Finite Group Databasebeta

← #17949709

Subgroups — #17949709

Order 4352.

568 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 114 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#14544676C211yesmaximal, characteristic
2#14544445C211yesmaximal, characteristic
3#14543603C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5#14531265C211yesmaximal, characteristic
6C22×#63242—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587578C411yescharacteristic
13C2×#10587578C411yescharacteristic
14C2×C42×D17C2211yescharacteristic
15#11762708C2211yescharacteristic
16#11762707C411yescharacteristic
17#11762705C2211yescharacteristic
18#11762683—21no—
19#11762680—21no—
20C4⋊C4×Dic17—21no—
21#11762660—21no—
22C17×#389C411yescharacteristic
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×#3624—22no—
27C22×#3624—22no—
28C2×#63242—24no—
29C2×#63242—42no—
30C2×#63242—42no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×#63227—21no—
36C2×#63227—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17C2311yescharacteristic
47#10587578—22no—
48#10587578—42no—
49#10587578C814yescomplemented
50C42×D17—42no—
51#10587543—42no—
52C17×#95—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C23×D17C22⋊C411yescharacteristic
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×Dic17C2×D411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C2×C4×D17—22no—
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—21no—
72C2×C4×D17—21no—
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—41no—
76C2×C4×D17C2×C812yes—
77C2×C4×D17M1612yes—
78C2×#3624—24no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84#63242—48no—
85C4⋊C4×C17—21no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17C2×C812yescomplemented
89C42×C17—41no—
90#63227—24no—
91#63227—42no—
92C4⋊Dic17—21no—
93C4⋊Dic17—42no—
94C4⋊Dic17—41no—
95C4⋊Dic17C2×C812yescomplemented
96#63225—42no—
97#63225—41no—
98#63225—41no—
99C4×Dic17—24no—
100C4×Dic17—41no—
101C4×Dic17—41no—
102C4×Dic17—41no—
103C4×Dic17—42no—
104#7579—171nomaximal, Sylow
105C23×C17C2×C22⋊C411yescharacteristic
106C22×D17—21no—
107C22×D17—21no—
108C22×D17—21no—
109C22×D17—21no—
110C22×D17C22⋊C812yes—
111C22×D17#9912yes—
112C22×D17#10112yes—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—22no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17#10412yes—
127C2×C4×C17C22×C811yescharacteristic
128C2×C4×C17C2×M1611yescharacteristic
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—22no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17C22⋊C812yes—
142C2×Dic17#13512yes—
143C2×Dic17AGL(1,ℤ/8)11yescharacteristic
144C2×Dic17#13711yescharacteristic
145C4×D17—24no—
146C4×D17—24no—
147C4×D17—24no—
148C4×D17—24no—
149C4×D17—44no—
150#3624—44no—
151#3624—82no—
152#2296—171no—
153#1817—171no—
154C2×#336—171no—
155#1271—172no—
156#1271—172no—
157C22×C17C2×C22⋊C811yescharacteristic
158C22×C17C2×#9911yescharacteristic
159C22×C17C2×#10111yescharacteristic
160C22×C17#41611yescharacteristic
161C22×C17#41711yescharacteristic
162C22×C17C2×#10411yescharacteristic
163C22×C17#42111yescharacteristic
164C2×D17—22no—
165C2×D17—22no—
166C2×D17—22no—
167C2×D17—22no—
168C2×D17—22no—
169C2×D17—22no—
170C2×D17#32414yes—
171C4×C17—21no—
172C4×C17—21no—
173C4×C17—22no—
174C4×C17—42no—
175C4×C17—41no—
176C4×C17—81no—
177Dic17—22no—
178Dic17—21no—
179Dic17—21no—
180Dic17—22no—
181Dic17—42no—
182Dic17—81no—
183C23×C8—341no—
184C2×#117—171no—
185C2×#117—171no—
186C22×C42—171no—
187C2×C22⋊C8—341no—
188#389—171no—
189#389—171no—
190#389—171no—
191#389#59511yescharacteristic
192#385—341no—
