Finite Group Databasebeta

← #26129108

Subgroups — #26129108

Order 12032.

346 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 63 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#20244549C211yesmaximal, characteristic
2#20242429C211yesmaximal, characteristic
3#20242361C211yesmaximal, characteristic
4#20241418C211yesmaximal, characteristic
5#20241415C211yesmaximal, characteristic
6#20241322C211yesmaximal, characteristic
7C47×#2799C211yesmaximal, characteristic
8#15689754C2211yescharacteristic
9#15689458—21no—
10#15689279—21no—
11#15689132—21no—
12#15688892C2211yescharacteristic
13#15688883—21no—
14#15688859—21no—
15#15688859—21no—
16#15688849C2211yescharacteristic
17#15688849C2211yescharacteristic
18C47×#400—21no—
19C47×#400C2211yescharacteristic
20C47×#398—21no—
21C47×#395C2211yescharacteristic
22C47×#391—21no—
23C47×#391—21no—
24C47×#391C2211yescharacteristic
25C47×#387—21no—
26C47×#387—21no—
27C2×#10625337—41no—
28C2×#10625329—21no—
29#13094023—41no—
30#13094015—41no—
31#13094003—21no—
32#13094003—21no—
33#13093983—21no—
34#13093957—41no—
35#13093948—41no—
36#13093948—41no—
37#13093948—41no—
38#13093947—41no—
39C22⋊(C47⋊C8)—21no—
40C22⋊(C47⋊C8)—21no—
41C22×Q8×C47D411yescharacteristic
42C22×Q8×C47D411yescharacteristic
43C2×C4⋊C4×C47—21no—
44C2×C4⋊C4×C47—21no—
45C2×C4⋊C4×C47—21no—
46C2×C4⋊C4×C47—21no—
47C2×C4⋊C4×C47D411yescharacteristic
48C2×C4⋊C4×C47C2311yescharacteristic
49C2×C42×C47—21no—
50C2×C42×C47D411yescharacteristic
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—
58#95×C47—21no—
59#95×C47—21no—
60#95×C47D411yescharacteristic
61#95×C47D411yescharacteristic
62C2×Q8×C47—21no—
63C2×Q8×C47—42no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C2×Q8×C47Q1612yescomplemented
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—21no—
73C22×C4×C47—21no—
74C22×C4×C47C2×D411yescharacteristic
75C22×C4×C47C2×D411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C2×Dic94—21no—
78C2×Dic94—21no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—42no—
81C4⋊C4×C47—42no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—81no—
84C4⋊C4×C47—81no—
85C42×C47—41no—
86C42×C47—81no—
87C22⋊Dic47—41no—
88#10625337—82no—
89C4⋊Dic47—21no—
90C4⋊Dic47—21no—
91C4⋊Dic47—21no—
92C4⋊Dic47—21no—
93Dic47⋊C4—41no—
94Dic47⋊C4—41no—
95C4×Dic47—41no—
96#10625329—41no—
97#10625329—41no—
98C2×C47⋊C8—41no—
99C2×C47⋊C8—41no—
100C2×C47⋊C8—41no—
101C23×C47#12011yescharacteristic
102Q8×C47—42no—
103Q8×C47—42no—
104Q8×C47—81no—
105Q8×C47—81no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
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—41no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—41no—
128C2×C4×C47—41no—
129C2×C4×C47C2×Q1611yescharacteristic
130C2×C4×C47#13711yescharacteristic
131C2×Dic47—41no—
132C2×Dic47—41no—
133Dic94—81no—
134Dic94—81no—
135C47⋊C8—81no—
136C47⋊C8—81no—
137C47⋊C8—81no—
138#12222—471nomaximal, Sylow
139C22×C47—21no—
140C22×C47—21no—
141C22×C47#45111yescharacteristic
142C22×C47#45511yescharacteristic
143C22×C47B(4,𝔽2)11yescharacteristic
144C4×C47—21no—
145C4×C47—21no—
146C4×C47—42no—
147C4×C47—41no—
148C4×C47—41no—
149C4×C47—42no—
150C4×C47—81no—
151C4×C47—81no—
152C4×C47—81no—
153C4×C47—81no—
