Finite Group Databasebeta

← #17949650

Subgroups — #17949650

Order 4352.

462 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C4×#10587544C211yesmaximal, characteristic
2#14544499C211yesmaximal, characteristic
3#14543604C212yesmaximal
4#14543603C212yesmaximal
5#14531259C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×C4×#3624—21no—
8C2×C4×#3624—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×Q8×Dic17C411yescharacteristic
13#11762716C2211yescharacteristic
14#11762707C411yescharacteristic
15#11762705C2211yescharacteristic
16#11762694—21no—
17#11762692—21no—
18C4⋊C4×Dic17C411yescharacteristic
19#11762660—21no—
20#11762654—21no—
21C42×Dic17C2211yescharacteristic
22C17×#391C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×#3624—22no—
25C22×#3624—22no—
26C2×C4⋊C4×C17C2×C411yescharacteristic
27C2×C42×C17C2×C411yescharacteristic
28Q8×Dic17—22no—
29Q8×Dic17C814yescomplemented
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×#63225—21no—
34C2×#63225—21no—
35C2×C4×Dic17—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17C2311yescharacteristic
43#10587544—24no—
44C4×#3624—24no—
45C4×#3624—42no—
46C17×#95—21no—
47C17×#95—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C2×Q8×C17—21no—
54C2×Q8×C17—41no—
55C2×Q8×C17—41no—
56C2×Q8×C17C2×C812yes—
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×#3624—24no—
69C2×#3624—24no—
70C2×#3624—42no—
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C4⋊C4×C17—41no—
75C42×C17—41no—
76C4⋊Dic17—21no—
77C4⋊Dic17—24no—
78C4⋊Dic17—41no—
79C4⋊Dic17C2×C812yes—
80#63225—42no—
81#63225—42no—
82C4×Dic17—22no—
83C4×Dic17—24no—
84C4×Dic17—22no—
85C4×Dic17—22no—
86C4×Dic17—41no—
87C4×Dic17—42no—
88C4×Dic17C2×C812yes—
89C4×Dic17M1612yes—
90C4×Dic17SD1612yes—
91C4×Dic17Q1612yes—
92#7562—171nomaximal, Sylow
93C23×C17C2×C22⋊C411yescharacteristic
94Q8×C17—42no—
95Q8×C17—42no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—22no—
103C2×C4×C17—41no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17#10312yes—
109C2×C4×C17#10412yes—
110C2×C4×C17C22×C811yescharacteristic
111C2×C4×C17C2×M1611yescharacteristic
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—22no—
119C2×Dic17—41no—
120C2×Dic17—41no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17C22⋊C812yes—
125C2×Dic17C22⋊C812yes—
126C2×Dic17#10312yes—
127C2×Dic17#10412yes—
128C2×Dic17C2×SD1611yescharacteristic
129C2×Dic17C2×Q1611yescharacteristic
130C2×Dic17#13512yes—
131#3624—44no—
132#3624—44no—
133#2295—171no—
134C4×C4⋊C8—171no—
135#1744—171no—
136#1272—172no—
137#1271—172no—
138C22×C17C2×C22⋊C811yescharacteristic
139C22×C17#41011yescharacteristic
140C22×C17#41311yescharacteristic
141C22×C17C2×#10311yescharacteristic
142C22×C17#41611yescharacteristic
143C22×C17C2×#10411yescharacteristic
144C22×C17C2×#10411yescharacteristic
145C4×C17—22no—
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—22no—
149C4×C17—42no—
150C4×C17—81no—
151Dic17—22no—
152Dic17—22no—
153Dic17—21no—
154Dic17—21no—
155Dic17—42no—
156Dic17—81no—
157Dic17Q8⋊C814yes—
158C2×C4×Q8—171no—
159C2×C4⋊C8—341no—
160C2×C4×C8—341no—
161C2×C4×C8—341no—
162#391—171no—
163#391#59511yescharacteristic
164#389—171no—
165#389—171no—
166#387—341no—
