Finite Group Databasebeta

← #16945000

Subgroups — #16945000

Order 4352.

402 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 69 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#14534549C211yesmaximal, characteristic
2#14528860C211yesmaximal, characteristic
3#14527115C211yesmaximal, characteristic
4#14526785C211yesmaximal, characteristic
5#14526784C211yesmaximal, characteristic
6#14526783C211yesmaximal, characteristic
7C17×#1344C211yesmaximal, characteristic
8C2×#10587574C411yescharacteristic, complemented
9C2×#10587564—21no—
10C2×#10587563C2211yescharacteristic
11#11762863—21no—
12#11762743—21no—
13#11762741C411yescharacteristic, complemented
14#11762740C411yescharacteristic, complemented
15#11762737—21no—
16#11762734C2211yescharacteristic
17#11762713—21no—
18#11762673C411yescharacteristic, complemented
19#11762670—21no—
20#11762576C2211yescharacteristic
21#11762575—21no—
22#11762534—21no—
23#11762533C2211yescharacteristic
24C17×#397C2211yescharacteristic
25C17×#342C2211yescharacteristic
26C17×#323C2211yescharacteristic
27C23×Dic17D411yescharacteristic
28C22×Dic34—21no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17—21no—
31C2×C22⋊C4×C17—21no—
32C2×C22⋊C4×C17C2311yescharacteristic
33C2×C22⋊Dic17—21no—
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17D411yescharacteristic
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225C2×C411yescharacteristic
42C2×#63225D411yescharacteristic
43C2×C4×Dic17C2×C411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45#10587574—41no—
46#10587574—41no—
47#10587564—42no—
48#10587564—42no—
49#10587563—41no—
50#10587563—41no—
51C17×C22⋊C8—21no—
52C17×#95—21no—
53C17×#95C2×C411yescharacteristic
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502—41no—
59#10587502—41no—
60#10587502C2×C411yescharacteristic
61C22⋊#3622—21no—
62#10587486—41no—
63#10587485—41no—
64C24×C17C2×D411yescharacteristic
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—41no—
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17C22⋊C411yescharacteristic
79C22×Dic17C22⋊C411yescharacteristic
80C22×Dic17C22⋊C411yescharacteristic
81C22×Dic17C22⋊C411yescharacteristic
82C2×Dic34—42no—
83C2×Dic34—42no—
84C2×Dic34—41no—
85C2×Dic34—41no—
86C2×C8×C17—41no—
87C4⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—41no—
93C22⋊Dic17—41no—
94C22⋊Dic17—41no—
95C22⋊Dic17—41no—
96C22⋊Dic17—41no—
97C22⋊Dic17—41no—
98C4⋊Dic17—42no—
99C4⋊Dic17—41no—
100#63225—41no—
101#63225—41no—
102#63225—42no—
103#63225—42no—
104#63225—42no—
105C4×Dic17—21no—
106C4×Dic17—21no—
107C4×Dic17—21no—
108C4×Dic17—21no—
109C2×#3622—41no—
110#11801—171nomaximal, Sylow
111C23×C17—21no—
112C23×C17—21no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17C2×C22⋊C411yescharacteristic
116C23×C17AGL(1,ℤ/8)11yescharacteristic
117C23×C17AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—81no—
129C2×C4×C17—81no—
130C2×C4×C17—81no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—81no—
151C2×Dic17—81no—
152C2×Dic17—81no—
153C2×Dic17#9911yescharacteristic
154C2×Dic17#9911yescharacteristic
155Dic34—82no—
156Dic34—82no—
157C8×C17—81no—
158#3622—81no—
159#2621—171no—
160#1820—171no—
161#1464—171no—
162#1344—171no—
163#1344—171no—
164#1344—171no—
165#1344D1711yescharacteristic
166C22×C17—21no—
167C22×C17—21no—
168C22×C17—21no—
169C22×C17—21no—
170C22×C17—41no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—41no—
174C22×C17—81no—
175C22×C17C2×#9911yescharacteristic
176C22×C17#41811yescharacteristic
177C22×C17#42111yescharacteristic
178C4×C17—41no—
179C4×C17—41no—
180C4×C17—81no—
181C4×C17—81no—
182C4×C17—161no—
183Dic17—21no—
184Dic17—21no—
185Dic17—41no—
