Finite Group Databasebeta

← #16947328

Subgroups — #16947328

Order 4352.

428 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 63 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#14530134C211yesmaximal, characteristic, complemented
2#14527791C211yesmaximal, characteristic, complemented
3#14527723C211yesmaximal, characteristic
4#14526790C211yesmaximal, characteristic, complemented
5#14526787C211yesmaximal, characteristic
6#14526728C211yesmaximal, characteristic, complemented
7C17×#2609C211yesmaximal, characteristic
8#11763440C2211yescharacteristic
9#11763139—21no—
10#11762960—21no—
11#11762849—21no—
12#11762578C2211yescharacteristic
13#11762569—21no—
14#11762545—21no—
15#11762545—21no—
16#11762535C2211yescharacteristic
17#11762535C2211yescharacteristic
18C2×C17×#117—21no—
19C17×#398C2211yescharacteristic
20C17×#396—21no—
21C17×#394C2211yescharacteristic
22C17×#388—21no—
23C17×#387—21no—
24C17×#386—21no—
25C17×#385—21no—
26C17×#382C2211yescharacteristic
27C22×D4×C17D411yescharacteristic
28C23×C4×C17—21no—
29C17×#117—42no—
30C17×#117—42no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17C2311yescharacteristic
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C22⋊C4×C17—21no—
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C42×C17—21no—
43C2×C42×C17D411yescharacteristic
44C2×#63229—41no—
45#10587612—21no—
46#10587610—21no—
47#10587610—21no—
48C2×#63223—21no—
49#10587576—41no—
50#10587574—41no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54C17×#95—21no—
55C17×#95D411yescharacteristic
56C17×#95D411yescharacteristic
57#10587503—41no—
58C22⋊#3622—21no—
59C22⋊#3622—21no—
60#10587481—41no—
61#10587479—41no—
62#10587479—41no—
63#10587478—41no—
64C24×C17—21no—
65C2×D4×C17—41no—
66C2×D4×C17—42no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—42no—
73C22×C4×C17—42no—
74C22×C4×C17—42no—
75C22×C4×C17C2×D411yescharacteristic
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C2×Dic34—21no—
79C2×Dic34—21no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—42no—
84C4⋊C4×C17—42no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—41no—
87C22⋊C4×C17—42no—
88C22⋊C4×C17—42no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—41no—
91C22⋊C4×C17—41no—
92C22⋊C4×C17—81no—
93C42×C17—21no—
94C42×C17—42no—
95C42×C17SD1612yescomplemented
96C22⋊Dic17—41no—
97#63229—82no—
98C4⋊Dic17—21no—
99C4⋊Dic17—21no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102#63225—41no—
103#63225—41no—
104C4×Dic17—41no—
105#63223—22no—
106#63223—41no—
107C2×#3622—41no—
108C2×#3622—41no—
109C2×#3622—41no—
110#12248—171nomaximal, Sylow
111C23×C17—22no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17#12011yescharacteristic
119D4×C17—42no—
120D4×C17—81no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—48no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17C2×SD1611yescharacteristic
144C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
145C2×Dic17—41no—
146C2×Dic17—41no—
147Dic34—81no—
148Dic34—81no—
149#3622—42no—
150#3622—81no—
151#3622—81no—
152#2609D1711yescharacteristic
153#2061—171no—
154#1611—171no—
155#1599—171no—
156#1346—171no—
157#1345—171no—
158#1324—171no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—42no—
162C22×C17—42no—
163C22×C17—42no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17#45011yescharacteristic
169C22×C17#45611yescharacteristic
170C22×C17B(4,𝔽2)11yescharacteristic
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—42no—
174C4×C17—42no—
175C4×C17—41no—
176C4×C17—41no—
177C4×C17—81no—
178C4×C17—81no—
179C4×C17—81no—
180Dic17—81no—
181Dic17—81no—
182#544—171no—
183C2×#117—21no—
184#483—341no—
185#450—341no—
186#428—341no—
187#398C2×D1711yescharacteristic
188#396—21no—
189#394C2×D1711yescharacteristic
190#388—21no—
191#387—21no—
192#386—21no—
193#385—21no—
194#382C2×D1711yescharacteristic
195#343—171no—
196#339—341no—
197#328—341no—
198#328—341no—
199#324—171no—
200#324—171no—
201C2×C17—21no—
202C2×C17—21no—
203C2×C17—82no—
204C2×C17#160411yescharacteristic
205C2×C17#219711yescharacteristic
206C2×C17#220411yescharacteristic
207C22×D4C22⋊D1711yescharacteristic
208C23×C4—21no—
209C2×SD16—681no—
210C2×M16—341no—
211C4⋊Q8—681no—
212#125—681no—
213C22⋊Q8—341no—
214C22⋊Q8—341no—
215#117—42no—
216#117—42no—
217#117—341no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4C22⋊D1711yescharacteristic
222C2×C4⋊C4C22⋊D1711yescharacteristic
223C2×C4⋊C4C22×D1711yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C42—21no—
230C2×C42C22⋊D1711yescharacteristic
231#107—681no—
232#106—681no—
233#106—681no—
234#103—681no—
235#102—681no—
236C22⋊C8—341no—
237C22⋊C8—341no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95C22⋊D1711yescharacteristic
243#95C22⋊D1711yescharacteristic
244C17#1224811yescharacteristic, complemented, Sylow
245C24—21no—
246C2×Q8—341no—
247C2×Q8—341no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×C22⋊D1711yescharacteristic
259C22×C4C2×C22⋊D1711yescharacteristic
260C22×C4C2×C22⋊D1711yescharacteristic
261SD16—1362no—
262M16—342no—
263M16—681no—
264C2×C8—681no—
265C2×C8—681no—
266C2×C8—681no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—341no—
275C4⋊C4—341no—
276C4⋊C4—341no—
277C4⋊C4—341no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—81no—
286C22⋊C4—681no—
287C42—21no—
288C42—42no—
289C42—681no—
290C42#6322912yescomplemented
291C23—22no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#1058763911yescharacteristic
299Q8—1361no—
300Q8—1361no—
301D4—42no—
302D4—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4#1058761811yescharacteristic
328C2×C4C2×#6322911yescharacteristic
329C8—682no—
330C8—1361no—
331C8—1361no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1176296011yescharacteristic
342C22#1176299211yescharacteristic
343C22#1176299611yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—1361no—
354C4—1361no—
355C2—21no—
356C2—21no—
357C2—82no—
358C2#1452775311yescharacteristic
359C2#1453077411yescharacteristic
360C2#1453081011yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2609C2
socleC22×C17B(4,𝔽2)
prime radicalC22×D4×C17D4
hypercenter#2609D17
hypocenterC17#12248
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2609

Complements

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

NH ≅ G/Ntype
1. #14530134C2semidirect
2. #14527791C2semidirect
4. #14526790C2semidirect
6. #14526728C2semidirect
95. C42×C17SD16semidirect
244. C17#12248semidirect
290. C42#63229semidirect

Maximal subgroups

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

subgroupindex
#145301342
#145277912
#145277232
#145267902
#145267872
#145267282
C17×#26092
#1224817