Finite Group Databasebeta

← #26105356

Subgroups — #26105356

Order 12032.

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

normal subgroups only

This group has 394 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
2#20244859C211yesmaximal, characteristic
3#20244796C211yesmaximal, characteristic
4#20241214C211yesmaximal, characteristic
5#20241211C211yesmaximal, characteristic
6#20241211C211yesmaximal, characteristic
7C47×#2734C211yesmaximal, characteristic
8C2×#13093957—21no—
9C2×#13093948—21no—
10#15688993C2211yescharacteristic
11#15688885—21no—
12#15688885—21no—
13#15688885—21no—
14#15688885—21no—
15#15688878C2211yescharacteristic
16#15688878C2211yescharacteristic
17#15688878C2211yescharacteristic
18C2×C4⋊Q8×C47—21no—
19C47×#398—21no—
20C47×#395—21no—
21C47×#395—21no—
22C47×#391C2211yescharacteristic
23C47×#389C2211yescharacteristic
24C47×#389C2211yescharacteristic
25C47×#387—21no—
26C47×#387—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#13093948—42no—
35C22×Q8×C47—21no—
36C22×Q8×C47D411yescharacteristic
37C4⋊Q8×C47—42no—
38C4⋊Q8×C47—42no—
39C2×C4⋊C4×C47—21no—
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×C42×C47—21no—
50C2×C42×C47C2311yescharacteristic
51#95×C47—21no—
52#95×C47—21no—
53#95×C47—21no—
54#95×C47—21no—
55#95×C47—21no—
56#95×C47—21no—
57#95×C47—21no—
58C2×Q8×C47—21no—
59C2×Q8×C47—42no—
60C2×Q8×C47—41no—
61C2×Q8×C47—41no—
62C2×Q8×C47—81no—
63C2×Q8×C47—81no—
64C2×Q8×C47D812yes—
65C22×C4×C47—21no—
66C22×C4×C47—21no—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—41no—
75C4⋊C4×C47—21no—
76C4⋊C4×C47—21no—
77C4⋊C4×C47—42no—
78C4⋊C4×C47—42no—
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—41no—
86C4⋊C4×C47—81no—
87C4⋊C4×C47SD1612yes—
88C4⋊C4×C47Q1612yes—
89C42×C47—41no—
90C42×C47—42no—
91C4⋊Dic47—42no—
92C4⋊Dic47—41no—
93C4⋊Dic47—41no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96C2×C47⋊C8—22no—
97C2×C47⋊C8—41no—
98C2×C47⋊C8—41no—
99C2×C47⋊C8—41no—
100C2×C47⋊C8—41no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C23×C47#12011yescharacteristic
106Q8×C47—42no—
107Q8×C47—81no—
108Q8×C47—84no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—22no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47—41no—
130C2×C4×C47—41no—
131C2×C4×C47—41no—
132C2×C4×C47—41no—
133C2×C4×C47C2×D811yescharacteristic
134C2×C4×C47C2×SD1611yescharacteristic
135C2×C4×C47C2×Q1611yescharacteristic
136C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C47#13711yescharacteristic
138C2×C4×C47#13711yescharacteristic
139C2×Dic47—81no—
140C2×Dic47—81no—
141C2×Dic47—81no—
142C47⋊C8—42no—
143C47⋊C8—81no—
144C47⋊C8—81no—
145#7794—471nomaximal, Sylow
146C22×C47#44711yescharacteristic
147C22×C47#44811yescharacteristic
148C22×C47#45111yescharacteristic
149C22×C47#45411yescharacteristic
150C22×C47#45511yescharacteristic
151C22×C47#45611yescharacteristic
152C22×C47#45811yescharacteristic
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—21no—
158C4×C47—21no—
159C4×C47—42no—
160C4×C47—42no—
161C4×C47—42no—
162C4×C47—81no—
163C4×C47—81no—
164C4×C47—81no—
165Dic47—161no—
166#2734D4711yescharacteristic
167#2099—471no—
168#2083—471no—
169#2072—471no—
170#1272—471no—
171#1271—471no—
172#1271—471no—
173C2×C47#160711yescharacteristic
174C2×C47#161111yescharacteristic
