Finite Group Databasebeta

← #17949643

Subgroups — #17949643

Order 4352.

538 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C4×#10587544C211yesmaximal, characteristic, complemented
2#14544499C211yesmaximal, characteristic, complemented
3#14543603C212yesmaximal, complemented
4#14543600C212yesmaximal
5#14531246C211yesmaximal, characteristic
6C2×#10587544—21no—
7C2×C4×#3624—21no—
8C2×C4×#3624—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×D4×Dic17C411yescharacteristic
13C2×#10587563—21no—
14#11762712C2211yescharacteristic
15#11762707C411yescharacteristic
16#11762705C2211yescharacteristic
17#11762703—21no—
18#11762697—21no—
19C4⋊C4×Dic17C411yescharacteristic
20C22⋊C4×Dic17—21no—
21C42×Dic17C2211yescharacteristic
22C17×#390C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×Dic17—21no—
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C22⋊C4×C17—21no—
29C2×C22⋊C4×C17—21no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C22⋊Dic17—21no—
32C2×C22⋊Dic17—21no—
33C2×C22⋊Dic17—21no—
34D4×Dic17—22no—
35D4×Dic17C814yescomplemented
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×#63225—21no—
39C2×#63225—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17C2311yescharacteristic
47#10587563—42no—
48#10587563—42no—
49#10587544—24no—
50C4×#3624—24no—
51C4×#3624—42no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C24×C17—21no—
56C2×D4×C17—21no—
57C2×D4×C17—41no—
58C2×D4×C17—41no—
59C2×D4×C17C2×C812yes—
60C22×C4×C17—21no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17—24no—
66C22×Dic17—21no—
67C22×Dic17—42no—
68C22×Dic17—42no—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C2×D411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73C2×#3624—24no—
74C2×#3624—24no—
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C4⋊C4×C17—41no—
80C22⋊C4×C17—42no—
81C22⋊C4×C17—42no—
82C42×C17—41no—
83C22⋊Dic17—24no—
84C22⋊Dic17—41no—
85C22⋊Dic17—41no—
86C22⋊Dic17—42no—
87C4⋊Dic17—21no—
88C4⋊Dic17—41no—
89C4⋊Dic17C2×C812yes—
90#63225—42no—
91#63225—41no—
92#63225—41no—
93C4×Dic17—22no—
94C4×Dic17—22no—
95C4×Dic17—22no—
96C4×Dic17—41no—
97C4×Dic17—42no—
98C4×Dic17C2×C812yes—
99C4×Dic17M1612yes—
100C4×Dic17D812yes—
101C4×Dic17SD1612yes—
102#7560—171nomaximal, Sylow
103C23×C17—22no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17C2×C22⋊C411yescharacteristic
111D4×C17—42no—
112D4×C17—42no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17#10212yes—
123C2×C4×C17#10412yes—
124C2×C4×C17C22×C811yescharacteristic
125C2×C4×C17C2×M1611yescharacteristic
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—22no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—48no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17C22⋊C812yes—
140C2×Dic17C22⋊C812yes—
141C2×Dic17#10212yes—
142C2×Dic17#10412yes—
143C2×Dic17C2×D811yescharacteristic
144C2×Dic17C2×SD1611yescharacteristic
145C2×Dic17#13512yes—
146#3624—44no—
147#3624—44no—
148#2293—171no—
149C4×C4⋊C8—171no—
150#1744—171no—
151#1271—172no—
152#1270—172no—
153C22×C17—42no—
154C22×C17—42no—
155C22×C17—42no—
156C22×C17—81no—
157C22×C17—81no—
158C22×C17—81no—
159C22×C17—81no—
160C22×C17C2×C22⋊C811yescharacteristic
161C22×C17#41011yescharacteristic
162C22×C17#41311yescharacteristic
163C22×C17C2×#10211yescharacteristic
164C22×C17#41611yescharacteristic
165C22×C17C2×#10411yescharacteristic
166C22×C17C2×#10411yescharacteristic
167C4×C17—22no—
168C4×C17—21no—
169C4×C17—21no—
170C4×C17—22no—
171C4×C17—81no—
172Dic17—22no—
173Dic17—22no—
174Dic17—21no—
175Dic17—21no—
176Dic17—42no—
177Dic17—81no—
178Dic17D4⋊C814yes—
179C2×C4×D4—171no—
180C2×#117—341no—
181C2×C4⋊C8—341no—
182C2×C4×C8—341no—
