Finite Group Databasebeta

← #16931994

Subgroups — #16931994

Order 4352.

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

normal subgroups only

This group has 490 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#14537813C211yesmaximal, characteristic
2#14527411C212yesmaximal, complemented
3#14527125C211yesmaximal, characteristic
4#14526640C212yesmaximal, complemented
5C17×#2566C211yesmaximal, characteristic
6#11762881—24no—
7#11762866C2212yes—
8#11762866C2212yes—
9#11762773—21no—
10C2×C22⋊#3622—21no—
11#11762551—24no—
12#11762533—21no—
13#11762533—21no—
14C2×C17×#125C2211yescharacteristic
15C2×C4×D4×C17C2211yescharacteristic
16C17×#396—21no—
17C17×#396—21no—
18C17×#390C411yescharacteristic
19C17×#390C411yescharacteristic
20C17×#385—21no—
21C17×#385—21no—
22C17×#384C411yescharacteristic
23C17×#382C411yescharacteristic
24C4×C4⋊C4×C17C2211yescharacteristic
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C17×#125—22no—
29C17×#125—22no—
30C4×D4×C17—22no—
31C4×D4×C17—22no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17C2×C411yescharacteristic
38C2×C4⋊C4×C17C2×C411yescharacteristic
39C2×C4⋊C4×C17C2×C411yescharacteristic
40C2×C22⋊C4×C17—21no—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—21no—
43C2×C22⋊C4×C17—21no—
44C2×C22⋊C4×C17D411yescharacteristic
45C2×C22⋊C4×C17D411yescharacteristic
46C2×C42×C17C2×C411yescharacteristic
47C2×C42×C17C2×C411yescharacteristic
48C2×C42×C17C2311yescharacteristic
49C2×#63223—22no—
50C22×#3622—22no—
51C17×#95—21no—
52C17×#95—21no—
53C17×#95—21no—
54C22⋊#3622—22no—
55C22⋊#3622—22no—
56C22⋊#3622—22no—
57C22⋊#3622—22no—
58C4⋊#3622—22no—
59C4⋊#3622—22no—
60#10587474—22no—
61#10587474D414yes—
62C24×C17C22⋊C411yescharacteristic
63C24×C17C22⋊C411yescharacteristic
64C2×D4×C17—22no—
65C2×D4×C17—22no—
66C2×D4×C17—41no—
67C22×C4×C17—21no—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—81no—
76C22×C4×C17—81no—
77C22×C4×C17C22⋊C411yescharacteristic
78C22×C4×C17C22⋊C411yescharacteristic
79C22×C4×C17C22×C411yescharacteristic
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C4⋊C4×C17—22no—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—22no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—42no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—81no—
90C4⋊C4×C17—81no—
91C22⋊C4×C17—22no—
92C22⋊C4×C17—22no—
93C22⋊C4×C17—42no—
94C22⋊C4×C17—42no—
95C22⋊C4×C17—81no—
96C22⋊C4×C17—81no—
97C42×C17—41no—
98C42×C17—41no—
99C42×C17C2×D412yes—
100C42×C17#4112yes—
101#63223—82no—
102C2×#3622—24no—
103C2×#3622—42no—
104C2×#3622—82no—
105#9449—171nomaximal, Sylow
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17#10112yes—
116C23×C17C2×C22⋊C411yescharacteristic
117D4×C17—44no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—22no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—81no—
141C2×C4×C17—81no—
142C2×C4×C17C4×D412yes—
143C2×C4×C17#12011yescharacteristic
144C2×C4×C17#12112yes—
145C2×C4×C17#12311yescharacteristic
146#3622—44no—
147#3622—82no—
148#2973—171no—
149#2566D1711yescharacteristic
150#1534—172no—
151#1466—171no—
152#1285—172no—
153C22×C17—21no—
154C22×C17—21no—
155C22×C17—22no—
156C22×C17—42no—
157C22×C17—42no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—81no—
164C22×C17—81no—
165C22×C17#38611yescharacteristic
166C22×C17C2×#10111yescharacteristic
167C22×C17#41311yescharacteristic
168C4×C17—22no—
169C4×C17—22no—
170C4×C17—21no—
171C4×C17—21no—
172C4×C17—42no—
173C4×C17—41no—
174C4×C17—41no—
175C4×C17—81no—
176C4×C17—81no—
177C4×C17—81no—
178C2×#125C2×D1711yescharacteristic
179C2×C4×D4C2×D1711yescharacteristic
180#435—344no—
181#432—172no—
182#432—172no—
183#407—341no—
184C2×C22⋊C8—341no—
185#396—21no—
186#396—21no—
187#390Dic1711yescharacteristic
188#390Dic1711yescharacteristic
189#385—21no—
190#385—21no—
191#384Dic1711yescharacteristic
192#382Dic1711yescharacteristic
193C4×C4⋊C4C2×D1711yescharacteristic
194#330—344no—
195#323—341no—
196#323—341no—
197C2×C17—21no—
198C2×C17—21no—
199C2×C17—42no—
200C2×C17—81no—
201C2×C17#188411yescharacteristic
202C2×C17#190111yescharacteristic
203C2×C17#191111yescharacteristic
204C22×D4C2×Dic1711yescharacteristic
205C23×C4C22⋊D1711yescharacteristic
206C23×C4C22⋊D1711yescharacteristic
207C2×M16—342no—
208C22×C8—342no—
209#125—22no—
210#125—22no—
211C4×D4—22no—
212C4×D4—22no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C2×Dic1711yescharacteristic
219C2×C4⋊C4C2×Dic1711yescharacteristic
220C2×C4⋊C4C2×Dic1711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4C22⋊D1711yescharacteristic
226C2×C22⋊C4C22⋊D1711yescharacteristic
227C2×C42C2×Dic1711yescharacteristic
228C2×C42C2×Dic1711yescharacteristic
229C2×C42C22×D1711yescharacteristic
230C4⋊C8—342no—
231C4⋊C8—342no—
232C22⋊C8—342no—
233C22⋊C8—342no—
234C22⋊C8—342no—
235C22⋊C8—342no—
236#97—174no—
237#97—342no—
238#95—21no—
239#95—21no—
240#95—21no—
241C17#944911yescharacteristic, complemented, Sylow
242C24C22⋊Dic1711yescharacteristic
243C24C22⋊Dic1711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—41no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4C22⋊Dic1711yescharacteristic
258C22×C4C22⋊Dic1711yescharacteristic
259C22×C4C22×Dic1711yescharacteristic
260C22×C4C2×C22⋊D1711yescharacteristic
261C22×C4C2×C22⋊D1711yescharacteristic
262M16—1362no—
263C2×C8—344no—
264C2×C8—682no—
265C2×C8—1362no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C42—41no—
282C42—41no—
283C42D4×D1712yes—
284C42#6325412yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1058750712yes—
295C23C2×C22⋊Dic1711yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4D4×Dic1712yes—
322C2×C4#1058762111yescharacteristic
323C2×C4#1058762311yescharacteristic
324C2×C4#1058762512yes—
325C8—684no—
326C8—1362no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1176268911yescharacteristic
340C22C2×#1058750711yescharacteristic
341C22#1176279311yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#1452916811yescharacteristic
357C2#1452925311yescharacteristic
358C2#1452931711yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2566C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC2×C42×C17C23
hypercenter#2566D17
hypocenterC17#9449
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2566

Complements

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

NH ≅ G/Ntype
2. #14527411C2semidirect
4. #14526640C2semidirect
241. C17#9449semidirect

Maximal subgroups

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

subgroupindex
#145378132
#145274112
#145271252
#145266402
C17×#25662
#944917