Finite Group Databasebeta

← #17949658

Subgroups — #17949658

Order 4352.

550 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 118 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×#10587543C211yesmaximal, characteristic
2C4×#10587543C211yesmaximal, characteristic
3#14543603C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5#14531265C211yesmaximal, characteristic
6C2×#10587543—21no—
7C2×#10587543—21no—
8C2×C4×#3624—21no—
9C2×C4×#3624—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587578C411yescharacteristic
13C2×#10587578C411yescharacteristic
14C2×C42×D17C2211yescharacteristic
15#11762708C2211yescharacteristic
16#11762707C411yescharacteristic
17#11762705C2211yescharacteristic
18#11762683—21no—
19#11762680—21no—
20C4⋊C4×Dic17—21no—
21#11762660—21no—
22C17×#389C411yescharacteristic
23C22×C4×D17C2×C411yescharacteristic
24C22×C4×D17D411yescharacteristic
25C22×C4×D17D411yescharacteristic
26C22×#3624—22no—
27C22×#3624—22no—
28C2×C4⋊C4×C17—21no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C42×C17C2×C411yescharacteristic
32C2×#63227—21no—
33C2×#63227—21no—
34C2×C4⋊Dic17—21no—
35C2×C4⋊Dic17C2×C411yescharacteristic
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×#63225—21no—
38C2×#63225—21no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17C2311yescharacteristic
44#10587578—22no—
45#10587578—42no—
46#10587578C814yescomplemented
47C42×D17—42no—
48#10587543—24no—
49#10587543—24no—
50C4×#3624—42no—
51C4×#3624—42no—
52C17×#95—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56C23×D17C22⋊C411yescharacteristic
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22×C411yescharacteristic
62C22×Dic17—21no—
63C22×Dic17—21no—
64C22×Dic17C22⋊C411yescharacteristic
65C22×Dic17C2×D411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C2×C4×D17—22no—
68C2×C4×D17—22no—
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×D17C2×C812yes—
76C2×C4×D17M1612yes—
77C2×#3624—24no—
78C2×#3624—24no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17C2×C812yes—
87C42×C17—41no—
88#63227—24no—
89#63227—42no—
90C4⋊Dic17—21no—
91C4⋊Dic17—42no—
92C4⋊Dic17—41no—
93C4⋊Dic17C2×C812yes—
94#63225—42no—
95#63225—42no—
96C4×Dic17—24no—
97C4×Dic17—22no—
98C4×Dic17—22no—
99C4×Dic17—41no—
100C4×Dic17—42no—
101#7564—171nomaximal, Sylow
102C23×C17C2×C22⋊C411yescharacteristic
103C22×D17—21no—
104C22×D17—21no—
105C22×D17—21no—
106C22×D17—21no—
107C22×D17—21no—
108C22×D17—21no—
109C22×D17C22⋊C812yes—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—22no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17#10412yes—
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—22no—
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17C22⋊C812yes—
137C2×Dic17#9912yes—
138C2×Dic17#10112yes—
139C2×Dic17#13512yes—
140C2×Dic17#13512yes—
141C4×D17—24no—
142C4×D17—24no—
143C4×D17—44no—
144C4×D17—44no—
145#3624—44no—
146#3624—44no—
147#2296—171no—
148C4×C22⋊C8—171no—
149C4×C22⋊C8—171no—
150#1271—172no—
151#1271—172no—
152C22×C17C2×C22⋊C811yescharacteristic
153C22×C17C2×#9911yescharacteristic
154C22×C17C2×#10111yescharacteristic
155C22×C17#41611yescharacteristic
156C22×C17#41611yescharacteristic
157C22×C17C2×#10411yescharacteristic
158C22×C17C2×#10411yescharacteristic
159C2×D17—22no—
160C2×D17—22no—
161C2×D17—22no—
162C2×D17—41no—
163C2×D17—41no—
164C2×D17—41no—
165C2×D17—41no—
166C4×C17—21no—
167C4×C17—21no—
168C4×C17—22no—
169C4×C17—22no—
170C4×C17—42no—
171C4×C17—81no—
172Dic17—22no—
173Dic17—22no—
174Dic17—42no—
175Dic17—81no—
176Dic17#32414yes—
177C2×#117—171no—
178C2×#117—171no—
179C22×C42—171no—
180C2×C22⋊C8—341no—
181C2×C22⋊C8—341no—
182C2×C4×C8—341no—
