Finite Group Databasebeta

← #26114313

Subgroups — #26114313

Order 12032.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#20252436C211yesmaximal, characteristic
2#20243478C211yesmaximal, characteristic, complemented
3#20243468C211yesmaximal, characteristic, complemented
4#20241931C211yesmaximal, characteristic
5#20241295C211yesmaximal, characteristic, complemented
6#20241264C211yesmaximal, characteristic, complemented
7C47×#3092C211yesmaximal, characteristic
8#15689201—21no—
9#15689195—21no—
10#15689195—21no—
11D4×C47⋊C8—21no—
12#15689184—21no—
13#15689090—21no—
14#15689087C2211yescharacteristic
15#15689087C2211yescharacteristic
16C2×C22⋊(C47⋊C8)C2211yescharacteristic
17#15688983C2211yescharacteristic
18#15688892—21no—
19#15688892—21no—
20#15688885—21no—
21#15688885—21no—
22#15688860—21no—
23#15688857—21no—
24C47×#525C2211yescharacteristic
25C22×C4⋊C4×C47C2211yescharacteristic
26C47×#484—21no—
27C47×#483C412yescomplemented
28C47×#481—21no—
29C47×#480C412yescomplemented
30C47×#428—21no—
31C47×#417—21no—
32C47×#407C2211yescharacteristic
33C2×#10625329—21no—
34C2×#10625329—21no—
35C2×#10625329D411yescharacteristic
36C2×#10625329D411yescharacteristic
37C22×C47⋊C8—21no—
38C22×C47⋊C8D411yescharacteristic
39C22×C47⋊C8D411yescharacteristic
40C2×C4⋊Dic47—21no—
41C2×C22⋊Dic47—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)—21no—
44C22⋊(C47⋊C8)—21no—
45C22⋊(C47⋊C8)—21no—
46C22⋊(C47⋊C8)—21no—
47C22⋊(C47⋊C8)—21no—
48C22⋊(C47⋊C8)—21no—
49C22⋊(C47⋊C8)—21no—
50C22⋊(C47⋊C8)—21no—
51C22⋊(C47⋊C8)—21no—
52#13093935—21no—
53C4×C47⋊C8—21no—
54C47⋊#95—21no—
55C2×#41×C47D411yescharacteristic
56C23×C4×C47—21no—
57C23×C4×C47C2311yescharacteristic
58C2×M16×C47—21no—
59#123×C47—21no—
60C22⋊Q8×C47C2×C411yescharacteristic
61C22⋊Q8×C47D411yescharacteristic
62#121×C47C2×C411yescharacteristic
63#121×C47D411yescharacteristic
64#120×C47—21no—
65C4×D4×C47—21no—
66C4×D4×C47—21no—
67#117×C47D411yescharacteristic
68C2×C4⋊C4×C47—21no—
69C2×C4⋊C4×C47—21no—
70C2×C4⋊C4×C47—21no—
71C2×C4⋊C4×C47—21no—
72C2×C4⋊C4×C47—21no—
73C2×C4⋊C4×C47C2×C412yes—
74#107×C47—22no—
75#106×C47—22no—
76#103×C47—22no—
77#102×C47—22no—
78C22⋊C8×C47—21no—
79C22⋊C8×C47C2×C412yes—
80C24×C47#4111yescharacteristic
81#41×C47—41no—
82#41×C47—41no—
83C2×Q8×C47C22⋊C411yescharacteristic
84C2×D4×C47—21no—
85C2×D4×C47—21no—
86C2×D4×C47—21no—
87C2×D4×C47C22⋊C411yescharacteristic
88C22×C4×C47—21no—
89C22×C4×C47—22no—
90C22×C4×C47—41no—
91C22×C4×C47—41no—
92C22×C4×C47—41no—
93C22×C4×C47—41no—
94C22×C4×C47—82no—
95C22×C4×C47C22×C411yescharacteristic
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2×D411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47#4111yescharacteristic
101C22×Dic47—41no—
102M16×C47—41no—
103M16×C47—41no—
104C2×C8×C47—22no—
105C4⋊C4×C47—21no—
106C4⋊C4×C47—21no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—22no—
109C4⋊C4×C47—41no—
110C4⋊C4×C47—41no—
111C4⋊C4×C47C22⋊C411yescharacteristic
112C4⋊C4×C47C22⋊C411yescharacteristic
113C22⋊C4×C47—21no—
114C22⋊C4×C47—21no—
115C22⋊C4×C47—21no—
116C22⋊C4×C47—21no—
117C22⋊C4×C47—21no—
