Finite Group Databasebeta

← #26105335

Subgroups — #26105335

Order 12032.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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#20244946C211yesmaximal, characteristic, complemented
2#20244864C211yesmaximal, characteristic, complemented
3#20244791C211yesmaximal, characteristic
4#20241211C211yesmaximal, characteristic, complemented
5#20241211C211yesmaximal, characteristic, complemented
6#20241209C211yesmaximal, characteristic
7C47×#2683C211yesmaximal, characteristic
8C2×#13093955—21no—
9C2×#13093945—21no—
10#15688993C2211yescharacteristic
11#15688885—21no—
12#15688885—21no—
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C2×C22⋊Q8×C47—21no—
19C2×C22⋊Q8×C47—21no—
20C47×#396—21no—
21C47×#396—21no—
22C47×#395—21no—
23C47×#390C2211yescharacteristic
24C47×#389C2211yescharacteristic
25C47×#389C2211yescharacteristic
26C47×#386—21no—
27C22×C47⋊C8—21no—
28C22×C47⋊C8—21no—
29C22×C47⋊C8—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47—21no—
32C2×C4⋊Dic47—21no—
33#13093955—42no—
34#13093945—42no—
35C22×Q8×C47—21no—
36C22×D4×C47D411yescharacteristic
37C23×C4×C47—21no—
38C22⋊Q8×C47—42no—
39C22⋊Q8×C47—42no—
40C22⋊Q8×C47—42no—
41C22⋊Q8×C47—42no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47D411yescharacteristic
50C2×C4⋊C4×C47D411yescharacteristic
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C22⋊C4×C47—21no—
56C2×C42×C47C2311yescharacteristic
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60C24×C47—21no—
61C2×Q8×C47—42no—
62C2×Q8×C47—42no—
63C2×Q8×C47—81no—
64C2×D4×C47—21no—
65C2×D4×C47—41no—
66C2×D4×C47—41no—
67C2×D4×C47SD1612yes—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47—81no—
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—41no—
80C4⋊C4×C47—21no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—42no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C4⋊C4×C47—41no—
90C4⋊C4×C47—41no—
91C4⋊C4×C47D812yes—
92C4⋊C4×C47SD1612yes—
93C22⋊C4×C47—42no—
94C22⋊C4×C47—42no—
95C22⋊C4×C47—41no—
96C22⋊C4×C47—41no—
97C22⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C22⋊C4×C47—81no—
100C42×C47—41no—
101C4⋊Dic47—42no—
102C4⋊Dic47—41no—
103C4⋊Dic47—41no—
104C4⋊Dic47—41no—
105C4⋊Dic47—41no—
106C2×C47⋊C8—22no—
107C2×C47⋊C8—41no—
108C2×C47⋊C8—41no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C2×C47⋊C8—41no—
113C2×C47⋊C8—41no—
114C2×C47⋊C8—41no—
115C23×C47—22no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47—41no—
120C23×C47—41no—
121C23×C47—41no—
122C23×C47#12011yescharacteristic
123Q8×C47—84no—
124D4×C47—42no—
125D4×C47—81no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—22no—
133C2×C4×C47—22no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—41no—
142C2×C4×C47—41no—
143C2×C4×C47—41no—
144C2×C4×C47—41no—
145C2×C4×C47—41no—
146C2×C4×C47—41no—
147C2×C4×C47—84no—
148C2×C4×C47C2×D811yescharacteristic
149C2×C4×C47C2×SD1611yescharacteristic
150C2×C4×C47C2×SD1611yescharacteristic
151C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C47#13711yescharacteristic
154C2×Dic47—81no—
155C2×Dic47—81no—
156C2×Dic47—81no—
157C47⋊C8—42no—
158C47⋊C8—81no—
159C47⋊C8—81no—
160#7791—471nomaximal, Sylow
161C22×C47—42no—
162C22×C47—42no—
163C22×C47—42no—
164C22×C47—81no—
165C22×C47—81no—
166C22×C47—81no—
167C22×C47—81no—
168C22×C47#44711yescharacteristic
169C22×C47#44811yescharacteristic
170C22×C47#45011yescharacteristic
171C22×C47#45411yescharacteristic
172C22×C47#45511yescharacteristic
173C22×C47#45611yescharacteristic
174C22×C47#45811yescharacteristic
175C4×C47—21no—
176C4×C47—21no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—21no—
180C4×C47—21no—
181C4×C47—42no—
182C4×C47—42no—
183C4×C47—81no—
184C4×C47—81no—
185C4×C47—81no—
186Dic47—161no—
187#2683D4711yescharacteristic
