Finite Group Databasebeta

← #26132551

Subgroups — #26132551

Order 12032.

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

normal subgroups only

This group has 614 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#20243316C211yesmaximal, characteristic
3#20241514C214yesmaximal, complemented
4C47×#1785C211yesmaximal, characteristic
5C2×#13093983C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7C2×#13093943C2212yes—
8C2×#13093941—21no—
9#15689087—21no—
10#15689030C2212yes—
11#15689026C412yes—
12#15689003—21no—
13#15689003—21no—
14#15688983—21no—
15#15688983—21no—
16C22×#41×C47C2211yescharacteristic
17C2×#101×C47C2212yes—
18C2×#100×C47—21no—
19C47×#407—21no—
20C23×Dic47—21no—
21C2×#10625329—22no—
22#13093983—42no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—42no—
26C2×C4⋊Dic47C2×C412yes—
27C2×C22⋊Dic47—21no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47—22no—
30C2×C4×Dic47C2×C412yes—
31#13093943—24no—
32#13093941—42no—
33C22⋊(C47⋊C8)—42no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C2×#41×C47—24no—
37C2×#41×C47—42no—
38C2×#41×C47—42no—
39C22×Q8×C47C2311yescharacteristic
40C22×D4×C47C2×C411yescharacteristic
41C22×D4×C47D411yescharacteristic
42C22×D4×C47D411yescharacteristic
43C23×C4×C47C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C2×M16×C47—22no—
47#101×C47—24no—
48#100×C47—42no—
49C22⋊C8×C47—42no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C22⋊C411yescharacteristic
52C24×C47C22⋊C411yescharacteristic
53#41×C47—416no—
54C2×Q8×C47—41no—
55C2×Q8×C47—41no—
56C2×Q8×C47#4114yes—
57C2×D4×C47—22no—
58C2×D4×C47—22no—
59C2×D4×C47—22no—
60C2×D4×C47—22no—
61C2×D4×C47—22no—
62C2×D4×C47—41no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C2×D4×C47—41no—
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×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—22no—
80C22×Dic47—22no—
81C22×Dic47—42no—
82C22×Dic47—82no—
83M16×C47—42no—
84M16×C47—42no—
85C2×C8×C47—42no—
86C22⋊Dic47—42no—
87C22⋊Dic47—82no—
88C22⋊Dic47—82no—
89C4⋊Dic47—42no—
90C4⋊Dic47—42no—
91C4⋊Dic47—42no—
92C4×Dic47—42no—
93#10625329—42no—
94#10625329—42no—
95C2×C47⋊C8—42no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—41no—
105C23×C47—41no—
106C23×C47—41no—
107C23×C47—41no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47#9912yes—
111C23×C47C2×C22⋊C411yescharacteristic
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—44no—
126C2×C4×C47—44no—
127C2×C4×C47#9912yes—
128C2×C4×C47#11712yes—
129C2×C4×C47#12312yes—
130C2×C4×C47#12312yes—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—84no—
138C8×C47—82no—
139C47⋊C8—82no—
140#12863—471nomaximal, Sylow
141C22×C47—21no—
142C22×C47—21no—
143C22×C47—22no—
144C22×C47—42no—
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—42no—
148C22×C47—41no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—41no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—81no—
157C22×C47—81no—
158C22×C47#38111yescharacteristic
159C22×C47C2×#9911yescharacteristic
160C22×C47C2×#9911yescharacteristic
161C4×C47—22no—
162C4×C47—22no—
163C4×C47—22no—
164C4×C47—22no—
165Dic47—82no—
166Dic47—82no—
167#2311—471no—
168#1785—471no—
169#1785D4711yescharacteristic
