Finite Group Databasebeta

← #17950152

Subgroups — #17950152

Order 4352.

640 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 94 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#14544511C211yesmaximal, characteristic, complemented
3#14543600C214yesmaximal, complemented
4#14537126C211yesmaximal, characteristic
5C2×#10587543—21no—
6C2×#10587543—21no—
7C2×#10587542—21no—
8#11763675—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C2×#10587624—21no—
12C2×#10587624—21no—
13C2×#10587624C412yes—
14C2×C42×D17C2211yescharacteristic
15#11762712—21no—
16#11762712—21no—
17#11762712C2212yes—
18#11762679C411yescharacteristic
19C2×C17×#127C411yescharacteristic
20C22×D4×C17—21no—
21C22×D4×C17—21no—
22C22×D4×C17C2×C412yes—
23C22×C22⋊D17—22no—
24C22×C4×D17C2×C411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C17×#127—42no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—22no—
32C2×C22⋊Dic17—22no—
33C2×C22⋊Dic17—22no—
34#10587624—44no—
35#10587624—44no—
36#10587624—44no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C412yes—
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17C2311yescharacteristic
43C42×D17—42no—
44#10587543—24no—
45#10587543—42no—
46#10587542—42no—
47C24×C17—22no—
48C23×D17C22⋊C411yescharacteristic
49C2×D4×C17—24no—
50C2×D4×C17—42no—
51C2×D4×C17—42no—
52C2×D4×C17—42no—
53C2×D4×C17—42no—
54C2×D4×C17—42no—
55C2×D4×C17—42no—
56C2×D4×C17—42no—
57C2×D4×C17—42no—
58C22×C4×C17C22⋊C411yescharacteristic
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C2×C22⋊D17—44no—
62C2×C22⋊D17—44no—
63C2×C22⋊D17—44no—
64C22×Dic17—22no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—22no—
73C2×C4×D17—22no—
74C2×C4×D17—41no—
75C2×C4×D17—41no—
76C2×C4×D17—41no—
77C2×#3624—24no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C42×C17—22no—
84C22⋊Dic17—44no—
85C22⋊Dic17—44no—
86C22⋊Dic17—44no—
87C4⋊Dic17—42no—
88C4⋊Dic17—42no—
89C4⋊Dic17—42no—
90C4×Dic17—22no—
91C4×Dic17—41no—
92C4×Dic17—41no—
93#7704—171nomaximal, Sylow
94C23×C17—42no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17—42no—
99C23×C17—42no—
100C23×C17—42no—
101C23×C17C2×C22⋊C411yescharacteristic
102C22×D17—21no—
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17#10012yes—
109D4×C17—84no—
110D4×C17—84no—
111D4×C17—84no—
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#9912yes—
119C2×C4×C17#10412yes—
120C2×C4×C17#13112yes—
121C22⋊D17—88no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17AGL(1,ℤ/8)12yes—
133C2×Dic17AGL(1,ℤ/8)12yes—
134C4×D17—44no—
135C4×D17—44no—
136C4×D17—44no—
137#3624—44no—
138#3624—82no—
139#2856—171no—
140#1817—171no—
141#1749—171no—
142#1270—174no—
143C22×C17—82no—
144C22×C17—82no—
145C22×C17—82no—
146C22×C17—82no—
147C22×C17—82no—
148C22×C17—82no—
149C22×C17—82no—
150C22×C17#40811yescharacteristic
151C22×C17C2×#9911yescharacteristic
152C22×C17C2×#10011yescharacteristic
153C22×C17#41811yescharacteristic
154C22×C17#41811yescharacteristic
155C22×C17C2×#10411yescharacteristic
156C22×C17#42111yescharacteristic
157C2×D17—22no—
158C2×D17—22no—
159C2×D17—22no—
160C2×D17—41no—
161C2×D17—41no—
162C2×D17—41no—
163C2×D17—41no—
164C4×C17—22no—
165C4×C17—22no—
166C4×C17—22no—
167Dic17—22no—
168Dic17—22no—
169Dic17—41no—
170Dic17—82no—
171C2×#127#59511yescharacteristic
172C2×#121—172no—
173C2×#121—341no—
174C2×#121—341no—
175C22×C42—171no—
176C2×C22⋊C8—341no—
177C2×C22⋊C8—341no—
178C2×#97—341no—
179#390—172no—