118C42×C47—21no—
119C22⋊Dic47—81no—
120C4⋊Dic47—41no—
121C4⋊Dic47—41no—
122#10625329—41no—
123#10625329—41no—
124#10625329—41no—
125#10625329—41no—
126#10625329—81no—
127C2×C47⋊C8—21no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—21no—
131C2×C47⋊C8—41no—
132C2×C47⋊C8—41no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—81no—
135C23×C47—21no—
136C23×C47—21no—
137C23×C47—21no—
138C23×C47—41no—
139C23×C47—41no—
140C23×C47C4×D411yescharacteristic
141C23×C47#12111yescharacteristic
142C23×C47#12311yescharacteristic
143Q8×C47—41no—
144D4×C47—41no—
145D4×C47—41no—
146D4×C47—41no—
147D4×C47—81no—
148D4×C47—81no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—21no—
155C2×C4×C47—21no—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—41no—
159C2×C4×C47—41no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—41no—
163C2×C4×C47—42no—
164C2×C4×C47—82no—
165C2×C4×C47—82no—
166C2×C4×C47—82no—
167C2×C4×C47C2×C22⋊C411yescharacteristic
168C2×C4×C47C4×D411yescharacteristic
169C2×C4×C47#12011yescharacteristic
170C2×C4×C47#12111yescharacteristic
171C2×Dic47—81no—
172C2×Dic47—81no—
173C2×Dic47—81no—
174C8×C47—42no—
175C47⋊C8—41no—
176C47⋊C8—41no—
177C47⋊C8—41no—
178C47⋊C8—41no—
179C47⋊C8—81no—
180#9521—471nomaximal, Sylow
181C22×C47—21no—
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—42no—
190C22×C47—42no—
191C22×C47—41no—
192C22×C47—41no—
193C22×C47—41no—
194C22×C47#38611yescharacteristic
195C22×C47#47111yescharacteristic
196C22×C47#47311yescharacteristic
197C4×C47—21no—
198C4×C47—21no—
199C4×C47—21no—
200C4×C47—41no—
201C4×C47—41no—
202C4×C47—41no—
203C4×C47—82no—
204C4×C47#45811yescharacteristic
205C4×C47#45811yescharacteristic
206Dic47—161no—
207#3092D4711yescharacteristic
208#2976—471no—
209#1824—471no—
210#1821—471no—
211#1513—471no—
212#1305—471no—
213#1293—471no—
214C2×C47—21no—
215C2×C47—21no—
216C2×C47—41no—
217C2×C47—41no—
218C2×C47—81no—
219C2×C47#189011yescharacteristic
220C2×C47#190311yescharacteristic
221C2×C47#190811yescharacteristic
222#525C2×D4711yescharacteristic
223C22×C4⋊C4C2×D4711yescharacteristic
224#484—21no—
225#483Dic4712yescomplemented
226#481—21no—
227#480Dic4712yescomplemented
228#436—941no—
229#435—941no—
230#435—941no—
231C8×D4—941no—
232#433—941no—
233#428—21no—
234#417—21no—
235#408—941no—
236#407—471no—
237#407—471no—
238#407C2×D4711yescharacteristic
239C2×C22⋊C8—471no—
240#380—471no—
241#343—941no—
242#343—941no—
243#340—941no—
244#340—941no—
245#328—941no—
246#327—941no—
247C47#952111yescharacteristic, complemented, Sylow
248C2×#41C22⋊D4711yescharacteristic
249C23×C4—21no—
250C23×C4C22×D4711yescharacteristic
251C2×M16—21no—
252C2×M16—471no—
253C2×M16—471no—
254C2×M16—941no—
255C2×M16—941no—
256C22×C8—471no—
257C22×C8—471no—
258C22×C8—941no—
259#123—21no—
260C22⋊Q8C2×Dic4711yescharacteristic
261C22⋊Q8C22⋊D4711yescharacteristic
262#121C2×Dic4711yescharacteristic
263#121C22⋊D4711yescharacteristic
264#120—21no—
265C4×D4—21no—
266C4×D4—21no—
267#117C22⋊D4711yescharacteristic
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—941no—
274C2×C4⋊C4C2×Dic4712yes—
