Finite Group Databasebeta

← #17950238

Subgroups — #17950238

Order 4352.

494 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544675C211yesmaximal, characteristic
2#14544638C211yesmaximal, characteristic
3#14543603C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5#14537134C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7C2×C22⋊#3624—21no—
8C2×C22⋊#3624—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587574C411yescharacteristic
13C2×#10587564—21no—
14C22×C4×Dic17C2211yescharacteristic
15#11762713—21no—
16#11762708C2211yescharacteristic
17#11762706—21no—
18#11762705C2211yescharacteristic
19#11762678C411yescharacteristic
20#11762669C411yescharacteristic
21#11762667—21no—
22C17×#380C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17D411yescharacteristic
25C23×Dic17D411yescharacteristic
26C22×Dic34—21no—
27C22×#3624—22no—
28C22×#3624—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊Dic17—21no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17C2×C411yescharacteristic
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×#63225—21no—
37C2×#63225—21no—
38C2×#63225—21no—
39C2×#63225—21no—
40C2×C4×Dic17—42no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45#10587574—42no—
46#10587564—42no—
47#10587564—42no—
48C22⋊#3624—42no—
49C22⋊#3624—42no—
50C4⋊#3624—42no—
51C17×#95—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C24×C17C22⋊C411yescharacteristic
56C22×C4×C17—22no—
57C22×C4×C17—22no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—41no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—41no—
68C22×Dic17—41no—
69C22×Dic17—41no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17D812yes—
73C22×Dic17SD1612yes—
74C22×Dic17C2×D411yescharacteristic
75C22×Dic17C2×D411yescharacteristic
76C2×Dic34—42no—
77C2×Dic34—42no—
78C2×Dic34—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84C2×#3624—42no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—41no—
87C22⋊C4×C17—42no—
88C22⋊Dic17—42no—
89C4⋊Dic17—22no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93#63225—42no—
94#63225—42no—
95#63225—42no—
96#63225—41no—
97#63225—41no—
98C4×Dic17—22no—
99C4×Dic17—22no—
100C4×Dic17—41no—
101C4×Dic17—41no—
102#7728—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17#10212yes—
110C23×C17C2×C22⋊C411yescharacteristic
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—44no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17#10112yes—
123C2×C4×C17#13112yes—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—44no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17#9912yes—
142C2×Dic17#10212yes—
143C2×Dic17C2×D811yescharacteristic
144C2×Dic17C2×SD1611yescharacteristic
145C2×Dic17#13712yes—
146Dic34—84no—
147#3624—82no—
148#3624—82no—
149#2857—171no—
150#1817—171no—
151#1797—171no—
152#1271—172no—
153#1271—172no—
154C22×C17—22no—
155C22×C17—22no—
156C22×C17—22no—
157C22×C17—41no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17#40811yescharacteristic
162C22×C17C2×#9911yescharacteristic
163C22×C17C2×#10111yescharacteristic
164C22×C17C2×#10211yescharacteristic
165C22×C17#41911yescharacteristic
166C22×C17#42111yescharacteristic
167C22×C17#42111yescharacteristic
168C4×C17—22no—
169C4×C17—22no—
170C4×C17—81no—
171C4×C17—81no—
172Dic17—22no—
173Dic17—22no—
174Dic17—21no—
175Dic17—22no—
176Dic17—21no—
177Dic17—81no—
178Dic17—81no—
179C2×C4⋊Q8—171no—
180C2×C22⋊Q8—341no—
181C22×C42—171no—
182C2×C4⋊C8—341no—
183C2×C22⋊C8—341no—
184C2×C22⋊C8—341no—
185#391—341no—
186#389—171no—
187#389—171no—
188#389—341no—
189#384—171no—
190#380—171no—
191#380—341no—
192#380#59511yescharacteristic
193#336—172no—
194#336—172no—
195#336—341no—
196C2×C17—42no—
197C2×C17#147211yescharacteristic
198C2×C17#148011yescharacteristic
199C2×C17#149711yescharacteristic
200C2×C17#151111yescharacteristic
201C2×C17#151911yescharacteristic
202C2×C17#153711yescharacteristic
203C2×C17#154611yescharacteristic
204C22×Q8—341no—
205C23×C4—171no—
206C23×C4—171no—
207C23×C4C2×#59511yescharacteristic
208C22×C8—342no—
209C22×C8—342no—
210C4⋊Q8—682no—
211C22⋊Q8—682no—
212C22⋊Q8—682no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4C2×#59511yescharacteristic
221C2×C4⋊C4C2×#59511yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—341no—
224C2×C42—171no—
225C2×C42—171no—
226C2×C42—171no—
227C2×C42—171no—
228C2×C42—682no—
229C4⋊C8—682no—
230C22⋊C8—682no—
231C22⋊C8—682no—
232#95—21no—
233#95—341no—
234#95—341no—
235#95—341no—
236C17#772811yescharacteristic, complemented, Sylow
237C24C22⋊#59511yescharacteristic
238C2×Q8—682no—
239C2×Q8—682no—
240C2×Q8—682no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—171no—
247C22×C4—171no—
248C22×C4—171no—
249C22×C4—171no—
250C22×C4—172no—
251C22×C4—172no—
252C22×C4—342no—
253C22×C4—342no—
254C22×C4—341no—
255C22×C4—341no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4—681no—
259C22×C4C22⋊#59511yescharacteristic
260C22×C4C22×#59511yescharacteristic
261C2×C8—682no—
262C2×C8—682no—
263C2×C8—682no—
264C2×C8—682no—
265C2×C8—682no—
266C2×C8—682no—
267C4⋊C4—22no—
268C4⋊C4—41no—
269C4⋊C4—342no—
270C4⋊C4—681no—
271C4⋊C4—682no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—682no—
277C4⋊C4—682no—
278C22⋊C4—42no—
279C22⋊C4—682no—
280C42—342no—
281C42—342no—
282C42—681no—
283C42—681no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23#1058754712yes—
291C23C2×C22⋊#59511yescharacteristic
292Q8—1364no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—172no—
305C2×C4—171no—
306C2×C4—171no—
307C2×C4—172no—
308C2×C4—172no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—682no—
318C2×C4—682no—
319C2×C4—684no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4#1058754512yes—
327C2×C4#1058767812yes—
328C8—1362no—
329C8—1362no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22#1176378211yescharacteristic
338C22D17⋊#9911yescharacteristic
339C22C2×#1058754511yescharacteristic
340C22C2×#1058754711yescharacteristic
341C22#1176380211yescharacteristic
342C22#1176380611yescharacteristic
343C22#1176380611yescharacteristic
344C4—22no—
345C4—22no—
346C4—81no—
347C4—81no—
348C4—342no—
349C4—342no—
350C4—341no—
351C4—342no—
352C4—341no—
353C4—1361no—
354C4—1361no—
355C2—42no—
356C2#1454406311yescharacteristic
357C2#1454408911yescharacteristic
358C2#1454414611yescharacteristic
359C2#1454419311yescharacteristic
360C2#1454421911yescharacteristic
361C2#1454427511yescharacteristic
362C2#1454429311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#380C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#380#595
hypocenterC17#7728
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #14543603C2semidirect
4. #14543602C2semidirect
236. C17#7728semidirect

Maximal subgroups

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

subgroupindex
#145446752
#145446382
#145436032
#145436022
#145371342
#772817