Finite Group Databasebeta

← #16931738

Subgroups — #16931738

Order 4352.

570 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 109 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#14537824C211yesmaximal, characteristic
2#14533733C211yesmaximal, characteristic
3#14528773C211yesmaximal, characteristic
4#14527470C211yesmaximal, characteristic, complemented
5#14527468C211yesmaximal, characteristic, complemented
6#14526687C211yesmaximal, characteristic, complemented
7C17×#1303C211yesmaximal, characteristic, complemented
8C2×#10587555C2211yescharacteristic
9C2×#10587583—22no—
10C2×C4×D68C2211yescharacteristic
11#11762888—22no—
12#11762886—22no—
13#11762862C2211yescharacteristic
14#11762833C2211yescharacteristic
15C2×#10587494—21no—
16#11762771—22no—
17#11762746C414yes—
18#11762711—21no—
19#11762679—21no—
20#11762577—21no—
21#11762556—22no—
22#11762554—22no—
23C17×#431C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C4×C4⋊C4×C17C2211yescharacteristic
26C22×D68D411yescharacteristic
27C22×C4×D17D412yes—
28C22×Dic34—21no—
29C2×M16×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C42×C17—21no—
32C2×C42×C17C2311yescharacteristic
33C2×#63227—24no—
34C2×#63227C2×C412yes—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—24no—
37C2×C4⋊Dic17C2×C412yes—
38C2×C4⋊Dic17D411yescharacteristic
39C2×#63223—21no—
40C2×#63218—24no—
41#10587583—48no—
42C4×D68—21no—
43C4×D68—21no—
44C4×D68—21no—
45C4×D68—21no—
46#10587555—21no—
47#10587555—21no—
48#10587555—21no—
49#10587555—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8—21no—
52C17×C4⋊C8D412yes—
53C17×C22⋊C8—21no—
54C4×C8×C17—21no—
55C17×#95C2×C412yes—
56C22⋊#3622—21no—
57#10587494—42no—
58#10587491—24no—
59C4⋊#3622—21no—
60C4⋊#3622—21no—
61C4⋊#3622D412yes—
62C4×#3622—21no—
63C23×D17C22⋊C412yes—
64C22×C4×C17—22no—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—22no—
69C22×Dic17C22⋊C412yes—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68—21no—
74C2×D68—21no—
75C2×D68—21no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—44no—
79C2×Dic34—44no—
80C2×Dic34—41no—
81C2×Dic34—41no—
82M16×C17—41no—
83M16×C17—41no—
84C2×C8×C17—22no—
85C2×C8×C17—41no—
86C4⋊C4×C17—24no—
87C42×C17—41no—
88C42×C17—41no—
89C42×C17C2×D411yescharacteristic
90C42×C17C2×D411yescharacteristic
91C42×C17#4111yescharacteristic
92C42×C17#4111yescharacteristic
93#63227—22no—
94#63227—22no—
95#63227—48no—
96C4⋊Dic17—21no—
97C4⋊Dic17—21no—
98C4⋊Dic17—22no—
99C4⋊Dic17—22no—
100C4⋊Dic17—48no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103#63223—41no—
104#63223—41no—
105C2×#3622—22no—
106C2×#3622—41no—
107#63218—84no—
108#9403—171nomaximal, Sylow
109C23×C17C2×C22⋊C411yescharacteristic
110C22×D17—22no—
111C22×D17—22no—
112C22×D17—22no—
113C22×D17—42no—
114C22×D17—42no—
115C2×C4×C17—22no—
116C2×C4×C17—24no—
117C2×C4×C17—41no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17C4×D411yescharacteristic
121C2×C4×C17C4×D411yescharacteristic
122C2×C4×C17#12011yescharacteristic
123C2×C4×C17#12111yescharacteristic
124C2×C4×C17#12111yescharacteristic
125C2×C4×C17#12311yescharacteristic
126C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C17#13711yescharacteristic
128C2×C4×C17#13711yescharacteristic
129C2×C4×C17#13711yescharacteristic
130C2×Dic17—22no—
131C2×Dic17—42no—
132C2×Dic17—44no—
133C2×Dic17—42no—
134D68—41no—
135D68—41no—
136D68—41no—
137D68—41no—
138C4×D17—44no—
139C4×D17—82no—
140Dic34—82no—
141Dic34—82no—
142C8×C17—42no—
143C8×C17—81no—
144#3622—42no—
145#3622—81no—
146#2975—171no—
147#2542—171no—
148#1800—171no—
149#1549—171no—
150#1549—171no—
151#1303—171no—
152#1303D1711yescharacteristic, complemented
153C22×C17—21no—
154C22×C17—21no—
155C22×C17#38611yescharacteristic
156C22×C17#41711yescharacteristic
157C22×C17#41911yescharacteristic
158C2×D17—44no—
159C2×D17—42no—
160C2×D17—42no—
161C2×D17—42no—
162C2×D17—82no—
163C4×C17—21no—
164C4×C17—21no—
165C4×C17—21no—
166C4×C17—44no—
167C4×C17#45312yes—
168C4×C17#46811yescharacteristic
169C4×C17#46811yescharacteristic
