Finite Group Databasebeta

← #17950340

Subgroups — #17950340

Order 4352.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763827C211yesmaximal, characteristic
2#14545231C211yesmaximal, characteristic
3#14543607C214yesmaximal, complemented
4#14537415C211yesmaximal, characteristic
5#11763827—21no—
6#11763827—21no—
7#11763827C412yescomplemented
8#11763827C2212yes—
9C2×#10587544—21no—
10C2×#10587543—21no—
11C2×C4×#3624C412yescomplemented
12#11763754—21no—
13#11763753—21no—
14#11763752—21no—
15#11763751—21no—
16C4×C4⋊#595C412yes—
17C4×C22⋊#595C412yes—
18#11763745C2212yes—
19C2×C42×D17C2211yescharacteristic
20#11762864—21no—
21#11762864—21no—
22#11762860C412yescomplemented
23#11762860C2212yes—
24#11762832—21no—
25#11762772—21no—
26C4×#63223C411yescharacteristic, complemented
27C4×M16×C17C411yescharacteristic, complemented
28C22×C4×D17C2×C411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×C4×D17D411yescharacteristic
31C2×C22⋊#595—22no—
32C22×#3624—22no—
33C2×C4⋊#595—22no—
34C2×C4×#595C2×C414yes—
35C2×M16×C17—21no—
36C2×C42×C17C2×C411yescharacteristic
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17Q811yescharacteristic
40C2×#63223—21no—
41C42×D17C2×C411yescharacteristic
42C42×D17C2×C411yescharacteristic
43C42×D17C2×C411yescharacteristic
44C42×D17D411yescharacteristic
45C42×D17D411yescharacteristic
46C42×D17D411yescharacteristic
47C42×D17Q811yescharacteristic
48C42×D17C2311yescharacteristic
49(C2×#595)⋊C4—22no—
50(C2×#595)⋊C4—22no—
51(C2×#595)⋊C4—22no—
52(C2×#595)⋊C4—22no—
53#10587544—22no—
54#10587544—22no—
55#10587543—22no—
56#10587543—22no—
57C4×#3624—22no—
58C4×#3624C2×C414yes—
59C17×#97—21no—
60C4×C8×C17C2×C412yes—
61#10587491—22no—
62#10587491—22no—
63#10587485—22no—
64#10587485—22no—
65#10587474—21no—
66C4×#3622C2×C412yes—
67C23×D17C22⋊C411yescharacteristic
68C22×#595—24no—
69C22×C4×C17C4211yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C4⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C4⋊C411yescharacteristic
74C22×Dic17C4⋊C411yescharacteristic
75C2×C4×D17—24no—
76C2×C4×D17—42no—
77C2×C4×D17—42no—
78C2×C4×D17C4211yescharacteristic
79C2×C4×D17C22⋊C411yescharacteristic
80C2×C4×D17C22⋊C411yescharacteristic
81C2×C4×D17C22⋊C411yescharacteristic
82C2×C4×D17C22⋊C411yescharacteristic
83C2×C4×D17C22⋊C411yescharacteristic
84C2×C4×D17C4⋊C411yescharacteristic
85C2×C4×D17C4⋊C411yescharacteristic
86C2×C4×D17C4⋊C411yescharacteristic
87C2×C4×D17C22×C411yescharacteristic
88C2×C4×D17C2×D411yescharacteristic
89C2×C4×D17C2×D411yescharacteristic
90C22⋊#595—44no—
91C2×#3624—24no—
92C2×#3624—44no—
93C4⋊#595—44no—
94C4×#595—28no—
95M16×C17—42no—
96C2×C8×C17—22no—
97C42×C17C4211yescharacteristic
98C42×C17C22⋊C411yescharacteristic
99C42×C17C4⋊C411yescharacteristic
100C42×C17C22×C411yescharacteristic
101C4×Dic17C4211yescharacteristic
102C4×Dic17C22⋊C411yescharacteristic
103C4×Dic17C22⋊C411yescharacteristic
104C4×Dic17C22⋊C411yescharacteristic
