Finite Group Databasebeta

← #26132539

Subgroups — #26132539

Order 12032.

642 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 99 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#20245980C211yesmaximal, characteristic
2#20243306C211yesmaximal, characteristic
3#20241512C214yesmaximal, complemented
4C47×#1782C211yesmaximal, characteristic
5C2×#13093983C411yescharacteristic, complemented
6C22×C4⋊Dic47C411yescharacteristic, complemented
7C2×#13093939—21no—
8C2×#13093939C2212yes—
9#15689030C412yescomplemented
10#15689026C2212yes—
11#15689003—21no—
12#15689003—21no—
13#15688986—21no—
14#15688983—21no—
15#15688983—21no—
16C22×#41×C47C2211yescharacteristic
17C2×#99×C47—21no—
18C2×#99×C47C2212yes—
19C47×#381—21no—
20C23×Dic47—21no—
21#13093983—42no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—42no—
25C2×C4⋊Dic47C2×C412yes—
26C2×C22⋊Dic47—21no—
27C2×C22⋊Dic47—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C2×C4×Dic47C2×C412yes—
31#13093939—24no—
32#13093939—42no—
33C47⋊#95—22no—
34C47⋊#95—22no—
35C47⋊#95—42no—
36C2×#41×C47—24no—
37C2×#41×C47—42no—
38C2×#41×C47—42no—
39C22×Q8×C47C2×C411yescharacteristic
40C22×D4×C47D411yescharacteristic
41C22×D4×C47D411yescharacteristic
42C22×D4×C47C2311yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C2×C22⋊C4×C47—22no—
47#99×C47—24no—
48#99×C47—42no—
49#95×C47—42no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C24×C47C2×D411yescharacteristic
53#41×C47—416no—
54C2×Q8×C47—22no—
55C2×Q8×C47—41no—
56C2×Q8×C47—41no—
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47—41no—
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C2×D4×C47#4114yes—
66C22×C4×C47—22no—
67C22×C4×C47—22no—
68C22×C4×C47—22no—
69C22×C4×C47—22no—
70C22×C4×C47—22no—
71C22×C4×C47—22no—
72C22×C4×C47—22no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—42no—
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22⋊C411yescharacteristic
79C22×C4×C47C22×C411yescharacteristic
80C22×Dic47—22no—
81C22×Dic47—22no—
82C22×Dic47—42no—
83C22×Dic47—42no—
84C22×Dic47—82no—
85C22⋊C4×C47—42no—
86C22⋊C4×C47—42no—
87C22⋊Dic47—42no—
88C22⋊Dic47—42no—
89C22⋊Dic47—42no—
90C22⋊Dic47—82no—
91C22⋊Dic47—82no—
92C4⋊Dic47—42no—
93C4⋊Dic47—42no—
94C4⋊Dic47—42no—
95C4×Dic47—42no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47#9912yes—
109C23×C47C2×C22⋊C411yescharacteristic
110C23×C47#12312yes—
111C23×C47#12312yes—
112Q8×C47—44no—
113D4×C47—44no—
114D4×C47—44no—
115D4×C47—44no—
116C2×C4×C47—28no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—44no—
128C2×C4×C47—44no—
129C2×C4×C47—82no—
130C2×C4×C47—82no—
131C2×C4×C47—82no—
132C2×C4×C47#9912yes—
133C2×C4×C47#11712yes—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—84no—
141C2×Dic47—82no—
142C2×Dic47—82no—
143C2×Dic47—82no—
144#12859—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—22no—
148C22×C47—22no—
149C22×C47—22no—
150C22×C47—42no—
151C22×C47—42no—
152C22×C47—42no—
153C22×C47—42no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—41no—
159C22×C47—41no—
160C22×C47—81no—
161C22×C47—81no—
162C22×C47#38111yescharacteristic
163C22×C47C2×#9911yescharacteristic
164C22×C47C2×#9911yescharacteristic
165C4×C47—22no—
166C4×C47—22no—
167C4×C47—41no—
168C4×C47—41no—
169C4×C47—162no—
170Dic47—82no—
171Dic47—82no—
172Dic47—162no—
173#2311—471no—
174#1782—471no—
175#1782D4711yescharacteristic
176#1393—474no—
