Finite Group Databasebeta

← #16945374

Subgroups — #16945374

Order 4352.

460 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 107 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#14528776C211yesmaximal, characteristic
2#14527102C211yesmaximal, characteristic
3#14526790C214yesmaximal, complemented
4C17×#2486C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587504—21no—
7C2×C22⋊#3622—21no—
8#11762578C2212yes—
9#11762535C2212yes—
10#11762533—21no—
11C2×C17×#123—21no—
12C2×C4×D4×C17C2211yescharacteristic
13C22×C4⋊C4×C17C411yescharacteristic
14C17×#398C2212yes—
15C17×#397C412yes—
16C17×#389—21no—
17C17×#381—21no—
18C17×#380—21no—
19C17×#380C411yescharacteristic
20C22×D4×C17D411yescharacteristic
21C23×C4×C17—21no—
22C23×C4×C17C2×C411yescharacteristic
23C23×C4×C17D411yescharacteristic
24C17×#123—42no—
25C17×#123—42no—
26C4×D4×C17—41no—
27C4×D4×C17—41no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—41no—
32C2×C4⋊C4×C17—41no—
33C2×C4⋊C4×C17C2×C412yes—
34C2×C4⋊C4×C17C2311yescharacteristic
35C2×C22⋊C4×C17—22no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17C2×C411yescharacteristic
38C2×C22⋊C4×C17D411yescharacteristic
39C2×C42×C17D411yescharacteristic
40C2×#63223—22no—
41C22×#3622—41no—
42C17×#95—22no—
43C17×#95—22no—
44C17×#95C2×C412yes—
45#10587504—42no—
46C22⋊#3622—22no—
47C22⋊#3622—41no—
48C4⋊#3622—41no—
49C4⋊#3622—41no—
50C24×C17C22⋊C411yescharacteristic
51C24×C17C22⋊C411yescharacteristic
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C2×D4×C17D812yes—
55C2×D4×C17SD1612yes—
56C22×C4×C17—21no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—22no—
61C22×C4×C17—22no—
62C22×C4×C17—42no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17—42no—
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C4⋊C4×C17—41no—
73C4⋊C4×C17—42no—
74C4⋊C4×C17—42no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C22⋊C4×C17—41no—
80C22⋊C4×C17—41no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—44no—
83C42×C17—21no—
84C42×C17—21no—
85#63223—42no—
86#63223—42no—
87C2×#3622—42no—
88C2×#3622—42no—
89C2×#3622—82no—
90#11881—171nomaximal, Sylow
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—41no—
96C23×C17—41no—
97C23×C17—41no—
98C23×C17—41no—
99C23×C17#10212yes—
100C23×C17#10212yes—
101C23×C17C2×C22⋊C411yescharacteristic
102D4×C17—42no—
103D4×C17—42no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—22no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—42no—
111C2×C4×C17—42no—
112C2×C4×C17—42no—
113C2×C4×C17—41no—
114C2×C4×C17—41no—
115C2×C4×C17—41no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—81no—
123C2×C4×C17—84no—
124C2×C4×C17#9911yescharacteristic
125C2×C4×C17#9911yescharacteristic
126C2×C4×C17C2×D811yescharacteristic
127C2×C4×C17C2×SD1611yescharacteristic
128#3622—82no—
129#3622—82no—
130#2486D1711yescharacteristic
131#1800—171no—
132#1459—171no—
133#1346—174no—
134C22×C17—21no—
135C22×C17—21no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—42no—
139C22×C17—42no—
140C22×C17—42no—
141C22×C17—41no—
142C22×C17—41no—
143C22×C17—41no—
144C22×C17—41no—
145C22×C17—81no—
146C22×C17C2×#9911yescharacteristic
147C22×C17C2×#10211yescharacteristic
148C22×C17#42111yescharacteristic
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—21no—
152C4×C17—41no—
153C4×C17—41no—
154C4×C17—41no—
155C4×C17—82no—
156C4×C17—82no—
157C4×C17#35212yes—
158C2×#123—21no—
159C2×C4×D4C2×D1711yescharacteristic
160C22×C4⋊C4Dic1711yescharacteristic
161#423—341no—
162C2×#100—341no—
163C2×C22⋊C8—341no—
164#398C2×D1712yes—
165#397Dic1712yes—
166#389—21no—
167#381—21no—
168#380—21no—
169#380Dic1711yescharacteristic
170#343—172no—
171#324—172no—
172#323—341no—
173C2×C17—21no—
174C2×C17—21no—
175C2×C17—42no—
176C2×C17—42no—
177C2×C17#151911yescharacteristic
178C2×C17C2×#35211yescharacteristic
179C2×C17#213511yescharacteristic
180C22×D4C22⋊D1711yescharacteristic
181C23×C4—21no—
182C23×C4C2×Dic1711yescharacteristic
183C23×C4C22⋊D1711yescharacteristic
184C2×M16—342no—
185C22×C8—681no—
186#123—42no—
187#123—42no—
188C4×D4—41no—
189C4×D4—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C2×Dic1712yes—
196C2×C4⋊C4C22×D1711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4C2×Dic1711yescharacteristic
200C2×C22⋊C4C22⋊D1711yescharacteristic
201C2×C42C22⋊D1711yescharacteristic
202C4⋊C8—681no—
203C4⋊C8—681no—
204#100—682no—
205C22⋊C8—342no—
206C22⋊C8—681no—
207#95—22no—
208#95—22no—
209#95C2×Dic1712yes—
210C17#1188111yescharacteristic, complemented, Sylow
211C24C22⋊Dic1711yescharacteristic
212C24C22⋊Dic1711yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#6322812yes—
216C2×D4#6322912yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
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—
237C2×C8—1362no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1058750312yes—
260C23#1058750312yes—
261C23C2×C22⋊Dic1711yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—84no—
284C2×C4#1058750511yescharacteristic
285C2×C4#1058750511yescharacteristic
286C2×C4C2×#6322811yescharacteristic
287C2×C4C2×#6322911yescharacteristic
288C8—1362no—
289C8—1362no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1058750511yescharacteristic
303C22C2×#1058750311yescharacteristic
304C22#1176282611yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—82no—
312C4—82no—
313C4#1176259912yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#1452735211yescharacteristic
319C2C2×#1176259911yescharacteristic
320C2#1453047911yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
FittingC17×#2486C2
socleC22×C17C2×#99
prime radicalC22×D4×C17D4
cosocleC2×C4⋊C4×C17C23
hypercenter#2486D17
hypocenterC17#11881
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #14526790C2semidirect
210. C17#11881semidirect

Maximal subgroups

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

subgroupindex
#145287762
#145271022
#145267902
C17×#24862
#1188117