Finite Group Databasebeta

← #16930397

Subgroups — #16930397

Order 4352.

690 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 115 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#14534518C211yesmaximal, characteristic
2#14527351C212yesmaximal, complemented
3C2×#11762533C211yesmaximal, characteristic, complemented
4#14526651C212yesmaximal, complemented
5C17×#2312C211yesmaximal, characteristic, complemented
6C2×#10587616C2211yescharacteristic
7C2×#10587639—21no—
8C2×C22⋊#3622C2212yes—
9#11762734C412yescomplemented
10#11762725—22no—
11#11762718C412yescomplemented
12#11762688—21no—
13#11762669C2211yescharacteristic
14#11762542—24no—
15#11762533—22no—
16C22×C22⋊C4×C17C2211yescharacteristic
17C17×#388—24no—
18C17×#380C2212yes—
19C17×#379—21no—
20C4×C22⋊C4×C17—21no—
21C25×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C22×C22⋊D17—21no—
25C22×C22⋊D17—22no—
26C22×D68C2×C411yescharacteristic
27C2×C4⋊C4×C17—22no—
28C2×C22⋊C4×C17—22no—
29C2×C22⋊C4×C17—22no—
30C2×C22⋊C4×C17—22no—
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17—22no—
33C2×C22⋊C4×C17C2×C412yes—
34C2×C22⋊C4×C17D412yes—
35C2×C42×C17—22no—
36#10587639—44no—
37C2×C22⋊Dic17—22no—
38C2×C22⋊Dic17—21no—
39#10587616—22no—
40#10587616—22no—
41C2×#63227—22no—
42C2×#63227—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17C2×C411yescharacteristic
45C22×#3622—22no—
46C17×#95—22no—
47C17×#95D414yescomplemented
48#10587502—22no—
49#10587502C2×C412yes—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C24×C17—21no—
53C24×C17—21no—
54C24×C17—22no—
55C24×C17—22no—
56C24×C17—81no—
57C24×C17—81no—
58C24×C17C22⋊C412yes—
59C24×C17C2×D411yescharacteristic
60C23×D17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—24no—
64C22×C4×C17—81no—
65C22×C4×C17—81no—
66C22×C4×C17C22⋊C412yes—
67C22×C4×C17C22×C411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D412yes—
70C22×C4×C17#4112yes—
71C2×C22⋊D17—44no—
72C2×C22⋊D17—41no—
73C2×C22⋊D17—41no—
74C2×C22⋊D17—42no—
75C2×C22⋊D17—42no—
76C2×C22⋊D17—82no—
77C2×C22⋊D17—82no—
78C22×Dic17—21no—
79C22×Dic17—22no—
80C2×D68—22no—
81C2×D68—81no—
82C4⋊C4×C17—82no—
83C22⋊C4×C17—24no—
84C22⋊C4×C17—24no—
85C22⋊C4×C17—42no—
86C22⋊C4×C17—42no—
87C22⋊C4×C17—82no—
88C22⋊C4×C17—82no—
89C42×C17—82no—
90C22⋊Dic17—44no—
91C22⋊Dic17—42no—
92#63227—42no—
93#63227—82no—
94C4⋊Dic17—22no—
95C4⋊Dic17—81no—
96C2×#3622—42no—
97C2×#3622—82no—
98#9205—171nomaximal, Sylow
99C23×C17—22no—
100C23×C17—22no—
101C23×C17—22no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—42no—
105C23×C17—42no—
106C23×C17—42no—
107C23×C17—42no—
108C23×C17—42no—
109C23×C17—42no—
110C23×C17—81no—
111C23×C17—81no—
112C23×C17—84no—
113C23×C17—84no—
114C23×C17C2×C22⋊C411yescharacteristic
115C23×C17#12011yescharacteristic
116C23×C17#12311yescharacteristic
117C23×C17#13512yes—
118C23×C17AGL(1,ℤ/8)12yes—
119C22×D17—42no—
120C22×D17—42no—
121C22×D17—41no—
122C22×D17—41no—
123C22×D17—41no—
124C2×C4×C17—22no—
125C2×C4×C17—42no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—44no—
132C2×C4×C17—44no—
133C2×C4×C17—44no—
134C2×C4×C17—81no—
135C2×C4×C17C4×D412yes—
136C2×C4×C17#12112yes—
137C22⋊D17—88no—
138C22⋊D17—82no—
139C22⋊D17—82no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—42no—
145C2×Dic17—41no—
146D68—82no—
147#3622—82no—
148#2617—171no—
149#2312D1711yescharacteristic, complemented
150#1519—172no—
151C2×#323—171no—
152#1289—172no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—42no—
160C22×C17—42no—
161C22×C17—44no—
162C22×C17—42no—
163C22×C17—42no—
164C22×C17—42no—
165C22×C17—82no—
166C22×C17—84no—
167C22×C17—84no—
168C22×C17#38611yescharacteristic
169C22×C17#41611yescharacteristic
170C22×C17#41811yescharacteristic
171C22×C17#44912yes—
172C22×C17#48212yes—
173C2×D17—82no—
174C2×D17—82no—
