Finite Group Databasebeta

← #16934825

Subgroups — #16934825

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#14539537C211yesmaximal, characteristic
2#14533272C211yesmaximal, characteristic
3#14529770C211yesmaximal, characteristic, complemented
4#14528874C211yesmaximal, characteristic, complemented
5#14527353C211yesmaximal, characteristic
6#14526630C211yesmaximal, characteristic, complemented
7C2×C17×#484C211yesmaximal, characteristic, complemented
8C2×#10587616C2211yescharacteristic
9#11763137—21no—
10#11763127—21no—
11#11763040—22no—
12#11762959—21no—
13#11762948—21no—
14C2×C22⋊#3622C2211yescharacteristic
15#11762735—22no—
16#11762710—21no—
17#11762688—22no—
18#11762674—21no—
19#11762669C2211yescharacteristic
20C4×C22⋊Dic17C2211yescharacteristic
21#11762576—22no—
22#11762571—22no—
23#11762544—22no—
24C2×C17×C22⋊Q8C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#484—22no—
27C17×#484—22no—
28C2×C17×#107—21no—
29C2×C17×#103—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×Q8×C17D411yescharacteristic
32C23×C4×C17—21no—
33C23×C4×C17C2311yescharacteristic
34C22×C22⋊D17—21no—
35C22×D68D411yescharacteristic
36C22×C8×C17—21no—
37C17×C22⋊Q8—21no—
38C17×C22⋊Q8—21no—
39C17×C22⋊Q8—21no—
40C17×C22⋊Q8—21no—
41C2×C4⋊C4×C17—21no—
42C2×C4⋊C4×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17D412yes—
48C2×C4⋊C4×C17D411yescharacteristic
49C2×C22⋊C4×C17—21no—
50C2×C22⋊Dic17—21no—
51C2×C22⋊Dic17D412yes—
52C2×#63230—22no—
53#10587616—21no—
54#10587616—21no—
55#10587616—21no—
56#10587616—21no—
57C2×#63227—24no—
58C2×#63227—21no—
59C2×C4⋊Dic17—21no—
60C2×C4⋊Dic17D411yescharacteristic
61C2×C4×Dic17—21no—
62C22×#3622D412yes—
63C17×#107—24no—
64C17×#103—24no—
65C17×C22⋊C8—21no—
66C17×C22⋊C8D412yes—
67#10587506—22no—
68#10587502—24no—
69#10587502—21no—
70#10587502—21no—
71C22⋊#3622—21no—
72C22⋊#3622—21no—
73#10587480—22no—
74#10587479—22no—
75C24×C17C2×D411yescharacteristic
76C23×D17—21no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—42no—
80C22×C4×C17—22no—
81C22×C4×C17—21no—
82C22×C4×C17—21no—
83C22×C4×C17—42no—
84C22×C4×C17—41no—
85C22×C4×C17—41no—
86C22×C4×C17—82no—
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×C4×C17#4111yescharacteristic
93C2×C22⋊D17—41no—
94C2×C22⋊D17—41no—
95C2×C22⋊D17—42no—
96C2×C22⋊D17—42no—
97C22×Dic17—21no—
98C22×Dic17—22no—
99C22×Dic17—41no—
100C2×D68—21no—
101C2×D68—21no—
102C2×D68—42no—
103C2×C8×C17—22no—
104C2×C8×C17—42no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17—21no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—22no—
109C4⋊C4×C17—41no—
110C4⋊C4×C17—41no—
111C4⋊C4×C17—41no—
112C4⋊C4×C17—41no—
113C22⋊C4×C17—41no—
114C22⋊C4×C17—41no—
115C22⋊Dic17—22no—
116C22⋊Dic17—22no—
117C22⋊Dic17—42no—
118#63230—82no—
119#63227—41no—
120#63227—41no—
121#63227—84no—
122C4⋊Dic17—21no—
123C4⋊Dic17—21no—
124C4⋊Dic17—42no—
125C4×Dic17—42no—
126C2×#3622—22no—
127C2×#3622—42no—
128#9958—171nomaximal, Sylow
129C23×C17—41no—
130C23×C17—41no—
131C23×C17#12011yescharacteristic
132C23×C17#12111yescharacteristic
133C23×C17#12711yescharacteristic
134C23×C17#13512yes—
135C23×C17#13712yes—
136C22×D17—42no—
137C22×D17—42no—
138C22×D17—41no—
139C22×D17—42no—
140Q8×C17—42no—
141Q8×C17—42no—
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—44no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—42no—
152C2×C4×C17—82no—
153C2×C4×C17—82no—
154C2×C4×C17—81no—
155C2×C4×C17—81no—
156C2×C4×C17#12011yescharacteristic
157C2×C4×C17#12011yescharacteristic
158C2×C4×C17#12111yescharacteristic
159C2×C4×C17#12111yescharacteristic
160C22⋊D17—82no—
161C22⋊D17—81no—
162C22⋊D17—81no—
163C2×Dic17—42no—
164C2×Dic17—42no—
165C2×Dic17—41no—
166C2×Dic17—42no—
167C2×Dic17—42no—
168C2×Dic17—81no—
169C2×Dic17—81no—
170C2×Dic17—81no—
171D68—42no—
172D68—81no—
173C8×C17—42no—
174#3622—42no—
175#3187—171no—
176C2×#484D1711yescharacteristic, complemented
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#39211yescharacteristic
191C22×C17#45212yes—
