Finite Group Databasebeta

← #26134921

Subgroups — #26134921

Order 12032.

484 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 103 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#20247705C211yesmaximal, characteristic
2C4×#13093941C211yesmaximal, characteristic
3#20241505C214yesmaximal
4C47×#1799C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic, complemented
6C2×#13093941C2212yes—
7C4×#10625329—21no—
8#15689062—21no—
9#15689059—21no—
10#15689021—21no—
11#15689003C2212yes—
12#15688995C412yescomplemented
13#15688983—21no—
14#15688983C411yescharacteristic, complemented
15#15688892—21no—
16C2×C4×D4×C47C2211yescharacteristic
17C2×#99×C47C2212yes—
18C47×#380—21no—
19C47×#342—21no—
20C23×Dic47C2×C412yes—
21C2×#10625329—22no—
22C2×C4⋊Dic47—21no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47—41no—
27C2×Dic47⋊C4—22no—
28C2×C22⋊Dic47—22no—
29C2×C22⋊Dic47C2×C412yes—
30#13093941—22no—
31#13093941Q814yes—
32#13093935—41no—
33C4×C47⋊C8—41no—
34C47⋊#95—22no—
35C47⋊#95—21no—
36C47⋊#95—22no—
37C22×D4×C47C2311yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C4×D4×C47—41no—
41C4×D4×C47—41no—
42C2×C4⋊C4×C47—41no—
43C2×C4⋊C4×C47D411yescharacteristic
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48#99×C47—24no—
49#95×C47—41no—
50C24×C47C22×C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47C2×Q812yes—
55C2×D4×C47#4112yes—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—42no—
71C22×Dic47—82no—
72C4⋊C4×C47—41no—
73C4⋊C4×C47—82no—
74C22⋊C4×C47—41no—
75C22⋊C4×C47—41no—
76C22⋊C4×C47—42no—
77C22⋊C4×C47—42no—
78C42×C47—21no—
79C42×C47—21no—
80C22⋊Dic47—42no—
81C22⋊Dic47—82no—
82C4⋊Dic47—42no—
83C4⋊Dic47—42no—
84C4⋊Dic47—41no—
85C4⋊Dic47—41no—
86Dic47⋊C4—82no—
87#10625329—24no—
88#10625329—42no—
89C2×C47⋊C8—42no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47C2×C22⋊C411yescharacteristic
99C23×C47C4×Q812yes—
100C23×C47#12411yescharacteristic
101C23×C47C4⋊Q811yescharacteristic
102D4×C47—42no—
103D4×C47—42no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—81no—
117C2×C4×C47—82no—
118C2×C4×C47—82no—
119C2×C4×C47—82no—
120C2×C4×C47#9911yescharacteristic
121C2×C4×C47#9911yescharacteristic
122C2×C4×C47C22⋊Q812yes—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—84no—
130C47⋊C8—44no—
131#13292—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—22no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—42no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—81no—
146C22×C47#39111yescharacteristic
147C22×C47C2×#9911yescharacteristic
148C22×C47#41011yescharacteristic
149C4×C47—21no—
150C4×C47—22no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—162no—
155C4×C47#35212yes—
156Dic47—82no—
157Dic47—82no—
158#2487—471no—
159#1799D4711yescharacteristic
160C4×#100—471no—
161#1390—474no—
162C2×C47—21no—
163C2×C47—21no—
164C2×C47—41no—
165C2×C47—42no—
166C2×C47—41no—
167C2×C47#199611yescharacteristic
168C2×C47C2×#35211yescharacteristic
169C2×C47#212211yescharacteristic
170C2×C4×D4C2×D4711yescharacteristic
171C22×C4⋊C4—471no—
172C2×#100—472no—
173C2×#99C2×D4712yes—
174C4×M16—941no—
175#400—941no—
176#399—941no—
177#389—941no—
178#386—472no—
179#385—472no—
180#380—21no—
181#380—471no—
182#380—941no—
183#343—941no—
184#342—21no—
185C47#1329211yescharacteristic, complemented, Sylow
186C22×D4C22×D4711yescharacteristic
187C23×C4—472no—
188C23×C4C4×D4711yescharacteristic
189C23×C4C22⋊D4711yescharacteristic
190C2×M16—942no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—942no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4—1881no—
199C2×C4⋊C4—1881no—
200C2×C4⋊C4D18811yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—472no—
203C2×C22⋊C4—942no—
204C2×C22⋊C4C4×D4711yescharacteristic
205C2×C22⋊C4C22⋊D4711yescharacteristic
206C2×C42D18811yescharacteristic
207#100—474no—
208#100—942no—
209#99—24no—
210#97—1881no—
211C4×C8—1881no—
212#95—41no—
213#95—941no—
214#95—942no—
215#95—942no—
216C24C2×C4×D4711yescharacteristic
217C24C2×C22⋊D4711yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1062535212yes—
221C2×D4Q8×D4712yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—942no—
230C22×C4—942no—
231C22×C4—3762no—
232C22×C4—1882no—
233C22×C4(C2×D47)⋊C411yescharacteristic
234C22×C4(C2×D47)⋊C411yescharacteristic
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4C2×D18811yescharacteristic
238M16—944no—
239M16—1882no—
240C2×C8—1882no—
241C4⋊C4—41no—
242C4⋊C4—82no—
243C4⋊C4—3762no—
244C4⋊C4—1882no—
245C4⋊C4—1882no—
246C4⋊C4—1881no—
247C4⋊C4—1881no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—3762no—
253C22⋊C4—1882no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D47)⋊C411yescharacteristic
265C23#1309398912yes—
266C23#1309401111yescharacteristic
267C23#1309402211yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—3764no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4#1309393811yescharacteristic
294C2×C4#1309393811yescharacteristic
295C2×C4#1309400412yes—
296C8—1884no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22#1568902811yescharacteristic
312C22C2×#1309393811yescharacteristic
313C22#1568909611yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—162no—
320C4—3762no—
321C4—3762no—
322C4#1568891212yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2024437211yescharacteristic
329C2C2×#1568891211yescharacteristic
330C2#2024503711yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
FittingC47×#1799C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenter#1799D47
hypocenterC47#13292
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
5. C22×C4⋊Dic47C4semidirect
12. #15688995C4semidirect
14. #15688983C4semidirect
185. C47#13292semidirect

Maximal subgroups

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

subgroupindex
#202477052
C4×#130939412
#202415052
C47×#17992
#1329247