Finite Group Databasebeta

← #16934826

Subgroups — #16934826

Order 4352.

594 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 101 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, complemented
2#14533277C211yesmaximal, characteristic
3#14529770C211yesmaximal, characteristic, complemented
4#14528875C211yesmaximal, characteristic, complemented
5#14527352C211yesmaximal, characteristic
6#14526631C211yesmaximal, characteristic, complemented
7C17×#3187C211yesmaximal, characteristic
8C22×C4⋊Dic17C2211yescharacteristic
9#11763142—22no—
10#11763142—22no—
11C2×#10587478—21no—
12C2×#10587506—21no—
13C2×C22⋊#3622C2211yescharacteristic
14#11762734—22no—
15#11762712—21no—
16#11762689—22no—
17#11762674—21no—
18#11762669C2211yescharacteristic
19C4×C22⋊Dic17C2211yescharacteristic
20#11762576—22no—
21#11762572—22no—
22#11762543—22no—
23C2×C17×C22⋊Q8C2211yescharacteristic
24C2×C17×#121C2211yescharacteristic
25C17×#483—21no—
26C17×#481—21no—
27C17×#465—22no—
28C17×#450—21no—
29C17×#448—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17D411yescharacteristic
34C23×C4×C17C2311yescharacteristic
35C23×Dic17—21no—
36C2×SD16×C17—22no—
37C22×C8×C17D412yes—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C17×C22⋊Q8—21no—
42C17×#121—21no—
43C17×#121—21no—
44C17×#121—21no—
45C17×#121—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⋊Dic17—24no—
53C2×C22⋊Dic17D412yes—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17—21no—
56C2×C4⋊Dic17—21no—
57C2×C4⋊Dic17—21no—
58C2×C4⋊Dic17—21no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17D412yes—
61C2×C4×Dic17—21no—
62C22×#3622—21no—
63C17×#106—22no—
64C17×#103—22no—
65C17×#102—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×C17—21no—
76C24×C17C2×D411yescharacteristic
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—42no—
80C2×D4×C17—21no—
81C2×D4×C17—21no—
82C2×D4×C17—42no—
83C2×D4×C17—41no—
84C2×D4×C17—41no—
85C2×D4×C17—42no—
86C2×D4×C17—42no—
87C22×C4×C17—21no—
88C22×C4×C17—21no—
89C22×C4×C17—21no—
90C22×C4×C17—42no—
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17#4111yescharacteristic
97C22×Dic17—22no—
98C22×Dic17—22no—
99C22×Dic17—41no—
100C22×Dic17—41no—
101C22×Dic17—41no—
102C22×Dic17—82no—
103SD16×C17—82no—
104C2×C8×C17—22no—
105C2×C8×C17—42no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—21no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—41no—
111C4⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—41no—
114C22⋊C4×C17—41no—
115C22⋊C4×C17—41no—
116C22⋊Dic17—22no—
117C22⋊Dic17—22no—
118C22⋊Dic17—42no—
119C22⋊Dic17—84no—
120C4⋊Dic17—22no—
121C4⋊Dic17—22no—
122C4⋊Dic17—41no—
123C4⋊Dic17—41no—
124C4⋊Dic17—42no—
125C4×Dic17—42no—
126C2×#3622—22no—
127C2×#3622—42no—
128#9958—171nomaximal, Sylow
129C23×C17—41no—
130C23×C17—41no—
131C23×C17—42no—
132C23×C17—42no—
133C23×C17—41no—
134C23×C17—42no—
135C23×C17#12011yescharacteristic
136C23×C17#12111yescharacteristic
137C23×C17#12711yescharacteristic
138C23×C17#13512yes—
139C23×C17#13712yes—
140Q8×C17—42no—
141Q8×C17—42no—
142D4×C17—42no—
143D4×C17—81no—
144D4×C17—82no—
145D4×C17—81no—
146D4×C17—81no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—42no—
157C2×C4×C17—81no—
158C2×C4×C17—81no—
159C2×C4×C17#12011yescharacteristic
160C2×C4×C17#12011yescharacteristic
161C2×C4×C17#12111yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×Dic17—42no—
164C2×Dic17—44no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168C2×Dic17—82no—
169C2×Dic17—82no—
170C2×Dic17—81no—
171C2×Dic17—81no—
172C2×Dic17—81no—
173C8×C17—42no—
174#3622—42no—
175#3187D1711yescharacteristic
176C2×#484—171no—
177#2504—171no—
178#1996—171no—
179#1824—171no—
180#1519—171no—
181#1282—171no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—21no—
186C22×C17—21no—
187C22×C17—21no—
188C22×C17—42no—
189C22×C17—42no—
190C22×C17—84no—
191C22×C17—81no—
192C22×C17—81no—
193C22×C17—81no—
194C22×C17#39211yescharacteristic
195C22×C17#45212yes—
196C22×C17#46412yes—
197C22×C17#49511yescharacteristic
198C22×C17#49711yescharacteristic
