Finite Group Databasebeta

← #16932218

Subgroups — #16932218

Order 4352.

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

normal subgroups only

This group has 514 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#14534350C211yesmaximal, characteristic
2#14527228C212yesmaximal, complemented
3#14527118C211yesmaximal, characteristic, complemented
4#14526649C212yesmaximal, complemented
5C17×#2921C211yesmaximal, characteristic, complemented
6C2×C4×D68C2211yescharacteristic
7#11762866C2212yes—
8#11762748C411yescharacteristic
9#11762745C411yescharacteristic
10#11762726C411yescharacteristic
11#11762718C411yescharacteristic
12#11762700—21no—
13#11762688—21no—
14#11762685—21no—
15#11762680—21no—
16#11762677C2211yescharacteristic
17#11762538—24no—
18#11762535—21no—
19#11762533—21no—
20C2×C17×#117C2211yescharacteristic
21C17×#433—24no—
22C17×#423—21no—
23C17×#407—21no—
24C4×M16×C17C2212yes—
25C23×C4×C17D411yescharacteristic
26C22×C22⋊D17—21no—
27C22×D68C2×C411yescharacteristic
28C22×C4×D17—21no—
29C2×M16×C17—22no—
30C17×#117—22no—
31C17×#117—22no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C22⋊C4×C17C2×C411yescharacteristic
35C2×C22⋊C4×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C42×C17C2311yescharacteristic
38C2×C22⋊Dic17—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63227—21no—
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×#63225—21no—
45C4×D68—22no—
46C4×D68—22no—
47C17×C4⋊C8—24no—
48C17×C22⋊C8—24no—
49C17×#97—22no—
50C4×C8×C17D414yescomplemented
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502C2×C411yescharacteristic
56#10587502C2×C411yescharacteristic
57C22⋊#3622—22no—
58C4⋊#3622—22no—
59#10587474—22no—
60C24×C17C22⋊C411yescharacteristic
61C23×D17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—41no—
65C22×C4×C17—41no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C2×C22⋊D17—42no—
73C2×C22⋊D17—42no—
74C2×C22⋊D17—81no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—21no—
78C2×D68—22no—
79C2×D68—42no—
80C2×C4×D17—42no—
81C2×C4×D17—81no—
82C2×C4×D17—81no—
83M16×C17—42no—
84M16×C17—42no—
85C2×C8×C17—24no—
86C4⋊C4×C17—22no—
87C4⋊C4×C17—22no—
88C22⋊C4×C17—22no—
89C22⋊C4×C17—22no—
90C42×C17—21no—
91C42×C17—21no—
92C42×C17C2×D412yes—
93C42×C17#4112yes—
94C22⋊Dic17—81no—
95#63227—42no—
96#63227—42no—
97#63227—41no—
98#63227—41no—
99#63227—81no—
100C4⋊Dic17—22no—
101#63225—81no—
102C2×#3622—42no—
103#9494—171nomaximal, Sylow
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17C2×C22⋊C411yescharacteristic
110C22×D17—22no—
111C22×D17—41no—
112C22×D17—41no—
113C22×D17—41no—
114C22×D17—41no—
115C22×D17—41no—
116C22×D17—41no—
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—21no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—42no—
127C2×C4×C17—42no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17C4×D412yes—
131C2×C4×C17#12011yescharacteristic
132C2×C4×C17#12112yes—
133C2×C4×C17#12311yescharacteristic
134C22⋊D17—84no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144D68—24no—
145D68—42no—
146C4×D17—84no—
147C8×C17—44no—
148#3622—82no—
149#2921D1711yescharacteristic, complemented
150#2598—171no—
151#1490—172no—
152#1464—171no—
153#1288—172no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—42no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—81no—
161C22×C17#38611yescharacteristic
162C22×C17#42111yescharacteristic
163C22×C17#42111yescharacteristic
164C2×D17—42no—
165C2×D17—42no—
166C2×D17—42no—
167C2×D17—81no—
168C2×D17—81no—
169C2×D17—81no—
170C2×D17—81no—
171C4×C17—22no—
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—22no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—41no—
178Dic17—81no—
179Dic17—81no—
