Finite Group Databasebeta

← #17956569

Subgroups — #17956569

Order 4352.

592 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 114 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#14544641C211yesmaximal, characteristic
2#14543960C211yesmaximal, characteristic
3#14543657C214yesmaximal
4#14537137C211yesmaximal, characteristic
5C2×#10587544—21no—
6C2×#10587543C2212yes—
7C2×(C2×#595)⋊C4—21no—
8C2×(C2×#595)⋊C4—21no—
9C2×(C2×#595)⋊C4C2212yes—
10#11763675—21no—
11C2×#10587555C411yescharacteristic, complemented
12C2×#10587583—21no—
13C2×C42×D17C2211yescharacteristic
14#11762714—21no—
15#11762708—21no—
16#11762708C412yescomplemented
17#11762666—21no—
18#11762666C2212yes—
19C17×#384C411yescharacteristic, complemented
20C23×#595—22no—
21C22×C4×D17D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×C4×D17C2311yescharacteristic
24C22×Dic34—21no—
25C22×#3624—22no—
26C2×C4⋊C4×C17—21no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17C2×C412yes—
29C2×C42×C17C2×C411yescharacteristic
30C2×#63227—22no—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17C2×C412yes—
34C2×C4×Dic17C2×C411yescharacteristic
35C2×C4×Dic17D411yescharacteristic
36C2×C4×Dic17D411yescharacteristic
37#10587583—42no—
38#10587583—42no—
39C42×D17—42no—
40#10587555—42no—
41(C2×#595)⋊C4—22no—
42(C2×#595)⋊C4—42no—
43(C2×#595)⋊C4—42no—
44(C2×#595)⋊C4Q814yescomplemented
45#10587544—42no—
46#10587543—24no—
47#10587502—22no—
48#10587502—22no—
49C23×D17C2×D411yescharacteristic
50C22×#595—24no—
51C22×#595—42no—
52C22×#595—42no—
53C22×#595—42no—
54C22×#595—42no—
55C22×#595—42no—
56C22×#595—42no—
57C22×C4×C17—22no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—22no—
62C22×Dic17C22⋊C411yescharacteristic
63C22×Dic17C22⋊C411yescharacteristic
64C22×Dic17C2×D411yescharacteristic
65C2×C4×D17—22no—
66C2×C4×D17—22no—
67C2×C4×D17—41no—
68C2×C4×D17—41no—
69C2×C4×D17—41no—
70C2×C4×D17—41no—
71C2×C4×D17—41no—
72C2×C4×D17D812yes—
73C2×C4×D17SD1612yes—
74C2×C4×D17C2×Q812yes—
75C2×C4×D17#4112yes—
76C2×Dic34—42no—
77C2×Dic34—42no—
78C2×Dic34—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C4⋊C4×C17—42no—
83C4⋊C4×C17—42no—
84C4⋊C4×C17—42no—
85C42×C17—22no—
86#63227—42no—
87#63227—42no—
88C4⋊Dic17—42no—
89C4⋊Dic17—44no—
90C4⋊Dic17—42no—
91C4×Dic17—22no—
92C4×Dic17—41no—
93C4×Dic17—41no—
94#9912—171nomaximal, Sylow
95C23×C17C2×C22⋊C411yescharacteristic
96C22×D17—21no—
97C22×D17—21no—
98C22×D17—21no—
99C22×D17—21no—
100C22×D17C22⋊Q812yes—
101C22×D17C2×D811yescharacteristic
102C22×D17C2×SD1611yescharacteristic
103C22×D17#13712yes—
104C2×#595—44no—
105C2×#595—44no—
106C2×#595—44no—
107C2×#595—82no—
108C2×#595—82no—
109C2×#595—82no—
110C2×#595—82no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17#9912yes—
121C2×C4×C17#10212yes—
122C2×C4×C17C4×Q812yes—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17#10212yes—
134C2×Dic17#12411yescharacteristic
135C2×Dic17C4⋊Q811yescharacteristic
136C4×D17—42no—
137C4×D17—42no—
138C4×D17—42no—
139C4×D17—42no—
140C4×D17—44no—
141Dic34—84no—
142#3624—82no—
143#2857—171no—
144#1797—171no—
145#1434—171no—
146#1294—174no—
147C22×C17#39111yescharacteristic
148C22×C17C2×#9911yescharacteristic
149C22×C17#41011yescharacteristic
150C22×C17C2×#10211yescharacteristic
151C22×C17#41911yescharacteristic
152C22×C17#42111yescharacteristic
153C22×C17#42111yescharacteristic
154C2×D17—22no—
155C2×D17—22no—
156C2×D17—22no—
157C2×D17—22no—
158C2×D17—21no—
159C2×D17—21no—
160C2×D17—21no—
161C2×D17—21no—
162C2×D17#47412yes—
163C2×D17#47512yes—
164#595—84no—
165C4×C17—22no—
166C4×C17—22no—
167C4×C17—22no—
168C4×C17—82no—
169Dic17—21no—
170Dic17—21no—
171Dic17—22no—
172Dic17—41no—
173Dic17—82no—
174C2×C4⋊Q8—171no—
175C2×C22⋊Q8—341no—
176C22×C42—171no—
