Finite Group Databasebeta

← #26112036

Subgroups — #26112036

Order 12032.

630 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 143 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#20251727C211yesmaximal, characteristic
2C2×#15688989C211yesmaximal, characteristic
3C2×#15688861C212yesmaximal
4#20241268C212yesmaximal
5C47×#1817C211yesmaximal, characteristic
6C2×#13094003C2211yescharacteristic
7C22×C4⋊Dic47—21no—
8#15689029C412yescomplemented
9#15689021C412yescomplemented
10#15689007—22no—
11#15688996—21no—
12#15688989—24no—
13#15688986—21no—
14#15688983C2211yescharacteristic
15C2×C47⋊#95—21no—
16C2×C47⋊#95C2212yes—
17#15688861—24no—
18C22×C42×C47C2211yescharacteristic
19C2×C22⋊C8×C47C2212yes—
20C47×#336—22no—
21C22×Dic94C2×C411yescharacteristic
22C23×Dic47—22no—
23#13094003—22no—
24#13094003—22no—
25C2×C4⋊Dic47—24no—
26C2×C4⋊Dic47—42no—
27C2×C4⋊Dic47—42no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C412yes—
30C2×Dic47⋊C4—22no—
31C2×Dic47⋊C4—21no—
32C2×C22⋊Dic47—21no—
33C2×C22⋊Dic47—21no—
34C47⋊#95—24no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—22no—
39C47⋊#95—21no—
40C47⋊#95D414yes—
41C23×C4×C47D411yescharacteristic
42C23×C4×C47D411yescharacteristic
43C23×C4×C47C2311yescharacteristic
44C22×C8×C47—22no—
45C2×C42×C47—22no—
46C2×C42×C47—22no—
47C2×C42×C47C2×C412yes—
48C2×C42×C47D412yes—
49C22⋊C8×C47—22no—
50C22⋊C8×C47—22no—
51C24×C47C2×D411yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C412yes—
61C22×C4×C47C22⋊C412yes—
62C22×C4×C47SD1612yes—
63C22×C4×C47Q1612yes—
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D412yes—
67C22×C4×C47#4112yes—
68C22×Dic47—22no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C22×Dic47—24no—
72C22×Dic47—44no—
73C22×Dic47—44no—
74C22×Dic47—42no—
75C22×Dic47—42no—
76C2×Dic94—22no—
77C2×Dic94—42no—
78C2×C8×C47—44no—
79C2×C8×C47—42no—
80C42×C47—24no—
81C42×C47—24no—
82C22⋊Dic47—41no—
83C22⋊Dic47—41no—
84C22⋊Dic47—41no—
85C22⋊Dic47—41no—
86C4⋊Dic47—22no—
87C4⋊Dic47—48no—
88C4⋊Dic47—42no—
89Dic47⋊C4—44no—
90Dic47⋊C4—42no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47C2×C22⋊C411yescharacteristic
94C23×C47#12011yescharacteristic
95C23×C47#12311yescharacteristic
96C23×C47C2×SD1611yescharacteristic
97C23×C47C2×Q1611yescharacteristic
98C23×C47AGL(1,ℤ/8)11yescharacteristic
99C23×C47#13711yescharacteristic
100C2×C4×C47—22no—
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—22no—
106C2×C4×C47—22no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—42no—
115C2×C4×C47#10314yes—
116C2×C4×C47C4×D412yes—
117C2×C4×C47#12112yes—
118C2×Dic47—44no—
119C2×Dic47—44no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—44no—
128C2×Dic47—84no—
129C2×Dic47—84no—
130Dic94—82no—
131C8×C47—82no—
132#9152—471nomaximal, Sylow
133C22×C47—21no—
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—22no—
138C22×C47—22no—
139C22×C47—22no—
140C22×C47—22no—
141C22×C47#38611yescharacteristic
142C22×C47#41012yes—
143C22×C47C2×#10311yescharacteristic
144C22×C47#41711yescharacteristic
145C22×C47#42112yes—
146C22×C47#45011yescharacteristic
147C22×C47#45111yescharacteristic
148C22×C47#48311yescharacteristic
149C22×C47#48411yescharacteristic
150C4×C47—24no—
151C4×C47—22no—
152C4×C47—22no—
153C4×C47—42no—
154Dic47—84no—
155Dic47—81no—
156Dic47—81no—
157Dic47—82no—
158#2855—471no—
159C2×#382—471no—
160#1817D4711yescharacteristic
161C2×#328—472no—
162#1294—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#151212yes—
