Finite Group Databasebeta

← #16931678

Subgroups — #16931678

Order 4352.

538 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 117 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#14537807C211yesmaximal, characteristic
2#14533953C211yesmaximal, characteristic
3#14528773C211yesmaximal, characteristic
4#14527466C211yesmaximal, characteristic, complemented
5#14527464C211yesmaximal, characteristic, complemented
6#14526687C211yesmaximal, characteristic, complemented
7C17×#1303C211yesmaximal, characteristic, complemented
8C2×#10587555C2211yescharacteristic
9C2×C4×D68C2211yescharacteristic
10C8×D68—22no—
11#11762874—22no—
12#11762862C2211yescharacteristic
13#11762833C2211yescharacteristic
14C2×#10587494—21no—
15C22⋊C8×D17—22no—
16#11762722C414yes—
17#11762711—21no—
18#11762675—21no—
19#11762666—22no—
20#11762577—21no—
21#11762557—22no—
22#11762557—22no—
23C17×#431C2211yescharacteristic
24C17×#423C2211yescharacteristic
25C4×C4⋊C4×C17C2211yescharacteristic
26C22×D68D411yescharacteristic
27C22×C4×D17D412yes—
28C22×Dic34C2×C412yes—
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—21no—
37C2×C4⋊Dic17D411yescharacteristic
38C2×#63223—21no—
39C2×C8×D17—24no—
40C4×D68—21no—
41C4×D68—21no—
42C4×D68—21no—
43C4×D68—21no—
44#10587555—21no—
45#10587555—21no—
46#10587555—21no—
47#10587555—21no—
48C17×C4⋊C8—21no—
49C17×C4⋊C8—21no—
50C17×C4⋊C8D412yes—
51C17×C22⋊C8—21no—
52C4×C8×C17—21no—
53C17×#95C2×C412yes—
54#10587502—24no—
55C22⋊#3622—21no—
56#10587494—42no—
57#10587491—24no—
58C4⋊#3622—21no—
59C4⋊#3622—21no—
60C4⋊#3622D412yes—
61C4×#3622—21no—
62C23×D17C22⋊C412yes—
63C22×C4×C17—22no—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—22no—
68C22×Dic17C22⋊C412yes—
69C2×D68—21no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—21no—
73C2×D68SD1612yes—
74C2×D68Q1612yes—
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—44no—
78C2×Dic34—22no—
79C2×Dic34—22no—
80C2×Dic34—82no—
81M16×C17—41no—
82M16×C17—41no—
83C2×C8×C17—22no—
84C2×C8×C17—41no—
85C4⋊C4×C17—42no—
86C42×C17—41no—
87C42×C17—41no—
88C42×C17C2×D411yescharacteristic
89C42×C17C2×D411yescharacteristic
90C42×C17#4111yescharacteristic
91C42×C17#4111yescharacteristic
92#63227—22no—
93#63227—22no—
94#63227—84no—
95C4⋊Dic17—21no—
96C4⋊Dic17—21no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101#63223—41no—
102#63223—41no—
103C2×#3622—22no—
104C2×#3622—41no—
105C8×D17—84no—
106#9391—171nomaximal, Sylow
107C23×C17C2×C22⋊C411yescharacteristic
108C22×D17—22no—
109C22×D17—22no—
110C22×D17—42no—
111C22×D17—42no—
112C22×D17#10314yes—
113C2×C4×C17—22no—
114C2×C4×C17—41no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17C4×D411yescharacteristic
119C2×C4×C17C4×D411yescharacteristic
120C2×C4×C17#12011yescharacteristic
121C2×C4×C17#12111yescharacteristic
122C2×C4×C17#12111yescharacteristic
123C2×C4×C17#12311yescharacteristic
124C2×C4×C17C2×SD1611yescharacteristic
125C2×C4×C17C2×Q1611yescharacteristic
126C2×C4×C17#13511yescharacteristic
127C2×C4×C17#13511yescharacteristic
128C2×Dic17—22no—
129C2×Dic17—42no—
130C2×Dic17—44no—
131C2×Dic17—42no—
132D68—22no—
133D68—22no—
134D68—22no—
135D68—22no—
136C4×D17—44no—
137C4×D17—82no—
138Dic34—82no—
139Dic34—82no—
140C8×C17—42no—
141C8×C17—81no—
142#3622—42no—
143#3622—81no—
144#2972—171no—
145#2563—171no—
146#1800—171no—
147#1548—171no—
148#1548—171no—
149#1303—171no—
150#1303D1711yescharacteristic, complemented
151C22×C17—21no—
152C22×C17—21no—
153C22×C17#38611yescharacteristic
154C22×C17C2×#10311yescharacteristic
155C22×C17#41611yescharacteristic
156C2×D17—24no—
157C2×D17—42no—
158C2×D17—42no—
159C2×D17—44no—
160C2×D17—82no—
161C4×C17—21no—
162C4×C17—21no—
163C4×C17—21no—
164C4×C17—82no—
165C4×C17#45512yes—
166C4×C17#46511yescharacteristic
167C4×C17#46511yescharacteristic
168C4×C17#46711yescharacteristic
169C4×C17#46711yescharacteristic