275C2×C22⋊C4—941no—
276#107—22no—
277#106—22no—
278C4⋊C8—941no—
279C4⋊C8—941no—
280#103—22no—
281#102—22no—
282C22⋊C8—21no—
283C22⋊C8—941no—
284C22⋊C8—941no—
285C22⋊C8—941no—
286C22⋊C8—941no—
287C22⋊C8—941no—
288C22⋊C8—941no—
289C22⋊C8—941no—
290C22⋊C8—941no—
291C22⋊C8C2×Dic4712yes—
292#97—941no—
293C4×C8—941no—
294#95—941no—
295C24#1062535211yescharacteristic
296#41—41no—
297#41—41no—
298C2×Q8C22⋊Dic4711yescharacteristic
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4C22⋊Dic4711yescharacteristic
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—82no—
310C22×C4—1881no—
311C22×C4D4×D4711yescharacteristic
312C22×C4D4×D4711yescharacteristic
313C22×C4#1062535211yescharacteristic
314C22×C4C22×Dic4711yescharacteristic
315C22×C4C2×C22⋊D4711yescharacteristic
316C22×C4C2×C22⋊D4711yescharacteristic
317M16—41no—
318M16—41no—
319M16—3761no—
320M16—1881no—
321M16—1881no—
322M16—1881no—
323M16—1881no—
324C2×C8—22no—
325C2×C8—941no—
326C2×C8—941no—
327C2×C8—941no—
328C2×C8—941no—
329C2×C8—3761no—
330C2×C8—1881no—
331C2×C8—1881no—
332C2×C8—1881no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—1881no—
340C4⋊C4—1881no—
341C4⋊C4C22⋊Dic4711yescharacteristic
342C4⋊C4C22⋊Dic4711yescharacteristic
343C22⋊C4—21no—
344C22⋊C4—21no—
345C22⋊C4—21no—
346C22⋊C4—21no—
347C22⋊C4—21no—
348C22⋊C4—3761no—
349C42—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23D4×Dic4711yescharacteristic
356C23#1309399611yescharacteristic
357C23#1309400511yescharacteristic
358Q8—41no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—81no—
363D4—81no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—3761no—
383C2×C4—3761no—
384C2×C4—3761no—
385C2×C4C2×C22⋊Dic4711yescharacteristic
386C2×C4D4×Dic4711yescharacteristic
387C2×C4#1309399211yescharacteristic
388C2×C4#1309399611yescharacteristic
389C8—42no—
390C8—3761no—
391C8—1881no—
392C8—1881no—
393C8—1881no—
394C8—1881no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22#1568900311yescharacteristic
409C22#1568938811yescharacteristic
410C22#1568939611yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—82no—
418C4—7521no—
419C4#1568931211yescharacteristic
420C4#1568931211yescharacteristic
421C2—21no—
422C2—21no—
423C2—41no—
424C2—41no—
425C2—81no—
426C2#2024380211yescharacteristic
427C2#2024387011yescharacteristic
428C2#2024390511yescharacteristic
subgroupG/H
centerC22#15689003
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#3092C2
socleC22×C47#386
prime radicalC47×#525C22
cosocleC23×C4×C47C23
hypercenter#3092D47
hypocenterC47#9521
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3092

Complements

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

NH ≅ G/Ntype
2. #20243478C2semidirect
3. #20243468C2semidirect
5. #20241295C2semidirect
6. #20241264C2semidirect
27. C47×#483C4semidirect
29. C47×#480C4semidirect
225. #483Dic47semidirect
227. #480Dic47semidirect
247. C47#9521semidirect

Maximal subgroups

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

subgroupindex
#202524362
#202434782
#202434682
#202419312
#202412952
#202412642
C47×#30922
#952147