177C2×C47—21no—
178C2×C47—21no—
179C2×C47—42no—
180C2×C47—42no—
181C2×C47—42no—
182C2×C47#181911yescharacteristic
183C2×C47#212811yescharacteristic
184C2×C47#212911yescharacteristic
185C22×#41C2×D4711yescharacteristic
186C2×#117—471no—
187C22×C4⋊C4—471no—
188C2×#99—21no—
189C2×#99—472no—
190C2×#99—941no—
191C2×#99C2×D4712yes—
192#391—472no—
193#390—472no—
194#386—941no—
195#386—941no—
196#381—21no—
197#381—941no—
198#380—941no—
199#380—941no—
200C47#1285911yescharacteristic, complemented, Sylow
201C2×#41—24no—
202C2×#41—42no—
203C2×#41—42no—
204C22×Q8C4×D4711yescharacteristic
205C22×D4D18811yescharacteristic
206C22×D4C22⋊D4711yescharacteristic
207C22×D4C22×D4711yescharacteristic
208C23×C4—941no—
209C23×C4C4×D4711yescharacteristic
210C23×C4D18811yescharacteristic
211C23×C4C22⋊D4711yescharacteristic
212#117—1882no—
213C2×C4⋊C4—472no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4—941no—
216C2×C4⋊C4—1882no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—942no—
219C2×C22⋊C4—942no—
220C2×C22⋊C4—941no—
221C2×C22⋊C4—942no—
222C2×C42—472no—
223#99—24no—
224#99—42no—
225#99—944no—
226#99—1882no—
227#95—42no—
228#95—942no—
229#95—942no—
230#95—1882no—
231C24(C2×D47)⋊C411yescharacteristic
232C24C2×D18811yescharacteristic
233C24C2×C22⋊D4711yescharacteristic
234#41—416no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4#1062535214yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—942no—
258C22×C4—942no—
259C22×C4—3762no—
260C22×C4—1882no—
261C22×C4—1882no—
262C22×C4(C2×D47)⋊C411yescharacteristic
263C22×C4(C2×D47)⋊C411yescharacteristic
264C22×C4(C2×D47)⋊C411yescharacteristic
265C22×C4C2×C4×D4711yescharacteristic
266C4⋊C4—1882no—
267C4⋊C4—1882no—
268C4⋊C4—1882no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—3762no—
272C22⋊C4—3762no—
273C22⋊C4—1882no—
274C22⋊C4—1882no—
275C22⋊C4—1882no—
276C42—1882no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23#1309393812yes—
290C23C2×(C2×D47)⋊C411yescharacteristic
291C23#1309400512yes—
292C23#1309400812yes—
293Q8—44no—
294D4—44no—
295D4—44no—
296D4—44no—
297C2×C4—28no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—3764no—
314C2×C4—3762no—
315C2×C4—3762no—
316C2×C4—3762no—
317C2×C4—1882no—
318C2×C4—1882no—
319C2×C4—1882no—
320C2×C4—1882no—
321C2×C4—1882no—
322C2×C4—1882no—
323C2×C4#1309393812yes—
324C2×C4#1309398012yes—
325C22—21no—
326C22—21no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—81no—
342C22#1568898411yescharacteristic
343C22C2×#1309393811yescharacteristic
344C22C2×#1309393811yescharacteristic
345C4—22no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—162no—
350C4—3762no—
351C4—3762no—
352C4—7522no—
353C2—21no—
354C2—21no—
355C2—42no—
356C2—42no—
357C2—42no—
358C2#2024346011yescharacteristic
359C2#2024505911yescharacteristic
360C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#1782C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×D4×C47C23
hypercenter#1782D47
hypocenterC47#12859
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1782

Complements

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

NH ≅ G/Ntype
3. #20241512C2semidirect
5. C2×#13093983C4semidirect
6. C22×C4⋊Dic47C4semidirect
9. #15689030C4semidirect
200. C47#12859semidirect

Maximal subgroups

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

subgroupindex
#202459802
#202433062
#202415122
C47×#17822
#1285947