Finite Group Databasebeta

← #17959541

Subgroups — #17959541

Order 4352.

384 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 70 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#14544650C211yesmaximal, characteristic
2#14544046C211yesmaximal, characteristic
3#14543784C212yesmaximal
4#14543769C212yesmaximal
5#14532077C211yesmaximal, characteristic
6#11763830—21no—
7#11763812—21no—
8C2×#10587545—21no—
9#11763689C2212yes—
10#11763655C2212yes—
11#11763654—21no—
12C2×C4×Dic34C411yescharacteristic
13C2×C4×Dic34C2211yescharacteristic
14#11762741—21no—
15#11762741C411yescharacteristic
16#11762741C2211yescharacteristic
17#11762730—21no—
18#11762730C411yescharacteristic
19#11762730C2211yescharacteristic
20C4×C4⋊Dic17—21no—
21C4×C4⋊Dic17—21no—
22C17×#376C411yescharacteristic
23C22×Dic34C2×C411yescharacteristic
24C22×Dic34D411yescharacteristic
25C2×#63247—22no—
26C2×C42×C17—21no—
27C2×C42×C17C2×C411yescharacteristic
28C2×C4⋊Dic17—21no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×C4⋊Dic17D411yescharacteristic
34C2×#63225—21no—
35C2×#63225C2×C411yescharacteristic
36C2×#63225C2311yescharacteristic
37C2×C4×Dic17—21no—
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C4×Dic34—41no—
41C4×Dic34—41no—
42C4×Dic34—41no—
43C4×Dic34—41no—
44C22⋊#3624—22no—
45#10587545—42no—
46#10587544—41no—
47#10587542—41no—
48C4⋊#3624—41no—
49C4⋊#3624—41no—
50C17×#95—21no—
51C17×#95C2×C411yescharacteristic
52C17×#95C2×C411yescharacteristic
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×Dic34—21no—
69C2×Dic34—21no—
70C2×Dic34—41no—
71C2×Dic34—41no—
72C2×Dic34—41no—
73C2×Dic34—42no—
74#63247—42no—
75#63247—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C42×C17—24no—
79C42×C17—21no—
80C42×C17—21no—
81C4⋊Dic17—41no—
82C4⋊Dic17—42no—
83C4⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—41no—
86C4⋊Dic17—41no—
87C4⋊Dic17—41no—
88#63225—41no—
89#63225—41no—
90#63225—42no—
91C4×Dic17—21no—
92C4×Dic17—21no—
93C4×Dic17—21no—
94C4×Dic17—21no—
95C4×Dic17—41no—
96C4×Dic17—41no—
97#11831—171nomaximal, Sylow
98C23×C17C2×C22⋊C411yescharacteristic
99C2×C4×C17—22no—
100C2×C4×C17—22no—
101C2×C4×C17—22no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17#9911yescharacteristic
112C2×C4×C17#9911yescharacteristic
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—22no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17AGL(1,ℤ/8)11yescharacteristic
129C2×Dic17#13711yescharacteristic
130Dic34—42no—
131Dic34—42no—
132Dic34—42no—
133Dic34—42no—
134#3624—82no—
135#3624—82no—
136#2399—171no—
137#1801—171no—
138#1467—171no—
139#1354—172no—
140#1346—172no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17C2×#9911yescharacteristic
144C22×C17#41711yescharacteristic
145C22×C17#42111yescharacteristic
146C4×C17—21no—
147C4×C17—42no—
148C4×C17—42no—
149C4×C17—41no—
150C4×C17—41no—
151C4×C17#35612yes—
152Dic17—21no—
153Dic17—21no—
154Dic17—41no—
155Dic17—42no—
156Dic17—41no—
157Dic17—41no—
158Dic17—81no—
159C2×C4×Q8—171no—
160C2×C4×Q8—171no—
161#432—341no—
162#423—341no—
163C2×#101—341no—
164#398—171no—
165#398—171no—
166#398—341no—
167#395—171no—
168#395—171no—
169#395—341no—
170C4×C4⋊C4—341no—
171C4×C4⋊C4—341no—
172#376#59511yescharacteristic
173#343—172no—
174#324—172no—
175#324—341no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17#151311yescharacteristic
179C2×C17#212011yescharacteristic
180C2×C17C2×#35611yescharacteristic
181C22×Q8—171no—
182C22×Q8—171no—
183C2×M16—342no—
184C4×Q8—681no—
185C4×Q8—681no—
186C4×Q8—681no—
187C4×Q8—681no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4—341no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—341no—
197C2×C42—21no—
198C2×C42—171no—
199C2×C42—171no—
200C2×C42—341no—
201C2×C42C2×#59511yescharacteristic
202C4⋊C8—681no—
203C4⋊C8—681no—
204C4⋊C8—681no—
205#101—682no—
206C22⋊C8—342no—
207#97—681no—
208#95—21no—
209#95—341no—
210#95—341no—
211#95—341no—
212#95—341no—
213#95C2×#59511yescharacteristic
214#95C2×#59511yescharacteristic
215C17#1183111yescharacteristic, complemented, Sylow
216C2×Q8—341no—
217C2×Q8—341no—
218C2×Q8—681no—
219C2×Q8—681no—
220C2×Q8—681no—
221C2×Q8—682no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—171no—
225C22×C4—171no—
226C22×C4—171no—
227C22×C4—171no—
228C22×C4—341no—
229C22×C4—341no—
230C22×C4C22⋊#59511yescharacteristic
231C22×C4C22⋊#59511yescharacteristic
232C22×C4C22×#59511yescharacteristic
233M16—682no—
234M16—682no—
235C2×C8—682no—
236C2×C8—682no—
237C4⋊C4—681no—
238C4⋊C4—681no—
239C4⋊C4—681no—
240C4⋊C4—681no—
241C4⋊C4—682no—
242C4⋊C4—681no—
243C4⋊C4—681no—
244C4⋊C4—682no—
245C4⋊C4—682no—
246C4⋊C4—682no—
247C42—24no—
248C42—21no—
249C42—21no—
250C42—341no—
251C42—341no—
252C42—341no—
253C42—341no—
254C42—681no—
255C42—681no—
256C23C2×C22⋊#59511yescharacteristic
257Q8—682no—
258Q8—682no—
259Q8—682no—
260Q8—682no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—171no—
274C2×C4—171no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—342no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4—681no—
290C2×C4#1058753911yescharacteristic
291C2×C4#1058753911yescharacteristic
292C8—1362no—
293C8—1362no—
294C22—21no—
295C22—21no—
296C22C2×#1058753911yescharacteristic
297C22#1176380011yescharacteristic
298C22#1176380611yescharacteristic
299C4—21no—
300C4—42no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—341no—
305C4—341no—
306C4—681no—
307C4—681no—
308C4—682no—
309C4—681no—
310C4—1361no—
311C4#1176372012yes—
312C2—21no—
313C2—21no—
314C2#1454420111yescharacteristic
315C2#1454503711yescharacteristic
316C2C2×#1176372011yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#376C4
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×#63225C23
hypercenter#376#595
hypocenterC17#11831
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
215. C17#11831semidirect

Maximal subgroups

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

subgroupindex
#145446502
#145440462
#145437842
#145437692
#145320772
#1183117