Finite Group Databasebeta

← #17958577

Subgroups — #17958577

Order 4352.

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

normal subgroups only

This group has 398 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#14544032C211yesmaximal, characteristic
2C2×#11763654C211yesmaximal, characteristic
3#14543724C212yesmaximal, complemented
4#14543715C212yesmaximal, complemented
5#14533269C211yesmaximal, characteristic
6#11763779—21no—
7C2×C22⋊#3624—21no—
8#11763654—21no—
9#11763654—21no—
10#11763654C2212yes—
11#11763654C2212yes—
12C2×#10587610C411yescharacteristic
13C22×C4⋊Dic17C2211yescharacteristic
14#11762749—21no—
15#11762749—21no—
16#11762715C411yescharacteristic
17#11762715C2211yescharacteristic
18#11762675C411yescharacteristic
19#11762675C2211yescharacteristic
20#11762670—21no—
21#11762670—21no—
22C4×C22⋊C4×C17C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17D411yescharacteristic
25C23×Dic17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×#63247—41no—
28C22×#3624—41no—
29C2×C22⋊C4×C17—21no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×C22⋊Dic17—21no—
33C2×C4⋊Dic17—41no—
34C2×C4⋊Dic17—41no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17D411yescharacteristic
38C2×C4⋊Dic17D411yescharacteristic
39C2×C4⋊Dic17C2311yescharacteristic
40#10587610—41no—
41#10587610—41no—
42C2×#63225—21no—
43C22⋊#3624—22no—
44C22⋊#3624—22no—
45C22⋊#3624—41no—
46C22⋊#3624—41no—
47C17×#95—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—22no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—41no—
63C22×C4×C17C22⋊C411yescharacteristic
64C22×C4×C17C22×C411yescharacteristic
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—42no—
72C22×Dic17—42no—
73C22×Dic17C22⋊C411yescharacteristic
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C2×D411yescharacteristic
76C22×Dic17C2×D411yescharacteristic
77C2×Dic34—22no—
78C2×Dic34—41no—
79C2×Dic34—41no—
80#63247—82no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—82no—
84C22⋊C4×C17—42no—
85C42×C17—22no—
86C42×C17—41no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17—41no—
93C4⋊Dic17—41no—
94#63225—42no—
95#11551—171nomaximal, Sylow
96C23×C17—22no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17#9911yescharacteristic
101C23×C17#9911yescharacteristic
102C23×C17C2×C22⋊C411yescharacteristic
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17#13112yes—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—42no—
122C2×Dic17—42no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—42no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—81no—
133C2×Dic17—81no—
134Dic34—82no—
135#3624—82no—
136#3624—82no—
137#2503—171no—
138#1463—171no—
139C2×#324—171no—
140#1324—172no—
141#1319—172no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17#35412yes—
152C22×C17#40811yescharacteristic
153C22×C17C2×#9911yescharacteristic
154C22×C17#41311yescharacteristic
155C4×C17—22no—
156C4×C17—41no—
157C4×C17—81no—
158C4×C17—81no—
159Dic17—41no—
160Dic17—41no—
161Dic17—41no—
162Dic17—41no—
163Dic17—81no—
164Dic17—81no—
165C2×C22⋊Q8—171no—
166C22×C4⋊C4—171no—
167#407—341no—
168C2×C22⋊C8—341no—
169#400—341no—
170#400—341no—
171#391—171no—
172#391—171no—
173#382—171no—
174#382—171no—
175#381—341no—
176#381—341no—
177C4×C22⋊C4#59511yescharacteristic
178#324—172no—
179#324—172no—
180#324—341no—
181#324—341no—
182C2×C17—21no—
183C2×C17—21no—
184C2×C17—22no—
185C2×C17—41no—
186C2×C17#146811yescharacteristic
187C2×C17C2×#35411yescharacteristic
188C2×C17#213311yescharacteristic
189C22×Q8—171no—
190C23×C4—171no—
191C23×C4—171no—
192C23×C4C2×#59511yescharacteristic
193C2×M16—681no—
194C22×C8—681no—
195C22⋊Q8—681no—
196C22⋊Q8—681no—
197C2×C4⋊C4—171no—
198C2×C4⋊C4—171no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—681no—
204C2×C4⋊C4—681no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—341no—
207C2×C42C2×#59511yescharacteristic
208C2×C42C2×#59511yescharacteristic
209C22⋊C8—342no—
210C22⋊C8—342no—
211C22⋊C8—681no—
212C22⋊C8—681no—
213#95—21no—
214#95—341no—
215#95—341no—
216#95—341no—
217#95—341no—
218#95—341no—
219#95—341no—
220#95—341no—
221#95—341no—
222C17#1155111yescharacteristic, complemented, Sylow
223C24C22⋊#59511yescharacteristic
224C2×Q8—342no—
225C2×Q8—681no—
226C2×Q8—681no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—171no—
236C22×C4—171no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—682no—
244C22×C4—682no—
245C22×C4C22⋊#59511yescharacteristic
246C22×C4C22×#59511yescharacteristic
247M16—1362no—
248C2×C8—682no—
249C2×C8—682no—
250C2×C8—1362no—
251C4⋊C4—681no—
252C4⋊C4—681no—
253C4⋊C4—681no—
254C4⋊C4—681no—
255C4⋊C4—681no—
256C4⋊C4—682no—
257C22⋊C4—42no—
258C22⋊C4—681no—
259C22⋊C4—681no—
260C42—22no—
261C42—41no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23#1058753911yescharacteristic
267C23#1058753911yescharacteristic
268C23C2×C22⋊#59511yescharacteristic
269Q8—1362no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—341no—
284C2×C4—341no—
285C2×C4—341no—
286C2×C4—341no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—682no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—681no—
298C2×C4—1361no—
299C2×C4—1361no—
300C2×C4#1058767812yes—
301C8—1362no—
302C8—1362no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22#1176371612yes—
313C22#1176378211yescharacteristic
314C22C2×#1058753911yescharacteristic
315C22#1176379611yescharacteristic
316C4—22no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—681no—
321C4—681no—
322C4—681no—
323C4—681no—
324C4—1361no—
325C4—1361no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2#1454405111yescharacteristic
331C2C2×#1176371611yescharacteristic
332C2#1454508111yescharacteristic
subgroupG/H
centerC22C2×#10587539
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC4×C22⋊C4×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC4×C22⋊C4#595
hypocenterC17#11551
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543724C2semidirect
4. #14543715C2semidirect
222. C17#11551semidirect

Maximal subgroups

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

subgroupindex
#145440322
C2×#117636542
#145437242
#145437152
#145332692
#1155117