Finite Group Databasebeta

← #26119264

Subgroups — #26119264

Order 12032.

544 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 113 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#20243479C212yesmaximal
3#20241657C211yesmaximal, characteristic
4#20241268C212yesmaximal
5C47×#1280C211yesmaximal, characteristic
6C2×#13094003—21no—
7C22×C4⋊Dic47C2211yescharacteristic
8#15689055—22no—
9#15689044—22no—
10#15688983—21no—
11#15688983C2212yes—
12C2×C47⋊#95—21no—
13C2×C47⋊#95—21no—
14C2×C47⋊#95C2211yescharacteristic
15#15688886—22no—
16#15688886—22no—
17C2×C22⋊C8×C47C2212yes—
18C2×#95×C47C2211yescharacteristic
19C22×Dic94—21no—
20C23×Dic47—21no—
21C23×Dic47—21no—
22#13094003—42no—
23#13094003—42no—
24C2×C4⋊Dic47—22no—
25C2×C4⋊Dic47—22no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47—21no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47D412yes—
31C2×C4⋊Dic47D412yes—
32C2×C4⋊Dic47Q812yescomplemented
33C2×Dic47⋊C4—22no—
34C2×C22⋊Dic47—22no—
35C47⋊#95—22no—
36C47⋊#95—22no—
37C47⋊#95—22no—
38C47⋊#95—42no—
39C47⋊#95—42no—
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C23×C4×C47C2311yescharacteristic
43C22×C8×C47—22no—
44C22⋊C8×C47—22no—
45C22⋊C8×C47—22no—
46#95×C47—22no—
47C24×C47C2×D411yescharacteristic
48C22×C4×C47—21no—
49C22×C4×C47—21no—
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—41no—
53C22×C4×C47—41no—
54C22×C4×C47—41no—
55C22×C4×C47—41no—
56C22×C4×C47—41no—
57C22×C4×C47—41no—
58C22×C4×C47SD1612yes—
59C22×C4×C47Q1612yes—
60C22×C4×C47C2×D412yes—
61C22×C4×C47C2×Q811yescharacteristic
62C22×C4×C47#4112yes—
63C22×C4×C47#4111yescharacteristic
64C22×Dic47—22no—
65C22×Dic47—22no—
66C22×Dic47—22no—
67C22×Dic47—22no—
68C22×Dic47—41no—
69C22×Dic47—41no—
70C22×Dic47—41no—
71C22×Dic47—41no—
72C22×Dic47—41no—
73C22×Dic47—41no—
74C22×Dic47—41no—
75C22×Dic47—41no—
76C22×Dic47—41no—
77C22×Dic47—41no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C2×Dic94—42no—
81C2×Dic94—81no—
82C2×Dic94—81no—
83C2×C8×C47—42no—
84C2×C8×C47—42no—
85C2×C8×C47—42no—
86C22⋊Dic47—42no—
87C22⋊Dic47—42no—
88C4⋊Dic47—24no—
89C4⋊Dic47—42no—
90C4⋊Dic47—42no—
91C4⋊Dic47—42no—
92C4⋊Dic47—42no—
93C4⋊Dic47—81no—
94C4⋊Dic47—81no—
95Dic47⋊C4—44no—
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—42no—
113C2×C4×C47—41no—
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—81no—
121C2×C4×C47—81no—
122C2×C4×C47C22⋊Q812yes—
123C2×C4×C47#12412yes—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—42no—
130C2×Dic47—42no—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—81no—
137C2×Dic47—81no—
138C2×Dic47—81no—
139C2×Dic47—81no—
140C2×Dic47—81no—
141C2×Dic47—81no—
142C2×Dic47—81no—
143C2×Dic47—81no—
144Dic94—84no—
145C8×C47—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—
174Dic47—82no—
175Dic47—82no—
176Dic47—82no—
177Dic47—82no—
178#2659—471no—
179#1824—472no—
180#1445—471no—
181#1294—472no—
182#1280D4711yescharacteristic
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⋊C4—471no—
196C2×C22⋊C8C2×D4712yes—
197#398—942no—
198#395—942no—
199#380—472no—
200#380—941no—
201C2×#95—471no—
202C2×#95—941no—
203C2×#95—941no—
204C2×#95C2×D4711yescharacteristic
205#340—942no—
206#340—942no—
207C47#1034311yescharacteristic, complemented, Sylow
208C22×Q8—941no—
209C23×C4—941no—
210C23×C4—941no—
