Finite Group Databasebeta

← #26105349

Subgroups — #26105349

Order 12032.

420 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 87 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#20244916C211yesmaximal, characteristic
2#20244864C211yesmaximal, characteristic, complemented
3#20244859C211yesmaximal, characteristic, complemented
4#20241211C211yesmaximal, characteristic, complemented
5#20241211C211yesmaximal, characteristic, complemented
6#20241209C211yesmaximal, characteristic
7C47×#2727C211yesmaximal, characteristic
8C2×#13093957—21no—
9C2×#13093957—21no—
10C2×#13093955—21no—
11C2×#13093945—21no—
12#15688993C2211yescharacteristic
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C2×#126×C47—21no—
19C47×#399—21no—
20C47×#396—21no—
21C47×#394—21no—
22C47×#390C2211yescharacteristic
23C47×#389—21no—
24C47×#389C2211yescharacteristic
25C47×#389C2211yescharacteristic
26C47×#388—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#13093957—42no—
34#13093957—42no—
35#13093955—42no—
36#13093945—42no—
37C22×D4×C47D411yescharacteristic
38#126×C47—42no—
39#126×C47—42no—
40C2×C4⋊C4×C47—21no—
41C2×C4⋊C4×C47—21no—
42C2×C4⋊C4×C47—21no—
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47D411yescharacteristic
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C22⋊C4×C47—21no—
50C2×C22⋊C4×C47—21no—
51C2×C22⋊C4×C47—21no—
52C2×C22⋊C4×C47—21no—
53C2×C22⋊C4×C47—21no—
54C2×C22⋊C4×C47—21no—
55C2×C42×C47—21no—
56C2×C42×C47C2311yescharacteristic
57#95×C47—21no—
58#95×C47—21no—
59#95×C47—21no—
60C24×C47—21no—
61C2×D4×C47—21no—
62C2×D4×C47—21no—
63C2×D4×C47—41no—
64C2×D4×C47—41no—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×Dic47—41no—
74C4⋊C4×C47—21no—
75C4⋊C4×C47—21no—
76C4⋊C4×C47—21no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C4⋊C4×C47—41no—
85C4⋊C4×C47—81no—
86C4⋊C4×C47D812yes—
87C22⋊C4×C47—42no—
88C22⋊C4×C47—42no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—41no—
91C22⋊C4×C47—41no—
92C22⋊C4×C47—41no—
93C22⋊C4×C47—81no—
94C22⋊C4×C47—81no—
95C42×C47—41no—
96C42×C47—42no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101C4⋊Dic47—41no—
102C4⋊Dic47—41no—
103C2×C47⋊C8—22no—
104C2×C47⋊C8—22no—
105C2×C47⋊C8—22no—
106C2×C47⋊C8—41no—
107C2×C47⋊C8—41no—
108C2×C47⋊C8—41no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111C2×C47⋊C8—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—41no—
115C23×C47—41no—
116C23×C47—41no—
117C23×C47—41no—
118C23×C47—41no—
119C23×C47#12011yescharacteristic
120D4×C47—42no—
121D4×C47—81no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—22no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47—41no—
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×C47C2×D811yescharacteristic
144C2×C4×C47#13511yescharacteristic
145C2×C4×C47#13511yescharacteristic
146C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C47#13711yescharacteristic
149C2×Dic47—81no—
150C2×Dic47—81no—
151C2×Dic47—81no—
152C47⋊C8—42no—
153C47⋊C8—42no—
154C47⋊C8—42no—
155#7793—471nomaximal, Sylow
156C22×C47—81no—
157C22×C47—81no—
158C22×C47—81no—
159C22×C47—81no—
160C22×C47—81no—
161C22×C47—81no—
162C22×C47—81no—
163C22×C47#44711yescharacteristic
164C22×C47#44911yescharacteristic
165C22×C47#45211yescharacteristic
166C22×C47#45311yescharacteristic
167C22×C47#45611yescharacteristic
168C22×C47#45611yescharacteristic
169C22×C47#45811yescharacteristic
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—21no—
174C4×C47—21no—
175C4×C47—21no—
176C4×C47—42no—
177C4×C47—81no—
178C4×C47—81no—
179C4×C47—81no—
180C4×C47—81no—