154Dic47—81no—
155Dic47—81no—
156#2799D4711yescharacteristic
157#2028—471no—
158#1617—471no—
159#1605—471no—
160#1354—471no—
161#1353—471no—
162#1319—471no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47#160211yescharacteristic
166C2×C47#219711yescharacteristic
167C2×C47#220311yescharacteristic
168#544—471no—
169#484—941no—
170#451—941no—
171#419—941no—
172#400—21no—
173#400C2×D4711yescharacteristic
174#398—21no—
175#395C2×D4711yescharacteristic
176#391—21no—
177#391—21no—
178#391C2×D4711yescharacteristic
179#387—21no—
180#387—21no—
181#343—471no—
182#339—941no—
183#328—941no—
184#328—941no—
185#324—471no—
186#324—471no—
187C47#1222211yescharacteristic, complemented, Sylow
188C22×Q8C22⋊D4711yescharacteristic
189C22×Q8C22⋊D4711yescharacteristic
190C2×Q16—1881no—
191C2×M16—941no—
192C4⋊Q8—1881no—
193#125—1881no—
194C22⋊Q8—941no—
195C22⋊Q8—941no—
196#117—941no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22⋊D4711yescharacteristic
202C2×C4⋊C4C22×D4711yescharacteristic
203C2×C42—21no—
204C2×C42C22⋊D4711yescharacteristic
205#107—1881no—
206#103—1881no—
207#103—1881no—
208#103—1881no—
209#103—1881no—
210C22⋊C8—941no—
211C22⋊C8—941no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95C22⋊D4711yescharacteristic
222#95C22⋊D4711yescharacteristic
223C2×Q8—21no—
224C2×Q8—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—941no—
229C2×Q8—941no—
230C2×Q8#1062533712yescomplemented
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4C2×C22⋊D4711yescharacteristic
238C22×C4C2×C22⋊D4711yescharacteristic
239C22×C4C2×C22⋊D4711yescharacteristic
240Q16—3762no—
241M16—1881no—
242M16—1881no—
243C2×C8—1881no—
244C2×C8—1881no—
245C2×C8—1881no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—941no—
253C4⋊C4—941no—
254C4⋊C4—941no—
255C4⋊C4—941no—
256C4⋊C4—1881no—
257C4⋊C4—1881no—
258C22⋊C4—1881no—
259C42—41no—
260C42—81no—
261C42—1881no—
262C23#1309399311yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—81no—
266Q8—81no—
267Q8—3761no—
268Q8—3761no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4C2×#1062533711yescharacteristic
295C2×C4#1309405511yescharacteristic
296C8—3761no—
297C8—3761no—
298C8—3761no—
299C22—21no—
300C22—21no—
301C22#1568927911yescharacteristic
302C22#1568930111yescharacteristic
303C22#1568931011yescharacteristic
304C4—21no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—42no—
309C4—42no—
310C4—81no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—3761no—
315C4—3761no—
316C2—21no—
317C2—21no—
318C2#2024234511yescharacteristic
319C2#2024537811yescharacteristic
320C2#2024540911yescharacteristic
subgroupG/H
centerC22#15689310
commutator, cosocleC2×C4⋊C4×C47C23
FrattiniC2×C4⋊C4C22×D47
FittingC47×#2799C2
socleC22×C47B(4,𝔽2)
prime radicalC23×C47#120
hypercenter#2799D47
hypocenterC47#12222
LCSG ▷ C2×C4⋊C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2799

Complements

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

NH ≅ G/Ntype
67. C2×Q8×C47Q16semidirect
187. C47#12222semidirect
230. C2×Q8#10625337semidirect

Maximal subgroups

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

subgroupindex
#202445492
#202424292
#202423612
#202414182
#202414152
#202413222
C47×#27992
#1222247