183C2×C4×C8—341no—
184#390—171no—
185#390#59511yescharacteristic
186#389—171no—
187#389—171no—
188#388—341no—
189#388—341no—
190C4×C4⋊C4—171no—
191C4×C22⋊C4—341no—
192C43—171no—
193#336—172no—
194#336—172no—
195#336—341no—
196C2×C17—82no—
197C2×C17#145811yescharacteristic
198C2×C17C2×D4⋊C811yescharacteristic
199C2×C17#147611yescharacteristic
200C2×C17#150011yescharacteristic
201C2×C17#150211yescharacteristic
202C2×C17#152411yescharacteristic
203C2×C17#154011yescharacteristic
204C22×D4C2×#59511yescharacteristic
205C23×C4—341no—
206C22×C8—342no—
207C22×C8—342no—
208C4×D4—174no—
209C4×D4—342no—
210#117—682no—
211#117—682no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4C2×#59511yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—341no—
220C2×C22⋊C4—341no—
221C2×C22⋊C4—341no—
222C2×C42—171no—
223C2×C42—171no—
224C2×C42—171no—
225C2×C42—171no—
226C2×C42—171no—
227C2×C42—171no—
228C2×C42—341no—
229C2×C42C2×#59511yescharacteristic
230C4⋊C8—344no—
231C4×C8—344no—
232C4×C8—682no—
233#95—341no—
234#95—341no—
235#95—341no—
236C17#756011yescharacteristic, complemented, Sylow
237C24—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4#6324212yes—
242C22×C4—21no—
243C22×C4—171no—
244C22×C4—171no—
245C22×C4—171no—
246C22×C4—171no—
247C22×C4—344no—
248C22×C4—341no—
249C22×C4—341no—
250C22×C4—682no—
251C22×C4—682no—
252C22×C4C22⋊#59511yescharacteristic
253C22×C4C22⋊#59511yescharacteristic
254C22×C4C22×#59511yescharacteristic
255C2×C8—344no—
256C2×C8—344no—
257C2×C8—682no—
258C2×C8—682no—
259C2×C8—682no—
260C2×C8—682no—
261C4⋊C4—41no—
262C4⋊C4—172no—
263C4⋊C4—341no—
264C4⋊C4—681no—
265C4⋊C4—681no—
266C4⋊C4—681no—
267C4⋊C4—682no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—344no—
271C22⋊C4—682no—
272C22⋊C4—681no—
273C22⋊C4—681no—
274C42—41no—
275C42—172no—
276C42—172no—
277C42—172no—
278C42—172no—
279C42—342no—
280C42—342no—
281C42—342no—
282C42—681no—
283C42—682no—
284C23—22no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊#59511yescharacteristic
292D4—42no—
293D4—42no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—172no—
304C2×C4—172no—
305C2×C4—172no—
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—342no—
317C2×C4—688no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4#1058754712yes—
324C2×C4#1058754812yes—
325C2×C4C2×#6324211yescharacteristic
326C2×C4D17⋊M1611yescharacteristic
327C8—684no—
328C8—684no—
329C22—42no—
330C22—42no—
331C22—42no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22#1176377711yescharacteristic
337C22#1176378911yescharacteristic
338C22#1176379611yescharacteristic
339C22C2×#1058754711yescharacteristic
340C22#1176379911yescharacteristic
341C22C2×#1058754811yescharacteristic
342C22D17⋊#10411yescharacteristic
343C4—22no—
344C4—21no—
345C4—21no—
346C4—22no—
347C4—81no—
348C4—174no—
349C4—342no—
350C4—341no—
351C4—341no—
352C4—342no—
353C4—682no—
354C4—1361no—
355C2—82no—
356C2#1454401611yescharacteristic
357C2#1454407011yescharacteristic
358C2#1454407611yescharacteristic
359C2#1454415611yescharacteristic
360C2#1454416411yescharacteristic
361C2#1454423511yescharacteristic
362C2#1454428211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#390C4
socleC23×C17C2×C22⋊C4
prime radicalC22×D4×C17C2×C4
cosocleC2×C4×Dic17C23
hypercenter#390#595
hypocenterC17#7560
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C4×#10587544C2semidirect
2. #14544499C2semidirect
3. #14543603C2semidirect
35. D4×Dic17C8semidirect
236. C17#7560semidirect

Maximal subgroups

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

subgroupindex
C4×#105875442
#145444992
#145436032
#145436002
#145312462
#756017