Finite Group Databasebeta

← #16934833

Subgroups — #16934833

Order 4352.

568 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 105 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×#11763142C211yesmaximal, characteristic
2#14533263C211yesmaximal, characteristic
3#14529770C211yesmaximal, characteristic
4#14528875C211yesmaximal, characteristic
5#14527356C211yesmaximal, characteristic
6#14526631C211yesmaximal, characteristic
7C17×#3189C211yesmaximal, characteristic
8C22×C4⋊Dic17C2211yescharacteristic
9#11763142—22no—
10#11763142—22no—
11C2×#10587478—21no—
12C2×#10587506—21no—
13C2×C22⋊#3622C2211yescharacteristic
14#11762739—22no—
15#11762716—21no—
16#11762681—22no—
17#11762678—21no—
18#11762669C2211yescharacteristic
19C4×C22⋊Dic17C2211yescharacteristic
20#11762576—22no—
21#11762572—22no—
22#11762546—22no—
23C2×C17×C22⋊Q8C2211yescharacteristic
24C2×C17×C22⋊Q8C2211yescharacteristic
25C17×#484—21no—
26C17×#484—21no—
27C17×#467—22no—
28C17×#451—21no—
29C17×#451—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C22×Q8×C17D411yescharacteristic
33C23×C4×C17C2311yescharacteristic
34C23×Dic17—21no—
35C2×Q16×C17—22no—
36C22×C8×C17D412yes—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C17×C22⋊Q8—21no—
43C17×C22⋊Q8—21no—
44C17×C22⋊Q8—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D411yescharacteristic
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C22⋊C4×C17—21no—
51C2×C22⋊Dic17—21no—
52C2×C22⋊Dic17D412yes—
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17—21no—
58C2×C4⋊Dic17—21no—
59C2×C4⋊Dic17D412yes—
60C2×#63225—24no—
61C2×C4×Dic17—21no—
62C22×#3622—21no—
63C17×#107—22no—
64C17×#103—22no—
65C17×#103—22no—
66C17×C22⋊C8—21no—
67C17×C22⋊C8—21no—
68#10587506—24no—
69#10587502—24no—
70#10587502—21no—
71#10587502—21no—
72C22⋊#3622—21no—
73C22⋊#3622D412yes—
74#10587478—24no—
75C24×C17C2×D411yescharacteristic
76C2×Q8×C17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—42no—
81C2×Q8×C17—42no—
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—42no—
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17#4111yescharacteristic
93C22×Dic17—22no—
94C22×Dic17—22no—
95C22×Dic17—41no—
96C22×Dic17—41no—
97C22×Dic17—41no—
98C22×Dic17—82no—
99Q16×C17—82no—
100C2×C8×C17—22no—
101C2×C8×C17—42no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17—41no—
108C4⋊C4×C17—41no—
109C4⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—41no—
114C22⋊Dic17—22no—
115C22⋊Dic17—42no—
116C22⋊Dic17Q1614yescomplemented
117C4⋊Dic17—22no—
118C4⋊Dic17—22no—
119C4⋊Dic17—41no—
120C4⋊Dic17—41no—
121C4⋊Dic17—42no—
122#63225—84no—
123C4×Dic17—42no—
124C2×#3622—22no—
125C2×#3622—42no—
126#9959—171nomaximal, Sylow
127C23×C17—41no—
128C23×C17—41no—
129C23×C17#12011yescharacteristic
130C23×C17#12111yescharacteristic
131C23×C17#12711yescharacteristic
132C23×C17C2×Q1612yes—
133C23×C17#13712yes—
134Q8×C17—42no—
135Q8×C17—42no—
136Q8×C17—42no—
137Q8×C17—81no—
138C2×C4×C17—22no—
139C2×C4×C17—22no—
140C2×C4×C17—22no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—42no—
145C2×C4×C17—42no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—81no—
151C2×C4×C17—81no—
152C2×C4×C17#12011yescharacteristic
153C2×C4×C17#12011yescharacteristic
154C2×C4×C17#12111yescharacteristic
155C2×C4×C17#12111yescharacteristic
156C2×Dic17—24no—
157C2×Dic17—42no—
158C2×Dic17—42no—
159C2×Dic17—44no—
160C2×Dic17—42no—
161C2×Dic17—82no—
162C2×Dic17—82no—
163C2×Dic17—81no—
164C2×Dic17—81no—
165C2×Dic17—81no—
166C8×C17—42no—
167#3622—42no—
168#3189D1711yescharacteristic
169C2×#484—171no—
170#2502—171no—
171#1996—171no—
172#1824—171no—
173#1520—171no—
174#1282—171no—
175C22×C17—21no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—42no—
182C22×C17—42no—
183C22×C17#39211yescharacteristic
184C22×C17#45112yes—
185C22×C17#46212yes—
186C22×C17#49411yescharacteristic
