Finite Group Databasebeta

← #16945370

Subgroups — #16945370

Order 4352.

764 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528776C211yesmaximal, characteristic, complemented
2#14527102C211yesmaximal, characteristic, complemented
3#14526787C214yesmaximal, complemented
4C17×#2408C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587504—21no—
7C2×C22⋊#3622—21no—
8#11762578C2212yescomplemented
9#11762535C2212yescomplemented
10#11762533—21no—
11C23×D4×C17C411yescharacteristic
12C2×C17×#121—21no—
13C2×C17×#120—21no—
14C2×C4×D4×C17C2211yescharacteristic
15C17×#394C2212yes—
16C17×#392C412yes—
17C17×#390—21no—
18C17×#380C411yescharacteristic
19C17×#379—21no—
20C25×C17—21no—
21C22×D4×C17—22no—
22C22×D4×C17—21no—
23C22×D4×C17—41no—
24C22×D4×C17—41no—
25C22×D4×C17—44no—
26C22×D4×C17C2×C412yes—
27C22×D4×C17D411yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17D411yescharacteristic
30C17×#121—44no—
31C17×#120—42no—
32C17×#120—42no—
33C4×D4×C17—41no—
34C4×D4×C17—41no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17C2311yescharacteristic
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17—22no—
40C2×C22⋊C4×C17—22no—
41C2×C22⋊C4×C17C2×C411yescharacteristic
42C2×C22⋊C4×C17D411yescharacteristic
43C2×C42×C17D411yescharacteristic
44C2×#63223—22no—
45C22×#3622—41no—
46C17×#95C2×C412yes—
47#10587504—42no—
48C22⋊#3622—22no—
49C22⋊#3622—41no—
50C4⋊#3622—41no—
51C4⋊#3622—41no—
52C24×C17—22no—
53C24×C17—42no—
54C24×C17—44no—
55C24×C17—42no—
56C24×C17—42no—
57C24×C17—82no—
58C24×C17C22⋊C411yescharacteristic
59C24×C17C22⋊C411yescharacteristic
60C2×D4×C17—24no—
61C2×D4×C17—41no—
62C2×D4×C17—41no—
63C2×D4×C17—42no—
64C2×D4×C17—42no—
65C2×D4×C17—48no—
66C2×D4×C17—44no—
67C2×D4×C17—44no—
68C2×D4×C17—44no—
69C2×D4×C17—44no—
70C2×D4×C17—44no—
71C2×D4×C17—41no—
72C2×D4×C17—41no—
73C2×D4×C17—82no—
74C2×D4×C17D812yes—
75C2×D4×C17SD1612yes—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—22no—
81C22×C4×C17—42no—
82C22×C4×C17—42no—
83C22×C4×C17C22⋊C411yescharacteristic
84C22×C4×C17C22⋊C411yescharacteristic
85C22×C4×C17C22×C411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C4⋊C4×C17—41no—
89C4⋊C4×C17—42no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—44no—
93C22⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—42no—
96C22⋊C4×C17—82no—
97C42×C17—21no—
98C42×C17—21no—
99#63223—42no—
100#63223—42no—
101C2×#3622—42no—
102C2×#3622—42no—
103C2×#3622—82no—
104#11880—171nomaximal, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—24no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—42no—
115C23×C17—42no—
116C23×C17—42no—
117C23×C17—48no—
118C23×C17—42no—
119C23×C17—42no—
120C23×C17—42no—
121C23×C17—84no—
122C23×C17—88no—
123C23×C17—88no—
124C23×C17—84no—
125C23×C17#10212yes—
126C23×C17#10212yes—
127C23×C17C2×C22⋊C411yescharacteristic
128D4×C17—42no—
129D4×C17—42no—
130D4×C17—44no—
131D4×C17—44no—
132D4×C17—44no—
133D4×C17—44no—
134D4×C17—88no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—22no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—42no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—81no—
151C2×C4×C17#9911yescharacteristic
152C2×C4×C17#9911yescharacteristic
153C2×C4×C17C2×D811yescharacteristic
154C2×C4×C17C2×SD1611yescharacteristic
155#3622—82no—
156#3622—82no—
157#2408D1711yescharacteristic
158#1800—171no—
159#1459—171no—
160#1345—174no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17—42no—
166C22×C17—42no—
167C22×C17—42no—
168C22×C17—44no—
169C22×C17—44no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—44no—
175C22×C17—81no—
176C22×C17—82no—
177C22×C17—82no—
178C22×C17—816no—
179C22×C17—82no—
180C22×C17—82no—
181C22×C17C2×#9911yescharacteristic
182C22×C17C2×#10211yescharacteristic
