Finite Group Databasebeta

← #26118550

Subgroups — #26118550

Order 12032.

462 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 101 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#20254336C211yesmaximal, characteristic
2#20253910C211yesmaximal, characteristic
3#20244362C211yesmaximal, characteristic
4#20241953C211yesmaximal, characteristic
5#20241952C211yesmaximal, characteristic
6#20241232C211yesmaximal, characteristic
7C47×#3230C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689533—21no—
10#15689533—21no—
11#15689529—21no—
12#15689529—21no—
13#15689502—21no—
14#15689473—21no—
15#15689330—22no—
16#15689279—22no—
17#15689176C2211yescharacteristic
18#15689159—22no—
19C2×#13093935C2211yescharacteristic
20#15688989C2211yescharacteristic
21#15688885—22no—
22#15688861—22no—
23#15688860—22no—
24C47×#533C2211yescharacteristic
25C2×C4⋊Q8×C47C2211yescharacteristic
26C47×#500—21no—
27C47×#487—21no—
28C47×#484—22no—
29C47×#477—22no—
30C47×#431C2211yescharacteristic
31C2×#10625337—24no—
32C22×C47⋊C8D412yes—
33#13094024D411yescharacteristic
34#13094024D411yescharacteristic
35#13094016—21no—
36#13094003—21no—
37#13094003—21no—
38C4×Dic94—21no—
39C4×Dic94—21no—
40#13093983—21no—
41C2×C4⋊Dic47—21no—
42#13093957—22no—
43#13093957—22no—
44#13093955—22no—
45#13093955—22no—
46C4⋊(C47⋊C8)—21no—
47#13093948—24no—
48C22⋊(C47⋊C8)D412yes—
49#13093935—21no—
50#13093935—21no—
51C4×C47⋊C8—21no—
52C47⋊#95—21no—
53C47⋊#95—21no—
54C22×Q8×C47—21no—
55C4⋊Q8×C47—21no—
56C4⋊Q8×C47—21no—
57C4⋊Q8×C47—21no—
58C4⋊Q8×C47D412yes—
59C4⋊Q8×C47D411yescharacteristic
60C4⋊Q8×C47D411yescharacteristic
61#125×C47—21no—
62C22⋊Q8×C47—21no—
63C22⋊Q8×C47—21no—
64C4×Q8×C47—21no—
65C4×Q8×C47—21no—
66#117×C47—21no—
67C2×C4⋊C4×C47—21no—
68C2×C4⋊C4×C47—21no—
69C2×C42×C47C2311yescharacteristic
70#107×C47—24no—
71C4⋊C8×C47—21no—
72#103×C47—24no—
73C22⋊C8×C47D412yes—
74C4×C8×C47—21no—
75C2×Q8×C47—21no—
76C2×Q8×C47—22no—
77C2×Q8×C47—21no—
78C2×Q8×C47—41no—
79C2×Q8×C47—82no—
80C22×C4×C47—41no—
81C22×C4×C47C2×D411yescharacteristic
82C22×C4×C47C2×D411yescharacteristic
83C22×C4×C47C2×D411yescharacteristic
84C22×Dic47—41no—
85C2×Dic94—21no—
86C2×Dic94—21no—
87C2×C8×C47—22no—
88C4⋊C4×C47—22no—
89C4⋊C4×C47—21no—
90C4⋊C4×C47—21no—
91C4⋊C4×C47—21no—
92C4⋊C4×C47—22no—
93C4⋊C4×C47—21no—
94C4⋊C4×C47—41no—
95C4⋊C4×C47—41no—
96C4⋊C4×C47—41no—
97C4⋊C4×C47—41no—
98C22⋊C4×C47—41no—
99C42×C47—41no—
100C42×C47C2×D411yescharacteristic
101C42×C47C2×D411yescharacteristic
102C42×C47C2×D411yescharacteristic
103C42×C47#4111yescharacteristic
104C22⋊Dic47—41no—
105#10625337—84no—
106C4⋊Dic47—21no—
107C4⋊Dic47—21no—
108C4⋊Dic47—21no—
109C4⋊Dic47—21no—
110C4⋊Dic47—42no—
111Dic47⋊C4—41no—
112Dic47⋊C4—41no—
113C4×Dic47—41no—
114C2×C47⋊C8—22no—
115C2×C47⋊C8—22no—
116C2×C47⋊C8—22no—
117C2×C47⋊C8Q1614yescomplemented
118C23×C47#12711yescharacteristic
119Q8×C47—44no—
120Q8×C47—42no—
121Q8×C47—81no—
122Q8×C47—82no—
123C2×C4×C47—41no—
124C2×C4×C47—41no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—42no—
128C2×C4×C47—82no—
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47#12011yescharacteristic
132C2×C4×C47#12111yescharacteristic
133C2×C4×C47#12111yescharacteristic
134C2×C4×C47#12111yescharacteristic
135C2×C4×C47C2×Q1612yes—
136C2×C4×C47#13712yes—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—81no—
140C2×Dic47—81no—
141C2×Dic47—81no—
142Dic94—42no—
143Dic94—81no—
144C8×C47—42no—
145C47⋊C8—24no—
146C47⋊C8—42no—
147#10224—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47#39211yescharacteristic
151C22×C47#49411yescharacteristic
152C22×C47#49711yescharacteristic
153C4×C47—21no—
154C4×C47—21no—
155C4×C47—21no—
156C4×C47—21no—
157C4×C47—82no—
158C4×C47—81no—
159C4×C47—81no—
160C4×C47#45112yes—
161C4×C47#46212yes—
162Dic47—81no—
