Finite Group Databasebeta

← #26134926

Subgroups — #26134926

Order 12032.

612 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 107 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#20246753C211yesmaximal, characteristic
2#20243376C211yesmaximal, characteristic
3#20241507C214yesmaximal
4C4×#100×C47C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic, complemented
6C2×#13094003—21no—
7C2×#13093939C2212yes—
8#15689043—21no—
9#15689035—21no—
10#15689029—21no—
11#15689003C2212yes—
12#15688995C412yescomplemented
13#15688983—21no—
14#15688983C411yescharacteristic, complemented
15#15688890—21no—
16C2×C4×D4×C47C2211yescharacteristic
17C2×#100×C47C2212yes—
18C4×M16×C47—21no—
19C47×#343—21no—
20C22×Dic94—21no—
21C22×Dic94—21no—
22C22×Dic94—44no—
23C22×Dic94—41no—
24C22×Dic94—41no—
25C23×Dic47C2×C412yes—
26#13094003—44no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—41no—
29C2×Dic47⋊C4—22no—
30C2×C22⋊Dic47—22no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47C2×C412yes—
33#13093939—24no—
34C47⋊#95—21no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—41no—
38C22×D4×C47C2311yescharacteristic
39C23×C4×C47C2×C411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C2×M16×C47—22no—
42C4×D4×C47—41no—
43C4×D4×C47—41no—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C42×C47D411yescharacteristic
48#100×C47—22no—
49#100×C47Q814yescomplemented
50#97×C47—41no—
51C4×C8×C47—41no—
52C24×C47C22×C411yescharacteristic
53C24×C47C2×D411yescharacteristic
54C2×D4×C47—41no—
55C2×D4×C47—41no—
56C2×D4×C47C2×Q812yes—
57C2×D4×C47#4112yes—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C2×D411yescharacteristic
69C22×Dic47—22no—
70C22×Dic47—22no—
71C22×Dic47—44no—
72C22×Dic47—42no—
73C22×Dic47—42no—
74C2×Dic94—24no—
75C2×Dic94—44no—
76C2×Dic94—42no—
77C2×Dic94—42no—
78C2×Dic94—48no—
79C2×Dic94—44no—
80C2×Dic94—41no—
81C2×Dic94—41no—
82C2×Dic94—44no—
83M16×C47—24no—
84M16×C47—42no—
85C2×C8×C47—42no—
86C4⋊C4×C47—41no—
87C22⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C42×C47—21no—
90C42×C47—21no—
91C22⋊Dic47—44no—
92C22⋊Dic47—42no—
93C22⋊Dic47—42no—
94C22⋊Dic47—42no—
95C4⋊Dic47—42no—
96C4⋊Dic47—82no—
97Dic47⋊C4—44no—
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×C47C2×C22⋊C411yescharacteristic
107C23×C47C4×Q812yes—
108C23×C47#12411yescharacteristic
109C23×C47C4⋊Q811yescharacteristic
110D4×C47—42no—
111D4×C47—42no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—22no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—81no—
125C2×C4×C47#9911yescharacteristic
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47C22⋊Q812yes—
128C2×Dic47—24no—
129C2×Dic47—42no—
130C2×Dic47—44no—
131C2×Dic47—42no—
132C2×Dic47—44no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—82no—
137C2×Dic47—82no—
138C2×Dic47—82no—
139Dic94—44no—
140Dic94—48no—
141Dic94—44no—
142C8×C47—44no—
143#13293—471nomaximal, Sylow
144C22×C47—21no—
145C22×C47—21no—
146C22×C47—21no—
147C22×C47—22no—
148C22×C47—21no—
149C22×C47—42no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—42no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—81no—
158C22×C47#39111yescharacteristic
159C22×C47C2×#9911yescharacteristic
160C22×C47#41011yescharacteristic
161C4×C47—21no—
162C4×C47—22no—
163C4×C47—41no—
164C4×C47—41no—
165C4×C47—41no—
166C4×C47AΣL(1,𝔽16)12yes—
167Dic47—44no—
168Dic47—82no—
169Dic47—162no—
170#2409—471no—
171#1799—471no—
172C4×#100D4711yescharacteristic
173#1390—474no—
174C2×C47—21no—
175C2×C47—21no—
176C2×C47—41no—
177C2×C47—42no—
178C2×C47—41no—
179C2×C47#199611yescharacteristic
180C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