188#2099—471no—
189#2084—471no—
190#2071—471no—
191#1271—471no—
192#1271—471no—
193#1270—471no—
194C2×C47—82no—
195C2×C47#160811yescharacteristic
196C2×C47#161011yescharacteristic
197C2×C47#161611yescharacteristic
198C2×C47#162711yescharacteristic
199C2×C47#163911yescharacteristic
200C2×C47#165111yescharacteristic
201C2×C47#166411yescharacteristic
202C2×C22⋊Q8—21no—
203C2×C22⋊Q8—21no—
204C2×#106—941no—
205C2×#102—941no—
206#396—21no—
207#396—21no—
208#395—21no—
209#390C2×D4711yescharacteristic
210#389C2×D4711yescharacteristic
211#389C2×D4711yescharacteristic
212#386—21no—
213#384—471no—
214#340—941no—
215#340—941no—
216#340—941no—
217#340—941no—
218#336—471no—
219#336—471no—
220#336—471no—
221C47#779111yescharacteristic, complemented, Sylow
222C22×Q8—21no—
223C22×D4C22⋊D4711yescharacteristic
224C23×C4—21no—
225C22×C8—941no—
226C22×C8—941no—
227C22×C8—941no—
228C22⋊Q8—42no—
229C22⋊Q8—42no—
230C22⋊Q8—42no—
231C22⋊Q8—42no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—941no—
237C2×C4⋊C4—941no—
238C2×C4⋊C4—941no—
239C2×C4⋊C4C22⋊D4711yescharacteristic
240C2×C4⋊C4C22⋊D4711yescharacteristic
241C2×C4⋊C4C22⋊D4711yescharacteristic
242C2×C4⋊C4C22⋊D4711yescharacteristic
243C2×C4⋊C4C22⋊D4711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C42C22×D4711yescharacteristic
250#106—1882no—
251#102—1882no—
252#95—21no—
253#95—21no—
254#95—21no—
255C24—21no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—81no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4#1062533612yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—1881no—
272C22×C4C2×C22⋊D4711yescharacteristic
273C22×C4C2×C22⋊D4711yescharacteristic
274C22×C4C2×C22⋊D4711yescharacteristic
275C2×C8—942no—
276C2×C8—1881no—
277C2×C8—1881no—
278C2×C8—1881no—
279C2×C8—1881no—
280C2×C8—1881no—
281C2×C8—1881no—
282C2×C8—1881no—
283C2×C8—1881no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—1882no—
296C4⋊C4—1881no—
297C4⋊C4—1881no—
298C4⋊C4—1881no—
299C4⋊C4—1881no—
300C4⋊C4#1062533412yes—
301C4⋊C4#1062533512yes—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—81no—
309C42—41no—
310C23—22no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#1309399311yescharacteristic
318Q8—84no—
319D4—42no—
320D4—81no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—84no—
343C2×C4—3761no—
344C2×C4—3761no—
345C2×C4—3761no—
346C2×C4C2×#1062533411yescharacteristic
347C2×C4C2×#1062533611yescharacteristic
348C2×C4C2×#1062533511yescharacteristic
349C2×C4#1309405011yescharacteristic
350C2×C4#1309405011yescharacteristic
351C2×C4#1309405511yescharacteristic
352C8—3761no—
353C8—3761no—
354C8—1882no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#1568925711yescharacteristic
363C22#1568926211yescharacteristic
364C22#1568927411yescharacteristic
365C22#1568929611yescharacteristic
366C22#1568930111yescharacteristic
367C22#1568930611yescharacteristic
368C22#1568931311yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—7521no—
381C2—82no—
382C2#2024237911yescharacteristic
383C2#2024239111yescharacteristic
384C2#2024242511yescharacteristic
385C2#2024249111yescharacteristic
386C2#2024256711yescharacteristic
387C2#2024265111yescharacteristic
388C2#2024272211yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2683C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2683D47
hypocenterC47#7791
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2683

Complements

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

NH ≅ G/Ntype
1. #20244946C2semidirect
2. #20244864C2semidirect
4. #20241211C2semidirect
5. #20241211C2semidirect
221. C47#7791semidirect

Maximal subgroups

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

subgroupindex
#202449462
#202448642
#202447912
#202412112
#202412092
C47×#26832
#779147