163Dic47—81no—
164Dic47—161no—
165#3230D4711yescharacteristic
166#3209—471no—
167#3162—471no—
168#1993—471no—
169#1518—471no—
170#1518—471no—
171#1281—471no—
172C2×C47—21no—
173C2×C47—21no—
174C2×C47#200811yescharacteristic
175C2×C47#200911yescharacteristic
176C2×C47#201611yescharacteristic
177#533—471no—
178#533C2×D4711yescharacteristic
179C2×C4⋊Q8C2×D4711yescharacteristic
180#501—941no—
181#501—941no—
182#501—941no—
183#501—941no—
184#500—21no—
185#494—941no—
186#487—21no—
187#487—941no—
188#484—22no—
189#477—22no—
190#462—942no—
191#451—942no—
192#431—471no—
193#431C2×D4711yescharacteristic
194#427—942no—
195C2×#97—471no—
196#382—471no—
197#340—942no—
198#328—942no—
199#328—942no—
200C47#1022411yescharacteristic, complemented, Sylow
201C22×Q8—21no—
202C2×Q16—944no—
203C22×C8—472no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—471no—
208C4⋊Q8—471no—
209C4⋊Q8C22⋊D4712yes—
210C4⋊Q8C22⋊D4711yescharacteristic
211C4⋊Q8C22⋊D4711yescharacteristic
212#125—21no—
213#125—941no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—941no—
217C22⋊Q8—941no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—941no—
221C4×Q8—941no—
222#117—21no—
223#117—941no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—941no—
227C2×C42C22×D4711yescharacteristic
228#107—24no—
229#107—942no—
230#107—942no—
231#106—942no—
232#106—942no—
233C4⋊C8—21no—
234C4⋊C8—941no—
235#103—24no—
236#103—944no—
237C22⋊C8—472no—
238C22⋊C8C22⋊D4712yes—
239#97—941no—
240#97—941no—
241C4×C8—21no—
242C4×C8—941no—
243#95—941no—
244#95—941no—
245C2×Q8—22no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—41no—
249C2×Q8—82no—
250C2×Q8—941no—
251C2×Q8—941no—
252C22×C4—41no—
253C22×C4—1881no—
254C22×C4D4×D4711yescharacteristic
255C22×C4D4×D4711yescharacteristic
256C22×C4C2×C22⋊D4711yescharacteristic
257Q16—3764no—
258C2×C8—22no—
259C2×C8—474no—
260C2×C8—942no—
261C2×C8—942no—
262C2×C8—942no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—22no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—21no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—941no—
274C4⋊C4—941no—
275C4⋊C4—941no—
276C4⋊C4—941no—
277C4⋊C4—1882no—
278C4⋊C4—1881no—
279C4⋊C4—1881no—
280C22⋊C4—41no—
281C22⋊C4—1881no—
282C42—41no—
283C42—1881no—
284C42D4×D4711yescharacteristic
285C42#1062535211yescharacteristic
286C42C2×C22⋊D4711yescharacteristic
287C42C2×C22⋊D4711yescharacteristic
288C23#1309402011yescharacteristic
289Q8—44no—
290Q8—42no—
291Q8—81no—
292Q8—82no—
293Q8—3761no—
294Q8—1882no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—82no—
301C2×C4—3761no—
302C2×C4—3761no—
303C2×C4—3761no—
304C2×C4—1881no—
305C2×C4—1881no—
306C2×C4#1309399211yescharacteristic
307C2×C4#1309399211yescharacteristic
308C2×C4#1309399311yescharacteristic
309C2×C4#1309399511yescharacteristic
310C2×C4#1309399611yescharacteristic
311C2×C4#1309399611yescharacteristic
312C2×C4Q16×D4712yes—
313C2×C4#1309405612yes—
314C8—42no—
315C8—944no—
316C8—1882no—
317C22—21no—
318C22—21no—
319C22#1568903111yescharacteristic
320C22#1568950111yescharacteristic
321C22#1568951611yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—3761no—
330C4—3761no—
331C4—7521no—
332C4#1568927712yes—
333C4#1568933112yes—
334C2—21no—
335C2—21no—
336C2#2024443911yescharacteristic
337C2#2024444811yescharacteristic
338C2#2024448611yescharacteristic
subgroupG/H
centerC22#15689031
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#3230C2
socleC22×C47#392
prime radicalC23×C47#127
hypercenter#3230D47
hypocenterC47#10224
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
117. C2×C47⋊C8Q16semidirect
200. C47#10224semidirect

Maximal subgroups

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

subgroupindex
#202543362
#202539102
#202443622
#202419532
#202419522
#202412322
C47×#32302
#1022447