211C23×C4D18811yescharacteristic
212C23×C4D18811yescharacteristic
213C23×C4C22×D4711yescharacteristic
214C22×C8—22no—
215C22⋊Q8—1882no—
216C22⋊Q8—1882no—
217C2×C4⋊C4—472no—
218C2×C4⋊C4—472no—
219C2×C4⋊C4—472no—
220C2×C4⋊C4—942no—
221C2×C4⋊C4—942no—
222C2×C4⋊C4—941no—
223C2×C4⋊C4—941no—
224C2×C4⋊C4—942no—
225C2×C4⋊C4—941no—
226C2×C4⋊C4—941no—
227C2×C22⋊C4—942no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230#95—22no—
231#95—942no—
232#95—942no—
233#95—942no—
234#95—1882no—
235#95—1882no—
236C24C2×D18811yescharacteristic
237C2×Q8—3761no—
238C2×Q8—3761no—
239C2×Q8—1882no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—942no—
251C22×C4—942no—
252C22×C4—942no—
253C22×C4—942no—
254C22×C4—1881no—
255C22×C4—1881no—
256C22×C4—1881no—
257C22×C4—1881no—
258C22×C4—1881no—
259C22×C4—1881no—
260C22×C4—1881no—
261C22×C4—1881no—
262C22×C4—1881no—
263C22×C4—1881no—
264C22×C4—1881no—
265C22×C4—1881no—
266C22×C4#1062532512yes—
267C22×C4Dic18812yes—
268C22×C4#1062535012yes—
269C22×C4D4×D4712yes—
270C22×C4#1062535211yescharacteristic
271C22×C4Q8×D4711yescharacteristic
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C4⋊C4—944no—
276C4⋊C4—3761no—
277C4⋊C4—3761no—
278C4⋊C4—1882no—
279C4⋊C4—1882no—
280C4⋊C4—1884no—
281C4⋊C4—1882no—
282C4⋊C4—1882no—
283C22⋊C4—1882no—
284C22⋊C4—1882no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1309399411yescharacteristic
290C23#1309400411yescharacteristic
291C23#1309401211yescharacteristic
292C23C2×#1062532511yescharacteristic
293C23C2×Dic18811yescharacteristic
294C23#1309405211yescharacteristic
295C23#1309405711yescharacteristic
296Q8—3764no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—3761no—
313C2×C4—3761no—
314C2×C4—3761no—
315C2×C4—3761no—
316C2×C4—3761no—
317C2×C4—3761no—
318C2×C4—3761no—
319C2×C4—3761no—
320C2×C4—1882no—
321C2×C4—1882no—
322C2×C4—1882no—
323C2×C4—1882no—
324C2×C4—1882no—
325C2×C4—1882no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4#1309400212yes—
333C2×C4#1309401012yes—
334C8—82no—
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#1568903611yescharacteristic
348C22#1568927611yescharacteristic
349C22#1568928111yescharacteristic
350C22#1568929311yescharacteristic
351C22#1568929811yescharacteristic
352C22#1568931411yescharacteristic
353C22#1568931411yescharacteristic
354C22#1568940911yescharacteristic
355C22#1568941911yescharacteristic
356C22#1568947411yescharacteristic
357C22#1568948311yescharacteristic
358C4—22no—
359C4—22no—
360C4—42no—
361C4—42no—
362C4—3762no—
363C4—3762no—
364C4—3762no—
365C4—3762no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—21no—
370C2#2024241011yescharacteristic
371C2#2024241511yescharacteristic
372C2#2024242711yescharacteristic
373C2#2024243211yescharacteristic
374C2#2024453011yescharacteristic
375C2#2024454511yescharacteristic
376C2#2024455611yescharacteristic
subgroupG/H
centerC23#13093994
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#1280C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#1280D47
hypocenterC47#10343
LCSG ▷ C23×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
32. C2×C4⋊Dic47Q8semidirect
207. C47#10343semidirect

Maximal subgroups

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

subgroupindex
#202495022
#202434792
#202416572
#202412682
C47×#12802
#1034347