Finite Group Databasebeta

← #16926078

Subgroups — #16926078

Order 4352.

998 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#11763464C211yesmaximal, characteristic
2#14527466C212yesmaximal, complemented
3#14527208C212yesmaximal, complemented
4#14527105C211yesmaximal, characteristic, complemented
5C17×#1800C211yesmaximal, characteristic, complemented
6C22×#63252C412yes—
7#11763464—24no—
8#11763464—24no—
9C2×#10587555C2211yescharacteristic
10C2×#10587569—22no—
11C2×#10587610C412yes—
12C2×#10587624—21no—
13C2×C4×D68C2211yescharacteristic
14C2×C4⋊C4×D17—21no—
15#11762862C2212yes—
16#11762557—24no—
17#11762536—24no—
18#11762533—22no—
19C2×C4×D4×C17C2211yescharacteristic
20C17×#423C2212yes—
21C2×C17×#100—21no—
22C17×#343—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C412yes—
25C22×C22⋊D17—22no—
26C2×#63252—24no—
27C2×#63252—24no—
28C2×#63252—48no—
29C22×D68C2×C411yescharacteristic
30C22×C4×D17—22no—
31C22×C4×D17—21no—
32C22×Dic34C2×C412yes—
33C2×M16×C17—22no—
34C4×D4×C17—22no—
35C4×D4×C17—22no—
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C22⋊C4×C17D412yes—
38C2×C42×C17C2311yescharacteristic
39C2×C22⋊Dic17—22no—
40#10587624—28no—
41C2×#63227—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17C2×C411yescharacteristic
45#10587610—24no—
46#10587610—24no—
47C2×#63225—22no—
48C4⋊C4×D17—28no—
49#10587569—216no—
50C4×D68—22no—
51C4×D68D414yes—
52#10587555—22no—
53#10587555D414yes—
54C17×C4⋊C8—22no—
55C17×C4⋊C8—22no—
56C17×#100—42no—
57C22⋊#3622—22no—
58C4⋊#3622—22no—
59C4×#3622—22no—
60C24×C17C22⋊C412yes—
61C23×D17—21no—
62C2×D4×C17—22no—
63C2×D4×C17—22no—
64C2×D4×C17—21no—
65C2×D4×C17—21no—
66C22×C4×C17—22no—
67C22×C4×C17—22no—
68C22×C4×C17—44no—
69C22×C4×C17C22⋊C412yes—
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C2×C22⋊D17—28no—
74C2×C22⋊D17—48no—
75C22×Dic17—22no—
76C22×Dic17—21no—
77#63252—48no—
78#63252—416no—
79#63252—48no—
80C2×D68—21no—
81C2×D68—24no—
82C2×D68C22⋊C412yes—
83C2×C4×D17—28no—
84C2×C4×D17—24no—
85C2×C4×D17—48no—
86C2×C4×D17—42no—
87C2×C4×D17—41no—
88C2×C4×D17—41no—
89C2×Dic34—22no—
90C2×Dic34—44no—
91C2×Dic34C22⋊C414yes—
92M16×C17—42no—
93M16×C17—42no—
94C2×C8×C17—42no—
95C4⋊C4×C17—22no—
96C22⋊C4×C17—24no—
97C42×C17C2×D412yes—
98C42×C17C2×D412yes—
99C22⋊Dic17—28no—
100#63227—24no—
101C4⋊Dic17—21no—
102C4⋊Dic17—24no—
103C4⋊Dic17—24no—
104C4⋊Dic17C22⋊C412yes—
105#63225—28no—
106C2×#3622—42no—
107#8393—171nomaximal, Sylow
108C23×C17—22no—
109C23×C17—22no—
110C23×C17—22no—
111C23×C17—42no—
112C23×C17—42no—
113C23×C17C2×C22⋊C411yescharacteristic
114C22×D17—24no—
115C22×D17—22no—
116C22×D17—42no—
117C22×D17—41no—
118C22×D17—41no—
119D4×C17—42no—
120D4×C17—42no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—41no—
124C2×C4×C17—44no—
125C2×C4×C17—44no—
126C2×C4×C17—42no—
127C2×C4×C17C2×C22⋊C411yescharacteristic
128C2×C4×C17C2×C22⋊C411yescharacteristic
129C2×C4×C17#12012yes—
130C2×C4×C17#12012yes—
131C2×C4×C17#13512yes—
132C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C17#13711yescharacteristic
134C22⋊D17—416no—
135C2×Dic17—22no—
136C2×Dic17—24no—
137C2×Dic17—44no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140D68—24no—
141D68—22no—
142D68#10014yescomplemented
143C4×D17—416no—
144C4×D17—44no—
145C4×D17—44no—
146Dic34—44no—
147Dic34—44no—
148C8×C17—82no—
149#3622—82no—
150C2×#547—171no—
151#1800D1711yescharacteristic, complemented
152#1548—172no—
153#1486—172no—
154#1460—171no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—42no—
158C22×C17—42no—
159C22×C17—42no—
160C22×C17—82no—
161C22×C17—82no—
162C22×C17#37911yescharacteristic
163C22×C17#41611yescharacteristic
164C22×C17#41711yescharacteristic
165C2×D17—24no—
166C2×D17—44no—
167C2×D17—44no—
168C2×D17—42no—
169C2×D17—42no—
170C4×C17—22no—
171C4×C17—21no—
172C4×C17—42no—
173C4×C17C2×#10012yes—
174C4×C17#44912yes—
175C4×C17#45212yes—
176Dic17—44no—
177Dic17—42no—
178C22×#41—172no—
179#547—344no—
