Finite Group Databasebeta

← #16932096

Subgroups — #16932096

Order 4352.

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

normal subgroups only

This group has 598 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
1#14537808C211yesmaximal, characteristic
2#14533092C211yesmaximal, characteristic
3#14528868C212yesmaximal, complemented
4#14526651C212yesmaximal, complemented
5C17×#1799C211yesmaximal, characteristic
6C2×#10587621—21no—
7C22×C4⋊Dic17C2211yescharacteristic
8D4×#3622—24no—
9#11762862—21no—
10C2×C22⋊#3622—21no—
11C2×C22⋊#3622C2212yes—
12#11762739C412yescomplemented
13#11762734C412yescomplemented
14#11762707—21no—
15#11762670—21no—
16#11762669—21no—
17#11762669C2211yescharacteristic
18#11762571—24no—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×#99—21no—
21C17×#380C2212yes—
22C17×#342—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—22no—
36C2×C22⋊Dic17—21no—
37#10587621—42no—
38#10587621—42no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—22no—
42C2×C4⋊Dic17—22no—
43C2×C4⋊Dic17C2×C412yes—
44C2×#63225—22no—
45C22×#3622—22no—
46C22×#3622D414yes—
47C17×#99—42no—
48C17×#95—22no—
49#10587502—22no—
50#10587502—22no—
51#10587502C2×C412yes—
52C22⋊#3622—22no—
53C22⋊#3622—22no—
54C22⋊#3622—22no—
55C4⋊#3622—22no—
56C4×#3622—22no—
57C24×C17C22⋊C411yescharacteristic
58C24×C17C2×D411yescharacteristic
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17D812yes—
62C2×D4×C17SD1612yes—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D412yes—
74C22×C4×C17#4112yes—
75C22×Dic17—22no—
76C22×Dic17—22no—
77C22×Dic17—42no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—42no—
81C4⋊C4×C17—22no—
82C4⋊C4×C17—44no—
83C22⋊C4×C17—22no—
84C22⋊C4×C17—22no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—42no—
87C42×C17—21no—
88C42×C17—21no—
89C22⋊Dic17—44no—
90C22⋊Dic17—41no—
91C22⋊Dic17—41no—
92C4⋊Dic17—24no—
93C4⋊Dic17—42no—
94C4⋊Dic17—41no—
95C4⋊Dic17—41no—
96#63225—44no—
97C2×#3622—24no—
98C2×#3622—44no—
99C2×#3622—82no—
100#9470—171nomaximal, Sylow
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17#10212yes—
108C23×C17C2×C22⋊C411yescharacteristic
109C23×C17#12112yes—
110C23×C17C2×D811yescharacteristic
111C23×C17C2×SD1611yescharacteristic
112C23×C17#13512yes—
113D4×C17—24no—
114D4×C17—24no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—24no—
118C2×C4×C17—22no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—41no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—81no—
127C2×C4×C17—82no—
128C2×C4×C17—82no—
129C2×C4×C17—82no—
130C2×C4×C17#10212yes—
131C2×C4×C17C4×D412yes—
132C2×C4×C17#12011yescharacteristic
133C2×C4×C17#12311yescharacteristic
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—82no—
141C2×Dic17—82no—
142#3622—44no—
143#2972—171no—
144#2486—171no—
145#1823—172no—
146#1799D1711yescharacteristic
147#1289—172no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—21no—
151C22×C17—22no—
152C22×C17—22no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—42no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—81no—
160C22×C17#38611yescharacteristic
161C22×C17C2×#10211yescharacteristic
162C22×C17#41611yescharacteristic
163C22×C17#46512yes—
164C22×C17#46612yes—
165C4×C17—21no—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—162no—
171C4×C17#45812yes—
172Dic17—82no—
173Dic17—82no—
174C2×#123—341no—
175C2×C4×D4C2×D1711yescharacteristic
