Finite Group Databasebeta

← #16937509

Subgroups — #16937509

Order 4352.

360 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 75 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#14534432C211yesmaximal, characteristic
2#14529729C211yesmaximal, characteristic
3#14527335C211yesmaximal, characteristic
4#14526677C211yesmaximal, characteristic
5#14526650C211yesmaximal, characteristic
6#14526641C211yesmaximal, characteristic
7C17×#1506C211yesmaximal, characteristic
8#11762866C2211yescharacteristic
9C4×#63223C2211yescharacteristic
10#11762751—21no—
11#11762730—21no—
12#11762715—21no—
13#11762713—21no—
14#11762696—21no—
15#11762693—21no—
16#11762677C2211yescharacteristic
17C4×C4⋊Dic17C2211yescharacteristic
18#11762568—21no—
19#11762568—21no—
20#11762546—21no—
21#11762546—21no—
22C17×#533C2211yescharacteristic
23C17×#432C2211yescharacteristic
24C17×#376C2211yescharacteristic
25C17×#328—21no—
26C17×#328—21no—
27C22×Dic34—21no—
28C17×C4⋊Q8Q811yescharacteristic, complemented
29C17×C4⋊Q8Q811yescharacteristic, complemented
30C17×#125—21no—
31C17×C22⋊Q8—21no—
32C17×C22⋊Q8—21no—
33C4×Q8×C17—21no—
34C4×Q8×C17—21no—
35C17×#117—21no—
36C2×C42×C17—21no—
37C2×C42×C17C2311yescharacteristic
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×#63223—21no—
45C17×C4⋊C8—21no—
46C17×C22⋊C8—21no—
47C17×#97—21no—
48C17×#95—21no—
49C17×#95D411yescharacteristic
50C17×#95D411yescharacteristic
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502D411yescharacteristic
59#10587502D411yescharacteristic
60C22⋊#3622—21no—
61C4⋊#3622—21no—
62#10587474—21no—
63#10587474—21no—
64C4×#3622—21no—
65C2×Q8×C17—21no—
66C2×Q8×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C2×Dic34—42no—
77C2×Dic34—42no—
78C2×Dic34—81no—
79C2×C8×C17—41no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—21no—
82C4⋊C4×C17—21no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C42×C17—42no—
88C42×C17—41no—
89C42×C17C2×Q811yescharacteristic
90C42×C17#4111yescharacteristic
91C42×C17#4111yescharacteristic
92C42×C17#4111yescharacteristic
93C4⋊Dic17—42no—
94C4⋊Dic17—81no—
95#63225—42no—
96#63225—42no—
97C4×Dic17—41no—
98C4×Dic17—41no—
99#63223—22no—
100#63223—41no—
101C2×#3622—41no—
102C2×#3622—41no—
103#10379—171nomaximal, Sylow
104C23×C17#12011yescharacteristic
105Q8×C17—81no—
106Q8×C17—81no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—22no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17C22⋊Q811yescharacteristic
120C2×C4×C17C22⋊Q811yescharacteristic
121C2×C4×C17C22⋊Q811yescharacteristic
122C2×C4×C17#12411yescharacteristic
123C2×C4×C17#12411yescharacteristic
124C2×C4×C17#12411yescharacteristic
125C2×Dic17—22no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137Dic34—84no—
138C8×C17—81no—
139#3622—42no—
140#3622—81no—
141#2605—171no—
142#1986—171no—
143#1515—171no—
144#1506D1711yescharacteristic
145#1300—171no—
146#1288—171no—
147#1285—171no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17#39311yescharacteristic
151C22×C17#45511yescharacteristic
152C22×C17#45611yescharacteristic
153C4×C17—21no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—42no—
160C4×C17—81no—
161C4×C17—81no—
162C4×C17—81no—
163Dic17—42no—
164Dic17—81no—
165Dic17—81no—
166Dic17—81no—
167#533C2×D1711yescharacteristic
168#432—171no—
169#432C2×D1711yescharacteristic
170C4×M16—171no—
171#400—341no—
172#395—341no—
173#391—341no—
174#391—341no—
175#387—341no—
176#387—341no—
177#383—171no—
178C4×C4⋊C4—171no—
179#376C2×D1711yescharacteristic
180#339—341no—
181#339—341no—
182#328—21no—
183#328—21no—
184#328—341no—
185#328—341no—
186C2×C17—21no—
187C2×C17—21no—
188C2×C17#202811yescharacteristic
189C2×C17#203111yescharacteristic
190C2×C17#203111yescharacteristic
191C22×Q8—341no—
192C2×M16—341no—
193C4⋊Q8Dic3411yescharacteristic, complemented
194C4⋊Q8Dic3411yescharacteristic, complemented
195#125—21no—
196C22⋊Q8—21no—
197C22⋊Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200#117—21no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C42—21no—
206C2×C42—171no—
207C2×C42—171no—
208C2×C42C22×D1711yescharacteristic
209C4⋊C8—21no—
210C4⋊C8—341no—
211C22⋊C8—21no—
212C22⋊C8—341no—
213#97—21no—
214#97—341no—
215#97—341no—
216C4×C8—341no—
217#95—21no—
218#95—171no—
219#95—171no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95C22⋊D1711yescharacteristic
228#95C22⋊D1711yescharacteristic
229C17#1037911yescharacteristic, complemented, Sylow
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—682no—
233C2×Q8—682no—
234C2×Q8—1361no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4D4×D1711yescharacteristic
242C22×C4D4×D1711yescharacteristic
243C22×C4C2×C22⋊D1711yescharacteristic
244M16—342no—
245M16—681no—
246C2×C8—41no—
247C2×C8—681no—
248C2×C8—681no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—682no—
256C4⋊C4—682no—
257C4⋊C4—682no—
258C4⋊C4—1361no—
259C22⋊C4—41no—
260C42—42no—
261C42—41no—
262C42—681no—
263C42—681no—
264C42C2×Dic3411yescharacteristic
265C42#6325211yescharacteristic
266C42#6325411yescharacteristic
267C42#6325411yescharacteristic
268C23#1058762311yescharacteristic
269Q8—81no—
270Q8—81no—
271Q8—1364no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—342no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4#1058756411yescharacteristic
297C2×C4#1058756411yescharacteristic
298C2×C4#1058757111yescharacteristic
299C2×C4#1058757111yescharacteristic
300C2×C4#1058761011yescharacteristic
301C2×C4#1058762211yescharacteristic
302C8—81no—
303C8—682no—
304C8—1361no—
305C22—21no—
306C22—21no—
307C22#1176272111yescharacteristic
308C22#1176298511yescharacteristic
309C22#1176299011yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—81no—
318C4—81no—
319C4—81no—
320C4—682no—
321C4—1361no—
322C4—1361no—
323C4—1361no—
324C2—21no—
325C2—21no—
326C2#1452994311yescharacteristic
327C2#1452995611yescharacteristic
328C2#1452995811yescharacteristic
subgroupG/H
centerC22#11762721
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1506C2
socleC22×C17#393
prime radicalC23×C17#120
hypercenter#1506D17
hypocenterC17#10379
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
28. C17×C4⋊Q8Q8semidirect
29. C17×C4⋊Q8Q8semidirect
193. C4⋊Q8Dic34semidirect
194. C4⋊Q8Dic34semidirect
229. C17#10379semidirect

Maximal subgroups

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

subgroupindex
#145344322
#145297292
#145273352
#145266772
#145266502
#145266412
C17×#15062
#1037917