180#547—344no—
181C2×C4⋊Q8—171no—
182C2×#123—342no—
183C2×C22⋊Q8—172no—
184C2×#121—341no—
185C2×C4×D4—171no—
186C2×C4×D4C2×D1711yescharacteristic
187C22×C4⋊C4—341no—
188#431—172no—
189#423C2×D1712yes—
190C2×#100—21no—
191#343—21no—
192#333—344no—
193D4⋊C8—344no—
194#323—342no—
195C2×C17—21no—
196C2×C17—21no—
197C2×C17—82no—
198C2×C17#178311yescharacteristic
199C2×C17#178811yescharacteristic
200C2×C17#179611yescharacteristic
201D17—44no—
202C2×#41—344no—
203C2×#41—344no—
204C2×#41—688no—
205C22×Q8—172no—
206C22×D4—171no—
207C22×D4—342no—
208C22×D4C22⋊D1711yescharacteristic
209C23×C4—341no—
210C23×C4—342no—
211C23×C4C4×D1712yes—
212C2×M16—22no—
213C4⋊Q8—174no—
214C4⋊Q8—342no—
215#123—3416no—
216C22⋊Q8—344no—
217C22⋊Q8—344no—
218#121—348no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—174no—
222C4×D4—342no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—348no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—342no—
228C2×C4⋊C4C22⋊D1711yescharacteristic
229C2×C22⋊C4—342no—
230C2×C22⋊C4—341no—
231C2×C22⋊C4D6812yes—
232C2×C42C22×D1711yescharacteristic
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—342no—
236#100—42no—
237C22⋊C8—342no—
238C4×C8—342no—
239C17#839311yescharacteristic, complemented, Sylow
240C24—341no—
241C24#6322712yes—
242#41—688no—
243#41—6816no—
244#41—688no—
245C2×Q8—174no—
246C2×Q8—342no—
247C2×Q8—684no—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—22no—
252C2×D4—172no—
253C2×D4—344no—
254C2×D4—348no—
255C2×D4—341no—
256C2×D4—688no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—44no—
260C22×C4—348no—
261C22×C4—341no—
262C22×C4—342no—
263C22×C4—344no—
264C22×C4—688no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4—682no—
268C22×C4#6322712yes—
269C22×C4C2×C4×D1711yescharacteristic
270C22×C4C2×D6811yescharacteristic
271C22×C4C2×C22⋊D1711yescharacteristic
272M16—42no—
273M16—42no—
274C2×C8—42no—
275C2×C8—682no—
276C4⋊C4—22no—
277C4⋊C4—172no—
278C4⋊C4—348no—
279C4⋊C4—341no—
280C4⋊C4—344no—
281C4⋊C4—344no—
282C22⋊C4—24no—
283C22⋊C4—348no—
284C22⋊C4—344no—
285C42D4×D1712yes—
286C42D4×D1712yes—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—42no—
291C23—42no—
292C23—344no—
293C23—342no—
294C23—682no—
295C23—681no—
296C23—681no—
297C23C2×#6322711yescharacteristic
298Q8—684no—
299Q8—684no—
300D4—42no—
301D4—42no—
302D4—174no—
303D4—344no—
304D4—342no—
305D4—6816no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—342no—
313C2×C4—344no—
314C2×C4—6816no—
315C2×C4—684no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—684no—
319C2×C4—684no—
320C2×C4C22⋊C4×D1711yescharacteristic
321C2×C4C22⋊C4×D1711yescharacteristic
322C2×C4#1058756812yes—
323C2×C4#1058761811yescharacteristic
324C2×C4#1058762312yes—
325C2×C4#1058762811yescharacteristic
326C2×C4#1058763612yes—
327C8—82no—
328C8—1362no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—82no—
335C22—82no—
336C22—344no—
337C22—684no—
338C22—684no—
339C22—682no—
340C22—682no—
341C22#1176266411yescharacteristic
342C22#1176280311yescharacteristic
343C22#1176280811yescharacteristic
344C4—21no—
345C4—22no—
346C4—42no—
347C4—684no—
348C4—682no—
349C4#100×D1712yes—
350C4#1176295212yes—
351C4#1176296912yes—
352C2—21no—
353C2—21no—
354C2—82no—
355C2—684no—
356C2#1452870311yescharacteristic
357C2#1452872211yescharacteristic
358C2#1452876011yescharacteristic
subgroupG/H
centerC22#11762664
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1800C2
socleC22×C17#379
prime radicalC2×#11763464C2
cosocleC2×C42×C17C23
hypercenter#1800D17
hypocenterC17#8393
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #14527466C2semidirect
3. #14527208C2semidirect
4. #14527105C2semidirect
5. C17×#1800C2semidirect
142. D68#100semidirect
151. #1800D17semidirect
239. C17#8393semidirect

Maximal subgroups

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

subgroupindex
C2×#117634642
#145274662
#145272082
#145271052
C17×#18002
#839317