Finite Group Databasebeta

← #26119263

Subgroups — #26119263

Order 12032.

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

normal subgroups only

This group has 544 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#20243478C212yesmaximal
3#20241267C212yesmaximal
4#20241231C211yesmaximal, characteristic
5C47×#1445C211yesmaximal, characteristic
6C2×#13094003—21no—
7C2×C22⋊(C47⋊C8)C2212yes—
8#15689055—22no—
9#15689044—22no—
10#15688983—21no—
11#15688983C2212yes—
12#15688885—22no—
13#15688885—22no—
14C22×C4⋊C4×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—
30C22⋊(C47⋊C8)—22no—
31C22⋊(C47⋊C8)—22no—
32C47⋊#95—22no—
33C47⋊#95—22no—
34C23×C4×C47—21no—
35C23×C4×C47—21no—
36C23×C4×C47D411yescharacteristic
37C23×C4×C47D411yescharacteristic
38C23×C4×C47C2311yescharacteristic
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—22no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47Q812yescomplemented
43#95×C47—22no—
44#95×C47—22no—
45#95×C47—42no—
46#95×C47—42no—
47C24×C47C2×D411yescharacteristic
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47—41no—
59C22×C4×C47—41no—
60C22×C4×C47—41no—
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×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47—41no—
70C22×C4×C47—41no—
71C22×C4×C47—41no—
72C22×C4×C47SD1612yes—
73C22×C4×C47Q1612yes—
74C22×C4×C47C2×D412yes—
75C22×C4×C47C2×Q811yescharacteristic
76C22×C4×C47#4112yes—
77C22×C4×C47#4111yescharacteristic
78C22×Dic47—22no—
79C22×Dic47—22no—
80C2×Dic94—42no—
81C2×Dic94—81no—
82C2×Dic94—81no—
83C4⋊C4×C47—24no—
84C4⋊C4×C47—42no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C4⋊Dic47—42no—
88C4⋊Dic47—42no—
89C4⋊Dic47—42no—
90C4⋊Dic47—81no—
91C4⋊Dic47—81no—
92Dic47⋊C4—44no—
93C2×C47⋊C8—42no—
94C2×C47⋊C8—42no—
95C2×C47⋊C8—42no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47#12011yescharacteristic
101C23×C47C22⋊Q811yescharacteristic
102C23×C47#12411yescharacteristic
103C23×C47C2×SD1611yescharacteristic
104C23×C47C2×Q1611yescharacteristic
105C23×C47AGL(1,ℤ/8)11yescharacteristic
106C23×C47#13711yescharacteristic
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—81no—
127C2×C4×C47—81no—
128C2×C4×C47—81no—
129C2×C4×C47—81no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47—81no—
133C2×C4×C47—81no—
134C2×C4×C47—81no—
135C2×C4×C47—81no—
136C2×C4×C47C22⋊Q812yes—
137C2×C4×C47#12412yes—
138C2×Dic47—42no—
139C2×Dic47—42no—
140C2×Dic47—42no—
141C2×Dic47—42no—
142C2×Dic47—42no—
143C2×Dic47—42no—
144Dic94—84no—
145C47⋊C8—82no—
146#10343—471nomaximal, Sylow
147C22×C47—21no—
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—21no—
151C22×C47—21no—
152C22×C47—21no—
153C22×C47—21no—
154C22×C47—21no—
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47#39311yescharacteristic
160C22×C47#45011yescharacteristic
161C22×C47#45111yescharacteristic
162C22×C47#45311yescharacteristic
163C22×C47#45411yescharacteristic
164C22×C47#45811yescharacteristic
165C22×C47#45811yescharacteristic
166C22×C47#47511yescharacteristic
167C22×C47#47711yescharacteristic
168C22×C47#48711yescharacteristic
169C22×C47#48911yescharacteristic
170C4×C47—22no—
171C4×C47—22no—
172C4×C47—42no—
173C4×C47—42no—
174C4×C47—82no—
175C4×C47—82no—
176Dic47—82no—
177Dic47—82no—
178#2659—471no—
179#1824—472no—
180#1445D4711yescharacteristic
181#1294—472no—
182#1280—471no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47#161311yescharacteristic
