Finite Group Databasebeta

← #16945070

Subgroups — #16945070

Order 4352.

386 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 79 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#14534597C211yesmaximal, characteristic
2#14527114C211yesmaximal, characteristic
3#14526813C212yesmaximal, complemented
4#14526810C212yesmaximal, complemented
5C17×#1821C211yesmaximal, characteristic
6C2×#10587610C411yescharacteristic
7#11762773—21no—
8#11762748C411yescharacteristic
9#11762748C2211yescharacteristic
10#11762742—21no—
11#11762742—21no—
12#11762729C411yescharacteristic
13#11762729C2211yescharacteristic
14#11762682—21no—
15#11762672—21no—
16#11762669C411yescharacteristic
17#11762535—21no—
18#11762535C2212yes—
19C22×C4⋊C4×C17C2211yescharacteristic
20C17×#407—21no—
21C17×#343—21no—
22C17×#343C2212yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17D411yescharacteristic
26C22×Dic34C2×C411yescharacteristic
27C2×M16×C17—22no—
28C2×C4⋊C4×C17—41no—
29C2×C4⋊C4×C17—41no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C4⋊C4×C17C2311yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38#10587610—41no—
39#10587610—41no—
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63223—41no—
44C17×C22⋊C8—41no—
45C17×C22⋊C8—41no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54C22⋊#3622—22no—
55C22⋊#3622—41no—
56C24×C17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—41no—
65C22×C4×C17—81no—
66C22×C4×C17—81no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22×C411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C2×Dic34—41no—
78C2×Dic34—81no—
79M16×C17—42no—
80M16×C17—42no—
81C2×C8×C17—42no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—41no—
86C22⋊Dic17—42no—
87C22⋊Dic17—41no—
88C22⋊Dic17—41no—
89C4⋊Dic17—41no—
90C4⋊Dic17—81no—
91#63225—42no—
92#63225—81no—
93#63225—81no—
94#63223—82no—
95C2×#3622—42no—
96#11817—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17#9911yescharacteristic
102C23×C17#9911yescharacteristic
103C23×C17C2×C22⋊C411yescharacteristic
104C2×C4×C17—21no—
105C2×C4×C17—21no—
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—42no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17—41no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—81no—
121C2×C4×C17—81no—
122C2×C4×C17—81no—
123C2×C4×C17#13712yes—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—42no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135Dic34—82no—
136C8×C17—82no—
137#3622—82no—
138#2627—171no—
139#1821D1711yescharacteristic
140#1463—171no—
141#1354—172no—
142#1353—172no—
143C22×C17—21no—
144C22×C17—21no—
145C22×C17—21no—
146C22×C17—21no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17—42no—
152C22×C17—42no—
153C22×C17C2×#9911yescharacteristic
154C22×C17#41911yescharacteristic
155C22×C17#42111yescharacteristic
156C4×C17—22no—
157C4×C17—41no—
158C4×C17—41no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162Dic17—81no—
163Dic17—81no—
164Dic17—81no—
165Dic17—81no—
166C2×C22⋊Q8—171no—
167C22×C4⋊C4C2×D1711yescharacteristic
168#407—21no—
169#407—341no—
170#400—171no—
171#400—171no—
172#398—341no—
173#398—341no—
174#395—171no—
175#395—171no—
176#385—341no—
177#381—341no—
178#380—171no—
179#343—21no—
180#343C2×D1712yes—
181#324—172no—
182#324—341no—
183C2×C17—21no—
184C2×C17—21no—
185C2×C17—41no—
186C2×C17—41no—
187C2×C17#151111yescharacteristic
188C2×C17#212111yescharacteristic
189C2×C17#213311yescharacteristic
190C22×Q8—171no—
191C23×C4C4×D1711yescharacteristic
192C23×C4D6811yescharacteristic
193C23×C4C22⋊D1711yescharacteristic
194C2×M16—22no—
195C2×M16—681no—
196C22⋊Q8—681no—
197C22⋊Q8—681no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—341no—
202C2×C4⋊C4—341no—
203C2×C4⋊C4—341no—
204C2×C4⋊C4—341no—
205C2×C4⋊C4C4×D1711yescharacteristic
206C2×C4⋊C4D6811yescharacteristic
207C2×C4⋊C4C22⋊D1711yescharacteristic
208C2×C4⋊C4C22×D1711yescharacteristic
209C2×C22⋊C4—341no—
210C2×C22⋊C4—341no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—342no—
214C22⋊C8—681no—
215#95—171no—
216#95—171no—
217#95—341no—
218#95—341no—
219#95—341no—
220#95—341no—
221#95—341no—
222#95—341no—
223C17#1181711yescharacteristic, complemented, Sylow
224C24#6322711yescharacteristic
225C2×Q8—681no—
226C2×Q8—1361no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—341no—
238C22×C4—341no—
239C22×C4—341no—
240C22×C4—341no—
241C22×C4#6322711yescharacteristic
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4C2×C4×D1711yescharacteristic
245C22×C4C2×D6811yescharacteristic
246C22×C4C2×C22⋊D1711yescharacteristic
247M16—42no—
248M16—42no—
249M16—1362no—
250C2×C8—42no—
251C2×C8—682no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—681no—
257C4⋊C4—682no—
258C4⋊C4—1361no—
259C4⋊C4—1361no—
260C4⋊C4—1361no—
261C22⋊C4—682no—
262C22⋊C4—681no—
263C22⋊C4—681no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23#1058747711yescharacteristic
269C23#1058747711yescharacteristic
270C23C2×#6322711yescharacteristic
271Q8—1362no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—681no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4—681no—
295C2×C4—681no—
296C2×C4—681no—
297C2×C4—682no—
298C2×C4—681no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4#1058763712yes—
303C8—82no—
304C8—1362no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—42no—
314C22—42no—
315C22C2×#1058747711yescharacteristic
316C22#1176281711yescharacteristic
317C22#1176282711yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—1361no—
325C4—1361no—
326C4—1361no—
327C4—1361no—
328C2—21no—
329C2—21no—
330C2—41no—
331C2—41no—
332C2#1452731611yescharacteristic
333C2#1453043111yescharacteristic
334C2#1453047111yescharacteristic
subgroupG/H
centerC22C2×#10587477
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#1821C2
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#1821D17
hypocenterC17#11817
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #14526813C2semidirect
4. #14526810C2semidirect
223. C17#11817semidirect

Maximal subgroups

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

subgroupindex
#145345972
#145271142
#145268132
#145268102
C17×#18212
#1181717