Finite Group Databasebeta

← #17950226

Subgroups — #17950226

Order 4352.

634 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544676C211yesmaximal, characteristic, complemented
2#14544641C211yesmaximal, characteristic, complemented
3#14543601C214yesmaximal, complemented
4#14537130C211yesmaximal, characteristic
5C2×#10587544—21no—
6C2×#10587543—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C2×#10587560C411yescharacteristic
12C2×#10587581—21no—
13C2×#10587581—21no—
14C2×#10587581C412yes—
15C2×C42×D17C2211yescharacteristic
16#11762710—21no—
17#11762710—21no—
18#11762710C2212yes—
19C17×#384C411yescharacteristic
20C22×D68—22no—
21C22×D68—21no—
22C22×D68—21no—
23C22×D68C2×C412yes—
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17C2×C412yes—
32C2×C42×C17C2×C411yescharacteristic
33C2×#63227—22no—
34C2×#63227—22no—
35C2×#63227—22no—
36C2×C4×Dic17D411yescharacteristic
37C2×C4×Dic17D411yescharacteristic
38C2×C4×Dic17C2311yescharacteristic
39#10587581—44no—
40#10587581—44no—
41#10587581—42no—
42#10587581—42no—
43#10587560—42no—
44C42×D17—42no—
45#10587544—42no—
46#10587543—42no—
47#10587543—42no—
48C23×D17—22no—
49C23×D17C22⋊C411yescharacteristic
50C22×C4×C17—22no—
51C22×C4×C17C22⋊C411yescharacteristic
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C22×C411yescharacteristic
54C22×Dic17C22⋊C411yescharacteristic
55C22×Dic17C2×D411yescharacteristic
56C22×Dic17C2×D411yescharacteristic
57C2×D68—24no—
58C2×D68—42no—
59C2×D68—42no—
60C2×D68—42no—
61C2×D68—42no—
62C2×D68—42no—
63C2×D68—42no—
64C2×D68—44no—
65C2×D68—44no—
66C2×D68—44no—
67C2×D68—42no—
68C2×D68—42no—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×C4×D17—41no—
74C2×C4×D17—41no—
75C2×C4×D17—41no—
76C2×C4×D17—41no—
77C2×C4×D17D812yes—
78C2×C4×D17SD1612yes—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84C2×#3624—42no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—42no—
87C4⋊C4×C17—42no—
88C42×C17—22no—
89#63227—44no—
90#63227—42no—
91#63227—42no—
92#63227—44no—
93C4×Dic17—22no—
94C4×Dic17—41no—
95C4×Dic17—41no—
96#7724—171nomaximal, Sylow
97C23×C17C2×C22⋊C411yescharacteristic
98C22×D17—21no—
99C22×D17—21no—
100C22×D17—21no—
101C22×D17—21no—
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—42no—
105C22×D17—42no—
106C22×D17—42no—
107C22×D17—42no—
108C22×D17—42no—
109C22×D17—42no—
110C22×D17—42no—
111C22×D17#10212yes—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—42no—
119C2×C4×C17—42no—
120C2×C4×C17—42no—
121C2×C4×C17#9912yes—
122C2×C4×C17#10212yes—
123C2×C4×C17#13112yes—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17#10012yes—
131C2×Dic17C2×D811yescharacteristic
132C2×Dic17C2×SD1611yescharacteristic
133C2×Dic17AGL(1,ℤ/8)12yes—
134D68—84no—
135D68—84no—
136D68—84no—
137D68—88no—
138C4×D17—42no—
139C4×D17—42no—
140C4×D17—44no—
141C4×D17—44no—
142#3624—82no—
143#3624—82no—
144#2856—171no—
145#1817—171no—
146#1797—171no—
147#1270—174no—
148C22×C17#40811yescharacteristic
149C22×C17C2×#9911yescharacteristic
150C22×C17C2×#10011yescharacteristic
151C22×C17C2×#10211yescharacteristic
152C22×C17#41811yescharacteristic
153C22×C17#42111yescharacteristic
154C22×C17#42111yescharacteristic
155C2×D17—22no—
156C2×D17—22no—
157C2×D17—22no—
158C2×D17—41no—
159C2×D17—41no—
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—82no—
163C2×D17—82no—
164C2×D17—82no—
165C2×D17—82no—
166C2×D17—82no—
167C2×D17—82no—
168C2×D17—82no—
169C4×C17—22no—
170C4×C17—22no—
171C4×C17—22no—
172C4×C17—82no—
173Dic17—22no—
174Dic17—22no—
175Dic17—21no—
176Dic17—21no—
177C2×#127—171no—
178C2×#121—172no—
179C2×#121—341no—
180C2×#121—341no—
181C22×C42—171no—
182C2×C4⋊C8—341no—
183C2×C22⋊C8—341no—
