Finite Group Databasebeta

← #16938728

Subgroups — #16938728

Order 4352.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#14534898C211yesmaximal, characteristic
2#14528974C212yesmaximal
3#14527048C211yesmaximal, characteristic
4#14526664C212yesmaximal
5C17×#1280C211yesmaximal, characteristic
6C2×#10587610—21no—
7C22×C4⋊Dic17C2211yescharacteristic
8C22×#63225—21no—
9C2×#10587489—22no—
10#11762741—22no—
11#11762730—22no—
12#11762669—21no—
13#11762669C2212yes—
14C2×#10587502—21no—
15C2×#10587502C2211yescharacteristic
16#11762572—22no—
17C2×C17×C22⋊C8C2212yes—
18C2×C17×#95C2211yescharacteristic
19C23×C4×C17D411yescharacteristic
20C23×C4×C17D411yescharacteristic
21C23×C4×C17C2311yescharacteristic
22C23×Dic17—21no—
23C23×Dic17—21no—
24C22×Dic34—21no—
25C22×C8×C17—22no—
26C2×C22⋊Dic17—22no—
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—22no—
31C2×C4⋊Dic17—42no—
32C2×C4⋊Dic17D412yes—
33C2×C4⋊Dic17D412yes—
34#10587610—42no—
35#10587610—42no—
36C2×#63225—22no—
37C2×#63225—42no—
38C2×#63225—42no—
39C2×#63225—42no—
40C17×C22⋊C8—22no—
41C17×C22⋊C8Q814yescomplemented
42C17×#95—22no—
43#10587502—22no—
44#10587502—22no—
45#10587502—21no—
46#10587502—42no—
47#10587502Q812yescomplemented
48#10587489—44no—
49C24×C17C2×D411yescharacteristic
50C22×C4×C17—21no—
51C22×C4×C17—21no—
52C22×C4×C17—21no—
53C22×C4×C17—21no—
54C22×C4×C17—41no—
55C22×C4×C17—41no—
56C22×C4×C17—41no—
57C22×C4×C17—41no—
58C22×C4×C17—41no—
59C22×C4×C17—41no—
60C22×C4×C17D812yes—
61C22×C4×C17Q1612yes—
62C22×C4×C17C2×D412yes—
63C22×C4×C17C2×Q812yes—
64C22×C4×C17C2×Q811yescharacteristic
65C22×C4×C17#4111yescharacteristic
66C22×Dic17—22no—
67C22×Dic17—22no—
68C22×Dic17—22no—
69C22×Dic17—22no—
70C22×Dic17—22no—
71C22×Dic17—22no—
72C22×Dic17—41no—
73C22×Dic17—41no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C22×Dic17—41no—
77C22×Dic17—41no—
78C22×Dic17—41no—
79C22×Dic17—41no—
80C22×Dic17—41no—
81C22×Dic17—41no—
82C2×Dic34—42no—
83C2×Dic34—81no—
84C2×Dic34—81no—
85C2×C8×C17—24no—
86C2×C8×C17—42no—
87C2×C8×C17—42no—
88C22⋊Dic17—42no—
89C22⋊Dic17—42no—
90C4⋊Dic17—24no—
91C4⋊Dic17—42no—
92C4⋊Dic17—42no—
93C4⋊Dic17—42no—
94C4⋊Dic17—42no—
95C4⋊Dic17—81no—
96C4⋊Dic17—81no—
97#63225—44no—
98#63225—48no—
99#10567—171nomaximal, Sylow
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—21no—
103C23×C17—21no—
104C23×C17#12011yescharacteristic
105C23×C17C22⋊Q811yescharacteristic
106C23×C17C4⋊Q811yescharacteristic
107C23×C17C2×D811yescharacteristic
108C23×C17C2×Q1611yescharacteristic
109C23×C17AGL(1,ℤ/8)11yescharacteristic
110C23×C17#13711yescharacteristic
111C2×C4×C17—22no—
112C2×C4×C17—22no—
113C2×C4×C17—22no—
114C2×C4×C17—22no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—81no—
125C2×C4×C17—81no—
126C2×C4×C17C22⋊Q812yes—
127C2×C4×C17C4⋊Q812yes—
128C2×Dic17—24no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—42no—
143C2×Dic17—42no—
144C2×Dic17—81no—
145C2×Dic17—81no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148Dic34—84no—
149C8×C17—44no—
150#2659—171no—
151#1852—172no—
152#1444—171no—
153#1294—172no—
154#1280D1711yescharacteristic
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17—21no—
165C22×C17—21no—
166C22×C17—21no—
167C22×C17#39511yescharacteristic
168C22×C17#44711yescharacteristic
169C22×C17#45111yescharacteristic
170C22×C17#45311yescharacteristic
171C22×C17#45611yescharacteristic
172C22×C17B(4,𝔽2)11yescharacteristic
173C22×C17#45811yescharacteristic
174C22×C17#47411yescharacteristic
175C22×C17#47711yescharacteristic
176C22×C17#50011yescharacteristic
177C22×C17#50111yescharacteristic
178C4×C17—22no—
179C4×C17—22no—
180C4×C17—42no—
181C4×C17—42no—
182Dic17—44no—
183Dic17—82no—
184Dic17—82no—
185Dic17—82no—
186C2×C22⋊Q8—341no—
187C22×C4⋊C4—171no—
188C22×C4⋊C4—341no—
189C2×#107—342no—
190C2×C22⋊C8C2×D1712yes—