175C2×C47#161711yescharacteristic
176C2×C47#162711yescharacteristic
177C2×C47#164511yescharacteristic
178C2×C47#164911yescharacteristic
179C2×C47#166411yescharacteristic
180C2×C4⋊Q8—21no—
181C2×#107—941no—
182C2×#103—941no—
183#398—21no—
184#395—21no—
185#395—21no—
186#391C2×D4711yescharacteristic
187#389C2×D4711yescharacteristic
188#389C2×D4711yescharacteristic
189#387—21no—
190#387—21no—
191#384—471no—
192#340—941no—
193#340—941no—
194#340—941no—
195#340—941no—
196#336—471no—
197#336—471no—
198#336—471no—
199C47#779411yescharacteristic, complemented, Sylow
200C22×Q8—21no—
201C22×Q8C22⋊D4711yescharacteristic
202C22×C8—941no—
203C22×C8—941no—
204C22×C8—941no—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4—941no—
215C2×C4⋊C4C22⋊D4711yescharacteristic
216C2×C4⋊C4C22⋊D4711yescharacteristic
217C2×C4⋊C4C22⋊D4711yescharacteristic
218C2×C4⋊C4C22⋊D4711yescharacteristic
219C2×C4⋊C4C22⋊D4711yescharacteristic
220C2×C42—21no—
221C2×C42C22×D4711yescharacteristic
222#107—1882no—
223#103—1882no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C2×Q8—21no—
232C2×Q8—42no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—81no—
236C2×Q8—81no—
237C2×Q8#1062533412yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—1881no—
245C22×C4C2×C22⋊D4711yescharacteristic
246C22×C4C2×C22⋊D4711yescharacteristic
247C22×C4C2×C22⋊D4711yescharacteristic
248C2×C8—942no—
249C2×C8—1881no—
250C2×C8—1881no—
251C2×C8—1881no—
252C2×C8—1881no—
253C2×C8—1881no—
254C2×C8—1881no—
255C2×C8—1881no—
256C2×C8—1881no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—81no—
269C4⋊C4—1882no—
270C4⋊C4—1881no—
271C4⋊C4—1881no—
272C4⋊C4—1881no—
273C4⋊C4—1881no—
274C4⋊C4#1062533612yes—
275C4⋊C4#1062533712yes—
276C42—41no—
277C42—42no—
278C23#1309399311yescharacteristic
279Q8—42no—
280Q8—81no—
281Q8—84no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—3761no—
307C2×C4—3761no—
308C2×C4—3761no—
309C2×C4C2×#1062533411yescharacteristic
310C2×C4C2×#1062533611yescharacteristic
311C2×C4C2×#1062533711yescharacteristic
312C2×C4#1309405011yescharacteristic
313C2×C4#1309405511yescharacteristic
314C2×C4#1309405511yescharacteristic
315C8—3761no—
316C8—3761no—
317C8—1882no—
318C22#1568925711yescharacteristic
319C22#1568926211yescharacteristic
320C22#1568927911yescharacteristic
321C22#1568929611yescharacteristic
322C22#1568930111yescharacteristic
323C22#1568930611yescharacteristic
324C22#1568931311yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—42no—
333C4—42no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—7521no—
338C2#2024237411yescharacteristic
339C2#2024239611yescharacteristic
340C2#2024243011yescharacteristic
341C2#2024249111yescharacteristic
342C2#2024260911yescharacteristic
343C2#2024263711yescharacteristic
344C2#2024272211yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#2734C2
socle, prime radicalC23×C47#120
hypercenter#2734D47
hypocenterC47#7794
LCSG ▷ C2×C42×C47 ▷ C23×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2734

Complements

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

NH ≅ G/Ntype
199. C47#7794semidirect

Maximal subgroups

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

subgroupindex
#202449462
#202448592
#202447962
#202412142
#202412112
C47×#27342
#779447