Finite Group Databasebeta

← #26119628

Subgroups — #26119628

Order 12032.

576 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 117 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#20243696C212yesmaximal
2C2×#15688860C212yesmaximal
3#20241644C211yesmaximal, characteristic
4#20241231C211yesmaximal, characteristic
5C47×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C2×#13093948—22no—
8C2×C22⋊(C47⋊C8)C2212yes—
9C2×C47⋊#95—21no—
10C2×C47⋊#95—21no—
11C2×C47⋊#95C2211yescharacteristic
12#15688885—22no—
13#15688860—24no—
14C2×C22⋊Q8×C47C2212yes—
15C47×#398—22no—
16C47×#395—22no—
17C47×#380—21no—
18C47×#380—21no—
19C2×#95×C47C2211yescharacteristic
20C23×Dic47—21no—
21C23×Dic47—21no—
22C22×C47⋊C8—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—42no—
25#13093948—44no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—22no—
28C47⋊#95—21no—
29C47⋊#95—42no—
30C47⋊#95—42no—
31C47⋊#95D412yes—
32C22×Q8×C47D412yes—
33C23×C4×C47D411yescharacteristic
34C23×C4×C47D411yescharacteristic
35C23×C4×C47C2311yescharacteristic
36C22⋊Q8×C47—22no—
37C22⋊Q8×C47—22no—
38C22⋊Q8×C47—22no—
39C22⋊Q8×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47D412yes—
46C2×C22⋊C4×C47—22no—
47#95×C47—22no—
48#95×C47—22no—
49#95×C47—22no—
50C24×C47C2×D411yescharacteristic
51C2×Q8×C47—22no—
52C2×Q8×C47—22no—
53C2×Q8×C47—42no—
54C2×Q8×C47—42no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—41no—
62C22×C4×C47—41no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47SD1612yes—
68C22×C4×C47Q1612yes—
69C22×C4×C47C2×D412yes—
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4112yes—
72C22×C4×C47#4111yescharacteristic
73C22×Dic47—22no—
74C22×Dic47—22no—
75C22×Dic47—41no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C22×Dic47—41no—
81C22×Dic47—41no—
82C22×Dic47—41no—
83C22×Dic47—41no—
84C22×Dic47—41no—
85C22×Dic47—41no—
86C22×Dic47—41no—
87C4⋊C4×C47—22no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—42no—
90C4⋊C4×C47—42no—
91C4⋊C4×C47—81no—
92C4⋊C4×C47—81no—
93C4⋊C4×C47—81no—
94C4⋊C4×C47—81no—
95C22⋊C4×C47—42no—
96C22⋊C4×C47—42no—
97C4⋊Dic47—24no—
98C4⋊Dic47—42no—
99C2×C47⋊C8—42no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47#12011yescharacteristic
107C23×C47#12111yescharacteristic
108C23×C47#12411yescharacteristic
109C23×C47C2×SD1611yescharacteristic
110C23×C47C2×Q1611yescharacteristic
111C23×C47#13711yescharacteristic
112C23×C47#13711yescharacteristic
113Q8×C47—44no—
114Q8×C47—82no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—22no—
119C2×C4×C47—41no—
120C2×C4×C47—42no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—42no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—81no—
135C2×C4×C47—81no—
136C2×C4×C47#12112yes—
137C2×C4×C47#12312yes—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—42no—
144C2×Dic47—81no—
145C2×Dic47—81no—
146C2×Dic47—81no—
147C2×Dic47—81no—
148C2×Dic47—81no—
149C2×Dic47—81no—
150C2×Dic47—81no—
151C2×Dic47—81no—
152C47⋊C8—82no—
153#10407—471nomaximal, Sylow
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47—21no—
166C22×C47#39411yescharacteristic
167C22×C47#44811yescharacteristic
168C22×C47#45111yescharacteristic
169C22×C47#45411yescharacteristic
170C22×C47#45511yescharacteristic
171C22×C47B(4,𝔽2)11yescharacteristic
172C22×C47#45811yescharacteristic
173C22×C47#46111yescharacteristic
174C22×C47#46211yescharacteristic
175C22×C47#48711yescharacteristic
176C22×C47#49111yescharacteristic
177C4×C47—22no—
178C4×C47—22no—
179C4×C47—42no—
180C4×C47—42no—
181C4×C47—82no—
182C4×C47—82no—
183Dic47—82no—
184Dic47—82no—
185#2659D4711yescharacteristic
186#1872—472no—
187C2×#328—472no—
188#1442—471no—
189#1280—471no—