177C2×C4⋊C8—341no—
178C2×C22⋊C8—172no—
179#391—341no—
180#389—172no—
181#389—341no—
182#384#59511yescharacteristic, complemented
183#380—172no—
184#380—341no—
185C2×#95—172no—
186C2×#95—341no—
187C2×#95—341no—
188#336—341no—
189C2×C17#151111yescharacteristic
190C2×C17#151511yescharacteristic
191C2×C17#151711yescharacteristic
192C2×C17#151911yescharacteristic
193C2×C17#198811yescharacteristic
194C2×C17#199511yescharacteristic
195C2×C17#199611yescharacteristic
196D17—22no—
197D17—22no—
198C22×Q8—341no—
199C23×C4—171no—
200C23×C4—171no—
201C23×C4—171no—
202C23×C4—342no—
203C22×C8—342no—
204C4⋊Q8—682no—
205C22⋊Q8—682no—
206C22⋊Q8—682no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—172no—
210C2×C4⋊C4—342no—
211C2×C4⋊C4—341no—
212C2×C4⋊C4C2×#59512yes—
213C2×C22⋊C4—342no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42—171no—
217C2×C42—682no—
218C2×C42C2×#59511yescharacteristic
219C4⋊C8—682no—
220C22⋊C8—344no—
221#95—174no—
222#95—342no—
223#95—342no—
224#95—342no—
225#95—682no—
226#95—682no—
227C17#991211yescharacteristic, complemented, Sylow
228C24—171no—
229C2×Q8—682no—
230C2×Q8—682no—
231C2×Q8—682no—
232C22×C4—22no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—172no—
236C22×C4—172no—
237C22×C4—172no—
238C22×C4—172no—
239C22×C4—171no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—344no—
243C22×C4—342no—
244C22×C4—681no—
245C22×C4—681no—
246C22×C4—681no—
247C22×C4—681no—
248C22×C4—681no—
249C22×C4—682no—
250C22×C4—682no—
251C22×C4—682no—
252C22×C4—682no—
253C22×C4—682no—
254C22×C4—682no—
255C22×C4C22⋊#59511yescharacteristic
256C22×C4C22⋊#59511yescharacteristic
257C22×C4C22×#59511yescharacteristic
258C2×C8—682no—
259C2×C8—682no—
260C2×C8—682no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—682no—
265C4⋊C4—684no—
266C4⋊C4—682no—
267C22⋊C4—682no—
268C22⋊C4—682no—
269C42—22no—
270C42—342no—
271C42—681no—
272C42—681no—
273C23—172no—
274C23—171no—
275C23—171no—
276C23—172no—
277C23—341no—
278C23—341no—
279C23—341no—
280C23—341no—
281C23C2×C22⋊#59511yescharacteristic
282Q8—1364no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—171no—
293C2×C4—171no—
294C2×C4—172no—
295C2×C4—341no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—682no—
303C2×C4—682no—
304C2×C4—682no—
305C2×C4—682no—
306C2×C4—684no—
307C2×C4—684no—
308C2×C4—684no—
309C2×C4—684no—
310C2×C4—682no—
311C2×C4—682no—
312C2×C4—682no—
313C2×C4—1362no—
314C2×C4—1362no—
315C2×C4—1362no—
316C2×C4—1362no—
317C2×C4#1058753912yes—
318C2×C4#1058754712yes—
319C2×C4Q8×#59512yes—
320C8—1362no—
321C22—172no—
322C22—172no—
323C22—342no—
324C22—342no—
325C22—342no—
326C22—342no—
327C22—341no—
328C22—341no—
329C22—341no—
330C22—341no—
331C22(C2×#595)⋊Q811yescharacteristic
332C22C2×#1058753911yescharacteristic
333C22#1176378911yescharacteristic
334C22C2×#1058754711yescharacteristic
335C22#1176380211yescharacteristic
336C22#1176380711yescharacteristic
337C22#1176380811yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—82no—
342C4—341no—
343C4—341no—
344C4—342no—
345C4—681no—
346C4—1364no—
347C4—1362no—
348C2—342no—
349C2—342no—
350C2#1454419211yescharacteristic
351C2#1454420811yescharacteristic
352C2#1454421311yescharacteristic
353C2#1454422111yescharacteristic
354C2#1454496311yescharacteristic
355C2#1454497211yescharacteristic
356C2#1454497411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C2×D4
cosocleC22×C4×D17C23
hypercenter#384#595
hypocenterC17#9912
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
11. C2×#10587555C4semidirect
16. #11762708C4semidirect
19. C17×#384C4semidirect
44. (C2×#595)⋊C4Q8semidirect
182. #384#595semidirect
227. C17#9912semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145439602
#145436572
#145371372
#991217