Finite Group Databasebeta

← #16952630

Subgroups — #16952630

Order 4352.

628 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 107 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×#11762599C212yesmaximal, complemented
2C2×#11762596C212yesmaximal
3#14528776C211yesmaximal, characteristic, complemented
4C4×#10587505C211yesmaximal, characteristic, complemented
5C17×#2534C211yesmaximal, characteristic
6#11762833—21no—
7C2×#10587504C2212yes—
8C2×#10587505C2212yes—
9C4×C22⋊Dic17—21no—
10#11762599—24no—
11#11762596—24no—
12#11762578—21no—
13#11762576—21no—
14C2×C17×#123C2211yescharacteristic
15C2×C17×#120C2211yescharacteristic
16C2×C4×D4×C17C2211yescharacteristic
17C22×C22⋊C4×C17C411yescharacteristic, complemented
18C17×#396—21no—
19C17×#394—21no—
20C17×#388—21no—
21C17×#386—21no—
22C17×#381C411yescharacteristic, complemented
23C17×#380C411yescharacteristic, complemented
24C17×#379C411yescharacteristic, complemented
25C25×C17C2×C411yescharacteristic
26C22×D4×C17—21no—
27C22×D4×C17C2311yescharacteristic
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17C2×C411yescharacteristic
30C23×C4×C17D411yescharacteristic
31C17×#123—22no—
32C17×#123—22no—
33C17×#120—22no—
34C17×#120—22no—
35C4×D4×C17—41no—
36C4×D4×C17—41no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17—21no—
45C2×C22⋊C4×C17—21no—
46C2×C22⋊C4×C17—41no—
47C2×C22⋊C4×C17—41no—
48C2×C22⋊C4×C17C2×C411yescharacteristic
49C2×C22⋊C4×C17C2×C411yescharacteristic
50C2×C22⋊C4×C17C2×C411yescharacteristic
51C2×C22⋊C4×C17D411yescharacteristic
52C2×C42×C17D411yescharacteristic
53C2×C22⋊Dic17—22no—
54C2×C4×Dic17—41no—
55C2×#63223—22no—
56C17×#95—21no—
57C17×#95—21no—
58C17×#95—21no—
59#10587505—22no—
60#10587505D414yes—
61#10587504—22no—
62#10587504—22no—
63#10587502—41no—
64C4⋊#3622—41no—
65C4⋊#3622—41no—
66C24×C17—21no—
67C24×C17—22no—
68C24×C17—42no—
69C24×C17—41no—
70C24×C17—41no—
71C24×C17—81no—
72C24×C17C22×C411yescharacteristic
73C24×C17C2×D411yescharacteristic
74C2×D4×C17—41no—
75C2×D4×C17—41no—
76C2×D4×C17—42no—
77C2×D4×C17—42no—
78C2×D4×C17—81no—
79C2×D4×C17C2×D412yes—
80C2×D4×C17#4112yes—
81C22×C4×C17—21no—
82C22×C4×C17—21no—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17—22no—
89C22×C4×C17—41no—
90C22×C4×C17—41no—
91C22×C4×C17—81no—
92C22×C4×C17—81no—
93C22×C4×C17C22⋊C411yescharacteristic
94C22×C4×C17C22⋊C411yescharacteristic
95C22×C4×C17C22⋊C411yescharacteristic
96C22×C4×C17C22⋊C411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C22×Dic17—42no—
99C4⋊C4×C17—41no—
100C4⋊C4×C17—41no—
101C4⋊C4×C17—41no—
102C4⋊C4×C17—81no—
103C22⋊C4×C17—22no—
104C22⋊C4×C17—22no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C22⋊C4×C17—42no—
108C22⋊C4×C17—42no—
109C22⋊C4×C17—42no—
110C22⋊C4×C17—41no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊C4×C17—41no—
114C22⋊C4×C17—81no—
115C42×C17—21no—
116C42×C17—21no—
117C22⋊Dic17—24no—
118C22⋊Dic17—42no—
119C4×Dic17—82no—
120#63223—42no—
121#63223—42no—
122C2×#3622—42no—
123#13222—171nomaximal, Sylow
124C23×C17—21no—
125C23×C17—21no—
126C23×C17—21no—
127C23×C17—21no—
128C23×C17—22no—
129C23×C17—41no—
130C23×C17—41no—
131C23×C17—41no—
132C23×C17—41no—
133C23×C17—41no—
134C23×C17—41no—
135C23×C17—41no—
136C23×C17—42no—
137C23×C17—42no—
138C23×C17—41no—
139C23×C17—41no—
140C23×C17—41no—
141C23×C17—82no—
142C23×C17—84no—
143C23×C17—84no—
144C23×C17—82no—
145C23×C17C2×C22⋊C411yescharacteristic
146C23×C17C4×D412yes—
147C23×C17#12112yes—
148D4×C17—42no—
149D4×C17—42no—
150D4×C17—84no—
151C2×C4×C17—21no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—41no—
157C2×C4×C17—42no—
158C2×C4×C17—41no—
159C2×C4×C17—41no—
160C2×C4×C17—41no—
161C2×C4×C17—41no—
162C2×C4×C17—41no—
163C2×C4×C17—41no—
164C2×C4×C17—41no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—41no—
169C2×C4×C17—41no—
170C2×C4×C17—81no—
171C2×C4×C17—81no—
172C2×C4×C17—81no—
173C2×C4×C17—82no—
174C2×C4×C17#9911yescharacteristic
175C2×C4×C17#9911yescharacteristic
176C2×C4×C17#12411yescharacteristic
177C2×C4×C17#12711yescharacteristic
178C2×Dic17—44no—
179C2×Dic17—82no—
180C2×Dic17—82no—
181#3622—82no—
182#2534D1711yescharacteristic
183C2×#352—172no—
184C2×AΣL(1,𝔽16)—172no—
185#1800—171no—
186C4×#99—171no—
187C22×C17—21no—
188C22×C17—21no—
189C22×C17—22no—
190C22×C17—22no—
191C22×C17—42no—
192C22×C17—42no—
193C22×C17—42no—
194C22×C17—42no—
195C22×C17—42no—
