Finite Group Databasebeta

← #16929338

Subgroups — #16929338

Order 4352.

928 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#11762726C211yesmaximal, characteristic
2#14528714C212yesmaximal
3C2×#11762535C211yesmaximal, characteristic, complemented
4#14526662C212yesmaximal, complemented
5C17×#2313C211yesmaximal, characteristic, complemented
6C23×D68C2211yescharacteristic
7C22×#63227—21no—
8C22×#63227—21no—
9C2×#10587480—24no—
10C2×C22⋊#3622C2212yes—
11#11762726—22no—
12#11762726C414yescomplemented
13C2×#10587502C2211yescharacteristic
14#11762535—22no—
15C22×C4⋊C4×C17C2211yescharacteristic
16C17×#387—24no—
17C17×#381—21no—
18C17×#380C2212yes—
19C4×C22⋊C4×C17—21no—
20C24×D17—21no—
21C23×C4×C17D411yescharacteristic
22C23×C4×C17D411yescharacteristic
23C23×C4×C17C2311yescharacteristic
24C23×Dic17—21no—
25C22×D68—22no—
26C22×D68—22no—
27C22×D68—21no—
28C22×D68—44no—
29C22×D68C2×C412yes—
30C2×C4⋊C4×C17—22no—
31C2×C4⋊C4×C17—22no—
32C2×C4⋊C4×C17C2×C412yes—
33C2×C4⋊C4×C17D412yes—
34C2×C4⋊C4×C17D414yes—
35C2×C22⋊C4×C17—22no—
36C2×C42×C17—22no—
37C2×#63227—24no—
38C2×#63227—24no—
39C2×#63227—44no—
40C2×#63227—44no—
41C22×#3622—22no—
42C17×#95—22no—
43C17×#95—24no—
44C17×#95—22no—
45#10587502—21no—
46#10587502C2×C412yes—
47C22⋊#3622—22no—
48C22⋊#3622—22no—
49#10587480—48no—
50C24×C17C2×D411yescharacteristic
51C23×D17—24no—
52C23×D17—22no—
53C23×D17—42no—
54C23×D17—42no—
55C23×D17—42no—
56C23×D17—41no—
57C23×D17—41no—
58C23×D17—42no—
59C23×D17—42no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—22no—
65C22×C4×C17—21no—
66C22×C4×C17—24no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17C22⋊C412yes—
70C22×C4×C17C22⋊C412yes—
71C22×C4×C17C22×C411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×C4×C17C2×D412yes—
74C22×C4×C17#4112yes—
75C22×Dic17—22no—
76C22×Dic17—42no—
77C22×Dic17—42no—
78C22×Dic17—42no—
79C2×D68—28no—
80C2×D68—24no—
81C2×D68—44no—
82C2×D68—48no—
83C2×D68—44no—
84C2×D68—44no—
85C2×D68—42no—
86C4⋊C4×C17—24no—
87C4⋊C4×C17—24no—
88C4⋊C4×C17—44no—
89C22⋊C4×C17—42no—
90C22⋊C4×C17—42no—
91C42×C17—82no—
92#63227—48no—
93#63227—48no—
94C2×#3622—44no—
95C2×#3622—42no—
96#9018—171nomaximal, Sylow
97C23×C17—22no—
98C23×C17—22no—
99C23×C17C2×C22⋊C411yescharacteristic
100C23×C17#12011yescharacteristic
101C23×C17#12311yescharacteristic
102C23×C17#13512yes—
103C23×C17AGL(1,ℤ/8)12yes—
104C22×D17—24no—
105C22×D17—42no—
106C22×D17—44no—
107C22×D17—48no—
108C22×D17—48no—
109C22×D17—42no—
110C22×D17—42no—
111C22×D17—42no—
112C22×D17—42no—
113C22×D17—42no—
114C22×D17—44no—
115C22×D17—84no—
116C22×D17—84no—
117C22×D17—82no—
118C22×D17—82no—
119C22×D17—82no—
120C22×D17—82no—
121C2×C4×C17—24no—
122C2×C4×C17—22no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—22no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17—42no—
132C2×C4×C17—42no—
133C2×C4×C17—44no—
134C2×C4×C17—44no—
135C2×C4×C17—44no—
136C2×C4×C17—81no—
137C2×C4×C17C4×D412yes—
138C2×C4×C17#12112yes—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—42no—
142C2×Dic17—84no—
143D68—28no—
144D68—48no—
145D68—44no—
146#3622—82no—
147C2×#394—171no—
148#2313D1711yescharacteristic, complemented
149#1786—172no—
150C2×#324—171no—
151#1294—172no—
152C22×C17—24no—
153C22×C17—24no—
154C22×C17—21no—
155C22×C17—21no—
156C22×C17—21no—
157C22×C17—21no—
158C22×C17#38611yescharacteristic
159C22×C17#41611yescharacteristic
160C22×C17#41811yescharacteristic
161C22×C17#44912yes—
162C22×C17#45412yes—
163C22×C17B(4,𝔽2)12yes—
164C22×C17#48212yes—
165C2×D17—44no—
166C2×D17—44no—
167C2×D17—44no—
168C2×D17—82no—
169C2×D17—82no—
170C2×D17—88no—
171C2×D17—88no—
172C2×D17—82no—
173C2×D17—82no—
174C4×C17—22no—
175C4×C17—22no—
176C4×C17—42no—
177C4×C17—42no—
178C4×C17—84no—
179Dic17—82no—