167#387—341no—
168C4×C4⋊C4—171no—
169C4×C4⋊C4—341no—
170#376—341no—
171C43—171no—
172#336—172no—
173#336—172no—
174#336—341no—
175C2×C17#145811yescharacteristic
176C2×C17C2×Q8⋊C811yescharacteristic
177C2×C17#147611yescharacteristic
178C2×C17#150011yescharacteristic
179C2×C17#150311yescharacteristic
180C2×C17#152411yescharacteristic
181C2×C17#154211yescharacteristic
182C22×Q8C2×#59511yescharacteristic
183C22×C8—342no—
184C22×C8—342no—
185C4×Q8—174no—
186C4×Q8—342no—
187C2×C4⋊C4—171no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—341no—
190C2×C4⋊C4—341no—
191C2×C4⋊C4—341no—
192C2×C4⋊C4C2×#59511yescharacteristic
193C2×C42—171no—
194C2×C42—171no—
195C2×C42—171no—
196C2×C42—171no—
197C2×C42—171no—
198C2×C42—171no—
199C2×C42—341no—
200C2×C42—341no—
201C2×C42C2×#59511yescharacteristic
202C4⋊C8—344no—
203C4×C8—344no—
204C4×C8—682no—
205#95—21no—
206#95—21no—
207#95—341no—
208#95—341no—
209#95—341no—
210#95—341no—
211#95—341no—
212C17#756211yescharacteristic, complemented, Sylow
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8#6324212yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—171no—
220C22×C4—171no—
221C22×C4—171no—
222C22×C4—171no—
223C22×C4—341no—
224C22×C4—341no—
225C22×C4C22⋊#59511yescharacteristic
226C22×C4C22⋊#59511yescharacteristic
227C22×C4C22×#59511yescharacteristic
228C2×C8—344no—
229C2×C8—344no—
230C2×C8—682no—
231C2×C8—682no—
232C2×C8—682no—
233C2×C8—682no—
234C4⋊C4—41no—
235C4⋊C4—172no—
236C4⋊C4—344no—
237C4⋊C4—341no—
238C4⋊C4—681no—
239C4⋊C4—682no—
240C4⋊C4—682no—
241C42—41no—
242C42—172no—
243C42—172no—
244C42—172no—
245C42—172no—
246C42—342no—
247C42—342no—
248C42—344no—
249C42—342no—
250C42—681no—
251C42—682no—
252C23C2×C22⋊#59511yescharacteristic
253Q8—42no—
254Q8—42no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—172no—
268C2×C4—172no—
269C2×C4—172no—
270C2×C4—172no—
271C2×C4—171no—
272C2×C4—171no—
273C2×C4—172no—
274C2×C4—341no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—342no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4#1058754912yes—
287C2×C4#1058755012yes—
288C2×C4C2×#6324211yescharacteristic
289C2×C4D17⋊M1611yescharacteristic
290C8—684no—
291C8—684no—
292C22#1176377711yescharacteristic
293C22#1176378911yescharacteristic
294C22#1176379611yescharacteristic
295C22C2×#1058754911yescharacteristic
296C22#1176379911yescharacteristic
297C22D17⋊#10411yescharacteristic
298C22C2×#1058755011yescharacteristic
299C4—22no—
300C4—21no—
301C4—21no—
302C4—22no—
303C4—42no—
304C4—81no—
305C4—174no—
306C4—342no—
307C4—341no—
308C4—341no—
309C4—342no—
310C4—682no—
311C4—1361no—
312C2#1454401611yescharacteristic
313C2#1454407311yescharacteristic
314C2#1454407811yescharacteristic
315C2#1454415611yescharacteristic
316C2#1454416811yescharacteristic
317C2#1454423711yescharacteristic
318C2#1454428611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#391C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#391#595
hypocenterC17#7562
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
29. Q8×Dic17C8semidirect
212. C17#7562semidirect

Maximal subgroups

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

subgroupindex
C4×#105875442
#145444992
#145436042
#145436032
#145312592
#756217