Finite Group Databasebeta

← #16945140

Subgroups — #16945140

Order 4352.

384 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 91 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#14528780C211yesmaximal, characteristic
2#14527128C211yesmaximal, characteristic
3#14526814C212yesmaximal
4#14526790C212yesmaximal
5C17×#2399C211yesmaximal, characteristic
6#11762866—21no—
7#11762833—21no—
8C2×#10587507—21no—
9#11762578C2212yes—
10#11762535—21no—
11#11762535C2212yes—
12C2×C4×Q8×C17C411yescharacteristic
13C2×C4×Q8×C17C2211yescharacteristic
14C17×#398—21no—
15C17×#398C411yescharacteristic
16C17×#398C2211yescharacteristic
17C17×#395—21no—
18C17×#395C411yescharacteristic
19C17×#395C2211yescharacteristic
20C4×C4⋊C4×C17—21no—
21C4×C4⋊C4×C17—21no—
22C17×#376C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C22×Q8×C17D411yescharacteristic
25C4×Q8×C17—41no—
26C4×Q8×C17—41no—
27C4×Q8×C17—41no—
28C4×Q8×C17—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C4⋊C4×C17C2311yescharacteristic
38C2×C42×C17—21no—
39C2×C42×C17—21no—
40C2×C42×C17C2×C411yescharacteristic
41C2×C42×C17D411yescharacteristic
42C2×C42×C17D411yescharacteristic
43C2×#63223—22no—
44C17×#95—21no—
45C17×#95—21no—
46C17×#95—21no—
47C17×#95—21no—
48C17×#95—21no—
49C17×#95C2×C411yescharacteristic
50C17×#95C2×C411yescharacteristic
51#10587507—42no—
52C22⋊#3622—22no—
53C4⋊#3622—41no—
54C4⋊#3622—41no—
55C4⋊#3622—41no—
56#10587474—41no—
57C2×Q8×C17—21no—
58C2×Q8×C17—21no—
59C2×Q8×C17—41no—
60C2×Q8×C17—41no—
61C2×Q8×C17—42no—
62C2×Q8×C17—41no—
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D411yescharacteristic
74C4⋊C4×C17—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—41no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—42no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C42×C17—24no—
85C42×C17—21no—
86C42×C17—21no—
87C42×C17—21no—
88C42×C17—21no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—41no—
92C42×C17—41no—
93#63223—42no—
94#63223—42no—
95C2×#3622—42no—
96C2×#3622—42no—
97#11831—171nomaximal, Sylow
98C23×C17C2×C22⋊C411yescharacteristic
99Q8×C17—42no—
100Q8×C17—42no—
101Q8×C17—42no—
102Q8×C17—42no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—22no—
106C2×C4×C17—22no—
107C2×C4×C17—22no—
108C2×C4×C17—22no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17#9911yescharacteristic
132C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C17#13711yescharacteristic
134#3622—82no—
135#3622—82no—
136#2399D1711yescharacteristic
137#1801—171no—
138#1467—171no—
139#1354—172no—
140#1346—172no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17C2×#9911yescharacteristic
144C22×C17#41711yescharacteristic
145C22×C17#42111yescharacteristic
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—42no—
150C4×C17—42no—
151C4×C17—41no—
152C4×C17—41no—
153C4×C17—42no—
154C4×C17—41no—
155C4×C17—41no—
156C4×C17—41no—
157C4×C17—81no—
158C4×C17#35612yes—
159C2×C4×Q8Dic1711yescharacteristic
160C2×C4×Q8C2×D1711yescharacteristic
161#432—341no—
162#423—341no—
163C2×#101—341no—
164#398—21no—
165#398Dic1711yescharacteristic
166#398C2×D1711yescharacteristic
167#395—21no—
168#395Dic1711yescharacteristic
169#395C2×D1711yescharacteristic
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172#376Dic1711yescharacteristic
173#343—172no—
174#324—172no—
175#324—341no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17#151311yescharacteristic
179C2×C17#212011yescharacteristic
180C2×C17C2×#35611yescharacteristic
181C22×Q8C2×Dic1711yescharacteristic
182C22×Q8C22⋊D1711yescharacteristic
183C2×M16—342no—
184C4×Q8—41no—
185C4×Q8—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic1711yescharacteristic
194C2×C4⋊C4C2×Dic1711yescharacteristic
195C2×C4⋊C4C22⋊D1711yescharacteristic
196C2×C4⋊C4C22×D1711yescharacteristic
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C2×Dic1711yescharacteristic
200C2×C42C22⋊D1711yescharacteristic
201C2×C42C22⋊D1711yescharacteristic
202C4⋊C8—681no—
203C4⋊C8—681no—
204C4⋊C8—681no—
205#101—682no—
206C22⋊C8—342no—
207#97—681no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95C2×Dic1711yescharacteristic
214#95C2×Dic1711yescharacteristic
215C17#1183111yescharacteristic, complemented, Sylow
216C2×Q8—21no—
217C2×Q8—21no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—41no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4C22⋊Dic1711yescharacteristic
227C22×C4C22⋊Dic1711yescharacteristic
228C22×C4C22⋊Dic1711yescharacteristic
229C22×C4C22⋊Dic1711yescharacteristic
230C22×C4C22×Dic1711yescharacteristic
231C22×C4C2×C22⋊D1711yescharacteristic
232C22×C4C2×C22⋊D1711yescharacteristic
233M16—682no—
234M16—682no—
235C2×C8—682no—
236C2×C8—682no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C42—24no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—41no—
255C42—41no—
256C23C2×C22⋊Dic1711yescharacteristic
257Q8—42no—
258Q8—42no—
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4#1058750511yescharacteristic
289C2×C4#1058750511yescharacteristic
290C2×C4#1058763511yescharacteristic
291C2×C4#1058763711yescharacteristic
292C8—1362no—
293C8—1362no—
294C22—21no—
295C22—21no—
296C22C2×#1058750511yescharacteristic
297C22#1176281011yescharacteristic
298C22#1176282611yescharacteristic
299C4—21no—
300C4—21no—
301C4—21no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—42no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4#1176260812yes—
312C2—21no—
313C2—21no—
314C2#1452732711yescharacteristic
315C2#1453042511yescharacteristic
316C2C2×#1176260811yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2399C2
socleC22×C17C2×#99
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4⋊C4×C17C23
hypercenter#2399D17
hypocenterC17#11831
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2399

Complements

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

NH ≅ G/Ntype
215. C17#11831semidirect

Maximal subgroups

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

subgroupindex
#145287802
#145271282
#145268142
#145267902
C17×#23992
#1183117