181Dic47—161no—
182#2727D4711yescharacteristic
183#2093—471no—
184#2084—471no—
185#2083—471no—
186#1271—471no—
187#1271—471no—
188#1270—471no—
189C2×C47—161no—
190C2×C47#160911yescharacteristic
191C2×C47#161011yescharacteristic
192C2×C47#161811yescharacteristic
193C2×C47#162111yescharacteristic
194C2×C47#164711yescharacteristic
195C2×C47#165211yescharacteristic
196C2×C47#166011yescharacteristic
197C2×#126—21no—
198C2×#107—941no—
199C2×#107—941no—
200C2×#106—941no—
201C2×#102—941no—
202#399—21no—
203#396—21no—
204#394—21no—
205#390C2×D4711yescharacteristic
206#389—21no—
207#389C2×D4711yescharacteristic
208#389C2×D4711yescharacteristic
209#388—21no—
210#384—471no—
211#340—941no—
212#340—941no—
213#336—471no—
214#336—471no—
215#336—471no—
216C47#779311yescharacteristic, complemented, Sylow
217C22×D4C22⋊D4711yescharacteristic
218C22×C8—941no—
219C22×C8—941no—
220C22×C8—941no—
221#126—42no—
222#126—42no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—941no—
228C2×C4⋊C4—941no—
229C2×C4⋊C4—941no—
230C2×C4⋊C4C22⋊D4711yescharacteristic
231C2×C4⋊C4C22⋊D4711yescharacteristic
232C2×C4⋊C4C22⋊D4711yescharacteristic
233C2×C4⋊C4C22⋊D4711yescharacteristic
234C2×C4⋊C4C22⋊D4711yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C42—21no—
242C2×C42C22×D4711yescharacteristic
243#107—1882no—
244#107—1882no—
245#106—1882no—
246#102—1882no—
247#95—21no—
248#95—21no—
249#95—21no—
250C24—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1881no—
261C22×C4C2×C22⋊D4711yescharacteristic
262C22×C4C2×C22⋊D4711yescharacteristic
263C22×C4C2×C22⋊D4711yescharacteristic
264C2×C8—942no—
265C2×C8—942no—
266C2×C8—942no—
267C2×C8—1881no—
268C2×C8—1881no—
269C2×C8—1881no—
270C2×C8—1881no—
271C2×C8—1881no—
272C2×C8—1881no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—81no—
285C4⋊C4—1881no—
286C4⋊C4—1881no—
287C4⋊C4—1881no—
288C4⋊C4—1881no—
289C4⋊C4—1881no—
290C4⋊C4—1881no—
291C4⋊C4#1062533412yes—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C42—41no—
301C42—42no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1309399311yescharacteristic
310D4—42no—
311D4—81no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—3761no—
334C2×C4—3761no—
335C2×C4—3761no—
336C2×C4C2×#1062533411yescharacteristic
337C2×C4#1309404411yescharacteristic
338C2×C4#1309404411yescharacteristic
339C2×C4#1309404911yescharacteristic
340C2×C4#1309405011yescharacteristic
341C2×C4#1309405411yescharacteristic
342C8—1882no—
343C8—1882no—
344C8—1882no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#1568925711yescharacteristic
353C22#1568926811yescharacteristic
354C22#1568928511yescharacteristic
355C22#1568929111yescharacteristic
356C22#1568930611yescharacteristic
357C22#1568930611yescharacteristic
358C22#1568931311yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—7521no—
371C2—161no—
372C2#2024238511yescharacteristic
373C2#2024239111yescharacteristic
374C2#2024243611yescharacteristic
375C2#2024245311yescharacteristic
376C2#2024262311yescharacteristic
377C2#2024265811yescharacteristic
378C2#2024269911yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2727C2
socleC23×C47#120
prime radicalC22×D4×C47D4
hypercenter#2727D47
hypocenterC47#7793
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
2. #20244864C2semidirect
3. #20244859C2semidirect
4. #20241211C2semidirect
5. #20241211C2semidirect
216. C47#7793semidirect

Maximal subgroups

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

subgroupindex
#202449162
#202448642
#202448592
#202412112
#202412092
C47×#27272
#779347