180#390—341no—
181#390—341no—
182#384—171no—
183#336—172no—
184#336—172no—
185#336—341no—
186C2×C17—162no—
187C2×C17#147111yescharacteristic
188C2×C17#148411yescharacteristic
189C2×C17#149611yescharacteristic
190C2×C17#150711yescharacteristic
191C2×C17#151011yescharacteristic
192C2×C17#153011yescharacteristic
193C2×C17#155511yescharacteristic
194D17—42no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—342no—
198C22×D4C2×#59512yes—
199C23×C4—171no—
200C23×C4—171no—
201C23×C4—171no—
202C22×C8—342no—
203C22×C8—342no—
204#127—42no—
205#121—684no—
206#121—684no—
207#121—684no—
208C2×C4⋊C4—172no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—341no—
211C2×C22⋊C4—342no—
212C2×C22⋊C4—342no—
213C2×C22⋊C4—342no—
214C2×C42—171no—
215C2×C42—171no—
216C2×C42—171no—
217C2×C42—682no—
218C2×C42C2×#59511yescharacteristic
219C22⋊C8—344no—
220C22⋊C8—682no—
221#97—682no—
222C17#770411yescharacteristic, complemented, Sylow
223C24—22no—
224C24—171no—
225C2×D4—24no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—684no—
235C2×D4—684no—
236C2×D4—684no—
237C22×C4—171no—
238C22×C4—171no—
239C22×C4—171no—
240C22×C4—342no—
241C22×C4—342no—
242C22×C4—342no—
243C22×C4—342no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—681no—
248C22×C4—681no—
249C22×C4—681no—
250C22×C4C22⋊#59511yescharacteristic
251C22×C4C22⋊#59511yescharacteristic
252C22×C4C22×#59511yescharacteristic
253C2×C8—344no—
254C2×C8—682no—
255C2×C8—682no—
256C2×C8—682no—
257C2×C8—682no—
258C2×C8—682no—
259C4⋊C4—682no—
260C4⋊C4—682no—
261C4⋊C4—682no—
262C22⋊C4—684no—
263C22⋊C4—684no—
264C22⋊C4—684no—
265C42—22no—
266C42—342no—
267C42—681no—
268C42—681no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—172no—
277C23—341no—
278C23—341no—
279C23—341no—
280C23—341no—
281C23—341no—
282C23—341no—
283C23C2×C22⋊#59511yescharacteristic
284D4—84no—
285D4—84no—
286D4—84no—
287D4—1368no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—172no—
295C2×C4—172no—
296C2×C4—341no—
297C2×C4—341no—
298C2×C4—341no—
299C2×C4—341no—
300C2×C4—341no—
301C2×C4—341no—
302C2×C4—341no—
303C2×C4—684no—
304C2×C4—684no—
305C2×C4—684no—
306C2×C4—682no—
307C2×C4—682no—
308C2×C4—682no—
309C2×C4#1058754812yes—
310C2×C4#1058755112yes—
311C2×C4#1058767812yes—
312C8—684no—
313C8—1362no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—342no—
322C22—342no—
323C22—342no—
324C22—681no—
325C22—681no—
326C22—681no—
327C22—681no—
328C22#1176378311yescharacteristic
329C22C2×#1058755111yescharacteristic
330C22D17⋊#10011yescharacteristic
331C22#1176380111yescharacteristic
332C22#1176380111yescharacteristic
333C22C2×#1058754811yescharacteristic
334C22#1176380711yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—342no—
339C4—342no—
340C4—681no—
341C4—1362no—
342C2—162no—
343C2—682no—
344C2#1454405911yescharacteristic
345C2#1454410411yescharacteristic
346C2#1454414411yescharacteristic
347C2#1454418011yescharacteristic
348C2#1454418911yescharacteristic
349C2#1454425411yescharacteristic
350C2#1454431211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#127C4
socleC23×C17C2×C22⋊C4
prime radical#14537126C2
cosocleC2×C4×Dic17C23
hypercenterC2×#127#595
hypocenterC17#7704
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #14544676C2semidirect
2. #14544511C2semidirect
3. #14543600C2semidirect
222. C17#7704semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145445112
#145436002
#145371262
#770417