170#1394—474no—
171C2×C47—21no—
172C2×C47—21no—
173C2×C47—42no—
174C2×C47—81no—
175C2×C47—81no—
176C2×C47#181911yescharacteristic
177C2×C47#213411yescharacteristic
178C2×C47#213511yescharacteristic
179C22×#41C2×D4711yescharacteristic
180C2×#117—471no—
181C22×C4⋊C4—471no—
182C2×#101—472no—
183C2×#101C2×D4712yes—
184C2×#100—21no—
185C2×#100—941no—
186#407—21no—
187#407—941no—
188#391—472no—
189#390—472no—
190#386—941no—
191#386—941no—
192#380—941no—
193#380—941no—
194C47#1286311yescharacteristic, complemented, Sylow
195C2×#41—24no—
196C2×#41—42no—
197C2×#41—42no—
198C22×Q8C22×D4711yescharacteristic
199C22×D4C4×D4711yescharacteristic
200C22×D4D18811yescharacteristic
201C22×D4C22⋊D4711yescharacteristic
202C23×C4—941no—
203C23×C4C4×D4711yescharacteristic
204C23×C4D18811yescharacteristic
205C23×C4C22⋊D4711yescharacteristic
206C2×M16—22no—
207C2×M16—942no—
208#117—1882no—
209C2×C4⋊C4—472no—
210C2×C4⋊C4—941no—
211C2×C4⋊C4—941no—
212C2×C4⋊C4—1882no—
213C2×C22⋊C4—942no—
214C2×C22⋊C4—942no—
215C2×C22⋊C4—941no—
216C2×C42—472no—
217#101—24no—
218#101—944no—
219#100—42no—
220#100—1882no—
221C22⋊C8—42no—
222C22⋊C8—1882no—
223#95—942no—
224#95—942no—
225C24(C2×D47)⋊C411yescharacteristic
226C24(C2×D47)⋊C411yescharacteristic
227C24(C2×D47)⋊C411yescharacteristic
228#41—416no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8#1062535214yes—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—942no—
251C22×C4—942no—
252C22×C4—3762no—
253C22×C4—1882no—
254C22×C4(C2×D47)⋊C411yescharacteristic
255C22×C4C2×C4×D4711yescharacteristic
256C22×C4C2×D18811yescharacteristic
257C22×C4C2×C22⋊D4711yescharacteristic
258M16—42no—
259M16—42no—
260M16—1882no—
261M16—1882no—
262C2×C8—42no—
263C2×C8—1882no—
264C4⋊C4—1882no—
265C4⋊C4—1882no—
266C4⋊C4—1882no—
267C22⋊C4—3762no—
268C22⋊C4—3762no—
269C22⋊C4—1882no—
270C42—1882no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23#1309393812yes—
286C23C2×(C2×D47)⋊C411yescharacteristic
287Q8—44no—
288D4—44no—
289D4—44no—
290D4—44no—
291C2×C4—28no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—3764no—
303C2×C4—1882no—
304C2×C4—1882no—
305C2×C4—1882no—
306C2×C4—1882no—
307C2×C4—1882no—
308C2×C4—1882no—
309C2×C4#1309393812yes—
310C2×C4#1309398012yes—
311C2×C4#1309400512yes—
312C2×C4#1309400812yes—
313C8—82no—
314C8—3762no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22—81no—
332C22#1568898411yescharacteristic
333C22C2×#1309393811yescharacteristic
334C22C2×#1309393811yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—3762no—
340C4—3762no—
341C2—21no—
342C2—21no—
343C2—42no—
344C2—81no—
345C2—81no—
346C2#2024346011yescharacteristic
347C2#2024507911yescharacteristic
348C2#2024508211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC22×Q8C22×D47
FittingC47×#1785C2
socleC22×C47C2×#99
prime radicalC22×#41×C47C22
cosocleC22×Q8×C47C23
hypercenter#1785D47
hypocenterC47#12863
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ #1785

Complements

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

NH ≅ G/Ntype
3. #20241514C2semidirect
194. C47#12863semidirect

Maximal subgroups

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

subgroupindex
#202459802
#202433162
#202415142
C47×#17852
#1286347