Finite Group Databasebeta

← #26132545

Subgroups — #26132545

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#20245879C211yesmaximal, 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—
14C4×C22⋊Dic47—21no—
15C4×C22⋊Dic47—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⋊Dic47C2×C412yes—
24C2×C22⋊Dic47—22no—
25C2×C22⋊Dic47—22no—
26C2×C22⋊Dic47—21no—
27C2×C4×Dic47—21no—
28C2×C4×Dic47—21no—
29C2×C4×Dic47—42no—
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—44no—
82C22×Dic47—42no—
83M16×C47—42no—
84M16×C47—42no—
85C2×C8×C47—42no—
86C22⋊Dic47—44no—
87C22⋊Dic47—44no—
88C22⋊Dic47—42no—
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—42no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47#9912yes—
108C23×C47C2×C22⋊C411yescharacteristic
109Q8×C47—44no—
110D4×C47—44no—
111D4×C47—44no—
112D4×C47—44no—
113C2×C4×C47—28no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—44no—
123C2×C4×C47—44no—
124C2×C4×C47#9912yes—
125C2×C4×C47#11712yes—
126C2×C4×C47#12312yes—
127C2×C4×C47#12312yes—
128C2×Dic47—24no—
129C2×Dic47—42no—
130C2×Dic47—48no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C8×C47—82no—
136C47⋊C8—82no—
137#12861—471nomaximal, Sylow
138C22×C47—22no—
139C22×C47—22no—
140C22×C47—42no—
141C22×C47—42no—
142C22×C47—42no—
143C22×C47—42no—
144C22×C47—42no—
145C22×C47—41no—
146C22×C47—41no—
147C22×C47—42no—
148C22×C47—42no—
149C22×C47—81no—
150C22×C47—81no—
151C22×C47#38111yescharacteristic
152C22×C47C2×#9911yescharacteristic
153C22×C47C2×#9911yescharacteristic
154C4×C47—22no—
155C4×C47—22no—
156C4×C47—22no—
157C4×C47—22no—
158Dic47—44no—
159Dic47—82no—
160#2297—471no—
161#1785—471no—
162#1785D4711yescharacteristic
163#1394—474no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—42no—
167C2×C47—81no—
168C2×C47—81no—
169C2×C47#181911yescharacteristic
170C2×C47#213212yes—
171C22×#41C2×D4711yescharacteristic
172C2×#117—471no—
173C22×C42—471no—
174C2×#101—472no—
175C2×#101C2×D4712yes—
176C2×#100—21no—
177C2×#100—941no—
178#407—21no—
179#407—941no—
180#391—472no—
181#390—472no—
182#386—941no—
183#386—941no—
184C4×C22⋊C4—941no—
185C4×C22⋊C4—941no—
186C47#1286111yescharacteristic, complemented, Sylow
187C2×#41—24no—
188C2×#41—42no—
189C2×#41—42no—
190C22×Q8C22×D4711yescharacteristic
191C22×D4C4×D4711yescharacteristic
192C22×D4D18811yescharacteristic
193C22×D4C22⋊D4711yescharacteristic
194C23×C4—941no—
195C23×C4C4×D4711yescharacteristic
196C23×C4D18811yescharacteristic
197C23×C4C22⋊D4711yescharacteristic
198C2×M16—22no—
199C2×M16—942no—
200#117—1882no—
201C2×C4⋊C4—472no—
202C2×C22⋊C4—941no—
203C2×C22⋊C4—942no—
204C2×C22⋊C4—942no—
205C2×C42—472no—
206C2×C42—941no—
207C2×C42—941no—
208C2×C42—1882no—
209#101—24no—
210#101—944no—
211#100—42no—
212#100—1882no—
213C22⋊C8—42no—
214C22⋊C8—1882no—
215#95—942no—
216#95—942no—
217C24(C2×D47)⋊C411yescharacteristic
218C24(C2×D47)⋊C411yescharacteristic
219C24(C2×D47)⋊C411yescharacteristic
220#41—416no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8#1062535214yes—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—942no—
243C22×C4—942no—
244C22×C4—1884no—
245C22×C4—1882no—
246C22×C4(C2×D47)⋊C411yescharacteristic
247C22×C4C2×C4×D4711yescharacteristic
248C22×C4C2×D18811yescharacteristic
249C22×C4C2×C22⋊D4711yescharacteristic
250M16—42no—
251M16—42no—
252M16—1882no—
253M16—1882no—
254C2×C8—42no—
255C2×C8—1882no—
256C4⋊C4—1882no—
257C22⋊C4—1884no—
258C22⋊C4—1884no—
259C22⋊C4—1882no—
260C42—1882no—
261C42—1882no—
262C42—1882no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23#1309393812yes—
275C23C2×(C2×D47)⋊C411yescharacteristic
276Q8—44no—
277D4—44no—
278D4—44no—
279D4—44no—
280C2×C4—28no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—944no—
292C2×C4—1882no—
293C2×C4—1882no—
294C2×C4—1882no—
295C2×C4—1882no—
296C2×C4—1888no—
297C2×C4—1882no—
298C2×C4#1309393812yes—
299C2×C4#1309398012yes—
300C2×C4#1309400512yes—
301C2×C4#1309400812yes—
302C8—82no—
303C8—3762no—
304C22—22no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—42no—
314C22—42no—
315C22—81no—
316C22—81no—
317C22#1568898411yescharacteristic
318C22C2×#1309393811yescharacteristic
319C22C2×#1309393811yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—3762no—
325C4—1884no—
326C2—21no—
327C2—21no—
328C2—42no—
329C2—81no—
330C2—81no—
331C2#2024346011yescharacteristic
332C2#2024507112yes—
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#12861
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
186. C47#12861semidirect

Maximal subgroups

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

subgroupindex
#202458792
#202433162
#202415142
C47×#17852
#1286147