Finite Group Databasebeta

← #17948751

Subgroups — #17948751

Order 4352.

422 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 60 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#14544683C211yesmaximal, characteristic
2#14544034C211yesmaximal, characteristic
3#14527019C211yesmaximal, characteristic
4#11763779—21no—
5#11763779—21no—
6#11763689—21no—
7#11763689C412yescomplemented
8#11763655—21no—
9#11763655C412yescomplemented
10C22×#63225C2211yescharacteristic
11C22×C4×Dic17C411yescharacteristic
12C2×#10587502—21no—
13C2×#10587502—21no—
14C2×C17×#95C411yescharacteristic
15C23×C4×C17—21no—
16C23×C4×C17C2×C411yescharacteristic
17C23×Dic17—21no—
18C23×Dic17D411yescharacteristic
19C23×Dic17D411yescharacteristic
20C2×#63247—22no—
21C2×#63247—41no—
22C2×#63225—41no—
23C2×#63225—41no—
24C2×#63225C2×C411yescharacteristic
25C2×#63225C2×C411yescharacteristic
26C2×#63225D411yescharacteristic
27C2×#63225Q811yescharacteristic
28C2×C4×Dic17—42no—
29C2×C4×Dic17—41no—
30C22⋊#3624—22no—
31C22⋊#3624—41no—
32C22⋊#3624—41no—
33C22⋊#3624—41no—
34C17×#95—41no—
35#10587502—24no—
36#10587502—42no—
37C24×C17C22⋊C411yescharacteristic
38C22×C4×C17—21no—
39C22×C4×C17—21no—
40C22×C4×C17—22no—
41C22×C4×C17—41no—
42C22×C4×C17—41no—
43C22×C4×C17—41no—
44C22×C4×C17—41no—
45C22×C4×C17—41no—
46C22×C4×C17—41no—
47C22×C4×C17—41no—
48C22×C4×C17—41no—
49C22×C4×C17C4211yescharacteristic
50C22×C4×C17C4⋊C411yescharacteristic
51C22×Dic17—22no—
52C22×Dic17—22no—
53C22×Dic17—22no—
54C22×Dic17—21no—
55C22×Dic17—21no—
56C22×Dic17—21no—
57C22×Dic17—21no—
58C22×Dic17—22no—
59C22×Dic17—41no—
60C22×Dic17—41no—
61C22×Dic17—41no—
62C22×Dic17—41no—
63C22×Dic17—41no—
64C22×Dic17—41no—
65C22×Dic17—41no—
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C4⋊C411yescharacteristic
69C22×Dic17C4⋊C411yescharacteristic
70#63247—42no—
71#63247—42no—
72#63247—81no—
73#63247—81no—
74C2×#3624—42no—
75C2×#3624—42no—
76#63225—41no—
77#63225—41no—
78#63225—41no—
79#63225—41no—
80C4×Dic17—42no—
81C4×Dic17—42no—
82#7215—171nomaximal, Sylow
83C23×C17—21no—
84C23×C17—21no—
85C23×C17—21no—
86C23×C17—21no—
87C23×C17—21no—
88C23×C17—21no—
89C23×C17#9511yescharacteristic
90C23×C17#9911yescharacteristic
91C23×C17#9911yescharacteristic
92C2×C4×C17—21no—
93C2×C4×C17—21no—
94C2×C4×C17—42no—
95C2×C4×C17—41no—
96C2×C4×C17—41no—
97C2×C4×C17—41no—
98C2×C4×C17—41no—
99C2×C4×C17—42no—
100C2×C4×C17—42no—
101C2×C4×C17—42no—
102C2×C4×C17—81no—
103C2×C4×C17—81no—
104C2×C4×C17—81no—
105C2×C4×C17—81no—
106C2×Dic17—21no—
107C2×Dic17—21no—
108C2×Dic17—24no—
109C2×Dic17—21no—
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17—41no—
122C2×Dic17—42no—
123C2×Dic17—41no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—81no—
127C2×Dic17#10412yes—
128#3624—82no—
129#3624—82no—
130#1821—171no—
131#1463—171no—
132#1436—171no—
133C22×C17—21no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—21no—
137C22×C17—21no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17H(3,ℤ/4)11yescharacteristic
148C22×C17#33911yescharacteristic
149C22×C17#34211yescharacteristic
150C22×C17AΣL(1,𝔽16)11yescharacteristic
151C22×C17#35211yescharacteristic
152C22×C17#35511yescharacteristic
153C22×C17#35611yescharacteristic