105C4×Dic17C4⋊C411yescharacteristic
106C4×Dic17C4⋊C411yescharacteristic
107C4×Dic17C4⋊C411yescharacteristic
108C4×Dic17C4⋊C411yescharacteristic
109C4×Dic17C4⋊C411yescharacteristic
110C4×Dic17C22×C411yescharacteristic
111C4×Dic17C2×D411yescharacteristic
112C4×Dic17C2×Q811yescharacteristic
113#63223—42no—
114C2×#3622—22no—
115#7858—171nomaximal, Sylow
116C23×C17#9511yescharacteristic
117C22×D17—22no—
118C22×D17—22no—
119C22×D17—41no—
120C22×D17#9511yescharacteristic
121C22×D17C2×C22⋊C411yescharacteristic
122C2×#595—44no—
123C2×#595—44no—
124C2×#595—44no—
125C2×C4×C17—22no—
126C2×C4×C17—22no—
127C2×C4×C17—41no—
128C2×C4×C17#9511yescharacteristic
129C2×C4×C17#9511yescharacteristic
130C2×C4×C17#9511yescharacteristic
131C2×C4×C17C2×C4211yescharacteristic
132C2×C4×C17C2×C22⋊C411yescharacteristic
133C2×C4×C17C2×C4⋊C411yescharacteristic
134C2×Dic17—22no—
135C2×Dic17—22no—
136C2×Dic17—22no—
137C2×Dic17#9511yescharacteristic
138C2×Dic17#9511yescharacteristic
139C2×Dic17#9511yescharacteristic
140C2×Dic17C2×C22⋊C411yescharacteristic
141C2×Dic17C2×C4⋊C411yescharacteristic
142C2×Dic17C2×C4⋊C411yescharacteristic
143C4×D17—22no—
144C4×D17—22no—
145C4×D17—22no—
146C4×D17—22no—
147C4×D17—22no—
148C4×D17—41no—
149C4×D17—41no—
150C4×D17—41no—
151C4×D17—41no—
152C4×D17—41no—
153C4×D17—41no—
154C4×D17#13114yes—
155#3624—44no—
156C8×C17—42no—
157#3622—42no—
158#2920—171no—
159C2×#431—171no—
160#2283—171no—
161#1274—174no—
162C22×C17—21no—
163C22×C17—21no—
164C22×C17H(3,ℤ/4)12yes—
165C22×C17C2×#9511yescharacteristic
166C2×D17—21no—
167C2×D17—21no—
168C2×D17—21no—
169C2×D17—21no—
170C2×D17—21no—
171C2×D17—44no—
172C2×D17#40812yes—
173#595—84no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17#40512yes—
180Dic17—21no—
181Dic17—21no—
182Dic17—21no—
183Dic17—21no—
184Dic17H(3,ℤ/4)12yes—
185Dic17#42412yes—
186C22×C42—171no—
187#432—341no—
188#432—341no—
189#431—172no—
190#431—172no—
191#431—172no—
192#431—172no—
193#431—341no—
194#431—341no—
195#423—341no—
196C2×C4⋊C8—341no—
197#407—341no—
198C2×C22⋊C8—341no—
199C4×M16—171no—
200C4×M16#59511yescharacteristic, complemented
201C2×C4×C8—172no—
202#383—341no—
203#382—341no—
204#381—341no—
205#380—341no—
206C4×C4⋊C4—172no—
207C4×C22⋊C4—172no—
208#376—172no—
209C2×C17—22no—
210C2×C17#172911yescharacteristic
211C2×C17C2×H(3,ℤ/4)11yescharacteristic
212C2×C17C2×H(3,ℤ/4)11yescharacteristic
213D17—22no—
214D17—41no—
215C23×C4—171no—
216C23×C4—171no—
217C23×C4—171no—
218C2×M16—21no—
219C2×M16—341no—
220C22×C8—342no—
221C2×C4⋊C4—342no—
222C2×C22⋊C4—342no—
223C2×C42—171no—
224C2×C42—171no—
225C2×C42—171no—
226C2×C42—171no—
227C2×C42—171no—
228C2×C42—171no—
229C2×C42—171no—
230C2×C42—171no—
231C2×C42—171no—
232C2×C42—171no—
233C2×C42—171no—
234C2×C42—174no—
235C2×C42C2×#59511yescharacteristic
236C4⋊C8—342no—
237C4⋊C8—342no—
238C4⋊C8—342no—
239C4⋊C8—342no—
240C22⋊C8—342no—
241C22⋊C8—342no—
242C22⋊C8—342no—
243C22⋊C8—342no—
244#97—21no—
245#97—341no—
