Finite Group Databasebeta

← #17950195

Subgroups — #17950195

Order 4352.

464 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 96 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#14544641C211yesmaximal, characteristic
2#14544639C211yesmaximal, characteristic
3#14543605C212yesmaximal, complemented
4#14543603C212yesmaximal, complemented
5#14537122C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×C4⋊#3624—21no—
8C2×#10587543—21no—
9C2×#10587543—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587583C411yescharacteristic
13C2×C42×D17C2211yescharacteristic
14#11762715C411yescharacteristic
15#11762714C2211yescharacteristic
16#11762705C2211yescharacteristic
17#11762696—21no—
18#11762695—21no—
19#11762680—21no—
20#11762671—21no—
21#11762666C411yescharacteristic
22C17×#389C411yescharacteristic
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17C2×C411yescharacteristic
32C2×C42×C17C2×C411yescharacteristic
33C2×#63227—21no—
34C2×#63227—21no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—21no—
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17C2311yescharacteristic
41#10587583—42no—
42C42×D17—42no—
43#10587544—42no—
44#10587543—42no—
45#10587543—42no—
46C4⋊#3624—42no—
47C17×#95—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54C23×D17C22⋊C411yescharacteristic
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17C22⋊C411yescharacteristic
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22×C411yescharacteristic
60C22×Dic17—21no—
61C22×Dic17—21no—
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C2×D411yescharacteristic
64C22×Dic17C2×D411yescharacteristic
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17—21no—
68C2×C4×D17—21no—
69C2×C4×D17—41no—
70C2×C4×D17—41no—
71C2×C4×D17—41no—
72C2×C4×D17—41no—
73C2×C4×D17—41no—
74C2×C4×D17SD1612yes—
75C2×C4×D17Q1612yes—
76C2×Dic34—22no—
77C2×Dic34—41no—
78C2×Dic34—41no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84C2×#3624—42no—
85C4⋊C4×C17—41no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—42no—
88C42×C17—41no—
89#63227—42no—
90#63227—42no—
91C4⋊Dic17—22no—
92C4⋊Dic17—42no—
93#63225—41no—
94#63225—41no—
95C4×Dic17—22no—
96C4×Dic17—22no—
97C4×Dic17—41no—
98#7715—171nomaximal, Sylow
99C23×C17C2×C22⋊C411yescharacteristic
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17#10312yes—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17#10312yes—
121C2×C4×C17#13112yes—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17#9912yes—
135C2×Dic17C2×SD1611yescharacteristic
136C2×Dic17C2×Q1611yescharacteristic
137C2×Dic17AGL(1,ℤ/8)11yescharacteristic
138C2×Dic17#13711yescharacteristic
139C4×D17—42no—
140C4×D17—42no—
141C4×D17—44no—
142C4×D17—42no—
143C4×D17—42no—
144Dic34—82no—
145#3624—82no—
146#3624—82no—
147#2855—171no—
148#1797—171no—
149#1797—171no—
150#1272—172no—
151#1271—172no—
152C22×C17#40811yescharacteristic
153C22×C17C2×#9911yescharacteristic
154C22×C17#41311yescharacteristic
155C22×C17C2×#10311yescharacteristic
156C22×C17#41711yescharacteristic
157C22×C17#42111yescharacteristic
158C22×C17#42111yescharacteristic
159C2×D17—22no—
160C2×D17—22no—
161C2×D17—22no—
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—41no—
169C4×C17—81no—
170C4×C17—81no—
171Dic17—22no—
172Dic17—21no—
173Dic17—21no—
174Dic17—21no—
175Dic17—21no—
176Dic17—81no—
177Dic17—81no—
178C2×C22⋊Q8—171no—
179C22×C42—171no—
180C2×C4⋊C8—341no—
181C2×C4⋊C8—341no—