170C4×C17#47011yescharacteristic
171C4×C17#47011yescharacteristic
172Dic17—42no—
173Dic17—82no—
174C2×C4⋊Q8—171no—
175C2×C22⋊Q8—342no—
176C2×C4×D4—171no—
177#436—342no—
178#436—342no—
179#431—171no—
180#431C2×D1711yescharacteristic
181#423—171no—
182#423C2×D1711yescharacteristic
183C2×#100—341no—
184#407—342no—
185#399—174no—
186#390—341no—
187#384—341no—
188C4×C4⋊C4C2×D1711yescharacteristic
189#343—341no—
190#332—342no—
191#332—342no—
192C2×C17—21no—
193C2×C17—21no—
194C2×C17#188711yescharacteristic
195C2×C17#189411yescharacteristic
196C2×C17#191511yescharacteristic
197D17—82no—
198C22×Q8—341no—
199C22×D4—171no—
200C23×C4—172no—
201C2×M16—21no—
202C2×M16—344no—
203C2×M16—341no—
204C4⋊Q8—341no—
205C4⋊Q8—341no—
206C4⋊Q8—341no—
207C4⋊Q8—341no—
208C22⋊Q8—688no—
209C4×D4—341no—
210C4×D4—341no—
211C4×D4—341no—
212C4×D4—341no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—172no—
216C2×C4⋊C4—344no—
217C2×C4⋊C4—341no—
218C2×C22⋊C4—172no—
219C2×C22⋊C4—344no—
220C2×C42—21no—
221C2×C42C22×D1711yescharacteristic
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—172no—
225C4⋊C8—341no—
226C4⋊C8—341no—
227C4⋊C8D6812yes—
228#100—682no—
229C22⋊C8—21no—
230C22⋊C8—344no—
231C22⋊C8—341no—
232C4×C8—21no—
233C4×C8—341no—
234#95C4×D1712yes—
235C17#940311yescharacteristic, complemented, Sylow
236C24—172no—
237C2×Q8—684no—
238C2×Q8—681no—
239C2×Q8—681no—
240C2×D4—341no—
241C2×D4—341no—
242C2×D4—341no—
243C2×D4—341no—
244C2×D4—341no—
245C2×D4—341no—
246C22×C4—22no—
247C22×C4—172no—
248C22×C4—342no—
249C22×C4—342no—
250C22×C4—342no—
251C22×C4—684no—
252C22×C4C2×C4×D1711yescharacteristic
253C22×C4D4×D1711yescharacteristic
254C22×C4D4×D1711yescharacteristic
255M16—41no—
256M16—41no—
257M16—681no—
258M16—681no—
259M16—1364no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—342no—
263C2×C8—681no—
264C4⋊C4—24no—
265C4⋊C4—341no—
266C4⋊C4—341no—
267C4⋊C4—342no—
268C4⋊C4—342no—
269C4⋊C4—688no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C22⋊C4—342no—
273C22⋊C4—342no—
274C22⋊C4—688no—
275C42—41no—
276C42—41no—
277C42C2×D6811yescharacteristic
278C42#6325211yescharacteristic
279C42D4×D1711yescharacteristic
280C42#6325611yescharacteristic
281C23—342no—
282C23—342no—
283C23—342no—
284C23—682no—
285C23—682no—
286C23C22⋊C4×D1711yescharacteristic
287Q8—1362no—
288Q8—1362no—
289D4—681no—
290D4—681no—
291D4—681no—
292D4—681no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—342no—
299C2×C4—684no—
300C2×C4—682no—
301C2×C4—684no—
302C2×C4—682no—
303C2×C4—1362no—
304C2×C4C4×D6811yescharacteristic
305C2×C4#1058756811yescharacteristic
306C2×C4#1058757011yescharacteristic
307C2×C4#1058757911yescharacteristic
308C2×C4#1058758011yescharacteristic
309C2×C4#1058758111yescharacteristic
310C2×C4#1058759711yescharacteristic
311C2×C4#1058760111yescharacteristic
312C2×C4#1058760111yescharacteristic
313C2×C4#1058760411yescharacteristic
314C8—42no—
315C8—81no—
316C8—682no—
317C8—1361no—
318C22—21no—
319C22—21no—
320C22—684no—
321C22—682no—
322C22—682no—
323C22—682no—
324C22—1362no—
325C22#1176268711yescharacteristic
326C22#1176280711yescharacteristic
327C22#1176281611yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—44no—
332C4—682no—
333C4—1362no—
334C4#1176297812yes—
335C4#1176305411yescharacteristic
336C4#1176305611yescharacteristic
337C4#1176306611yescharacteristic
338C4#1176306811yescharacteristic
339C2—21no—
340C2—21no—
341C2—1362no—
342C2#1452918111yescharacteristic
343C2#1452921611yescharacteristic
344C2#1452933511yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1303C2
socleC22×C17#386
prime radicalC22×D68D4
cosocleC2×C42×C17C23
hypercenter#1303D17
hypocenterC17#9403
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
4. #14527470C2semidirect
5. #14527468C2semidirect
6. #14526687C2semidirect
7. C17×#1303C2semidirect
152. #1303D17semidirect
235. C17#9403semidirect

Maximal subgroups

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

subgroupindex
#145378242
#145337332
#145287732
#145274702
#145274682
#145266872
C17×#13032
#940317