168C2×C47#151812yes—
169C2×C47#187211yescharacteristic
170C2×C47#189711yescharacteristic
171C2×C47#189911yescharacteristic
172C2×C22⋊Q8—471no—
173C22×C4⋊C4—941no—
174C22×C42C2×D4711yescharacteristic
175C2×C22⋊C8C2×D4712yes—
176#391—472no—
177#389—472no—
178#387—942no—
179#385—941no—
180#382—944no—
181#381—941no—
182#380—471no—
183C2×#95—472no—
184C2×#95—941no—
185#336—22no—
186#328—944no—
187C47#915211yescharacteristic, complemented, Sylow
188C22×Q8—471no—
189C23×C4—942no—
190C23×C4D18811yescharacteristic
191C23×C4D18811yescharacteristic
192C23×C4C22×D4711yescharacteristic
193C22×C8—22no—
194C22⋊Q8—942no—
195C22⋊Q8—942no—
196C2×C4⋊C4—471no—
197C2×C4⋊C4—472no—
198C2×C4⋊C4—944no—
199C2×C4⋊C4—942no—
200C2×C4⋊C4—941no—
201C2×C4⋊C4—1882no—
202C2×C4⋊C4—1882no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4—941no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D4712yes—
208C2×C42C22⋊D4712yes—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—474no—
212#95—944no—
213#95—942no—
214#95—942no—
215#95—942no—
216#95—942no—
217#95—941no—
218C24C2×D18811yescharacteristic
219C2×Q8—942no—
220C2×Q8—1882no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—942no—
230C22×C4—941no—
231C22×C4—941no—
232C22×C4—944no—
233C22×C4—1884no—
234C22×C4—1884no—
235C22×C4—1882no—
236C22×C4—1882no—
237C22×C4#1062532512yes—
238C22×C4Dic18812yes—
239C22×C4(C2×D47)⋊C412yes—
240C22×C4(C2×D47)⋊C412yes—
241C22×C4C2×C4×D4711yescharacteristic
242C22×C4D4×D4712yes—
243C22×C4#1062535212yes—
244C22×C4C2×C22⋊D4711yescharacteristic
245C2×C8—44no—
246C2×C8—42no—
247C4⋊C4—942no—
248C4⋊C4—1882no—
249C4⋊C4—1884no—
250C4⋊C4—1888no—
251C4⋊C4—1882no—
252C22⋊C4—1881no—
253C22⋊C4—1881no—
254C22⋊C4—1881no—
255C22⋊C4—1881no—
256C42—24no—
257C42—24no—
258C23—22no—
259C23—22no—
260C23C2×(C2×D47)⋊C411yescharacteristic
261C23#1309399411yescharacteristic
262C23#1309400811yescharacteristic
263C23C2×#1062532511yescharacteristic
264C23C2×Dic18811yescharacteristic
265C23#1309405211yescharacteristic
266C23#1309405711yescharacteristic
267Q8—3762no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—3764no—
284C2×C4—3764no—
285C2×C4—1884no—
286C2×C4—1884no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1884no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1882no—
294C2×C4—1882no—
295C2×C4#1309394914yes—
296C2×C4(C22⋊D47)⋊C412yes—
297C2×C4#1309399612yes—
298C8—82no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#1568900011yescharacteristic
308C22#1568909612yes—
309C22C2×#1309394911yescharacteristic
310C22#1568912511yescharacteristic
311C22#1568913912yes—
312C22#1568927611yescharacteristic
313C22#1568928111yescharacteristic
314C22#1568945411yescharacteristic
315C22#1568945911yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—3764no—
321C4—3762no—
322C4—3761no—
323C4—3761no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#2024192712yes—
329C2#2024195312yes—
330C2#2024369311yescharacteristic
331C2#2024383411yescharacteristic
332C2#2024384411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C2×D4
cosocleC23×C4×C47C23
hypercenter#1817D47
hypocenterC47#9152
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
8. #15689029C4semidirect
9. #15689021C4semidirect
187. C47#9152semidirect

Maximal subgroups

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

subgroupindex
#202517272
C2×#156889892
C2×#156888612
#202412682
C47×#18172
#915247