Finite Group Databasebeta

← #26119255

Subgroups — #26119255

Order 12032.

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

normal subgroups only

This group has 544 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#20243472C212yesmaximal
2#20241655C211yesmaximal, characteristic
3#20241266C212yesmaximal
4#20241231C211yesmaximal, characteristic
5C47×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C2×C22⋊(C47⋊C8)C2212yes—
8C2×C47⋊#95—21no—
9C2×C47⋊#95—21no—
10C2×C47⋊#95C2211yescharacteristic
11#15688885—22no—
12#15688885—22no—
13C2×C22⋊Q8×C47—21no—
14C47×#398—22no—
15C47×#395—22no—
16C47×#380—21no—
17C47×#380C2212yes—
18C2×#95×C47C2211yescharacteristic
19C23×Dic47—21no—
20C23×Dic47—21no—
21C22×C47⋊C8—22no—
22C2×C4⋊Dic47—22no—
23C2×C4⋊Dic47—22no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47Q812yescomplemented
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—22no—
28C47⋊#95—22no—
29C47⋊#95—42no—
30C47⋊#95—42no—
31C22×Q8×C47—21no—
32C23×C4×C47D411yescharacteristic
33C23×C4×C47D411yescharacteristic
34C23×C4×C47C2311yescharacteristic
35C22⋊Q8×C47—42no—
36C22⋊Q8×C47—42no—
37C2×C4⋊C4×C47—22no—
38C2×C4⋊C4×C47—21no—
39C2×C4⋊C4×C47—21no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47D412yes—
42C2×C4⋊C4×C47D412yes—
43C2×C22⋊C4×C47—22no—
44#95×C47—22no—
45#95×C47—22no—
46#95×C47—22no—
47C24×C47C2×D411yescharacteristic
48C2×Q8×C47—42no—
49C2×Q8×C47—81no—
50C2×Q8×C47—81no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
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×C47D812yes—
64C22×C4×C47SD1612yes—
65C22×C4×C47C2×D412yes—
66C22×C4×C47C2×Q811yescharacteristic
67C22×C4×C47#4112yes—
68C22×C4×C47#4111yescharacteristic
69C22×Dic47—22no—
70C22×Dic47—22no—
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—
80C22×Dic47—41no—
81C22×Dic47—41no—
82C22×Dic47—41no—
83C4⋊C4×C47—42no—
84C4⋊C4×C47—44no—
85C4⋊C4×C47—42no—
86C4⋊C4×C47—42no—
87C4⋊C4×C47—81no—
88C4⋊C4×C47—81no—
89C22⋊C4×C47—42no—
90C22⋊C4×C47—42no—
91C4⋊Dic47—24no—
92C4⋊Dic47—42no—
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×D811yescharacteristic
104C23×C47C2×SD1611yescharacteristic
105C23×C47AGL(1,ℤ/8)11yescharacteristic
106C23×C47#13711yescharacteristic
107Q8×C47—84no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—42no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
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—42no—
127C2×C4×C47—81no—
128C2×C4×C47—81no—
129C2×C4×C47C22⋊Q812yes—
130C2×C4×C47#12412yes—
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—81no—
138C2×Dic47—81no—
139C2×Dic47—81no—
140C2×Dic47—81no—
141C2×Dic47—81no—
142C2×Dic47—81no—
143C2×Dic47—81no—
144C2×Dic47—81no—
145C47⋊C8—82no—
146#10342—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#44711yescharacteristic
161C22×C47#44811yescharacteristic
162C22×C47#45511yescharacteristic
163C22×C47#45611yescharacteristic
164C22×C47#45811yescharacteristic
165C22×C47#45811yescharacteristic
166C22×C47#47411yescharacteristic
167C22×C47#47611yescharacteristic
168C22×C47#48611yescharacteristic
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#2659D4711yescharacteristic
179#1823—472no—
180#1445—471no—
181#1294—472no—
182#1280—471no—
183C2×C47—21no—
184C2×C47—21no—
185C2×C47—21no—
186C2×C47—21no—
187C2×C47#160711yescharacteristic
188C2×C47#160811yescharacteristic
189C2×C47#161011yescharacteristic
190C2×C47#161111yescharacteristic
191C2×C47#202311yescharacteristic
192C2×C47#202811yescharacteristic
193C2×C47#203011yescharacteristic
194C2×C22⋊Q8—21no—
195C22×C4⋊C4—471no—
196C2×C22⋊C8—472no—
197#398—22no—
198#395—22no—
199#380—21no—
200#380C2×D4712yes—
201C2×#95—471no—
202C2×#95—941no—
203C2×#95—941no—
204C2×#95C2×D4711yescharacteristic
205#340—942no—
206#340—942no—
207C47#1034211yescharacteristic, complemented, Sylow
208C22×Q8—21no—
209C23×C4—941no—
210C23×C4—941no—
211C23×C4C22⋊D4711yescharacteristic
212C23×C4C22⋊D4711yescharacteristic
213C23×C4C22×D4711yescharacteristic
214C22×C8—942no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—472no—
222C2×C4⋊C4—942no—
223C2×C4⋊C4—942no—
224C2×C4⋊C4—941no—
225C2×C4⋊C4C22⋊D4712yes—
226C2×C4⋊C4C22⋊D4712yes—
227C2×C22⋊C4—22no—
228C22⋊C8—942no—
229C22⋊C8—942no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—942no—
234#95—1882no—
235#95—1882no—
236C24C2×C22⋊D4711yescharacteristic
237C2×Q8—42no—
238C2×Q8—81no—
239C2×Q8—81no—
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—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#1062533412yes—
267C22×C4#1062533612yes—
268C22×C4Q8×D4711yescharacteristic
269C22×C4#1062535412yes—
270C22×C4#1062535411yescharacteristic
271C22×C4C2×C22⋊D4712yes—
272C2×C8—1882no—
273C2×C8—1882no—
274C2×C8—1882no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—944no—
282C4⋊C4—1882no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1309399311yescharacteristic
290C23#1309400011yescharacteristic
291C23#1309401311yescharacteristic
292C23C2×#1062533411yescharacteristic
293C23C2×#1062533611yescharacteristic
294C23#1309405011yescharacteristic
295C23#1309405511yescharacteristic
296Q8—84no—
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—3761no—
319C2×C4—3761no—
320C2×C4—3761no—
321C2×C4—3761no—
322C2×C4—3761no—
323C2×C4—3761no—
324C2×C4—3761no—
325C2×C4—3761no—
326C2×C4—1882no—
327C2×C4—1882no—
328C2×C4—1882no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4#1309400012yes—
333C2×C4#1309401312yes—
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#1568903411yescharacteristic
348C22#1568925711yescharacteristic
349C22#1568926211yescharacteristic
350C22#1568930111yescharacteristic
351C22#1568930611yescharacteristic
352C22#1568931311yescharacteristic
353C22#1568931311yescharacteristic
354C22#1568940011yescharacteristic
355C22#1568941011yescharacteristic
356C22#1568947111yescharacteristic
357C22#1568948511yescharacteristic
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#2024237411yescharacteristic
371C2#2024237911yescharacteristic
372C2#2024239111yescharacteristic
373C2#2024239611yescharacteristic
374C2#2024452111yescharacteristic
375C2#2024454611yescharacteristic
376C2#2024455411yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2659C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#2659D47
hypocenterC47#10342
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
25. C2×C4⋊Dic47Q8semidirect
207. C47#10342semidirect

Maximal subgroups

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

subgroupindex
#202434722
#202416552
#202412662
#202412312
C47×#26592
#1034247