180Dic17—81no—
181C2×C4×D4—171no—
182C2×#117C2×D1711yescharacteristic
183#433—24no—
184#432—172no—
185#423—21no—
186#407—21no—
187C4×M16C2×D1712yes—
188#400—171no—
189#399—171no—
190#394—171no—
191#392—171no—
192#388—341no—
193#386—341no—
194#386—341no—
195#385—341no—
196#383—171no—
197D4⋊C8—344no—
198#324—341no—
199#323—341no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—81no—
203C2×C17#189111yescharacteristic
204C2×C17#189711yescharacteristic
205C2×C17#191011yescharacteristic
206D17—82no—
207C22×D4—171no—
208C22×D4—341no—
209C23×C4—341no—
210C23×C4C22⋊D1711yescharacteristic
211C2×M16—22no—
212C4×D4—342no—
213C4×D4—342no—
214#117—22no—
215#117—22no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4C4×D1711yescharacteristic
219C2×C4⋊C4D6811yescharacteristic
220C2×C22⋊C4—341no—
221C2×C22⋊C4—341no—
222C2×C22⋊C4—341no—
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C22⋊C4C4×D1711yescharacteristic
226C2×C22⋊C4D6811yescharacteristic
227C2×C42C22⋊D1711yescharacteristic
228C2×C42C22×D1711yescharacteristic
229C4⋊C8—24no—
230C4⋊C8—342no—
231C22⋊C8—24no—
232C22⋊C8—342no—
233#97—22no—
234#97—342no—
235C4×C8C22⋊D1714yescomplemented
236#95—171no—
237#95—171no—
238#95—341no—
239#95—341no—
240#95—341no—
241#95—341no—
242C17#949411yescharacteristic, complemented, Sylow
243C24—341no—
244C24#6322711yescharacteristic
245C2×D4—342no—
246C2×D4—682no—
247C2×D4—682no—
248C2×D4—682no—
249C2×D4—1361no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—341no—
255C22×C4—341no—
256C22×C4—341no—
257C22×C4—682no—
258C22×C4—1361no—
259C22×C4—1361no—
260C22×C4#6322711yescharacteristic
261C22×C4#6322711yescharacteristic
262C22×C4#6322711yescharacteristic
263C22×C4C2×C4×D1711yescharacteristic
264C22×C4C2×D6811yescharacteristic
265C22×C4C2×C22⋊D1711yescharacteristic
266M16—42no—
267M16—42no—
268C2×C8—24no—
269C2×C8—682no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—342no—
273C4⋊C4—1361no—
274C22⋊C4—22no—
275C22⋊C4—22no—
276C22⋊C4—682no—
277C22⋊C4—682no—
278C22⋊C4—681no—
279C22⋊C4—681no—
280C22⋊C4—1361no—
281C22⋊C4—1361no—
282C42—21no—
283C42—21no—
284C42#6325212yes—
285C42C2×C22⋊D1712yes—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—342no—
292C23—681no—
293C23—681no—
294C23—681no—
295C23—681no—
296C23—681no—
297C23—681no—
298C23C2×#6322711yescharacteristic
299D4—344no—
300D4—682no—
301D4—1364no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—1364no—
325C2×C4C4×C22⋊D1712yes—
326C2×C4#1058761511yescharacteristic
327C2×C4#1058761612yes—
328C2×C4#1058763911yescharacteristic
329C8—44no—
330C8—1362no—
331C22—21no—
332C22—21no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—682no—
339C22—682no—
340C22—682no—
341C22—1361no—
342C22—1361no—
343C22—1361no—
344C22—1361no—
345C22#1176268811yescharacteristic
346C22#1176282711yescharacteristic
347C22#1176282711yescharacteristic
348C4—22no—
349C4—22no—
350C4—21no—
351C4—21no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—1361no—
356C4—1361no—
357C4—1361no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—1362no—
362C2#1452920211yescharacteristic
363C2#1452923211yescharacteristic
364C2#1452931111yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2921C2
socleC22×C17#386
prime radical#11762718C4
cosocleC2×C42×C17C23
hypercenter#2921D17
hypocenterC17#9494
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #14527228C2semidirect
3. #14527118C2semidirect
4. #14526649C2semidirect
5. C17×#2921C2semidirect
50. C4×C8×C17D4semidirect
149. #2921D17semidirect
235. C4×C8C22⋊D17semidirect
242. C17#9494semidirect

Maximal subgroups

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

subgroupindex
#145343502
#145272282
#145271182
#145266492
C17×#29212
#949417