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—21no—
202C4×C17—81no—
203C4×C17—81no—
204C4×C17—81no—
205C4×C17#45812yes—
206Dic17—82no—
207Dic17—82no—
208Dic17—161no—
209C2×C22⋊Q8C2×D1711yescharacteristic
210C2×#121C2×D1711yescharacteristic
211C22×C4⋊C4—171no—
212#484—342no—
213#484—342no—
214#483—21no—
215#481—21no—
216#465—22no—
217#450—21no—
218#448—21no—
219C2×#107—341no—
220C2×#103—341no—
221C2×C22⋊C8—171no—
222C2×C22⋊C8C2×D1711yescharacteristic
223#397—342no—
224#390—341no—
225#386—342no—
226#382—341no—
227#380—171no—
228C4×C22⋊C4—171no—
229#342—342no—
230#340—342no—
231#327—342no—
232C2×C17—22no—
233C2×C17—21no—
234C2×C17—22no—
235C2×C17—21no—
236C2×C17—161no—
237C2×C17#200811yescharacteristic
238C2×C17#201511yescharacteristic
239C2×C17#202211yescharacteristic
240C22×Q8C22⋊D1711yescharacteristic
241C22×D4—21no—
242C22×D4C22⋊D1711yescharacteristic
243C23×C4—341no—
244C23×C4C22×D1711yescharacteristic
245C2×SD16—22no—
246C22×C8—341no—
247C22×C8C22⋊D1712yes—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252#121—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—172no—
258C2×C4⋊C4—341no—
259C2×C4⋊C4—341no—
260C2×C4⋊C4—341no—
261C2×C4⋊C4—341no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4C22⋊D1711yescharacteristic
265C2×C4⋊C4C22⋊D1711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—172no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4—344no—
271C2×C42—341no—
272#107—344no—
273#106—22no—
274#103—22no—
275#103—344no—
276#102—22no—
277C22⋊C8—21no—
278C22⋊C8—21no—
279C22⋊C8—172no—
280C22⋊C8—341no—
281#95—344no—
282#95—341no—
283#95—341no—
284C17#995811yescharacteristic, complemented, Sylow
285C24—21no—
286C24D4×D1711yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—342no—
302C22×C4—342no—
303C22×C4—681no—
304C22×C4—681no—
305C22×C4—681no—
306C22×C4—1362no—
307C22×C4D4×D1711yescharacteristic
308C22×C4D4×D1711yescharacteristic
309C22×C4#6325411yescharacteristic
310C22×C4C2×C22⋊D1711yescharacteristic
311C22×C4C2×C22⋊D1711yescharacteristic
312C22×C4C2×C22⋊D1711yescharacteristic
313SD16—82no—
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—342no—
317C2×C8—682no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—342no—
325C4⋊C4—342no—
326C4⋊C4—682no—
327C4⋊C4—681no—
328C4⋊C4—681no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—342no—
334C22⋊C4—342no—
335C22⋊C4—682no—
336C22⋊C4—1364no—
337C42—682no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23#1058760112yes—
345C23#1058760212yes—
346C23#1058762311yescharacteristic
347C23#1058762511yescharacteristic
348C23#1058762611yescharacteristic
349Q8—42no—
350Q8—42no—
351D4—42no—
352D4—81no—
353D4—81no—
354D4—82no—
355D4—81no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4—684no—
371C2×C4—682no—
372C2×C4—682no—
373C2×C4—1361no—
374C2×C4—1361no—
375C2×C4—1361no—
376C2×C4—1362no—
377C2×C4—1362no—
378C2×C4#1058762311yescharacteristic
379C2×C4#1058762411yescharacteristic
380C2×C4#1058762511yescharacteristic
381C2×C4#1058763911yescharacteristic
382C8—42no—
383C8—682no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—42no—
391C22—42no—
392C22—84no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22#1176271711yescharacteristic
397C22#1176296812yes—
398C22#1176302912yes—
399C22#1176319011yescharacteristic
400C22#1176320011yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—2721no—
408C4—1362no—
409C4—1362no—
410C4#1176299912yes—
411C2—21no—
412C2—22no—
413C2—21no—
414C2—22no—
415C2—161no—
416C2#1452983411yescharacteristic
417C2#1452987611yescharacteristic
418C2#1452991311yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#3187C2
socleC22×C17#392
prime radicalC2×C17×#121C22
hypercenter#3187D17
hypocenterC17#9958
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
1. C2×#11763142C2semidirect
3. #14529770C2semidirect
4. #14528875C2semidirect
6. #14526631C2semidirect
284. C17#9958semidirect

Maximal subgroups

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

subgroupindex
C2×#117631422
#145332772
#145297702
#145288752
#145273522
#145266312
C17×#31872
#995817