190C2×C47—21no—
191C2×C47—21no—
192C2×C47—21no—
193C2×C47—21no—
194C2×C47#159911yescharacteristic
195C2×C47#160211yescharacteristic
196C2×C47#160811yescharacteristic
197C2×C47#161711yescharacteristic
198C2×C47#204011yescharacteristic
199C2×C47#204311yescharacteristic
200C2×C47#204411yescharacteristic
201C2×C22⋊Q8C2×D4712yes—
202C22×C4⋊C4—471no—
203C2×#103—942no—
204C2×C22⋊C8—472no—
205#398—22no—
206#395—22no—
207#380—21no—
208#380—21no—
209C2×#95—471no—
210C2×#95—941no—
211C2×#95—941no—
212C2×#95C2×D4711yescharacteristic
213#340—942no—
214#328—944no—
215C47#1040711yescharacteristic, complemented, Sylow
216C22×Q8C22⋊D4712yes—
217C23×C4—941no—
218C23×C4—941no—
219C23×C4C22⋊D4711yescharacteristic
220C23×C4C22⋊D4711yescharacteristic
221C23×C4C22×D4711yescharacteristic
222C22×C8—942no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—942no—
233C2×C4⋊C4—1882no—
234C2×C4⋊C4C22⋊D4712yes—
235C2×C22⋊C4—22no—
236#103—1884no—
237C22⋊C8—942no—
238C22⋊C8—942no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—472no—
243#95—941no—
244#95—1882no—
245#95—1882no—
246C24C2×C22⋊D4711yescharacteristic
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—42no—
250C2×Q8—42no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—942no—
264C22×C4—942no—
265C22×C4—1881no—
266C22×C4—1881no—
267C22×C4—1881no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4—1881no—
271C22×C4—1881no—
272C22×C4—1881no—
273C22×C4—1881no—
274C22×C4—1881no—
275C22×C4—1881no—
276C22×C4—1881no—
277C22×C4#1062533612yes—
278C22×C4#1062533712yes—
279C22×C4D4×D4711yescharacteristic
280C22×C4#1062535212yes—
281C22×C4#1062535211yescharacteristic
282C22×C4C2×C22⋊D4712yes—
283C2×C8—1882no—
284C2×C8—1882no—
285C2×C8—1882no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—81no—
294C4⋊C4—944no—
295C4⋊C4—1882no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23#1309399311yescharacteristic
303C23#1309399511yescharacteristic
304C23#1309401111yescharacteristic
305C23C2×#1062533611yescharacteristic
306C23C2×#1062533711yescharacteristic
307C23#1309405511yescharacteristic
308C23#1309405511yescharacteristic
309Q8—44no—
310Q8—82no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—3761no—
333C2×C4—3761no—
334C2×C4—3761no—
335C2×C4—3761no—
336C2×C4—3761no—
337C2×C4—3761no—
338C2×C4—3761no—
339C2×C4—3761no—
340C2×C4—1882no—
341C2×C4—1882no—
342C2×C4—1882no—
343C2×C4—1882no—
344C2×C4—1882no—
345C2×C4—1882no—
346C2×C4#1309399612yes—
347C2×C4#1309400512yes—
348C8—3762no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22#1568903711yescharacteristic
362C22#1568926211yescharacteristic
363C22#1568927911yescharacteristic
364C22#1568929611yescharacteristic
365C22#1568930111yescharacteristic
366C22#1568931011yescharacteristic
367C22#1568931311yescharacteristic
368C22#1568932511yescharacteristic
369C22#1568933011yescharacteristic
370C22#1568947311yescharacteristic
371C22#1568949111yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—82no—
377C4—82no—
378C4—3762no—
379C4—3762no—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—21no—
384C2#2024232811yescharacteristic
385C2#2024234511yescharacteristic
386C2#2024237911yescharacteristic
387C2#2024243011yescharacteristic
388C2#2024461411yescharacteristic
389C2#2024463111yescharacteristic
390C2#2024463611yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2659C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#2659D47
hypocenterC47#10407
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
215. C47#10407semidirect

Maximal subgroups

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

subgroupindex
#202436962
C2×#156888602
#202416442
#202412312
C47×#26592
#1040747