Finite Group Databasebeta

← #16938827

Subgroups — #16938827

Order 4352.

374 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 97 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#14537056C211yesmaximal, characteristic
2#14528952C211yesmaximal, characteristic
3#14528918C211yesmaximal, characteristic
4#14528917C211yesmaximal, characteristic
5#14526666C211yesmaximal, characteristic
6#14526666C211yesmaximal, characteristic
7C17×#1295C211yesmaximal, characteristic
8C2×#10587489—21no—
9C2×#10587488—21no—
10C2×#10587474C2211yescharacteristic
11#11762751—21no—
12#11762751—21no—
13#11762748—21no—
14#11762730—21no—
15#11762730—21no—
16#11762729—21no—
17#11762679C2211yescharacteristic
18#11762675C2211yescharacteristic
19#11762675C2211yescharacteristic
20#11762572—21no—
21#11762572—21no—
22#11762571—21no—
23#11762571—21no—
24C2×C17×#97C2211yescharacteristic
25C2×C17×#97C2211yescharacteristic
26C17×#382C2211yescharacteristic
27C22×Dic34—21no—
28C22×C8×C17—21no—
29C22×C8×C17—21no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C42×C17C2311yescharacteristic
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17D411yescharacteristic
38C2×C4⋊Dic17D411yescharacteristic
39C2×C4⋊Dic17D411yescharacteristic
40C2×C4⋊Dic17D411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C22×#3622—21no—
44C17×#97—21no—
45C17×#97—21no—
46C17×#97Q812yescomplemented
47C17×#97Q812yescomplemented
48C17×#95—21no—
49C17×#95—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587489—42no—
59#10587488—42no—
60#10587474—21no—
61#10587474Q812yescomplemented
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—41no—
72C2×Dic34—81no—
73C2×Dic34—81no—
74C2×Dic34—81no—
75C2×C8×C17—22no—
76C2×C8×C17—22no—
77C2×C8×C17—22no—
78C2×C8×C17—22no—
79C2×C8×C17—41no—
80C2×C8×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C42×C17C2×Q811yescharacteristic
84C42×C17C2×Q811yescharacteristic
85C42×C17C2×Q811yescharacteristic
86C42×C17#4111yescharacteristic
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17—42no—
94C4⋊Dic17—42no—
95C4⋊Dic17—81no—
96#63225—81no—
97#63225—81no—
98C2×#3622—22no—
99C2×#3622—41no—
100C2×#3622—41no—
101#10584—171nomaximal, Sylow
102C23×C17#12011yescharacteristic
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17C22⋊Q811yescharacteristic
116C2×C4×C17C22⋊Q811yescharacteristic
117C2×C4×C17C22⋊Q811yescharacteristic
118C2×C4×C17C4⋊Q811yescharacteristic
119C2×C4×C17C4⋊Q811yescharacteristic
120C2×C4×C17C4⋊Q811yescharacteristic
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—81no—
134C2×Dic17—81no—
135C2×Dic17—81no—
136Dic34—84no—
137C8×C17—24no—
138C8×C17—42no—
139#3622—42no—
140#2844—171no—
141#1847—171no—
142#1836—171no—
143#1836—171no—
144#1295—171no—
145#1295—171no—
146#1295D1711yescharacteristic
147C22×C17#39511yescharacteristic
148C22×C17#45311yescharacteristic
149C22×C17#45411yescharacteristic
150C22×C17#45411yescharacteristic
151C22×C17#45511yescharacteristic
152C22×C17#45511yescharacteristic
153C22×C17#45611yescharacteristic
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—21no—
160C4×C17—81no—
161C4×C17—81no—
162Dic17—81no—
163Dic17—81no—
164Dic17—81no—
165Dic17—81no—
166Dic17—161no—
167C2×#107—341no—
168C2×#106—341no—
169C2×#97—171no—
170C2×#97C2×D1711yescharacteristic
171C2×#97C2×D1711yescharacteristic
172#400—341no—
173#400—341no—
174#400—341no—
175#395—341no—
176#395—341no—
177#395—341no—
178#384—171no—
179#382—171no—
180#382—171no—
181#382C2×D1711yescharacteristic
182#340—341no—
183#340—341no—
184#340—341no—
185#340—341no—
186C2×C17#165711yescharacteristic
187C2×C17#165911yescharacteristic
188C2×C17#166411yescharacteristic
189C2×C17#166411yescharacteristic
190C2×C17#206011yescharacteristic
191C2×C17#206111yescharacteristic
192C2×C17#206111yescharacteristic
193C22×Q8—341no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—341no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4D6811yescharacteristic
208C2×C4⋊C4D6811yescharacteristic
209C2×C42C22×D1711yescharacteristic
210#107—682no—
211#106—682no—
212#97—21no—
213#97—21no—
214#97—172no—
215#97—341no—
216#97Dic3412yescomplemented
217#97Dic3412yescomplemented
218#95—21no—
219#95—21no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95—341no—
228C17#1058411yescharacteristic, complemented, Sylow
229C2×Q8—1361no—
230C2×Q8—1361no—
231C2×Q8—1361no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—341no—
235C22×C4—341no—
236C22×C4—341no—
237C22×C4—341no—
238C22×C4—681no—
239C22×C4C2×D6811yescharacteristic
240C22×C4D4×D1711yescharacteristic
241C22×C4D4×D1711yescharacteristic
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—342no—
249C2×C8—681no—
250C2×C8—681no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—681no—
254C4⋊C4—681no—
255C4⋊C4—681no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—681no—
259C4⋊C4—682no—
260C4⋊C4—682no—
261C4⋊C4—1361no—
262C4⋊C4—1361no—
263C4⋊C4—1361no—
264C42C2×Dic3411yescharacteristic
265C42C2×Dic3411yescharacteristic
266C42#6325411yescharacteristic
267C42Q8×D1711yescharacteristic
268C23#1058756811yescharacteristic
269Q8—1364no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
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—1361no—
295C2×C4—1361no—
296C2×C4—1361no—
297C2×C4#1058755511yescharacteristic
298C2×C4#1058756411yescharacteristic
299C2×C4#1058756411yescharacteristic
300C2×C4#1058757411yescharacteristic
301C2×C4#1058757411yescharacteristic
302C2×C4#1058758311yescharacteristic
303C8—24no—
304C8—42no—
305C8—682no—
306C22#1176273011yescharacteristic
307C22#1176297811yescharacteristic
308C22#1176298311yescharacteristic
309C22#1176298411yescharacteristic
310C22#1176298811yescharacteristic
311C22#1176298911yescharacteristic
312C22#1176299311yescharacteristic
313C4—21no—
314C4—21no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—21no—
319C4—81no—
320C4—81no—
321C4—2721no—
322C4—1361no—
323C4—1361no—
324C4—1361no—
325C4—1361no—
326C2#1452808411yescharacteristic
327C2#1452809211yescharacteristic
328C2#1452811911yescharacteristic
329C2#1452812111yescharacteristic
330C2#1453012811yescharacteristic
331C2#1453013311yescharacteristic
332C2#1453013511yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1295C2
socle, prime radicalC23×C17#120
hypercenter#1295D17
hypocenterC17#10584
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
46. C17×#97Q8semidirect
47. C17×#97Q8semidirect
61. #10587474Q8semidirect
216. #97Dic34semidirect
217. #97Dic34semidirect
228. C17#10584semidirect

Maximal subgroups

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

subgroupindex
#145370562
#145289522
#145289182
#145289172
#145266662
C17×#12952
#1058417