Finite Group Databasebeta

← #17958637

Subgroups — #17958637

Order 4352.

394 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 72 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#14544047C211yesmaximal, characteristic
2#14544027C211yesmaximal, characteristic
3#14543715C212yesmaximal
4#14543714C212yesmaximal
5#14533770C211yesmaximal, characteristic
6#11763829—21no—
7#11763825—21no—
8#11763654—21no—
9#11763654—21no—
10#11763654C2212yes—
11#11763654C2212yes—
12C2×#10587630C411yescharacteristic
13C2×#10587630C411yescharacteristic
14C2×Q8×Dic17C2211yescharacteristic
15#11762742—21no—
16#11762742—21no—
17#11762716C2211yescharacteristic
18#11762715C2211yescharacteristic
19#11762704—21no—
20#11762704—21no—
21C4×#63225C411yescharacteristic
22C2×C17×C4⋊Q8C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×Q8×C17C2×C411yescharacteristic
25C22×Dic34C2×C411yescharacteristic
26C17×C4⋊Q8—42no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—21no—
29C2×C42×C17C2×C411yescharacteristic
30Q8×Dic17—41no—
31Q8×Dic17—41no—
32#10587630—42no—
33#10587630—41no—
34#10587630—41no—
35C2×C4⋊Dic17D411yescharacteristic
36C2×C4⋊Dic17D411yescharacteristic
37C2×C4⋊Dic17C2311yescharacteristic
38C2×#63225—21no—
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17C2×C411yescharacteristic
44C2×C4×Dic17C2×C411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C22⋊#3624—22no—
48C22⋊#3624—22no—
49#10587544—41no—
50#10587542—41no—
51C4⋊#3624—41no—
52C4×#3624—41no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58C2×Q8×C17—22no—
59C2×Q8×C17—22no—
60C2×Q8×C17—22no—
61C2×Q8×C17—41no—
62C2×Q8×C17—41no—
63C2×Q8×C17—41no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×Dic34—22no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×#3624—42no—
79C2×#3624—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—42no—
82C42×C17—41no—
83C4⋊Dic17—41no—
84C4⋊Dic17—41no—
85C4⋊Dic17—41no—
86#63225—24no—
87#63225—42no—
88#63225—42no—
89#63225—42no—
90#63225—41no—
91#63225—41no—
92C4×Dic17—21no—
93C4×Dic17—21no—
94C4×Dic17—21no—
95C4×Dic17—21no—
96C4×Dic17—21no—
97C4×Dic17—21no—
98C4×Dic17—41no—
99#11568—171nomaximal, Sylow
100C23×C17C2×C22⋊C411yescharacteristic
101Q8×C17—44no—
102Q8×C17—82no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17#10012yes—
117C2×C4×C17#13112yes—
118C2×Dic17—22no—
119C2×Dic17—22no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
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#9911yescharacteristic
132C2×Dic17#9911yescharacteristic
133Dic34—82no—
134#3624—82no—
135#3624—82no—
136#2546—171no—
137#1467—171no—
138#1461—171no—
139#1319—172no—
140#1319—172no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17#40811yescharacteristic
144C22×C17C2×#9911yescharacteristic
145C22×C17C2×#10011yescharacteristic
146C4×C17—22no—
147C4×C17—22no—
148C4×C17—42no—
149C4×C17—41no—
150C4×C17—81no—
151Dic17—21no—
152Dic17—42no—
153Dic17—41no—
154Dic17—41no—
155Dic17—41no—
156Dic17—81no—
157Dic17#35612yes—
158C2×C4⋊Q8—171no—
159C2×C4⋊Q8—171no—
160C2×C4⋊Q8#59511yescharacteristic
161C2×C4×Q8—171no—
162#432—341no—
163#431—341no—
164#398—341no—
165#398—341no—
166#391—171no—
167#391—171no—
168#389—341no—
169#389—341no—
170C4×C4⋊C4—171no—
171#324—172no—
172#324—172no—
173#324—341no—
174#324—341no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17#147111yescharacteristic
178C2×C17#212911yescharacteristic
179C2×C17C2×#35611yescharacteristic
180C22×Q8—171no—
181C22×Q8C2×#59511yescharacteristic
182C22×Q8C2×#59511yescharacteristic
183C4⋊Q8—42no—
184C4⋊Q8—681no—
185C4⋊Q8—681no—
186C4⋊Q8—682no—
187C4×Q8—681no—
188C4×Q8—681no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—171no—
194C2×C4⋊C4—341no—
195C2×C4⋊C4—341no—
196C2×C4⋊C4—341no—
197C2×C4⋊C4—341no—
198C2×C4⋊C4—341no—
199C2×C42—171no—
200C2×C42—171no—
201C2×C42—171no—
202C2×C42—171no—
203C2×C42C2×#59511yescharacteristic
204C4⋊C8—681no—
205C4⋊C8—681no—
206C22⋊C8—342no—
207C22⋊C8—342no—
208#97—681no—
209C4×C8—681no—
210#95—341no—
211#95—341no—
212#95—341no—
213#95—341no—
214#95—341no—
215C17#1156811yescharacteristic, complemented, Sylow
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—342no—
223C2×Q8—681no—
224C2×Q8—681no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—171no—
228C22×C4—171no—
229C22×C4—171no—
230C22×C4—171no—
231C22×C4—341no—
232C22×C4—341no—
233C22×C4C22⋊#59511yescharacteristic
234C22×C4C22⋊#59511yescharacteristic
235C22×C4C22×#59511yescharacteristic
236C2×C8—682no—
237C2×C8—682no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—344no—
241C4⋊C4—681no—
242C4⋊C4—681no—
243C4⋊C4—681no—
244C4⋊C4—682no—
245C4⋊C4—681no—
246C4⋊C4—681no—
247C4⋊C4—682no—
248C4⋊C4—682no—
249C42—41no—
250C42—341no—
251C42—341no—
252C42—341no—
253C42—341no—
254C42—341no—
255C42—341no—
256C42—681no—
257C23C2×C22⋊#59511yescharacteristic
258Q8—44no—
259Q8—82no—
260Q8—1362no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—171no—
275C2×C4—171no—
276C2×C4—342no—
277C2×C4—342no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—681no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—681no—
287C2×C4—681no—
288C2×C4—681no—
289C2×C4#1058755312yes—
290C2×C4#1058767812yes—
291C8—1362no—
292C8—1362no—
293C22—21no—
294C22—21no—
295C22#1176378211yescharacteristic
296C22D17⋊#9911yescharacteristic
297C22C2×#1058755311yescharacteristic
298C4—22no—
299C4—22no—
300C4—42no—
301C4—41no—
302C4—81no—
303C4—172no—
304C4—341no—
305C4—682no—
306C4—681no—
307C4—681no—
308C4—681no—
309C4—1361no—
310C2—21no—
311C2—21no—
312C2#1454406011yescharacteristic
313C2#1454506711yescharacteristic
314C2D17⋊#35611yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#11568
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
215. C17#11568semidirect

Maximal subgroups

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

subgroupindex
#145440472
#145440272
#145437152
#145437142
#145337702
#1156817