Finite Group Databasebeta

← #16949832

Subgroups — #16949832

Order 4352.

1018 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 1018 conjugacy classes of subgroups — too many to draw. Only its 123 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
1C22×#10587630C211yesmaximal, characteristic
2C2×#11762607C214yesmaximal, complemented
3#14528708C211yesmaximal, characteristic
4C17×#1784C211yesmaximal, characteristic
5C23×Dic34C411yescharacteristic
6C2×#10587630—28no—
7C2×#10587630—21no—
8C2×#10587630—21no—
9C2×#10587630C412yes—
10C2×#10587630C2212yes—
11C22×#63225—21no—
12C22×#63225—21no—
13C22×C4×Dic17C411yescharacteristic
14C2×#10587507—21no—
15C2×#10587507C2212yes—
16#11762773—21no—
17#11762607—28no—
18C23×Q8×C17C2211yescharacteristic
19C2×C17×#101—21no—
20C2×C17×#101C2212yes—
21C17×#407—21no—
22C22×Q8×C17—24no—
23C22×Q8×C17—42no—
24C22×Q8×C17—42no—
25C22×Q8×C17C2×C411yescharacteristic
26C22×Q8×C17D411yescharacteristic
27C22×Q8×C17D411yescharacteristic
28C22×Q8×C17C2311yescharacteristic
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17D411yescharacteristic
31C23×C4×C17D411yescharacteristic
32C23×Dic17—21no—
33C23×Dic17—21no—
34C22×Dic34—21no—
35C22×Dic34—24no—
36C22×Dic34—42no—
37C22×Dic34—42no—
38C22×Dic34C2×C412yes—
39C2×M16×C17—22no—
40#10587630—28no—
41#10587630—28no—
42#10587630—24no—
43#10587630—24no—
44#10587630—24no—
45#10587630D418yes—
46C2×#63225—28no—
47C2×#63225—28no—
48C2×#63225—22no—
49C2×#63225—22no—
50C2×#63225—22no—
51C2×#63225—22no—
52C2×C4×Dic17—24no—
53C2×C4×Dic17—21no—
54C2×C4×Dic17C2×C412yes—
55C2×#63223—22no—
56C17×#101—24no—
57C17×#101—42no—
58C17×C22⋊C8—42no—
59#10587507—24no—
60#10587507—42no—
61C22⋊#3622—42no—
62C24×C17C22⋊C411yescharacteristic
63C2×Q8×C17—22no—
64C2×Q8×C17—22no—
65C2×Q8×C17—22no—
66C2×Q8×C17—22no—
67C2×Q8×C17—22no—
68C2×Q8×C17—41no—
69C2×Q8×C17—41no—
70C2×Q8×C17—41no—
71C2×Q8×C17—41no—
72C2×Q8×C17—41no—
73C2×Q8×C17—41no—
74C2×Q8×C17—416no—
75C2×Q8×C17C2×D414yes—
76C22×C4×C17—22no—
77C22×C4×C17—22no—
78C22×C4×C17—22no—
79C22×C4×C17—22no—
80C22×C4×C17—22no—
81C22×C4×C17—22no—
82C22×C4×C17—22no—
83C22×C4×C17—41no—
84C22×C4×C17—41no—
85C22×C4×C17C22⋊C411yescharacteristic
86C22×C4×C17C22⋊C411yescharacteristic
87C22×C4×C17C22⋊C411yescharacteristic
88C22×C4×C17C22×C411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×Dic17—28no—
92C22×Dic17—24no—
93C22×Dic17—22no—
94C22×Dic17—24no—
95C22×Dic17—22no—
96C22×Dic17—42no—
97C22×Dic17—42no—
98C2×Dic34—24no—
99C2×Dic34—22no—
100C2×Dic34—42no—
101C2×Dic34—416no—
102C2×Dic34—42no—
103C2×Dic34—42no—
104C2×Dic34—42no—
105C2×Dic34C22⋊C414yes—
106M16×C17—42no—
107M16×C17—42no—
108C2×C8×C17—42no—
109#63225—28no—
110#63225—28no—
111#63225—28no—
112#63225—28no—
113C4×Dic17—24no—
114C4×Dic17—22no—
115C4×Dic17C22⋊C414yes—
116#63223—42no—
117#63223—42no—
118C2×#3622—42no—
119#12700—171nomaximal, Sylow
120C23×C17—24no—
121C23×C17—21no—
122C23×C17—21no—
123C23×C17#9912yes—
124C23×C17C2×C22⋊C411yescharacteristic
125Q8×C17—44no—
126Q8×C17—44no—
127Q8×C17—44no—
128Q8×C17—44no—
129C2×C4×C17—28no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—21no—
142C2×C4×C17—21no—
143C2×C4×C17—21no—
144C2×C4×C17—44no—
145C2×C4×C17—44no—
146C2×C4×C17#9912yes—
147C2×C4×C17C2×C22⋊C412yes—
148C2×C4×C17#12012yes—
149C2×C4×C17#12012yes—
150C2×Dic17—216no—
151C2×Dic17—28no—
152C2×Dic17—24no—
153C2×Dic17—24no—
154C2×Dic17—48no—
155C2×Dic17—42no—
156C2×Dic17—42no—
157Dic34—48no—
158Dic34—44no—
159Dic34—44no—
160C8×C17—82no—
161#3622—82no—
162C22×C4⋊Q8—171no—
163C2×#356—174no—
164#1784—171no—
165#1784D1711yescharacteristic
166C22×C17—24no—
167C22×C17—24no—
168C22×C17—22no—
169C22×C17—22no—
170C22×C17—22no—
171C22×C17#35614yescomplemented
