Finite Group Databasebeta

← #16934828

Subgroups — #16934828

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
1#14539541C211yesmaximal, characteristic
2C2×#11763144C211yesmaximal, characteristic, complemented
3#14529769C211yesmaximal, characteristic, complemented
4#14528874C211yesmaximal, characteristic, complemented
5#14527352C211yesmaximal, characteristic
6#14526632C211yesmaximal, characteristic, complemented
7C17×#2504C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763144—22no—
10#11763144—22no—
11#11763139—21no—
12#11763125—21no—
13#11763036—22no—
14#11762960—21no—
15#11762947—21no—
16C2×#10587478—21no—
17C2×#10587481—21no—
18C2×C22⋊#3622C2211yescharacteristic
19C2×C22⋊#3622C2211yescharacteristic
20#11762669C2211yescharacteristic
21#11762575—22no—
22#11762571—22no—
23#11762543—22no—
24C2×C17×#121C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#397—22no—
27C17×#390—21no—
28C17×#386—22no—
29C17×#382—21no—
30C4×C22⋊C4×C17C2211yescharacteristic
31C22×D4×C17—21no—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17—21no—
34C23×C4×C17C2311yescharacteristic
35C22×Dic34D411yescharacteristic
36C17×#121—21no—
37C17×#121—21no—
38C17×#121—21no—
39C17×#121—21no—
40C2×C4⋊C4×C17—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17D412yes—
46C2×C4⋊C4×C17D411yescharacteristic
47C2×C22⋊C4×C17—21no—
48C2×C22⋊C4×C17—24no—
49C2×C22⋊C4×C17D412yes—
50C2×C42×C17—21no—
51C2×C22⋊Dic17—21no—
52C2×C22⋊Dic17—21no—
53C2×#63229—22no—
54C2×C4⋊Dic17—21no—
55C2×C4⋊Dic17D411yescharacteristic
56#10587610—21no—
57#10587610—21no—
58#10587610—21no—
59#10587610—21no—
60C2×#63225—21no—
61C22×#3622—21no—
62C22×#3622D412yes—
63C17×#95—24no—
64C17×#95—21no—
65#10587503—22no—
66#10587502—21no—
67C22⋊#3622—21no—
68C22⋊#3622—21no—
69C22⋊#3622—21no—
70C22⋊#3622D412yes—
71#10587481—22no—
72#10587481—24no—
73#10587479—22no—
74#10587478—24no—
75C24×C17—21no—
76C24×C17C2×D411yescharacteristic
77C2×D4×C17—21no—
78C2×D4×C17—21no—
79C2×D4×C17—42no—
80C2×D4×C17—41no—
81C2×D4×C17—41no—
82C2×D4×C17—42no—
83C2×D4×C17—42no—
84C22×C4×C17—22no—
85C22×C4×C17—21no—
86C22×C4×C17—22no—
87C22×C4×C17—42no—
88C22×C4×C17—41no—
89C22×C4×C17—41no—
90C22×C4×C17—82no—
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—21no—
98C22×Dic17—21no—
99C22×Dic17—41no—
100C2×Dic34—21no—
101C2×Dic34—21no—
102C2×Dic34—42no—
103C4⋊C4×C17—22no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—22no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—41no—
108C4⋊C4×C17—41no—
109C22⋊C4×C17—22no—
110C22⋊C4×C17—22no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—84no—
114C42×C17—42no—
115C22⋊Dic17—41no—
116C22⋊Dic17—41no—
117C22⋊Dic17—42no—
118#63229—82no—
119C4⋊Dic17—21no—
120C4⋊Dic17—21no—
121C4⋊Dic17—42no—
122#63225—41no—
123#63225—41no—
124C2×#3622—22no—
125C2×#3622—22no—
126C2×#3622—42no—
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—
140D4×C17—42no—
141D4×C17—81no—
142D4×C17—82no—
143D4×C17—81no—
144D4×C17—81no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—44no—
154C2×C4×C17—41no—
155C2×C4×C17—42no—
156C2×C4×C17—42no—
157C2×C4×C17—42no—
158C2×C4×C17—82no—
159C2×C4×C17—82no—
160C2×C4×C17#12011yescharacteristic
161C2×C4×C17#12011yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—81no—
167C2×Dic17—81no—
168C2×Dic17—81no—
169C2×Dic17—81no—
170C2×Dic17—81no—
171Dic34—42no—
172Dic34—42no—
173#3622—42no—
174#3622—42no—
175#3187—171no—
176C2×#484—171no—
177#2504D1711yescharacteristic
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—82no—
203C4×C17—82no—
204C4×C17—81no—
205C4×C17#45812yes—
206Dic17—81no—
207Dic17—81no—
208Dic17—161no—
209C2×C22⋊Q8—171no—
210C2×#121C2×D1711yescharacteristic
211C22×C4⋊C4C2×D1711yescharacteristic
212#484—342no—
213#484—342no—
214#483—341no—
