17—22no—
143C22×C17—22no—
144C22×C17—42no—
145C22×C17—42no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—42no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17—81no—
154C22×C17—81no—
155C22×C17#38111yescharacteristic
156C22×C17C2×#9911yescharacteristic
157C22×C17C2×#9911yescharacteristic
158C4×C17—22no—
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—22no—
162Dic17—44no—
163Dic17—82no—
164C22×#41C2×D1711yescharacteristic
165C2×#117—171no—
166C22×C42—171no—
167C2×#101—172no—
168C2×#101C2×D1712yes—
169C2×#100—21no—
170C2×#100—341no—
171#407—21no—
172#407—341no—
173#391—172no—
174#390—172no—
175#386—341no—
176#386—341no—
177C4×C22⋊C4—341no—
178C4×C22⋊C4—341no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17—42no—
182C2×C17—81no—
183C2×C17—81no—
184C2×C17#181911yescharacteristic
185C2×C17#213212yes—
186C2×#41—24no—
187C2×#41—42no—
188C2×#41—42no—
189C22×Q8C22×D1711yescharacteristic
190C22×D4C4×D1711yescharacteristic
191C22×D4D6811yescharacteristic
192C22×D4C22⋊D1711yescharacteristic
193C23×C4—341no—
194C23×C4C4×D1711yescharacteristic
195C23×C4D6811yescharacteristic
196C23×C4C22⋊D1711yescharacteristic
197C2×M16—22no—
198C2×M16—342no—
199#117—682no—
200C2×C4⋊C4—172no—
201C2×C22⋊C4—341no—
202C2×C22⋊C4—342no—
203C2×C22⋊C4—342no—
204C2×C42—172no—
205C2×C42—341no—
206C2×C42—341no—
207C2×C42—682no—
208#101—24no—
209#101—344no—
210#100—42no—
211#100—682no—
212C22⋊C8—42no—
213C22⋊C8—682no—
214#95—342no—
215#95—342no—
216C17#1286111yescharacteristic, complemented, Sylow
217C24#6322711yescharacteristic
218C24#6322711yescharacteristic
219C24#6322711yescharacteristic
220#41—416no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8#6325414yes—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—684no—
245C22×C4—682no—
246C22×C4#6322711yescharacteristic
247C22×C4C2×C4×D1711yescharacteristic
248C22×C4C2×D6811yescharacteristic
249C22×C4C2×C22⋊D1711yescharacteristic
250M16—42no—
251M16—42no—
252M16—682no—
253M16—682no—
254C2×C8—42no—
255C2×C8—682no—
256C4⋊C4—682no—
257C22⋊C4—684no—
258C22⋊C4—684no—
259C22⋊C4—682no—
260C42—682no—
261C42—682no—
262C42—682no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23#1058747712yes—
275C23C2×#6322711yescharacteristic
276Q8—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280C2×C4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—344no—
292C2×C4—682no—
293C2×C4—688no—
294C2×C4—682no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4#1058747712yes—
299C2×C4#1058756312yes—
300C2×C4#1058757212yes—
301C2×C4#1058762112yes—
302C8—82no—
303C8—1362no—
304C22—22no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—81no—
316C22—81no—
317C22#1176267011yescharacteristic
318C22C2×#1058747711yescharacteristic
319C22C2×#1058747711yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—684no—
325C4—1362no—
326C2—21no—
327C2—21no—
328C2—42no—
329C2—81no—
330C2—81no—
331C2#1452885611yescharacteristic
332C2#1453046712yes—
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1785C2
socleC22×C17C2×#99
prime radicalC22×#41×C17C22
cosocleC22×Q8×C17C23
hypercenter#1785D17
hypocenterC17#12861
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #14526910C2semidirect
216. C17#12861semidirect

Maximal subgroups

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

subgroupindex
#145312752
#145287122
#145269102
C17×#17852
#1286117