Finite Group Databasebeta

← #17950049

Subgroups — #17950049

Order 4352.

488 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 92 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#14544511C211yesmaximal, characteristic
2C4×#10587543C211yesmaximal, characteristic
3#14543604C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5#14537125C211yesmaximal, characteristic
6C2×#10587543—21no—
7C2×#10587543—21no—
8C2×#10587542—21no—
9C2×C4×#3624—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587632C411yescharacteristic
13C2×C42×D17C2211yescharacteristic
14#11762716C2211yescharacteristic
15#11762708C2211yescharacteristic
16#11762707C411yescharacteristic
17#11762692—21no—
18#11762691—21no—
19#11762683—21no—
20#11762671—21no—
21#11762666C411yescharacteristic
22C17×#391C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C42×C17C2×C411yescharacteristic
31#10587632—42no—
32C2×#63227—21no—
33C2×#63227—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—21no—
38C2×#63225—21no—
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17C2311yescharacteristic
42C42×D17—42no—
43#10587543—24no—
44#10587543—24no—
45#10587542—42no—
46C4×#3624—42no—
47C17×#95—21no—
48C17×#95—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C23×D17C22⋊C411yescharacteristic
55C2×Q8×C17—22no—
56C2×Q8×C17—41no—
57C2×Q8×C17—41no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—41no—
75C2×C4×D17—41no—
76C2×C4×D17—41no—
77C2×#3624—24no—
78C2×#3624—24no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C4⋊C4×C17—22no—
84C42×C17—41no—
85#63227—42no—
86#63227—42no—
87C4⋊Dic17—42no—
88C4⋊Dic17—41no—
89C4⋊Dic17—41no—
90#63225—42no—
91#63225—42no—
92C4×Dic17—22no—
93C4×Dic17—22no—
94C4×Dic17—41no—
95#7675—171nomaximal, Sylow
96C23×C17C2×C22⋊C411yescharacteristic
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17#10412yes—
104Q8×C17—82no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
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#10412yes—
119C2×C4×C17#13112yes—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17#9912yes—
133C2×Dic17#13512yes—
134C2×Dic17#13712yes—
135C4×D17—44no—
136C4×D17—44no—
137C4×D17—44no—
138#3624—44no—
139#3624—44no—
140#2855—171no—
141#1749—171no—
142C4×C22⋊C8—171no—
143#1272—172no—
144#1271—172no—
145C22×C17#40811yescharacteristic
146C22×C17C2×#9911yescharacteristic
147C22×C17#41311yescharacteristic
148C22×C17#41611yescharacteristic
149C22×C17#41911yescharacteristic
150C22×C17C2×#10411yescharacteristic
151C22×C17C2×#10411yescharacteristic
152C2×D17—22no—
153C2×D17—22no—
154C2×D17—22no—
155C2×D17—41no—
156C2×D17—41no—
157C2×D17—41no—
158C2×D17—41no—
159C4×C17—22no—
160C4×C17—22no—
161C4×C17—41no—
162C4×C17—81no—
163C4×C17—81no—
164Dic17—22no—
165Dic17—22no—
166Dic17—22no—
167Dic17—81no—
168Dic17—81no—
169C2×C22⋊Q8—171no—
170C22×C42—171no—
171C2×C22⋊C8—341no—
172C2×C22⋊C8—341no—
173C2×#97—341no—
174C2×C4×C8—341no—
175#391—171no—
176#391#59511yescharacteristic
177#389—171no—
178#389—171no—
179#387—341no—
180#387—341no—
181#385—341no—
182#381—341no—
183#380—171no—
184#336—172no—
185#336—172no—
186C2×C17#146811yescharacteristic
187C2×C17#148511yescharacteristic
188C2×C17#149411yescharacteristic
189C2×C17#150111yescharacteristic
190C2×C17#150811yescharacteristic
191C2×C17#152411yescharacteristic
192C2×C17#155411yescharacteristic
193D17—42no—
194C22×Q8C2×#59511yescharacteristic
195C23×C4—171no—
196C23×C4—171no—
197C23×C4—171no—
198C22×C8—342no—
199C22×C8—342no—
200C22⋊Q8—682no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4C2×#59511yescharacteristic
207C2×C22⋊C4—341no—
208C2×C22⋊C4—341no—
209C2×C42—171no—
210C2×C42—171no—
211C2×C42—171no—
212C2×C42—682no—
213C2×C42C2×#59511yescharacteristic
214C22⋊C8—344no—
215C22⋊C8—344no—
216#97—682no—
217C4×C8—682no—
218#95—21no—
219#95—21no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224#95—341no—
225C17#767511yescharacteristic, complemented, Sylow
226C24—171no—
227C2×Q8—22no—
228C2×Q8—41no—
229C2×Q8—41no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—342no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—342no—
240C22×C4—342no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—681no—
244C22×C4—681no—
245C22×C4—681no—
246C22×C4C22⋊#59511yescharacteristic
247C22×C4C22⋊#59511yescharacteristic
248C22×C4C22×#59511yescharacteristic
249C2×C8—344no—
250C2×C8—344no—
251C2×C8—682no—
252C2×C8—682no—
253C2×C8—682no—
254C2×C8—682no—
255C4⋊C4—22no—
256C4⋊C4—681no—
257C4⋊C4—681no—
258C4⋊C4—682no—
259C4⋊C4—682no—
260C4⋊C4—682no—
261C22⋊C4—682no—
262C22⋊C4—682no—
263C42—41no—
264C42—342no—
265C42—342no—
266C42—681no—
267C23—172no—
268C23—341no—
269C23—341no—
270C23—341no—
271C23—341no—
272C23—341no—
273C23—341no—
274C23C2×C22⋊#59511yescharacteristic
275Q8—82no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—172no—
290C2×C4—172no—
291C2×C4—172no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—684no—
299C2×C4—684no—
300C2×C4—684no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4—681no—
306C2×C4—681no—
307C2×C4#1058755012yes—
308C2×C4#1058767812yes—
309C8—684no—
310C8—684no—
311C22—342no—
312C22—342no—
313C22—342no—
314C22—681no—
315C22—681no—
316C22—681no—
317C22—681no—
318C22#1176378311yescharacteristic
319C22D17⋊#9911yescharacteristic
320C22#1176379511yescharacteristic
321C22#1176379911yescharacteristic
322C22#1176380211yescharacteristic
323C22D17⋊#10411yescharacteristic
324C22C2×#1058755011yescharacteristic
325C4—22no—
326C4—22no—
327C4—41no—
328C4—81no—
329C4—81no—
330C4—342no—
331C4—342no—
332C4—342no—
333C4—1361no—
334C4—1361no—
335C2—682no—
336C2#1454405011yescharacteristic
337C2#1454410711yescharacteristic
338C2#1454413611yescharacteristic
339C2#1454416211yescharacteristic
340C2#1454418211yescharacteristic
341C2#1454423611yescharacteristic
342C2#1454430911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#391C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#7675
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
3. #14543604C2semidirect
4. #14543602C2semidirect
225. C17#7675semidirect

Maximal subgroups

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

subgroupindex
#145445112
C4×#105875432
#145436042
#145436022
#145371252
#767517