187C22×C17#49711yescharacteristic
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—81no—
192C4×C17—81no—
193C4×C17—81no—
194C4×C17—81no—
195C4×C17#45812yes—
196Dic17—44no—
197Dic17—82no—
198Dic17—161no—
199C2×C22⋊Q8C2×D1711yescharacteristic
200C2×C22⋊Q8C2×D1711yescharacteristic
201C22×C4⋊C4—171no—
202#484—21no—
203#484—21no—
204#484—342no—
205#484—342no—
206#467—22no—
207#451—21no—
208#451—21no—
209C2×#107—341no—
210C2×#103—341no—
211C2×C22⋊C8—171no—
212C2×C22⋊C8C2×D1711yescharacteristic
213#398—342no—
214#391—341no—
215#385—342no—
216#384—341no—
217#380—171no—
218C4×C22⋊C4—171no—
219#342—342no—
220#340—342no—
221#328—342no—
222C2×C17—22no—
223C2×C17—21no—
224C2×C17—22no—
225C2×C17—21no—
226C2×C17#200811yescharacteristic
227C2×C17#201611yescharacteristic
228C2×C17#202211yescharacteristic
229C22×Q8C22⋊D1711yescharacteristic
230C22×Q8C22⋊D1711yescharacteristic
231C23×C4—341no—
232C23×C4C22×D1711yescharacteristic
233C2×Q16—22no—
234C22×C8—341no—
235C22×C8C22⋊D1712yes—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—172no—
247C2×C4⋊C4—341no—
248C2×C4⋊C4—341no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4—344no—
254C2×C4⋊C4C22⋊D1711yescharacteristic
255C2×C4⋊C4C22⋊D1711yescharacteristic
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—172no—
259C2×C22⋊C4—341no—
260C2×C42—341no—
261#107—22no—
262#107—344no—
263#103—22no—
264#103—22no—
265#103—344no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C22⋊C8—172no—
269C22⋊C8—341no—
270#95—344no—
271#95—341no—
272#95—341no—
273C17#995911yescharacteristic, complemented, Sylow
274C24D4×D1711yescharacteristic
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8—42no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—342no—
287C22×C4—342no—
288C22×C4—681no—
289C22×C4—681no—
290C22×C4—681no—
291C22×C4—1362no—
292C22×C4D4×D1711yescharacteristic
293C22×C4D4×D1711yescharacteristic
294C22×C4#6325411yescharacteristic
295C22×C4C2×C22⋊D1711yescharacteristic
296C22×C4C2×C22⋊D1711yescharacteristic
297C22×C4C2×C22⋊D1711yescharacteristic
298Q16—82no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—342no—
302C2×C8—682no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—342no—
312C4⋊C4—342no—
313C4⋊C4—682no—
314C4⋊C4—681no—
315C4⋊C4—681no—
316C4⋊C4—1364no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—174no—
322C22⋊C4—342no—
323C22⋊C4—682no—
324C42—682no—
325C23—41no—
326C23—41no—
327C23Q16×D1712yes—
328C23#1058760412yes—
329C23#1058762311yescharacteristic
330C23#1058762511yescharacteristic
331C23#1058762611yescharacteristic
332Q8—42no—
333Q8—42no—
334Q8—42no—
335Q8—81no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—344no—
351C2×C4—682no—
352C2×C4—682no—
353C2×C4—684no—
354C2×C4—682no—
355C2×C4—1361no—
356C2×C4—1361no—
357C2×C4—1361no—
358C2×C4—1362no—
359C2×C4—1362no—
360C2×C4#1058762311yescharacteristic
361C2×C4#1058762411yescharacteristic
362C2×C4#1058762511yescharacteristic
363C2×C4#1058763911yescharacteristic
364C8—42no—
365C8—682no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22#1176271711yescharacteristic
375C22#1176296312yes—
376C22#1176301712yes—
377C22#1176318711yescharacteristic
378C22#1176320211yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—2721no—
387C4—684no—
388C4—1362no—
389C4#1176299912yes—
390C2—21no—
391C2—22no—
392C2—22no—
393C2—21no—
394C2#1452983411yescharacteristic
395C2#1452988211yescharacteristic
396C2#1452991311yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#3189C2
socleC22×C17#392
prime radicalC24×C17C2×D4
hypercenter#3189D17
hypocenterC17#9959
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
116. C22⋊Dic17Q16semidirect
273. C17#9959semidirect

Maximal subgroups

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

subgroupindex
C2×#117631422
#145332632
#145297702
#145288752
#145273562
#145266312
C17×#31892
#995917