172C22×C17#37911yescharacteristic
173C22×C17C2×#9911yescharacteristic
174C22×C17C2×#9911yescharacteristic
175C4×C17—22no—
176C4×C17—22no—
177C4×C17—22no—
178C4×C17—22no—
179C4×C17—41no—
180C4×C17—41no—
181Dic17—28no—
182Dic17—44no—
183C23×Q8—171no—
184C23×Q8C2×D1711yescharacteristic
185C2×C4⋊Q8—172no—
186C2×C4⋊Q8—172no—
187C2×C4⋊Q8—348no—
188C2×C4⋊Q8—341no—
189C2×C4⋊Q8—341no—
190C22×C4⋊C4—341no—
191C22×C4⋊C4—341no—
192C22×C42—171no—
193C2×#101—21no—
194C2×#101—172no—
195C2×#101—341no—
196C2×#101C2×D1712yes—
197#407—21no—
198#407—341no—
199#356—348no—
200C2×C17—21no—
201C2×C17—24no—
202C2×C17—21no—
203C2×C17#178111yescharacteristic
204C2×C17C2×#35612yes—
205C22×Q8—24no—
206C22×Q8—42no—
207C22×Q8—42no—
208C22×Q8—172no—
209C22×Q8—344no—
210C22×Q8—341no—
211C22×Q8—682no—
212C22×Q8—682no—
213C22×Q8C4×D1711yescharacteristic
214C22×Q8D6811yescharacteristic
215C22×Q8C22⋊D1711yescharacteristic
216C22×Q8C22×D1711yescharacteristic
217C23×C4—341no—
218C23×C4—341no—
219C23×C4C4×D1711yescharacteristic
220C23×C4D6811yescharacteristic
221C23×C4C22⋊D1711yescharacteristic
222C2×M16—22no—
223C2×M16—342no—
224C4⋊Q8—178no—
225C4⋊Q8—344no—
226C4⋊Q8—344no—
227C4⋊Q8—348no—
228C4⋊Q8—348no—
229C4⋊Q8—344no—
230C2×C4⋊C4—348no—
231C2×C4⋊C4—348no—
232C2×C4⋊C4—342no—
233C2×C4⋊C4—342no—
234C2×C4⋊C4—342no—
235C2×C4⋊C4—342no—
236C2×C42—172no—
237C2×C42—344no—
238C2×C42—341no—
239#101—24no—
240#101—42no—
241#101—344no—
242#101—682no—
243C22⋊C8—42no—
244C22⋊C8—682no—
245C17#1270011yescharacteristic, complemented, Sylow
246C24#6322711yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—416no—
259C2×Q8—174no—
260C2×Q8—344no—
261C2×Q8—342no—
262C2×Q8—6816no—
263C2×Q8—682no—
264C2×Q8—682no—
265C2×Q8—682no—
266C2×Q8—682no—
267C2×Q8D4×D1714yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—344no—
278C22×C4—348no—
279C22×C4—342no—
280C22×C4—344no—
281C22×C4—342no—
282C22×C4—682no—
283C22×C4—682no—
284C22×C4#6322711yescharacteristic
285C22×C4#6322711yescharacteristic
286C22×C4#6322711yescharacteristic
287C22×C4C2×C4×D1711yescharacteristic
288C22×C4C2×D6811yescharacteristic
289C22×C4C2×C22⋊D1711yescharacteristic
290M16—42no—
291M16—42no—
292M16—682no—
293M16—682no—
294C2×C8—42no—
295C2×C8—682no—
296C4⋊C4—348no—
297C4⋊C4—348no—
298C4⋊C4—348no—
299C4⋊C4—348no—
300C42—174no—
301C42—344no—
302C42—342no—
303C23—21no—
304C23—24no—
305C23—21no—
306C23#1058747712yes—
307C23C2×#6322711yescharacteristic
308Q8—44no—
309Q8—44no—
310Q8—44no—
311Q8—44no—
312Q8—684no—
313Q8—688no—
314Q8—684no—
315C2×C4—28no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—3416no—
333C2×C4—348no—
334C2×C4—344no—
335C2×C4—344no—
336C2×C4—688no—
337C2×C4—682no—
338C2×C4—682no—
339C2×C4#1058747712yes—
340C2×C4C22⋊C4×D1712yes—
341C2×C4#1058756812yes—
342C2×C4#1058762312yes—
343C8—82no—
344C8—1362no—
345C22—24no—
346C22—24no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22#1176260714yescomplemented
351C22#1176266411yescharacteristic
352C22C2×#1058747711yescharacteristic
353C22C2×#1058747711yescharacteristic
354C4—22no—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—41no—
359C4—41no—
360C4—348no—
361C4—684no—
362C2—21no—
363C2—21no—
364C2—24no—
365C2#1452869811yescharacteristic
366C2C2×#1176260712yes—
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
FittingC17×#1784C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC22×Q8×C17C23
hypercenter#1784D17
hypocenterC17#12700
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
2. C2×#11762607C2semidirect
171. C22×C17#356semidirect
245. C17#12700semidirect
350. C22#11762607semidirect

Maximal subgroups

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

subgroupindex
C22×#105876302
C2×#117626072
#145287082
C17×#17842
#1270017