183C22×C17#42111yescharacteristic
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—41no—
188C4×C17—41no—
189C4×C17—41no—
190C4×C17—82no—
191C4×C17AΣL(1,𝔽16)12yes—
192C23×D4Dic1711yescharacteristic
193C2×#121—21no—
194C2×#120—21no—
195C2×C4×D4C2×D1711yescharacteristic
196#423—341no—
197C2×#100—341no—
198C2×C22⋊C8—341no—
199#394C2×D1712yes—
200#392Dic1712yes—
201#390—21no—
202#380Dic1711yescharacteristic
203#379—21no—
204#343—172no—
205#324—172no—
206#323—341no—
207C2×C17—21no—
208C2×C17—21no—
209C2×C17—42no—
210C2×C17—42no—
211C2×C17—84no—
212C2×C17#151911yescharacteristic
213C2×C17C2×AΣL(1,𝔽16)11yescharacteristic
214C2×C17#213411yescharacteristic
215C25—21no—
216C22×D4—22no—
217C22×D4—21no—
218C22×D4—41no—
219C22×D4—41no—
220C22×D4—44no—
221C22×D4C2×Dic1712yes—
222C22×D4C22⋊D1711yescharacteristic
223C23×C4C2×Dic1711yescharacteristic
224C23×C4C22⋊D1711yescharacteristic
225C2×M16—342no—
226C22×C8—681no—
227#121—44no—
228#120—42no—
229#120—42no—
230C4×D4—41no—
231C4×D4—41no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4C22×D1711yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4C2×Dic1711yescharacteristic
239C2×C22⋊C4C22⋊D1711yescharacteristic
240C2×C42C22⋊D1711yescharacteristic
241C4⋊C8—681no—
242C4⋊C8—681no—
243#100—682no—
244C22⋊C8—342no—
245C22⋊C8—681no—
246#95C2×Dic1712yes—
247C17#1188011yescharacteristic, complemented, Sylow
248C24—22no—
249C24—42no—
250C24—44no—
251C24—42no—
252C24—42no—
253C24—82no—
254C24C22⋊Dic1711yescharacteristic
255C24C22⋊Dic1711yescharacteristic
256C2×D4—24no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—48no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—82no—
270C2×D4#6322812yes—
271C2×D4#6322912yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4C22⋊Dic1711yescharacteristic
280C22×C4C22⋊Dic1711yescharacteristic
281C22×C4C22×Dic1711yescharacteristic
282C22×C4C2×C22⋊D1711yescharacteristic
283C22×C4C2×C22⋊D1711yescharacteristic
284M16—682no—
285M16—682no—
286C2×C8—682no—
287C2×C8—682no—
288C2×C8—1362no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—44no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—82no—
298C42—21no—
299C42—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—24no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—48no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—84no—
317C23—88no—
318C23—88no—
319C23—84no—
320C23#1058750312yes—
321C23#1058750312yes—
322C23C2×C22⋊Dic1711yescharacteristic
323D4—42no—
324D4—42no—
325D4—44no—
326D4—44no—
327D4—44no—
328D4—44no—
329D4—88no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4#1058750511yescharacteristic
347C2×C4#1058750511yescharacteristic
348C2×C4C2×#6322811yescharacteristic
349C2×C4C2×#6322911yescharacteristic
350C8—1362no—
351C8—1362no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—44no—
366C22—81no—
367C22—82no—
368C22—82no—
369C22—816no—
370C22—82no—
371C22—82no—
372C22C2×#1058750511yescharacteristic
373C22C2×#1058750311yescharacteristic
374C22#1176282611yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—82no—
382C4#1176259612yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—84no—
388C2#1452735211yescharacteristic
389C2C2×#1176259611yescharacteristic
390C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
Fitting, prime radicalC17×#2408C2
socleC22×C17C2×#99
cosocleC2×C4⋊C4×C17C23
hypercenter#2408D17
hypocenterC17#11880
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #14528776C2semidirect
2. #14527102C2semidirect
3. #14526787C2semidirect
8. #11762578C22semidirect
9. #11762535C22semidirect
247. C17#11880semidirect

Maximal subgroups

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

subgroupindex
#145287762
#145271022
#145267872
C17×#24082
#1188017