215#481—341no—
216#465—342no—
217#450—341no—
218#448—341no—
219C2×#107—341no—
220C2×#103—341no—
221C2×C22⋊C8—171no—
222C2×C22⋊C8—171no—
223#397—22no—
224#390—21no—
225#386—22no—
226#382—21no—
227#380—171no—
228C4×C22⋊C4C2×D1711yescharacteristic
229#342—342no—
230#340—342no—
231#327—342no—
232C2×C17—21no—
233C2×C17—22no—
234C2×C17—22no—
235C2×C17—21no—
236C2×C17—161no—
237C2×C17#200811yescharacteristic
238C2×C17#201511yescharacteristic
239C2×C17#202211yescharacteristic
240C22×Q8—171no—
241C22×D4—21no—
242C22×D4C22⋊D1711yescharacteristic
243C23×C4—21no—
244C23×C4C22×D1711yescharacteristic
245C2×SD16—342no—
246C22×C8—172no—
247C22×C8—341no—
248C22⋊Q8—341no—
249C22⋊Q8—341no—
250C22⋊Q8—341no—
251C22⋊Q8—341no—
252#121—21no—
253#121—21no—
254#121—21no—
255#121—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—171no—
262C2×C4⋊C4—341no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4C22⋊D1712yes—
265C2×C4⋊C4C22⋊D1711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—341no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4C22⋊D1712yes—
271C2×C42—21no—
272#107—344no—
273#106—342no—
274#103—344no—
275#103—342no—
276#102—342no—
277C22⋊C8—172no—
278C22⋊C8—341no—
279C22⋊C8—341no—
280C22⋊C8—341no—
281#95—24no—
282#95—21no—
283#95—341no—
284C17#995811yescharacteristic, complemented, Sylow
285C24—21no—
286C24C2×C22⋊D1711yescharacteristic
287C2×Q8—341no—
288C2×Q8—341no—
289C2×Q8—682no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—42no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—341no—
305C22×C4—341no—
306C22×C4—681no—
307C22×C4D4×D1711yescharacteristic
308C22×C4D4×D1711yescharacteristic
309C22×C4D4×D1711yescharacteristic
310C22×C4#6325411yescharacteristic
311C22×C4C2×C22⋊D1711yescharacteristic
312C22×C4C2×C22⋊D1711yescharacteristic
313SD16—1362no—
314C2×C8—342no—
315C2×C8—342no—
316C2×C8—682no—
317C2×C8—682no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—341no—
325C4⋊C4—341no—
326C4⋊C4—682no—
327C4⋊C4—681no—
328C4⋊C4—681no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—84no—
334C22⋊C4—681no—
335C22⋊C4—681no—
336C22⋊C4—682no—
337C42—42no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23#1058762411yescharacteristic
345C23#1058762611yescharacteristic
346C23#1058763612yes—
347C23#1058763712yes—
348C23#1058763911yescharacteristic
349Q8—682no—
350Q8—682no—
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—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—681no—
372C2×C4—681no—
373C2×C4—1361no—
374C2×C4—1361no—
375C2×C4—1361no—
376C2×C4—1361no—
377C2×C4—1361no—
378C2×C4#1058762311yescharacteristic
379C2×C4#1058762311yescharacteristic
380C2×C4#1058762511yescharacteristic
381C2×C4#1058762511yescharacteristic
382C8—682no—
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#1176297112yes—
398C22#1176302812yes—
399C22#1176319111yescharacteristic
400C22#1176320411yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—82no—
406C4—81no—
407C4—2721no—
408C4—1361no—
409C4—1361no—
410C4#1176299812yes—
411C2—22no—
412C2—21no—
413C2—21no—
414C2—22no—
415C2—161no—
416C2#1452983711yescharacteristic
417C2#1452987811yescharacteristic
418C2#1452991211yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#2504C2
socleC22×C17#392
prime radicalC2×C17×#121C22
hypercenter#2504D17
hypocenterC17#9958
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2504

Complements

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

NH ≅ G/Ntype
2. C2×#11763144C2semidirect
3. #14529769C2semidirect
4. #14528874C2semidirect
6. #14526632C2semidirect
284. C17#9958semidirect

Maximal subgroups

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

subgroupindex
#145395412
C2×#117631442
#145297692
#145288742
#145273522
#145266322
C17×#25042
#995817