Finite Group Databasebeta

← #16943945

Subgroups — #16943945

Order 4352.

616 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 95 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#14527122C211yesmaximal, characteristic, complemented
2#14527102C211yesmaximal, characteristic, complemented
3#14526716C214yesmaximal, complemented
4C17×#2541C211yesmaximal, characteristic
5#11762773—21no—
6C2×C22⋊#3622—21no—
7#11762535C2212yes—
8#11762535C2212yes—
9#11762533—21no—
10#11762533—21no—
11C2×C17×#127C411yescharacteristic
12C2×C17×#127C412yes—
13C2×C17×#121—21no—
14C2×C17×#121—21no—
15C2×C4×D4×C17C2211yescharacteristic
16C17×#392—21no—
17C17×#392—21no—
18C17×#390C2212yes—
19C4×C4⋊C4×C17C411yescharacteristic
20C22×D4×C17—22no—
21C22×D4×C17—22no—
22C22×D4×C17—21no—
23C22×D4×C17—21no—
24C22×D4×C17C2×C411yescharacteristic
25C22×D4×C17C2×C412yes—
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C17×#127—41no—
29C17×#127—41no—
30C17×#127—44no—
31C17×#121—44no—
32C17×#121—44no—
33C4×D4×C17—41no—
34C4×D4×C17—41no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17C2311yescharacteristic
37C2×C22⋊C4×C17—22no—
38C2×C22⋊C4×C17—22no—
39C2×C22⋊C4×C17D411yescharacteristic
40C2×C22⋊C4×C17D411yescharacteristic
41C2×C42×C17C2×C411yescharacteristic
42C2×C42×C17C2×C412yes—
43C2×#63223—41no—
44C22×#3622—41no—
45C17×#95—21no—
46C22⋊#3622—22no—
47C22⋊#3622—22no—
48C22⋊#3622—41no—
49C22⋊#3622—41no—
50C24×C17—22no—
51C24×C17C22⋊C411yescharacteristic
52C24×C17C22⋊C411yescharacteristic
53C2×D4×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—24no—
56C2×D4×C17—41no—
57C2×D4×C17—42no—
58C2×D4×C17—42no—
59C2×D4×C17—42no—
60C2×D4×C17—42no—
61C2×D4×C17—42no—
62C2×D4×C17—44no—
63C2×D4×C17—44no—
64C2×D4×C17—44no—
65C2×D4×C17—44no—
66C2×D4×C17—41no—
67C2×D4×C17—41no—
68C2×D4×C17—82no—
69C2×D4×C17—82no—
70C2×D4×C17—82no—
71C22×C4×C17—22no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—21no—
76C22×C4×C17—42no—
77C22×C4×C17—42no—
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22⋊C411yescharacteristic
80C22×C4×C17C22×C411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D411yescharacteristic
83C4⋊C4×C17—24no—
84C4⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—44no—
88C22⋊C4×C17—44no—
89C42×C17—21no—
90C42×C17—21no—
91C42×C17—42no—
92#63223—82no—
93C2×#3622—42no—
94C2×#3622—42no—
95C2×#3622—82no—
96#11563—171nomaximal, Sylow
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—21no—
101C23×C17—21no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—42no—
107C23×C17—42no—
108C23×C17—42no—
109C23×C17—42no—
110C23×C17—42no—
111C23×C17—42no—
112C23×C17—42no—
113C23×C17#10012yes—
114C23×C17C2×C22⋊C411yescharacteristic
115D4×C17—44no—
116D4×C17—84no—
117D4×C17—82no—
118D4×C17—88no—
119D4×C17—88no—
120D4×C17—84no—
121D4×C17—82no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—22no—
125C2×C4×C17—24no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—81no—
137C2×C4×C17—81no—
138C2×C4×C17#9911yescharacteristic
139C2×C4×C17#9911yescharacteristic
140C2×C4×C17#13112yes—
141#3622—82no—
142#3622—82no—
143#2541D1711yescharacteristic
144#1465—171no—
145#1459—171no—
146#1318—174no—
147C22×C17—21no—
148C22×C17—21no—
149C22×C17—22no—
150C22×C17—42no—
151C22×C17—42no—
152C22×C17—41no—
153C22×C17—41no—
154C22×C17—41no—
155C22×C17—41no—
156C22×C17—41no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17—82no—
160C22×C17—82no—
161C22×C17—82no—
162C22×C17—82no—
163C22×C17—82no—
164C22×C17—82no—
