Finite Group Databasebeta

← #17950021

Subgroups — #17950021

Order 4352.

568 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 126 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#14544639C211yesmaximal, characteristic
2#14544445C211yesmaximal, characteristic
3#14543602C214yesmaximal, complemented
4#14537137C211yesmaximal, characteristic
5C22×#63242—21no—
6C2×C4⋊#3624—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C2×#10587555C411yescharacteristic
12C2×#10587583—21no—
13C2×C42×D17C2211yescharacteristic
14#11762714—21no—
15#11762708—21no—
16#11762708C2212yes—
17#11762666—21no—
18#11762666C412yes—
19C17×#384C411yescharacteristic
20C22×C4×D17C2×C411yescharacteristic
21C22×C4×D17D411yescharacteristic
22C22×C4×D17D411yescharacteristic
23C22×Dic34—21no—
24C22×#3624—22no—
25C22×#3624—22no—
26C2×#63242—24no—
27C2×#63242—42no—
28C2×#63242—42no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C42×C17C2×C411yescharacteristic
33C2×#63227—22no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17—22no—
36C2×C4⋊Dic17C2×C412yes—
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17C2311yescharacteristic
40#10587583—42no—
41#10587583—42no—
42C42×D17—42no—
43#10587555—42no—
44#10587543—42no—
45C4⋊#3624—42no—
46#10587502—22no—
47#10587502—22no—
48C23×D17C22⋊C411yescharacteristic
49C22×C4×C17—22no—
50C22×C4×C17C22⋊C411yescharacteristic
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22×C411yescharacteristic
53C22×Dic17—22no—
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17C2×D411yescharacteristic
56C22×Dic17C2×D411yescharacteristic
57C2×C4×D17—22no—
58C2×C4×D17—22no—
59C2×C4×D17—22no—
60C2×C4×D17—22no—
61C2×C4×D17—41no—
62C2×C4×D17—41no—
63C2×C4×D17—41no—
64C2×C4×D17D812yes—
65C2×C4×D17SD1612yes—
66C2×C4×D17SD1612yes—
67C2×C4×D17Q1612yes—
68C2×Dic34—42no—
69C2×Dic34—42no—
70C2×Dic34—42no—
71C2×#3624—24no—
72C2×#3624—42no—
73C2×#3624—42no—
74C2×#3624—42no—
75C2×#3624—42no—
76C2×#3624—42no—
77#63242—48no—
78C4⋊C4×C17—24no—
79C4⋊C4×C17—42no—
80C4⋊C4×C17—42no—
81C42×C17—22no—
82#63227—44no—
83C4⋊Dic17—42no—
84C4⋊Dic17—42no—
85C4⋊Dic17—42no—
86C4⋊Dic17—42no—
87C4×Dic17—22no—
88C4×Dic17—41no—
89C4×Dic17—41no—
90#7668—171nomaximal, Sylow
91C23×C17C2×C22⋊C411yescharacteristic
92C22×D17—21no—
93C22×D17—21no—
94C22×D17—21no—
95C22×D17—21no—
96C22×D17#10112yes—
97C22×D17#10212yes—
98C22×D17#10312yes—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—42no—
104C2×C4×C17—42no—
105C2×C4×C17—42no—
106C2×C4×C17#9912yes—
107C2×C4×C17#10212yes—
108C2×C4×C17#10312yes—
109C2×C4×C17#10412yes—
110C2×C4×C17#13112yes—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—42no—
119C2×Dic17—42no—
120C2×Dic17—42no—
121C2×Dic17C2×D811yescharacteristic
122C2×Dic17C2×SD1611yescharacteristic
123C2×Dic17C2×SD1611yescharacteristic
124C2×Dic17C2×Q1611yescharacteristic
125C4×D17—24no—
126C4×D17—24no—
127C4×D17—42no—
128C4×D17—42no—
129C4×D17—44no—
130Dic34—84no—
131#3624—44no—
132#3624—82no—
133#2857—171no—
134#1797—171no—
135C2×#336—171no—
136#1271—174no—
137C22×C17#40811yescharacteristic
138C22×C17C2×#9911yescharacteristic
139C22×C17C2×#10111yescharacteristic
140C22×C17C2×#10211yescharacteristic
141C22×C17C2×#10311yescharacteristic
142C22×C17C2×#10411yescharacteristic
143C22×C17#42111yescharacteristic
144C2×D17—22no—
145C2×D17—22no—
146C2×D17—22no—
147C2×D17—22no—
148C2×D17—22no—
149C2×D17—22no—
150C2×D17#33214yes—
151C4×C17—22no—
152C4×C17—22no—
153C4×C17—82no—
154C4×C17#32714yes—
155Dic17—21no—
156Dic17—21no—
157Dic17—21no—
158Dic17—21no—
159Dic17—41no—
160Dic17—82no—