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—82no—
157Dic17—22no—
158Dic17—44no—
159Dic17—41no—
160Dic17—41no—
161Dic17—41no—
162C22×C4⋊C4—171no—
163C22×C42—171no—
164#407—341no—
165#407—341no—
166C2×#95—341no—
167C2×#95—341no—
168C2×#95#59511yescharacteristic
169#343—172no—
170#343—341no—
171#324—172no—
172#324—341no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—21no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17#128411yescharacteristic
180C2×C17#139011yescharacteristic
181C2×C17#139411yescharacteristic
182C23×C4—21no—
183C23×C4—171no—
184C23×C4—171no—
185C23×C4—341no—
186C23×C4C2×#59511yescharacteristic
187C2×M16—342no—
188C2×M16—681no—
189C2×C4⋊C4—171no—
190C2×C4⋊C4—171no—
191C2×C4⋊C4—171no—
192C2×C4⋊C4—171no—
193C2×C4⋊C4—681no—
194C2×C4⋊C4—681no—
195C2×C42—681no—
196C2×C42—682no—
197C22⋊C8—342no—
198C22⋊C8—681no—
199C22⋊C8—681no—
200C22⋊C8—681no—
201#95—41no—
202#95—344no—
203#95—682no—
204C17#721511yescharacteristic, complemented, Sylow
205C24C22⋊#59511yescharacteristic
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—171no—
221C22×C4—342no—
222C22×C4—342no—
223C22×C4—342no—
224C22×C4—342no—
225C22×C4—341no—
226C22×C4—341no—
227C22×C4—341no—
228C22×C4—341no—
229C22×C4—681no—
230C22×C4—681no—
231C22×C4—681no—
232C22×C4—681no—
233C22×C4—681no—
234C22×C4—681no—
235C22×C4—681no—
236C22×C4C4×#59511yescharacteristic
237C22×C4C4⋊#59511yescharacteristic
238M16—682no—
239M16—682no—
240M16—1361no—
241M16—1361no—
242C2×C8—682no—
243C2×C8—682no—
244C4⋊C4—681no—
245C4⋊C4—681no—
246C4⋊C4—681no—
247C4⋊C4—681no—
248C42—682no—
249C42—682no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#1058753911yescharacteristic
257C23(C2×#595)⋊C411yescharacteristic
258C23#1058755111yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—172no—
274C2×C4—341no—
275C2×C4—341no—
276C2×C4—344no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—341no—
280C2×C4—341no—
281C2×C4—341no—
282C2×C4—682no—
283C2×C4—682no—
284C2×C4—682no—
285C2×C4—682no—
286C2×C4—682no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—681no—
290C2×C4—682no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—1361no—
295C8—1362no—
296C8—1362no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1176367611yescharacteristic
312C22#1176368011yescharacteristic
313C22#1176368711yescharacteristic
314C22#1176371011yescharacteristic
315C22#1176371211yescharacteristic
316C22#1176371711yescharacteristic
317C22#1176371911yescharacteristic
318C4—41no—
319C4—41no—
320C4—82no—
321C4—342no—
322C4—684no—
323C4—681no—
324C4—681no—
325C4—681no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#1454363211yescharacteristic
333C2#1454386911yescharacteristic
334C2#1454388211yescharacteristic
subgroupG/H
centerC22#11763687
commutatorC22×C4×C17C42
FrattiniC23×C4C2×#595
FittingC2×C17×#95C4
socleC22×C17#342
prime radicalC24×C17C22⋊C4
cosocleC22×#63225C22
hypercenterC2×#95#595
hypocenterC17#7215
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
7. #11763689C4semidirect
9. #11763655C4semidirect
204. C17#7215semidirect

Maximal subgroups

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

subgroupindex
#145446832
#145440342
#145270192
#721517