Finite Group Databasebeta

← #16947241

Subgroups — #16947241

Order 4352.

382 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 59 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#14527825C211yesmaximal, characteristic, complemented
3#14527723C211yesmaximal, characteristic
4#14526811C211yesmaximal, characteristic, complemented
5#14526787C211yesmaximal, characteristic
6#14526728C211yesmaximal, characteristic, complemented
7C17×#2782C211yesmaximal, characteristic
8#11763440C2211yescharacteristic
9#11763144—21no—
10#11762960—21no—
11#11762849—21no—
12#11762578C2211yescharacteristic
13#11762569—21no—
14#11762545—21no—
15#11762545—21no—
16#11762535C2211yescharacteristic
17#11762535C2211yescharacteristic
18C17×#400—21no—
19C17×#400C2211yescharacteristic
20C17×#397—21no—
21C17×#394C2211yescharacteristic
22C17×#388—21no—
23C17×#386—21no—
24C17×#385—21no—
25C17×#385—21no—
26C17×#382C2211yescharacteristic
27C22×D4×C17D411yescharacteristic
28C23×C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊C4×C17—21no—
35C2×C22⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊C4×C17—21no—
39C2×C42×C17D411yescharacteristic
40C2×#63229—41no—
41#10587612—21no—
42#10587610—21no—
43#10587610—21no—
44C2×#63223—21no—
45#10587576—41no—
46#10587574—41no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95—21no—
50C17×#95—21no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95D411yescharacteristic
54C17×#95D411yescharacteristic
55#10587503—41no—
56C22⋊#3622—21no—
57C22⋊#3622—21no—
58#10587481—41no—
59#10587479—41no—
60#10587478—41no—
61#10587478—41no—
62C24×C17—21no—
63C2×D4×C17—41no—
64C2×D4×C17—42no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—81no—
71C22×C4×C17—81no—
72C22×C4×C17—81no—
73C22×C4×C17C2×D411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×C4×C17C2×D411yescharacteristic
76C2×Dic34—21no—
77C2×Dic34—21no—
78C4⋊C4×C17—41no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—81no—
82C4⋊C4×C17—81no—
83C22⋊C4×C17—42no—
84C22⋊C4×C17—42no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—81no—
88C22⋊C4×C17—81no—
89C42×C17—21no—
90C42×C17—21no—
91C22⋊Dic17—41no—
92#63229—82no—
93C4⋊Dic17—21no—
94C4⋊Dic17—21no—
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97#63225—41no—
98#63225—41no—
99C4×Dic17—41no—
100#63223—22no—
101#63223—41no—
102C2×#3622—41no—
103C2×#3622—41no—
104C2×#3622—41no—
105#12235—171nomaximal, Sylow
106C23×C17—22no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17#12011yescharacteristic
114D4×C17—42no—
115D4×C17—81no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
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—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
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—84no—
138C2×C4×C17#13511yescharacteristic
139C2×C4×C17#13711yescharacteristic
140C2×Dic17—41no—
141C2×Dic17—41no—
142Dic34—81no—
143Dic34—81no—
144#3622—42no—
145#3622—81no—
146#3622—81no—
147#2782D1711yescharacteristic
148#2061—171no—
149#1617—171no—
150#1599—171no—
151#1353—171no—
152#1345—171no—
153#1324—171no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—42no—
157C22×C17—42no—
158C22×C17—42no—
159C22×C17—81no—
160C22×C17—81no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17#45211yescharacteristic
164C22×C17#45611yescharacteristic
165C22×C17B(4,𝔽2)11yescharacteristic
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—41no—
169C4×C17—41no—
170C4×C17—81no—
171C4×C17—81no—
172C4×C17—81no—
173C4×C17—81no—
174C4×C17—81no—
175Dic17—81no—
176Dic17—81no—
177#544—171no—
178#484—341no—
179#450—341no—
180#428—341no—
181#400—21no—
182#400C2×D1711yescharacteristic
183#397—21no—
184#394C2×D1711yescharacteristic
185#388—21no—
186#386—21no—
187#385—21no—
188#385—21no—
189#382C2×D1711yescharacteristic
190#343—171no—
191#339—341no—
192#328—341no—
193#328—341no—
194#324—171no—
195#324—171no—
196C2×C17—21no—
197C2×C17—21no—
198C2×C17—82no—
199C2×C17#160311yescharacteristic
200C2×C17#219711yescharacteristic
201C2×C17#220211yescharacteristic
202C22×D4C22⋊D1711yescharacteristic
203C23×C4—21no—
204C2×SD16—681no—
205C2×M16—341no—
206C4⋊Q8—681no—
207#125—681no—
208C22⋊Q8—341no—
209C22⋊Q8—341no—
210#117—341no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4C22⋊D1711yescharacteristic
214C2×C4⋊C4C22⋊D1711yescharacteristic
215C2×C4⋊C4C22×D1711yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C42C22⋊D1711yescharacteristic
222#107—681no—
223#107—681no—
224#106—681no—
225#103—681no—
226#102—681no—
227C22⋊C8—341no—
228C22⋊C8—341no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95C22⋊D1711yescharacteristic
236#95C22⋊D1711yescharacteristic
237C17#1223511yescharacteristic, complemented, Sylow
238C24—21no—
239C2×Q8—341no—
240C2×Q8—341no—
241C2×D4—41no—
242C2×D4—42no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C2×C22⋊D1711yescharacteristic
252C22×C4C2×C22⋊D1711yescharacteristic
253C22×C4C2×C22⋊D1711yescharacteristic
254SD16—1362no—
255M16—342no—
256M16—681no—
257C2×C8—681no—
258C2×C8—681no—
259C2×C8—681no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—81no—
264C4⋊C4—81no—
265C4⋊C4—341no—
266C4⋊C4—341no—
267C4⋊C4—341no—
268C4⋊C4—341no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—81no—
276C22⋊C4—81no—
277C22⋊C4—681no—
278C42—21no—
279C42—21no—
280C42—681no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#1058763911yescharacteristic
289Q8—1361no—
290Q8—1361no—
291D4—42no—
292D4—81no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—84no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4#1058763611yescharacteristic
318C2×C4#1058763711yescharacteristic
319C8—682no—
320C8—1361no—
321C8—1361no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#1176297111yescharacteristic
332C22#1176299211yescharacteristic
333C22#1176299611yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—1361no—
344C4—1361no—
345C2—21no—
346C2—21no—
347C2—82no—
348C2#1452774711yescharacteristic
349C2#1453077411yescharacteristic
350C2#1453080111yescharacteristic
subgroupG/H
centerC22#11762996
commutator, cosocleC2×C4⋊C4×C17C23
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2782C2
socleC22×C17B(4,𝔽2)
prime radicalC22×D4×C17D4
hypercenter#2782D17
hypocenterC17#12235
LCSG ▷ C2×C4⋊C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2782

Complements

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

NH ≅ G/Ntype
1. #14530134C2semidirect
2. #14527825C2semidirect
4. #14526811C2semidirect
6. #14526728C2semidirect
237. C17#12235semidirect

Maximal subgroups

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

subgroupindex
#145301342
#145278252
#145277232
#145268112
#145267872
#145267282
C17×#27822
#1223517