188C2×C47#161411yescharacteristic
189C2×C47#161611yescharacteristic
190C2×C47#161711yescharacteristic
191C2×C47#202411yescharacteristic
192C2×C47#202711yescharacteristic
193C2×C47#203011yescharacteristic
194C2×C22⋊Q8—941no—
195C22×C4⋊C4C2×D4711yescharacteristic
196C2×C22⋊C8—472no—
197#398—942no—
198#395—942no—
199#380—472no—
200#380—941no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95C2×D4711yescharacteristic
204C2×#95C2×D4711yescharacteristic
205#340—942no—
206#340—942no—
207C47#1034311yescharacteristic, complemented, Sylow
208C22×Q8—941no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22⋊D4711yescharacteristic
212C23×C4C22⋊D4711yescharacteristic
213C23×C4C22×D4711yescharacteristic
214C22×C8—942no—
215C22⋊Q8—1882no—
216C22⋊Q8—1882no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—472no—
221C2×C4⋊C4—472no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4—942no—
225C2×C4⋊C4—941no—
226C2×C4⋊C4Dic9412yescomplemented
227C2×C22⋊C4—942no—
228C22⋊C8—942no—
229C22⋊C8—942no—
230#95—22no—
231#95—22no—
232#95—42no—
233#95—42no—
234#95—942no—
235#95—942no—
236C24C2×C22⋊D4711yescharacteristic
237C2×Q8—3761no—
238C2×Q8—3761no—
239C2×Q8—1882no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—942no—
265C22×C4—942no—
266C22×C4#1062533612yes—
267C22×C4#1062533712yes—
268C22×C4C2×Dic9411yescharacteristic
269C22×C4#1062535011yescharacteristic
270C22×C4D4×D4712yes—
271C22×C4#1062535212yes—
272C2×C8—1882no—
273C2×C8—1882no—
274C2×C8—1882no—
275C4⋊C4—24no—
276C4⋊C4—42no—
277C4⋊C4—3761no—
278C4⋊C4—3761no—
279C4⋊C4—1882no—
280C4⋊C4—1882no—
281C4⋊C4—1884no—
282C4⋊C4—1882no—
283C22⋊C4—1882no—
284C22⋊C4—1882no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1309399211yescharacteristic
290C23#1309400311yescharacteristic
291C23#1309401111yescharacteristic
292C23C2×#1062533611yescharacteristic
293C23C2×#1062533711yescharacteristic
294C23#1309404911yescharacteristic
295C23#1309405411yescharacteristic
296Q8—3764no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4#1309400112yes—
333C2×C4#1309401012yes—
334C8—3762no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22#1568903511yescharacteristic
348C22#1568927311yescharacteristic
349C22#1568927811yescharacteristic
350C22#1568929011yescharacteristic
351C22#1568929511yescharacteristic
352C22#1568931211yescharacteristic
353C22#1568931211yescharacteristic
354C22#1568940811yescharacteristic
355C22#1568941811yescharacteristic
356C22#1568947311yescharacteristic
357C22#1568948211yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—82no—
363C4—82no—
364C4—3762no—
365C4—3762no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2024240711yescharacteristic
371C2#2024241211yescharacteristic
372C2#2024242411yescharacteristic
373C2#2024242911yescharacteristic
374C2#2024452911yescharacteristic
375C2#2024454411yescharacteristic
376C2#2024455511yescharacteristic
subgroupG/H
centerC23#13093992
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1445C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#1445D47
hypocenterC47#10343
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
42. C2×C4⋊C4×C47Q8semidirect
207. C47#10343semidirect
226. C2×C4⋊C4Dic94semidirect

Maximal subgroups

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

subgroupindex
#202495022
#202434782
#202412672
#202412312
C47×#14452
#1034347