246C4×C8—174no—
247C4×C8—172no—
248C4×C8—342no—
249C4×C8C2×#59512yes—
250#95—342no—
251#95—342no—
252#95—342no—
253#95—342no—
254C17#785811yescharacteristic, complemented, Sylow
255C24—171no—
256C22×C4—171no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—171no—
262C22×C4—171no—
263C22×C4—171no—
264C22×C4—171no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—171no—
268C22×C4—171no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—344no—
272C22×C4—344no—
273C22×C4—682no—
274C22×C4—682no—
275C22×C4C4×#59511yescharacteristic
276C22×C4C4⋊#59511yescharacteristic
277C22×C4C22⋊#59511yescharacteristic
278M16—42no—
279M16—682no—
280C2×C8—22no—
281C2×C8—344no—
282C2×C8—342no—
283C2×C8—684no—
284C4⋊C4—684no—
285C22⋊C4—684no—
286C42—171no—
287C42—171no—
288C42—171no—
289C42—171no—
290C42—171no—
291C42—171no—
292C42—171no—
293C42—171no—
294C42—171no—
295C42—171no—
296C42—171no—
297C42—171no—
298C42—348no—
299C42C4×#59511yescharacteristic
300C42C4⋊#59511yescharacteristic
301C42C22⋊#59511yescharacteristic
302C42C22×#59511yescharacteristic
303C23—171no—
304C23—171no—
305C23—342no—
306C23—342no—
307C23—681no—
308C23(C2×#595)⋊C411yescharacteristic
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—171no—
313C2×C4—171no—
314C2×C4—174no—
315C2×C4—171no—
316C2×C4—171no—
317C2×C4—171no—
318C2×C4—171no—
319C2×C4—342no—
320C2×C4—342no—
321C2×C4—342no—
322C2×C4—342no—
323C2×C4—342no—
324C2×C4—342no—
325C2×C4—342no—
326C2×C4—342no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—684no—
334C2×C4—684no—
335C2×C4—684no—
336C2×C4(C2×#595)⋊C411yescharacteristic
337C2×C4(C2×#595)⋊C411yescharacteristic
338C2×C4(C2×#595)⋊C411yescharacteristic
339C2×C4C2×C4×#59511yescharacteristic
340C2×C4C2×C4⋊#59511yescharacteristic
341C2×C4C2×C22⋊#59511yescharacteristic
342C8—42no—
343C8—682no—
344C8—684no—
345C22—21no—
346C22—21no—
347C22—172no—
348C22—341no—
349C22—341no—
350C22—341no—
351C22—341no—
352C22—341no—
353C22—684no—
354C22#1176367612yes—
355C22C2×(C2×#595)⋊C411yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—172no—
362C4—172no—
363C4—341no—
364C4—341no—
365C4—341no—
366C4—341no—
367C4—1364no—
368C4#1176377512yes—
369C2—22no—
370C2—342no—
371C2—681no—
372C2#1454445211yescharacteristic
373C2C2×#1176367611yescharacteristic
374C2#1454445611yescharacteristic
subgroupG/H
centerC2×C4C2×C4×#595
commutatorC2×C4×C17C2×C42
FrattiniC42C22×#595
FittingC4×M16×C17C4
socleC22×C17C2×#95
prime radicalC23×D17C22⋊C4
cosocleC42×D17C23
hypercenterC4×M16#595
hypocenterC17#7858
LCSG ▷ C2×C4×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
3. #14543607C2semidirect
7. #11763827C4semidirect
11. C2×C4×#3624C4semidirect
22. #11762860C4semidirect
26. C4×#63223C4semidirect
27. C4×M16×C17C4semidirect
200. C4×M16#595semidirect
254. C17#7858semidirect

Maximal subgroups

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

subgroupindex
C2×#117638272
#145452312
#145436072
#145374152
#785817