170Dic17—42no—
171Dic17—82no—
172C2×C4⋊Q8—171no—
173C2×C4×D4—171no—
174C8×D4—342no—
175C8×D4—342no—
176#431—171no—
177#431C2×D1711yescharacteristic
178#423—171no—
179#423C2×D1711yescharacteristic
180C2×#100—341no—
181C2×C22⋊C8—342no—
182#393—174no—
183#390—341no—
184#382—341no—
185#380—342no—
186C4×C4⋊C4C2×D1711yescharacteristic
187#343—341no—
188#333—342no—
189#333—342no—
190C2×C17—21no—
191C2×C17—21no—
192C2×C17#188511yescharacteristic
193C2×C17#189211yescharacteristic
194C2×C17#191411yescharacteristic
195D17—44no—
196C22×Q8—172no—
197C22×D4—171no—
198C23×C4—172no—
199C2×M16—21no—
200C2×M16—341no—
201C22×C8—344no—
202C4⋊Q8—341no—
203C4⋊Q8—341no—
204C4⋊Q8—341no—
205C4⋊Q8—341no—
206C4×D4—341no—
207C4×D4—341no—
208C4×D4—341no—
209C4×D4—341no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—341no—
213C2×C4⋊C4—341no—
214C2×C22⋊C4—172no—
215C2×C22⋊C4—344no—
216C2×C42—21no—
217C2×C42C22×D1711yescharacteristic
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—172no—
221C4⋊C8—341no—
222C4⋊C8—341no—
223C4⋊C8D6812yes—
224#100—682no—
225C22⋊C8—21no—
226C22⋊C8—344no—
227C22⋊C8—341no—
228C4×C8—21no—
229C4×C8—341no—
230#95—344no—
231#95C4×D1712yes—
232C17#939111yescharacteristic, complemented, Sylow
233C24—172no—
234C2×Q8—342no—
235C2×Q8—342no—
236C2×Q8—1362no—
237C2×D4—172no—
238C2×D4—172no—
239C2×D4—341no—
240C2×D4—341no—
241C2×D4—341no—
242C2×D4—341no—
243C22×C4—22no—
244C22×C4—172no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—342no—
248C22×C4—684no—
249C22×C4C2×C4×D1711yescharacteristic
250C22×C4D4×D1711yescharacteristic
251C22×C4D4×D1711yescharacteristic
252M16—41no—
253M16—41no—
254M16—681no—
255M16—681no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—342no—
259C2×C8—681no—
260C2×C8—1364no—
261C4⋊C4—42no—
262C4⋊C4—341no—
263C4⋊C4—341no—
264C4⋊C4—681no—
265C4⋊C4—681no—
266C4⋊C4—681no—
267C4⋊C4—681no—
268C22⋊C4—342no—
269C22⋊C4—342no—
270C22⋊C4—1364no—
271C42—41no—
272C42—41no—
273C42C2×D6811yescharacteristic
274C42#6325211yescharacteristic
275C42D4×D1711yescharacteristic
276C42#6325611yescharacteristic
277C23—174no—
278C23—342no—
279C23—342no—
280C23—682no—
281C23—682no—
282C23C22⋊C4×D1711yescharacteristic
283Q8—1362no—
284Q8—1362no—
285D4—342no—
286D4—342no—
287D4—342no—
288D4—342no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—342no—
295C2×C4—684no—
296C2×C4—682no—
297C2×C4—684no—
298C2×C4—682no—
299C2×C4—1362no—
300C2×C4C4×D6811yescharacteristic
301C2×C4#1058756811yescharacteristic
302C2×C4#1058757011yescharacteristic
303C2×C4#1058757911yescharacteristic
304C2×C4#1058758011yescharacteristic
305C2×C4#1058758111yescharacteristic
306C2×C4#1058759811yescharacteristic
307C2×C4SD16×D1711yescharacteristic
308C2×C4#1058760211yescharacteristic
309C2×C4Q16×D1711yescharacteristic
310C8—42no—
311C8—81no—
312C8—682no—
313C8—1361no—
314C22—21no—
315C22—21no—
316C22—344no—
317C22—684no—
318C22—682no—
319C22—682no—
320C22—1362no—
321C22#1176268711yescharacteristic
322C22#103×D1711yescharacteristic
323C22#1176280211yescharacteristic
324C4—21no—
325C4—21no—
326C4—21no—
327C4—82no—
328C4—682no—
329C4—1362no—
330C4#1176298812yes—
331C4#1176303711yescharacteristic
332C4#1176303911yescharacteristic
333C4#1176304911yescharacteristic
334C4#1176305111yescharacteristic
335C2—21no—
336C2—21no—
337C2—684no—
338C2#1452917111yescharacteristic
339C2#1452920611yescharacteristic
340C2#1452933011yescharacteristic
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#9391
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. #14527466C2semidirect
5. #14527464C2semidirect
6. #14526687C2semidirect
7. C17×#1303C2semidirect
150. #1303D17semidirect
232. C17#9391semidirect

Maximal subgroups

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

subgroupindex
#145378072
#145339532
#145287732
#145274662
#145274642
#145266872
C17×#13032
#939117