Finite Group Databasebeta

← #16929014

Subgroups — #16929014

Order 4352.

626 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#10587621C211yesmaximal, characteristic
2#14528850C212yesmaximal, complemented
3#14527102C211yesmaximal, characteristic
4#14526631C212yesmaximal, complemented
5C17×#2972C211yesmaximal, characteristic
6C2×#10587621C411yescharacteristic
7C22×C4×Dic17C2211yescharacteristic
8C2×C22⋊#3622C2212yes—
9#11762697—21no—
10#11762690—21no—
11#11762689C411yescharacteristic
12#11762681C411yescharacteristic
13#11762670C411yescharacteristic
14C4×#63225—21no—
15C4×C22⋊Dic17—21no—
16C4×C22⋊Dic17C2211yescharacteristic
17#11762563—24no—
18#11762535—21no—
19#11762533—21no—
20C2×C4×D4×C17C2211yescharacteristic
21C8×D4×C17—24no—
22C17×#431—21no—
23C2×C17×C22⋊C8—21no—
24C2×C17×C22⋊C8C2212yes—
25C22×D4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17C2311yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C22×C8×C17—22no—
31C22×C8×C17D414yes—
32C4×D4×C17—22no—
33C4×D4×C17—22no—
34C2×C4⋊C4×C17D411yescharacteristic
35C2×C22⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17D411yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17C2×C411yescharacteristic
41C2×C22⋊Dic17C2×C411yescharacteristic
42#10587621—22no—
43#10587621C814yescomplemented
44C2×#63225—21no—
45C2×#63225—21no—
46C2×C4×Dic17—21no—
47C2×C4×Dic17—22no—
48C2×C4×Dic17—21no—
49C2×C4×Dic17—21no—
50C2×C4×Dic17—22no—
51C22×#3622—22no—
52C17×C4⋊C8—22no—
53C17×C22⋊C8—22no—
54C17×C22⋊C8—22no—
55C17×C22⋊C8—22no—
56C4×C8×C17—22no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60#10587502—21no—
61#10587502—21no—
62C22⋊#3622—22no—
63C22⋊#3622—22no—
64C24×C17C22⋊C411yescharacteristic
65C24×C17C22×C411yescharacteristic
66C2×D4×C17—22no—
67C2×D4×C17—22no—
68C2×D4×C17C2×C812yes—
69C2×D4×C17M1612yes—
70C22×C4×C17—21no—
71C22×C4×C17—21no—
72C22×C4×C17—42no—
73C22×C4×C17—42no—
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D412yes—
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17#4112yes—
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—21no—
84C22×Dic17—22no—
85C22×Dic17—22no—
86C22×Dic17—21no—
87C22×Dic17—22no—
88C22×Dic17—21no—
89C22×Dic17—42no—
90C22×Dic17C2×C812yescomplemented
91C2×C8×C17—24no—
92C2×C8×C17—44no—
93C2×C8×C17—82no—
94C4⋊C4×C17—22no—
95C22⋊C4×C17—22no—
96C22⋊C4×C17—22no—
97C42×C17—21no—
98C42×C17—21no—
99C22⋊Dic17—24no—
100C22⋊Dic17—21no—
101C22⋊Dic17—22no—
102C22⋊Dic17—42no—
103C22⋊Dic17C2×C812yescomplemented
104#63225—24no—
105#63225—42no—
106C4×Dic17—42no—
107C4×Dic17—41no—
108C4×Dic17—42no—
109C4×Dic17—41no—
110C2×#3622—44no—
111C2×#3622—42no—
112#8966—171nomaximal, Sylow
113C23×C17—21no—
114C23×C17—21no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17C22⋊C812yes—
120C23×C17C2×C22⋊C411yescharacteristic
121C23×C17C4×D412yes—
122C23×C17C22×C811yescharacteristic
123C23×C17C2×M1611yescharacteristic
124C23×C17#13112yes—
125D4×C17—24no—
126D4×C17—24no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—24no—
130C2×C4×C17—22no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—41no—
134C2×C4×C17—42no—
135C2×C4×C17—42no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—81no—
139C2×C4×C17C22⋊C812yes—
140C2×C4×C17#12011yescharacteristic
141C2×C4×C17#12112yes—
142C2×C4×C17#12311yescharacteristic
143C2×Dic17—22no—
144C2×Dic17—24no—
145C2×Dic17—22no—
146C2×Dic17—22no—
147C2×Dic17—24no—
148C2×Dic17—41no—
149C2×Dic17—42no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—42no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C8×C17—44no—
160#3622—82no—
161#2972D1711yescharacteristic
162C4×#123—171no—
163#1817—172no—
164#1459—171no—
165#1282—172no—
166C22×C17—21no—
167C22×C17—22no—
168C22×C17—21no—
169C22×C17—22no—
170C22×C17—22no—
171C22×C17—41no—
172C22×C17—41no—
173C22×C17—42no—
174C22×C17—42no—
175C22×C17—41no—
176C22×C17—41no—
177C22×C17—81no—
178C22×C17#38611yescharacteristic
179C22×C17C2×C22⋊C811yescharacteristic
180C22×C17#40811yescharacteristic
181C22×C17C8×D412yes—
182C22×C17#43512yes—
183C4×C17—22no—
184C4×C17—22no—
185C4×C17—21no—
186C4×C17—41no—
187C4×C17—41no—
188C4×C17#45812yes—