191#398—342no—
192#395—342no—
193#380—172no—
194#380—341no—
195C2×#95—171no—
196C2×#95—341no—
197C2×#95C2×D1711yescharacteristic
198#340—342no—
199C2×C17—21no—
200C2×C17—21no—
201C2×C17—21no—
202C2×C17—21no—
203C2×C17#159511yescharacteristic
204C2×C17#160511yescharacteristic
205C2×C17#161011yescharacteristic
206C2×C17#161411yescharacteristic
207C2×C17#205711yescharacteristic
208C2×C17#206011yescharacteristic
209C2×C17#206211yescharacteristic
210C22×Q8—341no—
211C23×C4—341no—
212C23×C4—341no—
213C23×C4D6811yescharacteristic
214C23×C4D6811yescharacteristic
215C23×C4C22×D1711yescharacteristic
216C22×C8—22no—
217C22⋊Q8—682no—
218C22⋊Q8—682no—
219C2×C4⋊C4—172no—
220C2×C4⋊C4—172no—
221C2×C4⋊C4—342no—
222C2×C4⋊C4—342no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—682no—
227C2×C4⋊C4—682no—
228C2×C4⋊C4—682no—
229C2×C4⋊C4—682no—
230C2×C22⋊C4—342no—
231#107—684no—
232C22⋊C8—22no—
233C22⋊C8Dic3414yescomplemented
234#95—22no—
235#95—172no—
236#95—342no—
237#95—342no—
238#95—341no—
239#95—682no—
240C17#1056711yescharacteristic, complemented, Sylow
241C24C2×D6811yescharacteristic
242C2×Q8—682no—
243C2×Q8—1361no—
244C2×Q8—1361no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—342no—
256C22×C4—342no—
257C22×C4—342no—
258C22×C4—342no—
259C22×C4—342no—
260C22×C4—342no—
261C22×C4—681no—
262C22×C4—681no—
263C22×C4—681no—
264C22×C4—681no—
265C22×C4—681no—
266C22×C4—681no—
267C22×C4—681no—
268C22×C4—681no—
269C22×C4—681no—
270C22×C4—681no—
271C22×C4D13612yes—
272C22×C4Dic6812yes—
273C22×C4C2×Dic3412yes—
274C22×C4D4×D1712yes—
275C22×C4#6325411yescharacteristic
276C22×C4Q8×D1711yescharacteristic
277C2×C8—24no—
278C2×C8—42no—
279C2×C8—42no—
280C4⋊C4—344no—
281C4⋊C4—682no—
282C4⋊C4—682no—
283C4⋊C4—682no—
284C4⋊C4—684no—
285C4⋊C4—682no—
286C4⋊C4—688no—
287C4⋊C4—1361no—
288C4⋊C4—1361no—
289C22⋊C4—682no—
290C22⋊C4—682no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1058755511yescharacteristic
296C23#1058756811yescharacteristic
297C23#1058758311yescharacteristic
298C23C2×D13611yescharacteristic
299C23C2×Dic6811yescharacteristic
300C23#1058759411yescharacteristic
301C23#1058759511yescharacteristic
302Q8—1364no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—344no—
319C2×C4—682no—
320C2×C4—682no—
321C2×C4—682no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4—682no—
329C2×C4—682no—
330C2×C4—682no—
331C2×C4—682no—
332C2×C4—682no—
333C2×C4—682no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4—1361no—
337C2×C4—1361no—
338C2×C4#1058756412yes—
339C2×C4#1058757412yes—
340C8—44no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1176273011yescharacteristic
354C22#1176294511yescharacteristic
355C22#1176296711yescharacteristic
356C22#1176297911yescharacteristic
357C22#1176299411yescharacteristic
358C22#1176299711yescharacteristic
359C22#1176300011yescharacteristic
360C22#1176309011yescharacteristic
361C22#1176310511yescharacteristic
362C22#1176321411yescharacteristic
363C22#1176321811yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—684no—
369C4—1362no—
370C4—1362no—
371C4—1362no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#1452770411yescharacteristic
377C2#1452776011yescharacteristic
378C2#1452778911yescharacteristic
379C2#1452781111yescharacteristic
380C2#1453011311yescharacteristic
381C2#1453013011yescharacteristic
382C2#1453013811yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC17×#1280C2
socleC23×C17#120
prime radicalC24×C17C2×D4
hypercenter#1280D17
hypocenterC17#10567
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
41. C17×C22⋊C8Q8semidirect
47. #10587502Q8semidirect
233. C22⋊C8Dic34semidirect
240. C17#10567semidirect

Maximal subgroups

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

subgroupindex
#145348982
#145289742
#145270482
#145266642
C17×#12802
#1056717