181C2×C47#212311yescharacteristic
182C23×Q8—471no—
183C2×C22⋊Q8—941no—
184C2×C4×D4C2×D4711yescharacteristic
185C2×#100C2×D4712yes—
186C2×#99—472no—
187C4×M16—21no—
188#395—941no—
189#393—941no—
190#391—941no—
191#386—472no—
192#385—472no—
193#380—471no—
194#380—941no—
195#343—21no—
196#342—941no—
197C47#1329311yescharacteristic, complemented, Sylow
198C22×Q8—941no—
199C22×Q8—941no—
200C22×Q8—1881no—
201C22×Q8—1881no—
202C22×Q8—1884no—
203C22×D4C22×D4711yescharacteristic
204C23×C4—472no—
205C23×C4C4×D4711yescharacteristic
206C23×C4D18811yescharacteristic
207C2×M16—22no—
208C22⋊Q8—1884no—
209C4×D4—41no—
210C4×D4—41no—
211C2×C4⋊C4—942no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4—1881no—
214C2×C4⋊C4C22⋊D4711yescharacteristic
215C2×C22⋊C4—472no—
216C2×C22⋊C4—942no—
217C2×C22⋊C4—942no—
218C2×C22⋊C4C4×D4711yescharacteristic
219C2×C22⋊C4D18811yescharacteristic
220C2×C42C22⋊D4711yescharacteristic
221#100—22no—
222#100Dic9414yescomplemented
223#99—944no—
224#97—41no—
225C4×C8—41no—
226#95—941no—
227#95—942no—
228#95—942no—
229#95—1881no—
230C24C2×C4×D4711yescharacteristic
231C24C2×D18811yescharacteristic
232C2×Q8—944no—
233C2×Q8—1882no—
234C2×Q8—1882no—
235C2×Q8—1884no—
236C2×Q8—1884no—
237C2×Q8—1888no—
238C2×Q8—1884no—
239C2×Q8—1881no—
240C2×Q8—1881no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4C2×Dic9412yes—
244C2×D4#1062535012yes—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—942no—
252C22×C4—942no—
253C22×C4—1884no—
254C22×C4—1882no—
255C22×C4—1882no—
256C22×C4(C2×D47)⋊C411yescharacteristic
257C22×C4(C2×D47)⋊C411yescharacteristic
258C22×C4(C2×D47)⋊C411yescharacteristic
259C22×C4(C2×D47)⋊C411yescharacteristic
260C22×C4C2×C22⋊D4711yescharacteristic
261M16—24no—
262M16—42no—
263C2×C8—42no—
264C4⋊C4—41no—
265C4⋊C4—3762no—
266C4⋊C4—1882no—
267C4⋊C4—1884no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—1884no—
271C22⋊C4—1882no—
272C22⋊C4—1882no—
273C22⋊C4—1882no—
274C42—21no—
275C42—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×(C2×D47)⋊C411yescharacteristic
285C23C4×Dic9412yes—
286C23#1309401211yescharacteristic
287C23#1309402411yescharacteristic
288Q8—1884no—
289Q8—1888no—
290Q8—1884no—
291D4—42no—
292D4—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—944no—
307C2×C4—3762no—
308C2×C4—3762no—
309C2×C4—3762no—
310C2×C4—1882no—
311C2×C4—1882no—
312C2×C4—1882no—
313C2×C4—1882no—
314C2×C4—1884no—
315C2×C4—1882no—
316C2×C4—1884no—
317C2×C4#1309393811yescharacteristic
318C2×C4#1309393811yescharacteristic
319C2×C4#1309400312yes—
320C8—44no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—22no—
325C22—21no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22#1568902911yescharacteristic
336C22C2×#1309393811yescharacteristic
337C22#1568909611yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3762no—
344C4—1884no—
345C4—7522no—
346C4#1568891112yes—
347C2—21no—
348C2—21no—
349C2—41no—
350C2—42no—
351C2—41no—
352C2#2024437311yescharacteristic
353C2C2×#1568891111yescharacteristic
354C2#2024504311yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC24×C47C22×C4
FrattiniC22×D4C22×D47
FittingC4×#100×C47C2
socleC22×C47C2×#99
prime radical, cosocleC22×D4×C47C23
hypercenterC4×#100D47
hypocenterC47#13293
LCSG ▷ C24×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
5. C23×Dic94C4semidirect
12. #15688995C4semidirect
14. #15688983C4semidirect
49. #100×C47Q8semidirect
197. C47#13293semidirect
222. #100Dic94semidirect

Maximal subgroups

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

subgroupindex
#202467532
#202433762
#202415072
C4×#100×C472
#1329347