189Dic17—42no—
190Dic17—42no—
191Dic17—42no—
192Dic17—81no—
193C2×#123—171no—
194C2×C4×D4C2×D1711yescharacteristic
195C22×C42—171no—
196C8×D4—24no—
197#431—21no—
198C2×C22⋊C8—21no—
199C2×C22⋊C8—172no—
200C2×C22⋊C8C2×D1712yes—
201#388—341no—
202#387—341no—
203#386—171no—
204#385—171no—
205#381—171no—
206C4×C4⋊C4—341no—
207C4×C22⋊C4—171no—
208C4×C22⋊C4—341no—
209#336—344no—
210#324—341no—
211#323—341no—
212C2×C17—21no—
213C2×C17—21no—
214C2×C17—22no—
215C2×C17—22no—
216C2×C17—42no—
217C2×C17#186911yescharacteristic
218C2×C17#189011yescharacteristic
219C2×C17#189111yescharacteristic
220C22×D4C4×D1711yescharacteristic
221C23×C4—171no—
222C23×C4—171no—
223C23×C4C22⋊D1711yescharacteristic
224C23×C4C22×D1711yescharacteristic
225C22×C8—22no—
226C22×C8—342no—
227C22×C8C22⋊D1714yes—
228#123—174no—
229#123—342no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4D6811yescharacteristic
235C2×C22⋊C4—171no—
236C2×C22⋊C4—171no—
237C2×C22⋊C4—341no—
238C2×C22⋊C4—341no—
239C2×C22⋊C4C4×D1711yescharacteristic
240C2×C22⋊C4D6811yescharacteristic
241C2×C42—341no—
242C2×C42—342no—
243C2×C42—342no—
244C2×C42—341no—
245C2×C42—341no—
246C2×C42C22⋊D1711yescharacteristic
247C4⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—342no—
252C22⋊C8—342no—
253C4×C8—22no—
254#95—341no—
255#95—341no—
256#95—341no—
257#95—341no—
258#95—341no—
259C17#896611yescharacteristic, complemented, Sylow
260C24#6322711yescharacteristic
261C24C2×C4×D1711yescharacteristic
262C2×D4—22no—
263C2×D4—22no—
264C2×D4C8×D1712yes—
265C2×D4#6321812yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—172no—
271C22×C4—341no—
272C22×C4—342no—
273C22×C4—341no—
274C22×C4—341no—
275C22×C4—341no—
276C22×C4—342no—
277C22×C4—342no—
278C22×C4—341no—
279C22×C4—682no—
280C22×C4#6322711yescharacteristic
281C22×C4#6322711yescharacteristic
282C22×C4#6322711yescharacteristic
283C22×C4C2×D6811yescharacteristic
284C22×C4#6325212yes—
285C22×C4C2×C22⋊D1711yescharacteristic
286C22×C4C2×C22⋊D1712yes—
287C2×C8—24no—
288C2×C8—44no—
289C2×C8—82no—
290C2×C8—684no—
291C2×C8—682no—
292C4⋊C4—22no—
293C4⋊C4—344no—
294C4⋊C4—682no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—172no—
298C22⋊C4—344no—
299C22⋊C4—341no—
300C22⋊C4—342no—
301C22⋊C4—682no—
302C42—21no—
303C42—21no—
304C42—682no—
305C42—681no—
306C42—682no—
307C42—681no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23#1058749112yes—
315C23C2×C8×D1711yescharacteristic
316C23C2×#6321811yescharacteristic
317C23#1058758712yes—
318C23C2×#6322711yescharacteristic
319C23C4×C22⋊D1712yes—
320D4—24no—
321D4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—342no—
335C2×C4—344no—
336C2×C4—342no—
337C2×C4—342no—
338C2×C4—344no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—682no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4—682no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—681no—
350C2×C4#1058749112yes—
351C2×C4#1058761511yescharacteristic
352C2×C4#1058761612yes—
353C2×C4#1058763911yescharacteristic
354C8—44no—
355C8—1362no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22#1176268811yescharacteristic
369C22C2×#1058749111yescharacteristic
370C22#1176277511yescharacteristic
371C22C8×C22⋊D1712yes—
372C22#1176288012yes—
373C4—22no—
374C4—22no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—682no—
379C4—682no—
380C4—682no—
381C4—1361no—
382C4#1176299912yes—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—22no—
387C2—42no—
388C2#1452907611yescharacteristic
389C2#1452919711yescharacteristic
390C2#1452920211yescharacteristic
subgroupG/H
centerC2×C4#10587639
commutatorC23×C17C22×C8
FrattiniC23×C4C22×D17
FittingC17×#2972C2
socleC22×C17#386
prime radicalC22×D4×C17C2×C4
cosocleC23×C4×C17C23
hypercenter#2972D17
hypocenterC17#8966
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #14528850C2semidirect
4. #14526631C2semidirect
43. #10587621C8semidirect
90. C22×Dic17C2×C8semidirect
103. C22⋊Dic17C2×C8semidirect
259. C17#8966semidirect

Maximal subgroups

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

subgroupindex
C4×#105876212
#145288502
#145271022
#145266312
C17×#29722
#896617