193#385—341no—
194C4×C4⋊C4—341no—
195C4×C4⋊C4—341no—
196#336—172no—
197#336—172no—
198#336—341no—
199#336—341no—
200C2×C17C2×#32411yescharacteristic
201C2×C17#147611yescharacteristic
202C2×C17#147711yescharacteristic
203C2×C17#150411yescharacteristic
204C2×C17#151611yescharacteristic
205C2×C17#153111yescharacteristic
206C2×C17#155211yescharacteristic
207D17—24no—
208C23×C4—171no—
209C23×C4—171no—
210C23×C4—171no—
211C22×C8—342no—
212C22×C8—344no—
213C22×C8—342no—
214C22×C8—682no—
215C22×C8—682no—
216#117—174no—
217#117—342no—
218#117—682no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4C2×#59511yescharacteristic
226C2×C4⋊C4C2×#59511yescharacteristic
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C42—171no—
230C2×C42—171no—
231C2×C42—171no—
232C2×C42—341no—
233C2×C42—341no—
234C2×C42—682no—
235C2×C42C2×#59511yescharacteristic
236C22⋊C8—682no—
237#95—21no—
238#95—341no—
239#95—341no—
240#95—341no—
241C17#757911yescharacteristic, complemented, Sylow
242C24—171no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—171no—
246C22×C4—171no—
247C22×C4—171no—
248C22×C4—172no—
249C22×C4—172no—
250C22×C4—341no—
251C22×C4—341no—
252C22×C4—342no—
253C22×C4—342no—
254C22×C4—342no—
255C22×C4—342no—
256C22×C4—342no—
257C22×C4—342no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—681no—
261C22×C4C22⋊#59511yescharacteristic
262C22×C4C22⋊#59511yescharacteristic
263C22×C4C22×#59511yescharacteristic
264C2×C8—344no—
265C2×C8—682no—
266C2×C8—688no—
267C2×C8—682no—
268C2×C8—682no—
269C2×C8—682no—
270C2×C8—682no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—42no—
274C4⋊C4—172no—
275C4⋊C4—341no—
276C4⋊C4—681no—
277C4⋊C4—682no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C4⋊C4—682no—
281C4⋊C4#6324212yescomplemented
282C22⋊C4—344no—
283C22⋊C4—682no—
284C42—41no—
285C42—344no—
286C42—681no—
287C42—681no—
288C42—681no—
289C42—682no—
290C23—172no—
291C23—172no—
292C23—172no—
293C23—341no—
294C23—341no—
295C23—341no—
296C23—341no—
297C23C2×C22⋊#59511yescharacteristic
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—172no—
312C2×C4—172no—
313C2×C4—171no—
314C2×C4—171no—
315C2×C4—344no—
316C2×C4—344no—
317C2×C4—344no—
318C2×C4—344no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—342no—
326C2×C4—684no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4#1058754812yes—
333C2×C4C2×#6324211yescharacteristic
334C2×C4D17⋊M1611yescharacteristic
335C8—684no—
336C8—1362no—
337C22—174no—
338C22—342no—
339C22—342no—
340C22—342no—
341C22—342no—
342C22—342no—
343C22—342no—
344C22#1176377711yescharacteristic
345C22D17⋊#9911yescharacteristic
346C22D17⋊#10111yescharacteristic
347C22#1176379911yescharacteristic
348C22#1176380011yescharacteristic
349C22C2×#1058754811yescharacteristic
350C22#1176380811yescharacteristic
351C4—22no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—42no—
356C4—81no—
357C4—341no—
358C4—341no—
359C4—342no—
360C4—342no—
361C4—682no—
362C4—1361no—
363C2—344no—
364C2#1454401311yescharacteristic
365C2#1454407611yescharacteristic
366C2#1454408211yescharacteristic
367C2#1454416911yescharacteristic
368C2#1454421111yescharacteristic
369C2#1454425711yescharacteristic
370C2#1454430611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7579
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #14543603C2semidirect
4. #14543602C2semidirect
49. #10587578C8semidirect
88. C4⋊C4×C17C2×C8semidirect
95. C4⋊Dic17C2×C8semidirect
241. C17#7579semidirect
281. C4⋊C4#63242semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145444452
#145436032
#145436022
#145312652
#757917