180C23×D4—171no—
181C22×C4⋊C4C2×D1711yescharacteristic
182C22×C22⋊C4—341no—
183C22×C22⋊C4—341no—
184C2×#102—344no—
185C2×C22⋊C8—172no—
186#394—174no—
187#394—342no—
188#387—24no—
189#381—21no—
190#380C2×D1712yes—
191C4×C22⋊C4—21no—
192C2×#95—171no—
193#324—342no—
194C2×C17—21no—
195C2×C17—21no—
196C2×C17—22no—
197C2×C17#159714yes—
198C2×C17#187511yescharacteristic
199C2×C17#188811yescharacteristic
200C2×C17#188911yescharacteristic
201D17—84no—
202C25—341no—
203C22×D4—172no—
204C22×D4—341no—
205C22×D4—342no—
206C22×D4—342no—
207C22×D4—684no—
208C23×C4—341no—
209C23×C4C22⋊D1711yescharacteristic
210C23×C4C22⋊D1711yescharacteristic
211C23×C4C22×D1711yescharacteristic
212C22×C8—342no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4C4×D1712yes—
216C2×C4⋊C4D6812yes—
217C2×C4⋊C4C22⋊D1714yes—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—344no—
220C2×C22⋊C4—344no—
221C2×C22⋊C4—684no—
222C2×C22⋊C4—684no—
223C2×C42—22no—
224#102—688no—
225C22⋊C8—342no—
226C22⋊C8—342no—
227#95—22no—
228#95—24no—
229#95—22no—
230#95—172no—
231#95—341no—
232C17#901811yescharacteristic, complemented, Sylow
233C24—344no—
234C24—342no—
235C24—682no—
236C24—682no—
237C24—682no—
238C24—681no—
239C24—682no—
240C24—682no—
241C24—681no—
242C24C2×C22⋊D1711yescharacteristic
243C2×D4—348no—
244C2×D4—344no—
245C2×D4—684no—
246C2×D4—684no—
247C2×D4—688no—
248C2×D4—684no—
249C2×D4—682no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—342no—
260C22×C4—682no—
261C22×C4—682no—
262C22×C4—682no—
263C22×C4#6322712yes—
264C22×C4#6322712yes—
265C22×C4C2×C4×D1711yescharacteristic
266C22×C4C2×D6811yescharacteristic
267C22×C4#6325212yes—
268C22×C4C2×C22⋊D1712yes—
269C2×C8—684no—
270C2×C8—682no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—688no—
277C22⋊C4—688no—
278C42—82no—
279C23—22no—
280C23—22no—
281C23—344no—
282C23—682no—
283C23—684no—
284C23—682no—
285C23—682no—
286C23—688no—
287C23—688no—
288C23—682no—
289C23—684no—
290C23—682no—
291C23—682no—
292C23—1362no—
293C23—1364no—
294C23—1364no—
295C23—1362no—
296C23—1362no—
297C23—1362no—
298C23C2×#6322711yescharacteristic
299C23#1058761511yescharacteristic
300C23#1058763512yes—
301C23#1058763612yes—
302C23#1058763911yescharacteristic
303D4—348no—
304D4—688no—
305D4—684no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—81no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—1364no—
326C2×C4C4×C22⋊D1712yes—
327C2×C4#1058761612yes—
328C8—1362no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—24no—
335C22—684no—
336C22—684no—
337C22—684no—
338C22—1362no—
339C22—1362no—
340C22—1368no—
341C22—1362no—
342C22—1362no—
343C22—1368no—
344C22#1176268811yescharacteristic
345C22#1176280311yescharacteristic
346C22#1176281311yescharacteristic
347C22#1176295412yes—
348C22#1176298212yes—
349C22#1176299612yes—
350C22#1176313312yes—
351C4—22no—
352C4—22no—
353C4—42no—
354C4—42no—
355C4—84no—
356C4—1362no—
357C2—21no—
358C2—21no—
359C2—22no—
360C2—1364no—
361C2#1452771314yes—
362C2#1452910611yescharacteristic
363C2#1452918711yescharacteristic
364C2#1452919211yescharacteristic
subgroupG/H
centerC23#10587639
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2313C2
socleC23×C17#120
prime radicalC23×D68C22
cosocleC23×C4×C17C23
hypercenter#2313D17
hypocenterC17#9018
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2313

Complements

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

NH ≅ G/Ntype
3. C2×#11762535C2semidirect
4. #14526662C2semidirect
5. C17×#2313C2semidirect
12. #11762726C4semidirect
148. #2313D17semidirect
232. C17#9018semidirect

Maximal subgroups

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

subgroupindex
C2×#117627262
#145287142
C2×#117625352
#145266622
C17×#23132
#901817