161C23×C8—341no—
162C2×C4⋊Q8—171no—
163C2×C22⋊Q8—341no—
164C22×C42—171no—
165C2×C4⋊C8—341no—
166C2×C22⋊C8—341no—
167#391—341no—
168#389—172no—
169#389—341no—
170#384#59511yescharacteristic
171#380—172no—
172#380—341no—
173#336—172no—
174#336—172no—
175#336—341no—
176C2×C17#147211yescharacteristic
177C2×C17#148011yescharacteristic
178C2×C17#149311yescharacteristic
179C2×C17C2×#32711yescharacteristic
180C2×C17#152011yescharacteristic
181C2×C17#153111yescharacteristic
182C2×C17C2×#33211yescharacteristic
183D17—24no—
184C22×Q8—341no—
185C23×C4—171no—
186C23×C4—171no—
187C23×C4—171no—
188C22×C8—342no—
189C22×C8—344no—
190C22×C8—342no—
191C22×C8—682no—
192C22×C8—682no—
193C4⋊Q8—682no—
194C22⋊Q8—682no—
195C22⋊Q8—682no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—172no—
199C2×C4⋊C4—342no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4C2×#59512yes—
202C2×C22⋊C4—342no—
203C2×C42—171no—
204C2×C42—171no—
205C2×C42—171no—
206C2×C42—682no—
207C2×C42C2×#59511yescharacteristic
208C4⋊C8—682no—
209C22⋊C8—682no—
210#95—342no—
211#95—342no—
212C17#766811yescharacteristic, complemented, Sylow
213C24—171no—
214C2×Q8—682no—
215C2×Q8—682no—
216C2×Q8—682no—
217C22×C4—22no—
218C22×C4—171no—
219C22×C4—171no—
220C22×C4—172no—
221C22×C4—172no—
222C22×C4—172no—
223C22×C4—172no—
224C22×C4—171no—
225C22×C4—342no—
226C22×C4—342no—
227C22×C4—342no—
228C22×C4—342no—
229C22×C4—342no—
230C22×C4—681no—
231C22×C4—681no—
232C22×C4—681no—
233C22×C4C22⋊#59511yescharacteristic
234C22×C4C22⋊#59511yescharacteristic
235C22×C4C22×#59511yescharacteristic
236C2×C8—344no—
237C2×C8—682no—
238C2×C8—688no—
239C2×C8—682no—
240C2×C8—682no—
241C2×C8—682no—
242C2×C8—682no—
243C4⋊C4—24no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—682no—
247C4⋊C4—682no—
248C4⋊C4—682no—
249C4⋊C4—682no—
250C22⋊C4—684no—
251C42—22no—
252C42—342no—
253C42—681no—
254C42—681no—
255C23—172no—
256C23—172no—
257C23—172no—
258C23—341no—
259C23—341no—
260C23—341no—
261C23—341no—
262C23C2×C22⋊#59511yescharacteristic
263Q8—1364no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—171no—
272C2×C4—171no—
273C2×C4—171no—
274C2×C4—171no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—344no—
280C2×C4—344no—
281C2×C4—341no—
282C2×C4—341no—
283C2×C4—341no—
284C2×C4—682no—
285C2×C4—682no—
286C2×C4—684no—
287C2×C4—682no—
288C2×C4—682no—
289C2×C4—682no—
290C2×C4#1058753912yes—
291C2×C4#1058754712yes—
292C2×C4#1058754912yes—
293C2×C4#1058755012yes—
294C2×C4#1058767812yes—
295C8—684no—
296C8—1362no—
297C22—174no—
298C22—342no—
299C22—342no—
300C22—342no—
301C22—342no—
302C22—342no—
303C22—342no—
304C22#1176378311yescharacteristic
305C22C2×#1058753911yescharacteristic
306C22D17⋊#10111yescharacteristic
307C22C2×#1058754711yescharacteristic
308C22C2×#1058754911yescharacteristic
309C22C2×#1058755011yescharacteristic
310C22#1176380711yescharacteristic
311C4—22no—
312C4—22no—
313C4—82no—
314C4—341no—
315C4—341no—
316C4—341no—
317C4—341no—
318C4—681no—
319C4—1362no—
320C4#1176366114yes—
321C2—344no—
322C2#1454406211yescharacteristic
323C2#1454409011yescharacteristic
324C2#1454413311yescharacteristic
325C2C2×#1176366111yescharacteristic
326C2#1454422311yescharacteristic
327C2#1454425811yescharacteristic
328C2D17⋊#33211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#384#595
hypocenterC17#7668
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
3. #14543602C2semidirect
212. C17#7668semidirect

Maximal subgroups

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

subgroupindex
#145446392
#145444452
#145436022
#145371372
#766817