175C2×D17—81no—
176C2×D17—81no—
177C2×D17—81no—
178C4×C17—22no—
179C4×C17—22no—
180C4×C17—42no—
181C4×C17—84no—
182Dic17—82no—
183Dic17—81no—
184C2×#121—171no—
185C2×#120—341no—
186C22×C22⋊C4C2×D1711yescharacteristic
187C2×C22⋊C8—172no—
188#397—172no—
189#394—342no—
190#392—172no—
191#388—24no—
192#386—341no—
193#380—171no—
194#380C2×D1712yes—
195#379—21no—
196C4×C22⋊C4—21no—
197#327—344no—
198#323—342no—
199C2×C17—21no—
200C2×C17—22no—
201C2×C17—21no—
202C2×C17—22no—
203C2×C17—82no—
204C2×C17#187511yescharacteristic
205C2×C17#191111yescharacteristic
206C2×C17#191311yescharacteristic
207D17—161no—
208C25C22⋊D1711yescharacteristic
209C22×D4—171no—
210C22×D4—341no—
211C22×D4—342no—
212C23×C4C22⋊D1711yescharacteristic
213C23×C4C22×D1711yescharacteristic
214C22×C8—342no—
215#121—342no—
216#121—342no—
217#120—684no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—341no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—341no—
227C2×C22⋊C4—342no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4—342no—
230C2×C22⋊C4C4×D1712yes—
231C2×C22⋊C4D6812yes—
232C2×C42—22no—
233C22⋊C8—342no—
234C22⋊C8—342no—
235#95—22no—
236#95—172no—
237#95—342no—
238#95C22⋊D1714yescomplemented
239C17#920511yescharacteristic, complemented, Sylow
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—81no—
245C24—81no—
246C24—341no—
247C24#6322712yes—
248C24C2×C22⋊D1711yescharacteristic
249C2×D4—342no—
250C2×D4—684no—
251C2×D4—681no—
252C2×D4—682no—
253C2×D4—682no—
254C2×D4—681no—
255C2×D4—1361no—
256C2×D4—1362no—
257C2×D4—1362no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—342no—
264C22×C4—341no—
265C22×C4#6322712yes—
266C22×C4C2×C4×D1711yescharacteristic
267C22×C4C2×D6811yescharacteristic
268C22×C4#6325212yes—
269C22×C4C2×C22⋊D1712yes—
270C2×C8—682no—
271C2×C8—1362no—
272C4⋊C4—82no—
273C4⋊C4—342no—
274C4⋊C4—1361no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—82no—
280C22⋊C4—82no—
281C22⋊C4—684no—
282C22⋊C4—682no—
283C22⋊C4—682no—
284C22⋊C4—1362no—
285C42—82no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—81no—
298C23—81no—
299C23—84no—
300C23—84no—
301C23—682no—
302C23—682no—
303C23—681no—
304C23—681no—
305C23—681no—
306C23C2×#6322711yescharacteristic
307C23#1058761511yescharacteristic
308C23#1058763512yes—
309C23#1058763612yes—
310C23#1058763911yescharacteristic
311D4—1362no—
312D4—1368no—
313D4—1362no—
314D4—1362no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—81no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—682no—
331C2×C4—681no—
332C2×C4C4×C22⋊D1712yes—
333C2×C4#1058761612yes—
334C8—1362no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—82no—
348C22—84no—
349C22—84no—
350C22—1362no—
351C22—1362no—
352C22—1361no—
353C22—1361no—
354C22—1361no—
355C22#1176268811yescharacteristic
356C22#1176280311yescharacteristic
357C22#1176281311yescharacteristic
358C22#1176295412yes—
359C22#1176313312yes—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—84no—
364C4—1362no—
365C4—1361no—
366C2—22no—
367C2—21no—
368C2—21no—
369C2—22no—
370C2—82no—
371C2—2721no—
372C2#1452910611yescharacteristic
373C2#1452931611yescharacteristic
374C2#1452932611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2312C2
socleC22×C17#386
prime radical#14534518C2
cosocleC23×C4×C17C23
hypercenter#2312D17
hypocenterC17#9205
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #14527351C2semidirect
3. C2×#11762533C2semidirect
4. #14526651C2semidirect
5. C17×#2312C2semidirect
9. #11762734C4semidirect
11. #11762718C4semidirect
47. C17×#95D4semidirect
149. #2312D17semidirect
238. #95C22⋊D17semidirect
239. C17#9205semidirect

Maximal subgroups

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

subgroupindex
#145345182
#145273512
C2×#117625332
#145266512
C17×#23122
#920517