182C2×C22⋊C8—341no—
183C2×C22⋊C8—341no—
184#391—171no—
185#391—171no—
186#389—171no—
187#389#59511yescharacteristic
188#387—341no—
189#387—341no—
190#385—341no—
191#381—341no—
192#380—171no—
193#336—172no—
194#336—172no—
195C2×C17#146811yescharacteristic
196C2×C17#148111yescharacteristic
197C2×C17#149511yescharacteristic
198C2×C17#151311yescharacteristic
199C2×C17#152011yescharacteristic
200C2×C17#153511yescharacteristic
201C2×C17#155011yescharacteristic
202D17—42no—
203C22×Q8—171no—
204C23×C4—171no—
205C23×C4—171no—
206C23×C4—171no—
207C22×C8—342no—
208C22×C8—342no—
209C22⋊Q8—682no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4C2×#59511yescharacteristic
215C2×C4⋊C4C2×#59511yescharacteristic
216C2×C22⋊C4—341no—
217C2×C22⋊C4—341no—
218C2×C42—171no—
219C2×C42—171no—
220C2×C42—171no—
221C2×C42—682no—
222C2×C42C2×#59511yescharacteristic
223C4⋊C8—682no—
224C4⋊C8—682no—
225C22⋊C8—682no—
226C22⋊C8—682no—
227#95—21no—
228#95—341no—
229#95—341no—
230#95—341no—
231#95—341no—
232#95—341no—
233#95—341no—
234C17#771511yescharacteristic, complemented, Sylow
235C24—171no—
236C2×Q8—342no—
237C2×Q8—681no—
238C2×Q8—681no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—171no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—172no—
245C22×C4—172no—
246C22×C4—341no—
247C22×C4—342no—
248C22×C4—341no—
249C22×C4—342no—
250C22×C4—341no—
251C22×C4—341no—
252C22×C4—681no—
253C22×C4—681no—
254C22×C4—681no—
255C22×C4—681no—
256C22×C4—681no—
257C22×C4C22⋊#59511yescharacteristic
258C22×C4C22⋊#59511yescharacteristic
259C22×C4C22×#59511yescharacteristic
260C2×C8—682no—
261C2×C8—682no—
262C2×C8—682no—
263C2×C8—682no—
264C2×C8—682no—
265C2×C8—682no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C4⋊C4—342no—
270C4⋊C4—682no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C22⋊C4—682no—
274C22⋊C4—682no—
275C42—41no—
276C42—342no—
277C42—342no—
278C42—681no—
279C23—172no—
280C23—341no—
281C23—341no—
282C23—341no—
283C23—341no—
284C23—341no—
285C23—341no—
286C23C2×C22⋊#59511yescharacteristic
287Q8—1362no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—172no—
302C2×C4—171no—
303C2×C4—171no—
304C2×C4—171no—
305C2×C4—171no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—682no—
313C2×C4—682no—
314C2×C4—684no—
315C2×C4—682no—
316C2×C4—682no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4#1058754912yes—
324C2×C4#1058768012yes—
325C8—1362no—
326C8—1362no—
327C22—342no—
328C22—342no—
329C22—342no—
330C22—681no—
331C22—681no—
332C22—681no—
333C22—681no—
334C22#1176378411yescharacteristic
335C22D17⋊#9911yescharacteristic
336C22#1176379511yescharacteristic
337C22C2×#1058754911yescharacteristic
338C22#1176380011yescharacteristic
339C22#1176380611yescharacteristic
340C22#1176380711yescharacteristic
341C4—22no—
342C4—22no—
343C4—41no—
344C4—81no—
345C4—81no—
346C4—341no—
347C4—341no—
348C4—341no—
349C4—341no—
350C4—342no—
351C4—1361no—
352C4—1361no—
353C2—682no—
354C2#1454405211yescharacteristic
355C2#1454409411yescharacteristic
356C2#1454414211yescharacteristic
357C2#1454420011yescharacteristic
358C2#1454422211yescharacteristic
359C2#1454426811yescharacteristic
360C2#1454430111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7715
LCSG ▷ C22×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. #14543605C2semidirect
4. #14543603C2semidirect
234. C17#7715semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145446392
#145436052
#145436032
#145371222
#771517