196C22×C17—41no—
197C22×C17—41no—
198C22×C17—41no—
199C22×C17—41no—
200C22×C17—42no—
201C22×C17—81no—
202C22×C17—81no—
203C22×C17—81no—
204C22×C17—84no—
205C22×C17—84no—
206C22×C17—81no—
207C22×C17—81no—
208C22×C17#39011yescharacteristic
209C22×C17C2×#9911yescharacteristic
210C22×C17#41011yescharacteristic
211C4×C17—21no—
212C4×C17—21no—
213C4×C17—21no—
214C4×C17—21no—
215C4×C17—41no—
216C4×C17—41no—
217C4×C17—41no—
218C4×C17—81no—
219C4×C17—81no—
220C4×C17—81no—
221Dic17—84no—
222C2×#123C2×D1711yescharacteristic
223C2×#120C2×D1711yescharacteristic
224C2×C4×D4C2×D1711yescharacteristic
225C22×C22⋊C4Dic1711yescharacteristic, complemented
226#423—341no—
227C2×#100—172no—
228C2×#99—172no—
229#396—21no—
230#394—21no—
231#388—21no—
232#386—21no—
233#381Dic1711yescharacteristic, complemented
234#380Dic1711yescharacteristic, complemented
235#379Dic1711yescharacteristic, complemented
236C4×C22⋊C4—341no—
237#352—344no—
238AΣL(1,𝔽16)—344no—
239#343—341no—
240#342—341no—
241C2×C17—21no—
242C2×C17—21no—
243C2×C17—42no—
244C2×C17—42no—
245C2×C17—82no—
246C2×C17#196411yescharacteristic
247C2×C17#212011yescharacteristic
248C2×C17#212111yescharacteristic
249C25C2×Dic1711yescharacteristic
250C22×D4—21no—
251C22×D4C22×D1711yescharacteristic
252C23×C4C2×Dic1711yescharacteristic
253C23×C4C2×Dic1711yescharacteristic
254C23×C4C22⋊D1711yescharacteristic
255C2×M16—342no—
256#123—22no—
257#123—22no—
258#120—22no—
259#120—22no—
260C4×D4—41no—
261C4×D4—41no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C22⋊D1711yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—342no—
274C2×C22⋊C4C2×Dic1711yescharacteristic
275C2×C22⋊C4C2×Dic1711yescharacteristic
276C2×C22⋊C4C2×Dic1711yescharacteristic
277C2×C22⋊C4C22⋊D1711yescharacteristic
278C2×C42—681no—
279C2×C42C22⋊D1711yescharacteristic
280C4⋊C8—681no—
281C4⋊C8—681no—
282#100—342no—
283#100—342no—
284#99—174no—
285#99—342no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—681no—
290C17#1322211yescharacteristic, complemented, Sylow
291C24—21no—
292C24—22no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24—81no—
297C24C22×Dic1711yescharacteristic
298C24C2×C22⋊D1711yescharacteristic
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—81no—
304C2×D4D4×D1712yes—
305C2×D4#6325412yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—682no—
319C22×C4C22⋊Dic1711yescharacteristic
320C22×C4C22⋊Dic1711yescharacteristic
321C22×C4C22⋊Dic1711yescharacteristic
322C22×C4C22⋊Dic1711yescharacteristic
323C22×C4C2×C22⋊D1711yescharacteristic
324M16—682no—
325M16—682no—
326C2×C8—682no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—81no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—81no—
343C22⋊C4—344no—
344C22⋊C4—682no—
345C42—21no—
346C42—21no—
347C42—1362no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—82no—
366C23—84no—
367C23—84no—
368C23—82no—
369C23D4×Dic1712yes—
370C23#1058762412yes—
371C23C2×C22⋊Dic1711yescharacteristic
372D4—42no—
373D4—42no—
374D4—84no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—684no—
399C2×C4—1362no—
400C2×C4—1362no—
401C2×C4#1058750511yescharacteristic
402C2×C4#1058750511yescharacteristic
403C2×C4#1058762211yescharacteristic
404C2×C4#1058762611yescharacteristic
405C8—1362no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—84no—
424C22—84no—
425C22—81no—
426C22—81no—
427C22#1176271211yescharacteristic
428C22C2×#1058750511yescharacteristic
429C22#1176278311yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—1364no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—42no—
445C2—82no—
446C2#1452962111yescharacteristic
447C2#1453042511yescharacteristic
448C2#1453043011yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2534C2
socleC22×C17C2×#99
prime radicalC2×C17×#120C22
cosocleC22×D4×C17C23
hypercenter#2534D17
hypocenterC17#13222
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. C2×#11762599C2semidirect
3. #14528776C2semidirect
4. C4×#10587505C2semidirect
17. C22×C22⋊C4×C17C4semidirect
22. C17×#381C4semidirect
23. C17×#380C4semidirect
24. C17×#379C4semidirect
225. C22×C22⋊C4Dic17semidirect
233. #381Dic17semidirect
234. #380Dic17semidirect
235. #379Dic17semidirect
290. C17#13222semidirect

Maximal subgroups

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

subgroupindex
C2×#117625992
C2×#117625962
#145287762
C4×#105875052
C17×#25342
#1322217