Finite Group Databasebeta

← #26119253

Subgroups — #26119253

Order 12032.

684 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 115 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#20249502C211yesmaximal, characteristic
2#20243323C212yesmaximal
3#20241267C212yesmaximal
4#20241231C211yesmaximal, characteristic
5C47×#1446C211yesmaximal, characteristic
6C2×#13094003—21no—
7C2×#13093948—22no—
8C2×#13093948—22no—
9C2×C22⋊(C47⋊C8)C2212yes—
10#15689055—22no—
11#15689044—22no—
12#15688983—21no—
13#15688983C2212yes—
14C23×Q8×C47C2211yescharacteristic
15C2×#95×C47—21no—
16C2×#95×C47—21no—
17C2×#95×C47C2211yescharacteristic
18C2×#95×C47C2211yescharacteristic
19C22×Dic94—21no—
20C22×C47⋊C8—22no—
21#13094003—42no—
22#13094003—42no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—21no—
26C2×C4⋊Dic47D412yes—
27C2×C4⋊Dic47D412yes—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—22no—
30#13093948—44no—
31#13093948—44no—
32C22⋊(C47⋊C8)—22no—
33C22⋊(C47⋊C8)—22no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C22×Q8×C47—21no—
37C22×Q8×C47—22no—
38C22×Q8×C47—22no—
39C22×Q8×C47—42no—
40C22×Q8×C47—42no—
41C22×Q8×C47Q812yescomplemented
42C23×C4×C47—21no—
43C23×C4×C47—21no—
44C23×C4×C47D411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47C2311yescharacteristic
47#95×C47—22no—
48#95×C47—22no—
49#95×C47—42no—
50#95×C47—42no—
51C24×C47C2×D411yescharacteristic
52C2×Q8×C47—24no—
53C2×Q8×C47—24no—
54C2×Q8×C47—24no—
55C2×Q8×C47—44no—
56C2×Q8×C47—42no—
57C2×Q8×C47—42no—
58C2×Q8×C47—44no—
59C2×Q8×C47—48no—
60C2×Q8×C47—42no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—22no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—22no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—41no—
74C22×C4×C47—41no—
75C22×C4×C47—41no—
76C22×C4×C47—41no—
77C22×C4×C47—41no—
78C22×C4×C47—41no—
79C22×C4×C47—41no—
80C22×C4×C47—41no—
81C22×C4×C47—41no—
82C22×C4×C47—41no—
83C22×C4×C47—41no—
84C22×C4×C47—41no—
85C22×C4×C47SD1612yes—
86C22×C4×C47Q1612yes—
87C22×C4×C47C2×D412yes—
88C22×C4×C47C2×Q811yescharacteristic
89C22×C4×C47#4112yes—
90C22×C4×C47#4111yescharacteristic
91C22×Dic47—22no—
92C22×Dic47—22no—
93C2×Dic94—42no—
94C2×Dic94—81no—
95C2×Dic94—81no—
96C22⋊Dic47—42no—
97C22⋊Dic47—42no—
98C4⋊Dic47—42no—
99C4⋊Dic47—42no—
100C4⋊Dic47—42no—
101C4⋊Dic47—81no—
102C4⋊Dic47—81no—
103Dic47⋊C4—44no—
104C2×C47⋊C8—42no—
105C2×C47⋊C8—42no—
106C2×C47⋊C8—42no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47#12011yescharacteristic
112C23×C47C22⋊Q811yescharacteristic
113C23×C47#12411yescharacteristic
114C23×C47C2×SD1611yescharacteristic
115C23×C47C2×Q1611yescharacteristic
116C23×C47AGL(1,ℤ/8)11yescharacteristic
117C23×C47#13711yescharacteristic
118Q8×C47—28no—
119Q8×C47—44no—
120Q8×C47—44no—
121Q8×C47—44no—
122C2×C4×C47—22no—
123C2×C4×C47—22no—
124C2×C4×C47—24no—
125C2×C4×C47—22no—
126C2×C4×C47—22no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—42no—
145C2×C4×C47—81no—
146C2×C4×C47—81no—
147C2×C4×C47—81no—
148C2×C4×C47—81no—
149C2×C4×C47—81no—
150C2×C4×C47—81no—
151C2×C4×C47C22⋊Q812yes—
152C2×C4×C47#12412yes—
153C2×Dic47—42no—
154C2×Dic47—42no—
155C2×Dic47—42no—
156C2×Dic47—42no—
157C2×Dic47—42no—
158C2×Dic47—42no—
159Dic94—84no—
160C47⋊C8—82no—
161#10341—471nomaximal, Sylow
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47—21no—
167C22×C47—21no—
168C22×C47—21no—
169C22×C47—21no—
170C22×C47—21no—
171C22×C47—21no—
172C22×C47—21no—
173C22×C47—21no—
174C22×C47#39311yescharacteristic
175C22×C47#45011yescharacteristic
176C22×C47#45111yescharacteristic
177C22×C47#45511yescharacteristic
178C22×C47#45611yescharacteristic
179C22×C47B(4,𝔽2)11yescharacteristic
180C22×C47B(4,𝔽2)11yescharacteristic
181C22×C47#47511yescharacteristic
182C22×C47#47711yescharacteristic
183C22×C47#48711yescharacteristic
184C22×C47#48911yescharacteristic
185C4×C47—22no—
