Finite Group Databasebeta

← #26120678

Subgroups — #26120678

Order 12032.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#20243575C212yesmaximal
2#20241652C211yesmaximal, characteristic
3#20241266C212yesmaximal
4#20241231C211yesmaximal, characteristic
5C47×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic47C2211yescharacteristic
7C22×Dic47⋊C4—21no—
8C2×#13093957—22no—
9C2×C22⋊(C47⋊C8)C2212yes—
10C2×C47⋊#95—21no—
11C2×C47⋊#95C2211yescharacteristic
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—42no—
24C2×Dic47⋊C4—42no—
25C2×Dic47⋊C4—42no—
26C2×Dic47⋊C4—42no—
27#13093957—44no—
28C22⋊(C47⋊C8)—22no—
29C22⋊(C47⋊C8)Q814yescomplemented
30C47⋊#95—21no—
31C47⋊#95—42no—
32C47⋊#95Q812yescomplemented
33C22×Q8×C47—21no—
34C23×C4×C47D411yescharacteristic
35C23×C4×C47D411yescharacteristic
36C23×C4×C47C2311yescharacteristic
37C22⋊Q8×C47—42no—
38C22⋊Q8×C47—42no—
39C2×C4⋊C4×C47—22no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47D412yes—
44C2×C4⋊C4×C47D412yes—
45C2×C22⋊C4×C47—22no—
46#95×C47—22no—
47#95×C47—22no—
48#95×C47—22no—
49C24×C47C2×D411yescharacteristic
50C2×Q8×C47—42no—
51C2×Q8×C47—81no—
52C2×Q8×C47—81no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—21no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
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×C47D812yes—
66C22×C4×C47Q1612yes—
67C22×C4×C47C2×D412yes—
68C22×C4×C47C2×Q812yes—
69C22×C4×C47C2×Q811yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C22×Dic47—22no—
72C22×Dic47—22no—
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—
85C4⋊C4×C47—42no—
86C4⋊C4×C47—44no—
87C4⋊C4×C47—42no—
88C4⋊C4×C47—42no—
89C4⋊C4×C47—81no—
90C4⋊C4×C47—81no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—42no—
93C4⋊Dic47—24no—
94C4⋊Dic47—42no—
95Dic47⋊C4—48no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—42no—
98C2×C47⋊C8—42no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—21no—
102C23×C47—21no—
103C23×C47#12011yescharacteristic
104C23×C47C22⋊Q811yescharacteristic
105C23×C47C4⋊Q811yescharacteristic
106C23×C47C2×D811yescharacteristic
107C23×C47C2×Q1611yescharacteristic
108C23×C47AGL(1,ℤ/8)11yescharacteristic
109C23×C47#13711yescharacteristic
110Q8×C47—84no—
111C2×C4×C47—22no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—22no—
115C2×C4×C47—42no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—42no—
128C2×C4×C47—42no—
129C2×C4×C47—42no—
130C2×C4×C47—81no—
131C2×C4×C47—81no—
132C2×C4×C47C22⋊Q812yes—
133C2×C4×C47C4⋊Q812yes—
134C2×Dic47—24no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—42no—
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—
148C47⋊C8—44no—
149#10567—471nomaximal, Sylow
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—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47#39511yescharacteristic
163C22×C47#44711yescharacteristic
164C22×C47#45111yescharacteristic
165C22×C47#45311yescharacteristic
166C22×C47#45611yescharacteristic
167C22×C47B(4,𝔽2)11yescharacteristic
168C22×C47#45811yescharacteristic
169C22×C47#47411yescharacteristic
170C22×C47#47711yescharacteristic
171C22×C47#50011yescharacteristic
172C22×C47#50111yescharacteristic
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—42no—
176C4×C47—42no—
177C4×C47—82no—
178C4×C47—82no—
179Dic47—44no—
180Dic47—82no—
181#2659D4711yescharacteristic
182#1852—472no—
183#1444—471no—
184#1294—472no—
185#1280—471no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—21no—
189C2×C47—21no—
190C2×C47#159511yescharacteristic
191C2×C47#160511yescharacteristic