176C22×C4⋊C4—171no—
177C8×D4—344no—
178#431—341no—
179C2×#99—21no—
180C2×C22⋊C8—172no—
181C2×C22⋊C8—341no—
182#398—172no—
183#397—172no—
184#389—341no—
185#381—341no—
186#380—171no—
187#380—341no—
188#380C2×D1712yes—
189#342—21no—
190#340—344no—
191C2×C17—21no—
192C2×C17—22no—
193C2×C17—21no—
194C2×C17—22no—
195C2×C17—42no—
196C2×C17#189011yescharacteristic
197C2×C17#189311yescharacteristic
198C2×C17#190911yescharacteristic
199C22×D4C22⋊D1711yescharacteristic
200C23×C4—341no—
201C23×C4D6811yescharacteristic
202C23×C4C22×D1711yescharacteristic
203C22×C8—174no—
204C22×C8—342no—
205#123—682no—
206#123—682no—
207C4×D4—22no—
208C4×D4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—172no—
211C2×C4⋊C4—342no—
212C2×C4⋊C4—342no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—342no—
216C2×C4⋊C4C4×D1711yescharacteristic
217C2×C22⋊C4—22no—
218C2×C22⋊C4—341no—
219C2×C22⋊C4—342no—
220C2×C22⋊C4C4×D1711yescharacteristic
221C2×C22⋊C4C22⋊D1711yescharacteristic
222C2×C42D6811yescharacteristic
223C4⋊C8—342no—
224#99—42no—
225C22⋊C8—342no—
226C22⋊C8—342no—
227C22⋊C8—342no—
228C4×C8—342no—
229#95—22no—
230#95—172no—
231#95—342no—
232#95—342no—
233C17#947011yescharacteristic, complemented, Sylow
234C24#6322711yescharacteristic
235C24C2×C22⋊D1711yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#6322812yes—
239C2×D4#6323012yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—681no—
248C22×C4—682no—
249C22×C4—681no—
250C22×C4—682no—
251C22×C4#6322711yescharacteristic
252C22×C4#6322711yescharacteristic
253C22×C4#6322711yescharacteristic
254C22×C4C2×C4×D1711yescharacteristic
255C22×C4C2×D6811yescharacteristic
256C22×C4D4×D1712yes—
257C22×C4#6325412yes—
258C2×C8—344no—
259C2×C8—684no—
260C2×C8—1362no—
261C4⋊C4—22no—
262C4⋊C4—44no—
263C4⋊C4—344no—
264C4⋊C4—682no—
265C4⋊C4—684no—
266C4⋊C4—681no—
267C4⋊C4—681no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—684no—
273C22⋊C4—681no—
274C22⋊C4—681no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1058748012yes—
284C23C2×#6322711yescharacteristic
285C23C2×#6322811yescharacteristic
286C23#1058762512yes—
287C23C2×#6323011yescharacteristic
288C23#1058763612yes—
289D4—24no—
290D4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4—682no—
310C2×C4—682no—
311C2×C4—682no—
312C2×C4—1362no—
313C2×C4—1362no—
314C2×C4#1058748012yes—
315C2×C4#1058756712yes—
316C2×C4#1058756811yescharacteristic
317C2×C4#1058757211yescharacteristic
318C8—684no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1176268611yescharacteristic
332C22C2×#1058748011yescharacteristic
333C22#1176280311yescharacteristic
334C22#1176304012yes—
335C22#1176304312yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—162no—
342C4—1362no—
343C4—1362no—
344C4#1176300012yes—
345C2—22no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#1452919511yescharacteristic
351C2#1452921011yescharacteristic
352C2#1452930411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1799C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#1799D17
hypocenterC17#9470
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14528868C2semidirect
4. #14526651C2semidirect
12. #11762739C4semidirect
13. #11762734C4semidirect
233. C17#9470semidirect

Maximal subgroups

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

subgroupindex
#145378082
#145330922
#145288682
#145266512
C17×#17992
#947017