165C22×C17—82no—
166C22×C17#40811yescharacteristic
167C22×C17C2×#9911yescharacteristic
168C22×C17C2×#10011yescharacteristic
169C4×C17—21no—
170C4×C17—22no—
171C4×C17—44no—
172C4×C17—41no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17#35312yes—
176C2×#127Dic1711yescharacteristic
177C2×#127Dic1712yes—
178C2×#121—21no—
179C2×#121—21no—
180C2×C4×D4C2×D1711yescharacteristic
181#407—341no—
182C2×C22⋊C8—341no—
183#392—21no—
184#392—21no—
185#390C2×D1712yes—
186C4×C4⋊C4Dic1711yescharacteristic
187#324—172no—
188#324—172no—
189#323—341no—
190#323—341no—
191C2×C17—21no—
192C2×C17—21no—
193C2×C17—42no—
194C2×C17—81no—
195C2×C17—162no—
196C2×C17#147111yescharacteristic
197C2×C17C2×#35311yescharacteristic
198C2×C17#213411yescharacteristic
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—21no—
202C22×D4—21no—
203C22×D4C2×Dic1711yescharacteristic
204C22×D4C2×Dic1712yes—
205C23×C4C22⋊D1711yescharacteristic
206C23×C4C22⋊D1711yescharacteristic
207C2×M16—681no—
208C22×C8—681no—
209#127—41no—
210#127—41no—
211#127—44no—
212#121—44no—
213#121—44no—
214C4×D4—41no—
215C4×D4—41no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22×D1711yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4C22⋊D1711yescharacteristic
221C2×C22⋊C4C22⋊D1711yescharacteristic
222C2×C42C2×Dic1711yescharacteristic
223C2×C42C2×Dic1712yes—
224C22⋊C8—342no—
225C22⋊C8—342no—
226C22⋊C8—681no—
227C22⋊C8—681no—
228#95—21no—
229C17#1156311yescharacteristic, complemented, Sylow
230C24—22no—
231C24C22⋊Dic1711yescharacteristic
232C24C22⋊Dic1711yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—24no—
236C2×D4—41no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—42no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C22⋊Dic1711yescharacteristic
259C22×C4C22⋊Dic1711yescharacteristic
260C22×C4C22×Dic1711yescharacteristic
261C22×C4C2×C22⋊D1711yescharacteristic
262C22×C4C2×C22⋊D1711yescharacteristic
263M16—1362no—
264C2×C8—682no—
265C2×C8—682no—
266C2×C8—1362no—
267C4⋊C4—24no—
268C4⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C42—21no—
274C42—21no—
275C42—42no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23#1058750412yes—
293C23C2×C22⋊Dic1711yescharacteristic
294D4—44no—
295D4—84no—
296D4—82no—
297D4—88no—
298D4—88no—
299D4—84no—
300D4—82no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4#1058750511yescharacteristic
318C2×C4#1058750511yescharacteristic
319C2×C4#1058763412yes—
320C8—1362no—
321C8—1362no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—81no—
333C22—81no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22—82no—
341C22#1176277611yescharacteristic
342C22C2×#1058750511yescharacteristic
343C22C2×#1058750411yescharacteristic
344C4—21no—
345C4—22no—
346C4—44no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4#1176260212yes—
351C2—21no—
352C2—21no—
353C2—42no—
354C2—81no—
355C2—162no—
356C2#1452714411yescharacteristic
357C2C2×#1176260211yescharacteristic
358C2#1453047611yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC22×C4×C17C22×C4
FrattiniC2×C4⋊C4C22×D17
Fitting, prime radicalC17×#2541C2
socleC22×C17C2×#99
cosocleC2×C4⋊C4×C17C23
hypercenter#2541D17
hypocenterC17#11563
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. #14527122C2semidirect
2. #14527102C2semidirect
3. #14526716C2semidirect
229. C17#11563semidirect

Maximal subgroups

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

subgroupindex
#145271222
#145271022
#145267162
C17×#25412
#1156317