Finite Group Databasebeta

← #17949979

Subgroups — #17949979

Order 4352.

518 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 102 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#14543605C212yesmaximal, complemented
4#14543603C212yesmaximal, complemented
5#14537122C211yesmaximal, characteristic
6C22×#63242—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675—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×#63242—24no—
30C2×#63242—42no—
31C2×#63242—42no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C42×C17C2×C411yescharacteristic
36C2×#63227—21no—
37C2×#63227—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17C2311yescharacteristic
44#10587583—42no—
45C42×D17—42no—
46#10587543—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—22no—
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—41no—
72C2×C4×D17—41no—
73C2×C4×D17SD1612yes—
74C2×C4×D17Q1612yes—
75C2×Dic34—22no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×#3624—24no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84#63242—48no—
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—42no—
94C4×Dic17—41no—
95C4×Dic17—41no—
96C4×Dic17—41no—
97#7655—171nomaximal, Sylow
98C23×C17C2×C22⋊C411yescharacteristic
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17#9912yes—
104C22×D17#10312yes—
105C22×D17#10412yes—
106C2×C4×C17—21no—
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—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17#10312yes—
120C2×C4×C17#13112yes—
121C2×Dic17—21no—
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×Dic17C2×SD1611yescharacteristic
135C2×Dic17C2×Q1611yescharacteristic
136C2×Dic17#13512yes—
137C4×D17—24no—
138C4×D17—24no—
139C4×D17—44no—
140C4×D17—44no—
141Dic34—82no—
142#3624—44no—
143#3624—82no—
144#2855—171no—
145#1817—171no—
146C2×#336—171no—
147#1272—172no—
148#1271—172no—
149C22×C17#40811yescharacteristic
150C22×C17C2×#9911yescharacteristic
151C22×C17#41311yescharacteristic
152C22×C17C2×#10311yescharacteristic
153C22×C17#41611yescharacteristic
154C22×C17C2×#10411yescharacteristic
155C22×C17#42111yescharacteristic
156C2×D17—22no—
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—22no—
161C2×D17—22no—
162C2×D17#32814yes—
163C4×C17—22no—
164C4×C17—22no—
165C4×C17—41no—
166C4×C17—81no—
167C4×C17—81no—
168Dic17—21no—
169Dic17—21no—
170Dic17—22no—
171Dic17—41no—
172Dic17—81no—
173Dic17—81no—
174C23×C8—341no—
175C2×C22⋊Q8—171no—
176C22×C42—171no—
177C2×C22⋊C8—341no—
178#391—171no—
179#391—171no—
180#389—171no—
181#389#59511yescharacteristic
182#387—341no—
183#387—341no—
184#385—341no—
185#381—341no—
186#380—171no—
187#336—172no—
188#336—172no—
189#336—341no—
190#336—341no—
191C2×C17#146811yescharacteristic
192C2×C17#148111yescharacteristic
193C2×C17#149411yescharacteristic
194C2×C17C2×#32811yescharacteristic
195C2×C17#151611yescharacteristic
196C2×C17#153211yescharacteristic
197C2×C17#154211yescharacteristic
198D17—24no—
199C22×Q8—171no—
200C23×C4—171no—
201C23×C4—171no—
202C23×C4—171no—
203C22×C8—342no—
204C22×C8—344no—
205C22×C8—342no—
206C22×C8—682no—
207C22×C8—682no—
208C22⋊Q8—682no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—171no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4C2×#59511yescharacteristic
214C2×C4⋊C4C2×#59511yescharacteristic
215C2×C22⋊C4—341no—
216C2×C22⋊C4—341no—
217C2×C42—171no—
218C2×C42—171no—
219C2×C42—171no—
220C2×C42—682no—
221C2×C42C2×#59511yescharacteristic
222C22⋊C8—682no—
223#95—21no—
224#95—341no—
225#95—341no—
226#95—341no—
227#95—341no—
228#95—341no—
229#95—341no—
230C17#765511yescharacteristic, complemented, Sylow
231C24—171no—
232C2×Q8—342no—
233C2×Q8—681no—
234C2×Q8—681no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—172no—
240C22×C4—172no—
241C22×C4—171no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—342no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—681no—
251C22×C4—681no—
252C22×C4C22⋊#59511yescharacteristic
253C22×C4C22⋊#59511yescharacteristic
254C22×C4C22×#59511yescharacteristic
255C2×C8—344no—
256C2×C8—682no—
257C2×C8—688no—
258C2×C8—682no—
259C2×C8—682no—
260C2×C8—682no—
261C2×C8—682no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—342no—
266C4⋊C4—682no—
267C4⋊C4—682no—
268C22⋊C4—682no—
269C22⋊C4—682no—
270C42—41no—
271C42—681no—
272C42—681no—
273C42—681no—
274C23—172no—
275C23—172no—
276C23—172no—
277C23—341no—
278C23—341no—
279C23—341no—
280C23—341no—
281C23C2×C22⋊#59511yescharacteristic
282Q8—1362no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—171no—
297C2×C4—171no—
298C2×C4—172no—
299C2×C4—344no—
300C2×C4—344no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—684no—
309C2×C4—684no—
310C2×C4—681no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4#1058754912yes—
317C2×C4#1058768012yes—
318C8—684no—
319C8—1362no—
320C22—174no—
321C22—342no—
322C22—342no—
323C22—342no—
324C22—342no—
325C22—342no—
326C22—342no—
327C22#1176378411yescharacteristic
328C22D17⋊#9911yescharacteristic
329C22#1176379511yescharacteristic
330C22C2×#1058754911yescharacteristic
331C22#1176379911yescharacteristic
332C22D17⋊#10411yescharacteristic
333C22#1176380711yescharacteristic
334C4—22no—
335C4—22no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—341no—
340C4—341no—
341C4—342no—
342C4—681no—
343C4—1361no—
344C4—1361no—
345C2—344no—
346C2#1454405211yescharacteristic
347C2#1454409411yescharacteristic
348C2#1454413811yescharacteristic
349C2D17⋊#32811yescharacteristic
350C2#1454420911yescharacteristic
351C2#1454425911yescharacteristic
352C2#1454428511yescharacteristic
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#7655
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
230. C17#7655semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145444452
#145436052
#145436032
#145371222
#765517