186C4×C47—22no—
187C4×C47—42no—
188C4×C47—42no—
189C4×C47—44no—
190C4×C47—82no—
191Dic47—82no—
192Dic47—82no—
193#2659—471no—
194#1787—472no—
195#1446D4711yescharacteristic
196#1294—472no—
197#1280—471no—
198C2×C47—21no—
199C2×C47—21no—
200C2×C47—21no—
201C2×C47—21no—
202C2×C47#160111yescharacteristic
203C2×C47#160211yescharacteristic
204C2×C47#160411yescharacteristic
205C2×C47#160511yescharacteristic
206C2×C47#202411yescharacteristic
207C2×C47#202811yescharacteristic
208C2×C47#202911yescharacteristic
209C23×Q8C2×D4711yescharacteristic
210C2×C22⋊Q8—941no—
211C2×#103—942no—
212C2×#103—942no—
213C2×C22⋊C8—472no—
214#398—942no—
215#395—942no—
216#380—472no—
217#380—941no—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95C2×D4711yescharacteristic
221C2×#95C2×D4711yescharacteristic
222C47#1034111yescharacteristic, complemented, Sylow
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—941no—
229C22×Q8Dic9412yescomplemented
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C22⋊D4711yescharacteristic
233C23×C4C22⋊D4711yescharacteristic
234C23×C4C22×D4711yescharacteristic
235C22×C8—942no—
236C22⋊Q8—1882no—
237C22⋊Q8—1882no—
238C2×C4⋊C4—472no—
239C2×C4⋊C4—472no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—942no—
243C2×C4⋊C4—941no—
244C2×C22⋊C4—942no—
245#103—1884no—
246#103—1884no—
247C22⋊C8—942no—
248C22⋊C8—942no—
249#95—22no—
250#95—22no—
251#95—42no—
252#95—42no—
253#95—942no—
254#95—942no—
255C24C2×C22⋊D4711yescharacteristic
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—48no—
264C2×Q8—42no—
265C2×Q8—3761no—
266C2×Q8—3761no—
267C2×Q8—1882no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—942no—
293C22×C4—942no—
294C22×C4#1062533612yes—
295C22×C4#1062533712yes—
296C22×C4C2×Dic9411yescharacteristic
297C22×C4#1062535011yescharacteristic
298C22×C4D4×D4712yes—
299C22×C4#1062535212yes—
300C2×C8—1882no—
301C2×C8—1882no—
302C2×C8—1882no—
303C4⋊C4—3761no—
304C4⋊C4—3761no—
305C4⋊C4—1882no—
306C4⋊C4—1882no—
307C4⋊C4—1884no—
308C4⋊C4—1882no—
309C22⋊C4—1882no—
310C22⋊C4—1882no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#1309399211yescharacteristic
316C23#1309400311yescharacteristic
317C23#1309401111yescharacteristic
318C23C2×#1062533611yescharacteristic
319C23C2×#1062533711yescharacteristic
320C23#1309404911yescharacteristic
321C23#1309405411yescharacteristic
322Q8—28no—
323Q8—44no—
324Q8—44no—
325Q8—44no—
326Q8—3764no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—1882no—
357C2×C4—1882no—
358C2×C4—1882no—
359C2×C4—1882no—
360C2×C4—1882no—
361C2×C4—1882no—
362C2×C4#1309400112yes—
363C2×C4#1309401012yes—
364C8—3762no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22#1568903511yescharacteristic
378C22#1568927311yescharacteristic
379C22#1568927811yescharacteristic
380C22#1568930011yescharacteristic
381C22#1568930511yescharacteristic
382C22#1568930911yescharacteristic
383C22#1568930911yescharacteristic
384C22#1568940811yescharacteristic
385C22#1568941811yescharacteristic
386C22#1568947311yescharacteristic
387C22#1568948211yescharacteristic
388C4—22no—
389C4—22no—
390C4—44no—
391C4—42no—
392C4—42no—
393C4—82no—
394C4—3762no—
395C4—3762no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2#2024233911yescharacteristic
401C2#2024234411yescharacteristic
402C2#2024235611yescharacteristic
403C2#2024236111yescharacteristic
404C2#2024452911yescharacteristic
405C2#2024454911yescharacteristic
406C2#2024455211yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1446C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#1446D47
hypocenterC47#10341
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
41. C22×Q8×C47Q8semidirect
222. C47#10341semidirect
229. C22×Q8Dic94semidirect

Maximal subgroups

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

subgroupindex
#202495022
#202433232
#202412672
#202412312
C47×#14462
#1034147