184C2×C22⋊C8—341no—
185#390—172no—
186#390—341no—
187#390—341no—
188#384#59511yescharacteristic
189#336—172no—
190#336—172no—
191#336—341no—
192C2×C17#147111yescharacteristic
193C2×C17#148011yescharacteristic
194C2×C17#149611yescharacteristic
195C2×C17#151011yescharacteristic
196C2×C17#151911yescharacteristic
197C2×C17#153611yescharacteristic
198C2×C17#154511yescharacteristic
199D17—42no—
200D17—162no—
201C22×D4—172no—
202C22×D4—342no—
203C22×D4—341no—
204C22×D4—341no—
205C23×C4—171no—
206C23×C4—171no—
207C23×C4—171no—
208C22×C8—342no—
209C22×C8—342no—
210#127—682no—
211#121—682no—
212#121—682no—
213#121—684no—
214#121—684no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C2×#59512yes—
218C2×C22⋊C4—342no—
219C2×C22⋊C4—342no—
220C2×C22⋊C4—342no—
221C2×C42—171no—
222C2×C42—171no—
223C2×C42—171no—
224C2×C42—682no—
225C2×C42C2×#59511yescharacteristic
226C4⋊C8—682no—
227C22⋊C8—682no—
228C22⋊C8—682no—
229C17#772411yescharacteristic, complemented, Sylow
230C24—171no—
231C24—342no—
232C2×D4—344no—
233C2×D4—682no—
234C2×D4—682no—
235C2×D4—682no—
236C2×D4—682no—
237C2×D4—682no—
238C2×D4—682no—
239C2×D4—684no—
240C2×D4—684no—
241C2×D4—684no—
242C2×D4—682no—
243C2×D4—682no—
244C22×C4—22no—
245C22×C4—171no—
246C22×C4—171no—
247C22×C4—171no—
248C22×C4—172no—
249C22×C4—172no—
250C22×C4—342no—
251C22×C4—342no—
252C22×C4—342no—
253C22×C4—342no—
254C22×C4—681no—
255C22×C4—681no—
256C22×C4—681no—
257C22×C4—681no—
258C22×C4C22⋊#59511yescharacteristic
259C22×C4C22⋊#59511yescharacteristic
260C22×C4C22×#59511yescharacteristic
261C2×C8—682no—
262C2×C8—682no—
263C2×C8—682no—
264C2×C8—682no—
265C2×C8—682no—
266C2×C8—682no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C22⋊C4—682no—
271C22⋊C4—682no—
272C22⋊C4—684no—
273C22⋊C4—684no—
274C42—22no—
275C42—342no—
276C42—681no—
277C42—681no—
278C23—172no—
279C23—341no—
280C23—341no—
281C23—341no—
282C23—341no—
283C23—341no—
284C23—341no—
285C23—682no—
286C23—682no—
287C23—682no—
288C23—682no—
289C23—682no—
290C23—682no—
291C23—682no—
292C23C2×C22⋊#59511yescharacteristic
293D4—1364no—
294D4—1364no—
295D4—1364no—
296D4—1368no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—172no—
307C2×C4—171no—
308C2×C4—171no—
309C2×C4—172no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—682no—
317C2×C4—682no—
318C2×C4—684no—
319C2×C4—684no—
320C2×C4#1058753912yes—
321C2×C4#1058754712yes—
322C2×C4#1058768012yes—
323C8—1362no—
324C8—1362no—
325C22—342no—
326C22—342no—
327C22—342no—
328C22—681no—
329C22—681no—
330C22—681no—
331C22—681no—
332C22—1362no—
333C22—1362no—
334C22—1362no—
335C22—1362no—
336C22—1362no—
337C22—1362no—
338C22—1362no—
339C22#1176378411yescharacteristic
340C22C2×#1058753911yescharacteristic
341C22D17⋊#10011yescharacteristic
342C22C2×#1058754711yescharacteristic
343C22#1176380111yescharacteristic
344C22#1176380811yescharacteristic
345C22#1176380811yescharacteristic
346C4—22no—
347C4—22no—
348C4—22no—
349C4—82no—
350C4—342no—
351C4—341no—
352C4—341no—
353C4—342no—
354C2—2722no—
355C2—682no—
356C2#1454406111yescharacteristic
357C2#1454409111yescharacteristic
358C2#1454414511yescharacteristic
359C2#1454419111yescharacteristic
360C2#1454422111yescharacteristic
361C2#1454427411yescharacteristic
362C2#1454429211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#384C4
socleC23×C17C2×C22⋊C4
prime radical#14537130C2
cosocleC2×C4×Dic17C23
hypercenter#384#595
hypocenterC17#7724
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
1. #14544676C2semidirect
2. #14544641C2semidirect
3. #14543601C2semidirect
229. C17#7724semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145446412
#145436012
#145371302
#772417