192C2×C47#161011yescharacteristic
193C2×C47#161411yescharacteristic
194C2×C47#205711yescharacteristic
195C2×C47#206011yescharacteristic
196C2×C47#206211yescharacteristic
197C2×C22⋊Q8—21no—
198C22×C4⋊C4—471no—
199C22×C4⋊C4—941no—
200C2×#107—942no—
201C2×C22⋊C8—472no—
202#398—22no—
203#395—22no—
204#380—21no—
205#380C2×D4712yes—
206C2×#95—471no—
207C2×#95—941no—
208C2×#95C2×D4711yescharacteristic
209#340—942no—
210C47#1056711yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C23×C4—941no—
213C23×C4—941no—
214C23×C4C22⋊D4711yescharacteristic
215C23×C4C22⋊D4711yescharacteristic
216C23×C4C22×D4711yescharacteristic
217C22×C8—942no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—942no—
225C2×C4⋊C4—1882no—
226C2×C4⋊C4—1882no—
227C2×C4⋊C4—1882no—
228C2×C4⋊C4—1882no—
229C2×C4⋊C4C22⋊D4712yes—
230C2×C4⋊C4C22⋊D4712yes—
231C2×C22⋊C4—22no—
232#107—1884no—
233C22⋊C8—474no—
234C22⋊C8—942no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—472no—
239#95—941no—
240#95—1882no—
241C24C2×C22⋊D4711yescharacteristic
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—942no—
258C22×C4—942no—
259C22×C4—942no—
260C22×C4—942no—
261C22×C4—1881no—
262C22×C4—1881no—
263C22×C4—1881no—
264C22×C4—1881no—
265C22×C4—1881no—
266C22×C4—1881no—
267C22×C4—1881no—
268C22×C4—1881no—
269C22×C4—1881no—
270C22×C4—1881no—
271C22×C4#1062533412yes—
272C22×C4#1062533712yes—
273C22×C4Q8×D4712yes—
274C22×C4Q8×D4711yescharacteristic
275C22×C4#1062535411yescharacteristic
276C22×C4C2×C22⋊D4712yes—
277C2×C8—944no—
278C2×C8—1882no—
279C2×C8—1882no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—944no—
287C4⋊C4—1882no—
288C4⋊C4—1888no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1309399311yescharacteristic
296C23#1309400011yescharacteristic
297C23#1309402211yescharacteristic
298C23C2×#1062533411yescharacteristic
299C23C2×#1062533711yescharacteristic
300C23#1309405011yescharacteristic
301C23#1309405511yescharacteristic
302Q8—84no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—944no—
325C2×C4—3761no—
326C2×C4—3761no—
327C2×C4—3761no—
328C2×C4—3761no—
329C2×C4—1882no—
330C2×C4—1882no—
331C2×C4—1882no—
332C2×C4—1882no—
333C2×C4—1882no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4#1309400012yes—
339C2×C4#1309402212yes—
340C8—1884no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1568904211yescharacteristic
354C22#1568925711yescharacteristic
355C22#1568927911yescharacteristic
356C22#1568929111yescharacteristic
357C22#1568930611yescharacteristic
358C22#1568931011yescharacteristic
359C22#1568931311yescharacteristic
360C22#1568940011yescharacteristic
361C22#1568941511yescharacteristic
362C22#1568952611yescharacteristic
363C22#1568952911yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4—3762no—
371C4—1884no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2024230611yescharacteristic
377C2#2024236211yescharacteristic
378C2#2024239111yescharacteristic
379C2#2024241311yescharacteristic
380C2#2024471311yescharacteristic
381C2#2024473011yescharacteristic
382C2#2024474011yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC23×C4×C47C23
FrattiniC23×C4C22×D47
FittingC47×#2659C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenter#2659D47
hypocenterC47#10567
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
29. C22⋊(C47⋊C8)Q8semidirect
32. C47⋊#95Q8semidirect
210. C47#10567semidirect

Maximal subgroups

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

subgroupindex
#202435752
#202416522
#202412662
#202412312
C47×#26592
#1056747