183C2×C4×C8—341no—
184#389—171no—
185#389—171no—
186#389—171no—
187#389#59511yescharacteristic
188#385—341no—
189#385—341no—
190C4×C4⋊C4—341no—
191C4×C4⋊C4—341no—
192#336—172no—
193#336—172no—
194C2×C17C2×#32411yescharacteristic
195C2×C17#147611yescharacteristic
196C2×C17#147611yescharacteristic
197C2×C17#150111yescharacteristic
198C2×C17#150111yescharacteristic
199C2×C17#152611yescharacteristic
200C2×C17#154411yescharacteristic
201D17—42no—
202C23×C4—171no—
203C23×C4—171no—
204C23×C4—171no—
205C22×C8—342no—
206C22×C8—342no—
207#117—174no—
208#117—342no—
209#117—682no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—171no—
212C2×C4⋊C4—171no—
213C2×C4⋊C4—341no—
214C2×C4⋊C4—341no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4C2×#59511yescharacteristic
217C2×C4⋊C4C2×#59511yescharacteristic
218C2×C22⋊C4—341no—
219C2×C22⋊C4—341no—
220C2×C42—171no—
221C2×C42—171no—
222C2×C42—171no—
223C2×C42—341no—
224C2×C42—341no—
225C2×C42—682no—
226C2×C42C2×#59511yescharacteristic
227C22⋊C8—344no—
228C22⋊C8—344no—
229C4×C8—682no—
230C4×C8—682no—
231#95—21no—
232#95—341no—
233#95—341no—
234#95—341no—
235C17#756411yescharacteristic, complemented, Sylow
236C24—171no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—171no—
240C22×C4—171no—
241C22×C4—171no—
242C22×C4—172no—
243C22×C4—172no—
244C22×C4—342no—
245C22×C4—342no—
246C22×C4—342no—
247C22×C4—342no—
248C22×C4—341no—
249C22×C4—342no—
250C22×C4—342no—
251C22×C4—341no—
252C22×C4—681no—
253C22×C4—681no—
254C22×C4C22⋊#59511yescharacteristic
255C22×C4C22⋊#59511yescharacteristic
256C22×C4C22×#59511yescharacteristic
257C2×C8—344no—
258C2×C8—344no—
259C2×C8—682no—
260C2×C8—682no—
261C2×C8—682no—
262C2×C8—682no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—172no—
267C4⋊C4—341no—
268C4⋊C4—681no—
269C4⋊C4—682no—
270C4⋊C4—682no—
271C4⋊C4—682no—
272C4⋊C4#6324212yes—
273C22⋊C4—344no—
274C22⋊C4—682no—
275C42—41no—
276C42—342no—
277C42—342no—
278C42—344no—
279C42—681no—
280C42—682no—
281C23—172no—
282C23—341no—
283C23—341no—
284C23—341no—
285C23—341no—
286C23—341no—
287C23—341no—
288C23C2×C22⋊#59511yescharacteristic
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—172no—
302C2×C4—172no—
303C2×C4—172no—
304C2×C4—172no—
305C2×C4—172no—
306C2×C4—344no—
307C2×C4—344no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—342no—
313C2×C4—684no—
314C2×C4—684no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4#1058754812yes—
321C2×C4#1058755012yes—
322C2×C4C2×#6324211yescharacteristic
323C2×C4D17⋊M1611yescharacteristic
324C8—684no—
325C8—684no—
326C22—342no—
327C22—342no—
328C22—342no—
329C22—681no—
330C22—681no—
331C22—681no—
332C22—681no—
333C22#1176377711yescharacteristic
334C22D17⋊#9911yescharacteristic
335C22D17⋊#10111yescharacteristic
336C22#1176379911yescharacteristic
337C22#1176379911yescharacteristic
338C22C2×#1058754811yescharacteristic
339C22C2×#1058755011yescharacteristic
340C4—22no—
341C4—22no—
342C4—21no—
343C4—21no—
344C4—42no—
345C4—81no—
346C4—174no—
347C4—342no—
348C4—342no—
349C4—682no—
350C4—1361no—
351C2—682no—
352C2#1454401311yescharacteristic
353C2#1454407611yescharacteristic
354C2#1454407811yescharacteristic
355C2#1454416011yescharacteristic
356C2#1454416211yescharacteristic
357C2#1454424211yescharacteristic
358C2#1454429011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7564
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #14543603C2semidirect
4. #14543602C2semidirect
46. #10587578C8semidirect
235. C17#7564semidirect

Maximal subgroups

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

subgroupindex
C4×#105875432
#145436032
#145436022
#145312652
#756417