192C22×C17#46412yes—
193C22×C17#49511yescharacteristic
194C22×C17#49711yescharacteristic
195C2×D17—84no—
196C2×D17—81no—
197C2×D17—81no—
198C2×D17—81no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—21no—
202C4×C17—82no—
203C4×C17—81no—
204C4×C17—81no—
205C4×C17#45812yes—
206Dic17—82no—
207Dic17—81no—
208Dic17—161no—
209C2×C22⋊Q8C2×D1711yescharacteristic
210C2×#121—171no—
211C22×C4⋊C4C2×D1711yescharacteristic
212#484—22no—
213#484—22no—
214#483—341no—
215#481—341no—
216#465—342no—
217#450—341no—
218#448—341no—
219C2×#107—21no—
220C2×#103—21no—
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#200811yescharacteristic
237C2×C17#201511yescharacteristic
238C2×C17#202211yescharacteristic
239D17—161no—
240C22×Q8C22⋊D1711yescharacteristic
241C22×D4—171no—
242C22×D4—341no—
243C23×C4—21no—
244C23×C4C22×D1711yescharacteristic
245C2×SD16—342no—
246C22×C8—21no—
247C22×C8—172no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252#121—341no—
253#121—341no—
254#121—341no—
255#121—341no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—171no—
263C2×C4⋊C4—341no—
264C2×C4⋊C4C22⋊D1712yes—
265C2×C4⋊C4C22⋊D1711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—172no—
268C2×C22⋊C4—341no—
269C2×C22⋊C4—341no—
270C2×C22⋊C4—344no—
271C2×C42—341no—
272#107—24no—
273#106—342no—
274#103—24no—
275#103—342no—
276#102—342no—
277C22⋊C8—21no—
278C22⋊C8—341no—
279C22⋊C8—341no—
280C22⋊C8C22⋊D1712yes—
281#95—344no—
282#95—341no—
283#95—341no—
284C17#995811yescharacteristic, complemented, Sylow
285C24—341no—
286C24D4×D1711yescharacteristic
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—42no—
290C2×D4—341no—
291C2×D4—341no—
292C2×D4—682no—
293C2×D4—682no—
294C2×D4—682no—
295C2×D4—681no—
296C2×D4—681no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—341no—
305C22×C4—342no—
306C22×C4—681no—
307C22×C4D4×D1711yescharacteristic
308C22×C4D4×D1711yescharacteristic
309C22×C4#6325411yescharacteristic
310C22×C4C2×C22⋊D1711yescharacteristic
311C22×C4C2×C22⋊D1711yescharacteristic
312C22×C4C2×C22⋊D1711yescharacteristic
313SD16—1362no—
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—342no—
317C2×C8—682no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—341no—
327C4⋊C4—341no—
328C4⋊C4—682no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—342no—
332C22⋊C4—342no—
333C22⋊C4—681no—
334C22⋊C4—681no—
335C22⋊C4—682no—
336C22⋊C4—1364no—
337C42—682no—
338C23—41no—
339C23—41no—
340C23—682no—
341C23—682no—
342C23—682no—
343C23—681no—
344C23#1058760412yes—
345C23#1058760512yes—
346C23#1058762311yescharacteristic
347C23#1058762511yescharacteristic
348C23#1058762611yescharacteristic
349Q8—42no—
350Q8—42no—
351D4—682no—
352D4—1361no—
353D4—1361no—
354D4—1362no—
355D4—1361no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—682no—
371C2×C4—682no—
372C2×C4—682no—
373C2×C4—682no—
374C2×C4—681no—
375C2×C4—1361no—
376C2×C4—1361no—
377C2×C4—1361no—
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—1364no—
393C22—1361no—
394C22—1361no—
395C22—1361no—
396C22#1176271711yescharacteristic
397C22#1176297012yes—
398C22#1176303312yes—
399C22#1176319411yescharacteristic
400C22#1176320211yescharacteristic
401C4—21no—
402C4—21no—
403C4—21no—
404C4—82no—
405C4—81no—
406C4—81no—
407C4—2721no—
408C4—1362no—
409C4—1361no—
410C4#1176299812yes—
411C2—21no—
412C2—22no—
413C2—21no—
414C2—22no—
415C2—2721no—
416C2#1452983511yescharacteristic
417C2#1452987511yescharacteristic
418C2#1452991211yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#484C2
socleC22×C17#392
prime radicalC2×#10587616C22
hypercenterC2×#484D17
hypocenterC17#9958
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
3. #14529770C2semidirect
4. #14528874C2semidirect
6. #14526630C2semidirect
7. C2×C17×#484C2semidirect
176. C2×#484D17semidirect
284. C17#9958semidirect

Maximal subgroups

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

subgroupindex
#145395372
#145332722
#145297702
#145288742
#145273532
#145266302
C2×C17×#4842
#995817