186Dic17—41no—
187Dic17—41no—
188Dic17—81no—
189Dic17—81no—
190Dic17—161no—
191C2×C4⋊Q8—171no—
192C2×C22⋊Q8—341no—
193C2×#117—171no—
194#432—341no—
195#398—171no—
196#398—171no—
197#398—341no—
198#397—171no—
199#397—341no—
200#397C2×D1711yescharacteristic
201#391—341no—
202#382—171no—
203#381—341no—
204#342—171no—
205#342—341no—
206#342C2×D1711yescharacteristic
207#324—341no—
208#323—171no—
209#323C2×D1711yescharacteristic
210C2×C17—21no—
211C2×C17—21no—
212C2×C17—41no—
213C2×C17—41no—
214C2×C17#151011yescharacteristic
215C2×C17#212311yescharacteristic
216C2×C17#212911yescharacteristic
217C22×Q8—341no—
218C23×C4—171no—
219C4⋊Q8—681no—
220C4⋊Q8—681no—
221C22⋊Q8—682no—
222C22⋊Q8—682no—
223#117—681no—
224#117—681no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4C4×D1711yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—171no—
236C2×C22⋊C4—341no—
237C2×C22⋊C4—341no—
238C2×C22⋊C4C22×D1711yescharacteristic
239C2×C42—171no—
240C2×C42—171no—
241C4⋊C8—681no—
242C22⋊C8—21no—
243C22⋊C8—341no—
244#97—681no—
245#95—21no—
246#95—171no—
247#95—341no—
248#95—341no—
249#95—341no—
250#95—341no—
251#95—681no—
252#95—681no—
253#95C4×D1711yescharacteristic
254C17#1180111yescharacteristic, complemented, Sylow
255C24D4×D1711yescharacteristic
256C2×Q8—681no—
257C2×Q8—681no—
258C2×Q8—682no—
259C2×Q8—682no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—171no—
264C22×C4—171no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—341no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4—341no—
272C22×C4—341no—
273C22×C4—681no—
274C22×C4—681no—
275C22×C4C2×C4×D1711yescharacteristic
276C22×C4D4×D1711yescharacteristic
277C2×C8—41no—
278C2×C8—681no—
279C4⋊C4—41no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—682no—
284C4⋊C4—682no—
285C4⋊C4—682no—
286C4⋊C4—682no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—681no—
293C22⋊C4—681no—
294C22⋊C4—681no—
295C22⋊C4—681no—
296C22⋊C4—681no—
297C42—341no—
298C42—341no—
299C42—341no—
300C42—341no—
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23C22⋊C4×D1711yescharacteristic
306C23#1058759711yescharacteristic
307C23#1058760011yescharacteristic
308Q8—1362no—
309Q8—1362no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—171no—
324C2×C4—171no—
325C2×C4—341no—
326C2×C4—341no—
327C2×C4—341no—
328C2×C4—341no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—681no—
336C2×C4—681no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—1361no—
345C2×C4—1361no—
346C2×C4—1361no—
347C8—81no—
348C8—1361no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—81no—
358C22#99×D1711yescharacteristic
359C22#1176281211yescharacteristic
360C22#1176282411yescharacteristic
361C4—41no—
362C4—41no—
363C4—81no—
364C4—81no—
365C4—161no—
366C4—2721no—
367C4—341no—
368C4—341no—
369C4—681no—
370C4—681no—
371C4—681no—
372C4—1361no—
373C4—1361no—
374C2—21no—
375C2—21no—
376C2—41no—
377C2—41no—
378C2#1452731111yescharacteristic
379C2#1453043611yescharacteristic
380C2#1453045711yescharacteristic
subgroupG/H
centerC22#99×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C22⋊C4C22×D17
FittingC17×#1344C2
socleC22×C17C2×#99
prime radicalC24×C17C2×D4
cosocleC2×C22⋊C4×C17C23
hypercenter#1344D17
hypocenterC17#11801
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1344

Complements

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

NH ≅ G/Ntype
8. C2×#10587574C4semidirect
13. #11762741C4semidirect
14. #11762740C4semidirect
18. #11762673C4semidirect
254. C17#11801semidirect

Maximal subgroups

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

subgroupindex
#145345492
#145288602
#145271152
#145267852
#145267842
#145267832
C17×#13442
#1180117