Finite Group Databasebeta

← #26112590

Subgroups — #26112590

Order 12032.

452 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C2×#15689464C211yesmaximal, characteristic
2#20252046C211yesmaximal, characteristic
3#20241907C211yesmaximal, characteristic
4#20241753C211yesmaximal, characteristic
5#20241264C211yesmaximal, characteristic
6#20241236C211yesmaximal, characteristic
7C47×#1826C211yesmaximal, characteristic
8C2×#13094003C2211yescharacteristic
9#15689464—21no—
10#15689464—21no—
11#15689464—21no—
12#15689464—21no—
13#15689184—22no—
14#15689180—21no—
15#15689176—21no—
16C2×#13093957C411yescharacteristic, complemented
17C2×#13093955C411yescharacteristic, complemented
18#15689150—21no—
19C2×#13093947C411yescharacteristic, complemented
20C2×#13093947C411yescharacteristic, complemented
21#15689087C2211yescharacteristic
22C2×C22⋊(C47⋊C8)C2211yescharacteristic
23C2×C22⋊(C47⋊C8)C2211yescharacteristic
24#15689079—21no—
25#15688859—22no—
26#15688848—21no—
27#15688848—21no—
28C2×#117×C47C2211yescharacteristic
29C47×#407C2211yescharacteristic
30C4×C22⋊C4×C47C2211yescharacteristic
31C47×#339—21no—
32C47×#339—21no—
33C22×Dic94C2×C411yescharacteristic
34C2×#10625329—21no—
35C22×C47⋊C8—21no—
36C22×C47⋊C8C2×C411yescharacteristic
37C22×C47⋊C8C2×C411yescharacteristic
38#13094003—21no—
39#13094003—21no—
40#13094003—21no—
41#13094003—21no—
42C2×C4⋊Dic47C2×C411yescharacteristic
43C2×Dic47⋊C4—21no—
44C2×C22⋊Dic47—21no—
45#13093957—21no—
46#13093957—21no—
47#13093955—21no—
48#13093955—21no—
49C4⋊(C47⋊C8)—22no—
50C4⋊(C47⋊C8)—22no—
51#13093947—21no—
52#13093947—21no—
53#13093947—21no—
54#13093947—21no—
55C22⋊(C47⋊C8)—21no—
56C22⋊(C47⋊C8)—21no—
57C22⋊(C47⋊C8)—21no—
58C22⋊(C47⋊C8)—21no—
59C22⋊(C47⋊C8)D411yescharacteristic
60C22⋊(C47⋊C8)D411yescharacteristic
61C22⋊(C47⋊C8)D412yes—
62#13093935—22no—
63C4×C47⋊C8—22no—
64C23×C4×C47C2311yescharacteristic
65C2×M16×C47—21no—
66#117×C47—21no—
67#117×C47—21no—
68#117×C47D411yescharacteristic
69#117×C47D411yescharacteristic
70#117×C47D411yescharacteristic
71#117×C47D411yescharacteristic
72C2×C4⋊C4×C47C2×C411yescharacteristic
73C2×C4⋊C4×C47C2×C411yescharacteristic
74C2×C22⋊C4×C47—21no—
75C2×C22⋊C4×C47—21no—
76C2×C42×C47—21no—
77C2×C42×C47—21no—
78C22⋊C8×C47—21no—
79C22⋊C8×C47—21no—
80#95×C47—21no—
81C24×C47#4111yescharacteristic
82C22×C4×C47—21no—
83C22×C4×C47—21no—
84C22×C4×C47—41no—
85C22×C4×C47—41no—
86C22×C4×C47—41no—
87C22×C4×C47C22×C411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47#4111yescharacteristic
93C22×Dic47—21no—
94C22×Dic47—21no—
95C2×Dic94—21no—
96C2×Dic94—21no—
97C2×Dic94—41no—
98C2×Dic94—41no—
99M16×C47—41no—
100M16×C47—41no—
101C2×C8×C47—41no—
102C4⋊C4×C47—21no—
103C4⋊C4×C47—21no—
104C4⋊C4×C47C22⋊C411yescharacteristic
105C4⋊C4×C47C22⋊C411yescharacteristic
106C4⋊C4×C47C22⋊C411yescharacteristic
107C4⋊C4×C47C22⋊C411yescharacteristic
108C22⋊C4×C47—21no—
109C22⋊C4×C47—21no—
110C22⋊C4×C47—21no—
111C22⋊C4×C47—21no—
112C22⋊C4×C47—22no—
113C22⋊C4×C47—41no—
114C42×C47—21no—
115C42×C47—21no—
116C42×C47—21no—
117C42×C47—21no—
118C42×C47—41no—
119C42×C47—41no—
120C22⋊Dic47—41no—
121C22⋊Dic47—41no—
122C4⋊Dic47—21no—
123C4⋊Dic47—21no—
124Dic47⋊C4—41no—
125Dic47⋊C4—41no—
126#10625329—41no—
127#10625329—41no—
128C2×C47⋊C8—21no—
129C2×C47⋊C8—21no—
130C2×C47⋊C8—21no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—21no—
133C2×C47⋊C8—41no—
134C2×C47⋊C8—41no—
135C2×C47⋊C8—41no—
136C2×C47⋊C8—41no—
137C23×C47—21no—
138C23×C47—21no—
139C23×C47—41no—
140C23×C47—41no—
141C23×C47C4×D411yescharacteristic
142C23×C47#12111yescharacteristic
143C23×C47#12311yescharacteristic
144C2×C4×C47—21no—
145C2×C4×C47—21no—
146C2×C4×C47—21no—
147C2×C4×C47—21no—
148C2×C4×C47—21no—
149C2×C4×C47—21no—
150C2×C4×C47—21no—
151C2×C4×C47—21no—
152C2×C4×C47—21no—
153C2×C4×C47—21no—
154C2×C4×C47—41no—
155C2×C4×C47—41no—
156C2×C4×C47—41no—
157C2×C4×C47—41no—
158C2×C4×C47—41no—
159C2×C4×C47—42no—
160C2×C4×C47—81no—
161C2×C4×C47—81no—
162C2×C4×C47—81no—
163C2×C4×C47C2×C22⋊C411yescharacteristic
164C2×C4×C47C4×D411yescharacteristic
165C2×C4×C47#12011yescharacteristic
166C2×C4×C47#12111yescharacteristic
167C2×Dic47—41no—
168C2×Dic47—41no—
169C2×Dic47—41no—
170C2×Dic47—41no—
171C2×Dic47—81no—
172Dic94—41no—
173Dic94—41no—
174Dic94—81no—
175C8×C47—81no—
176C47⋊C8—41no—
177C47⋊C8—41no—
178C47⋊C8—42no—
179C47⋊C8—41no—
180C47⋊C8—41no—
181#9240—471nomaximal, Sylow
182C22×C47—21no—
183C22×C47—21no—
184C22×C47—21no—
185C22×C47—21no—
186C22×C47—21no—
187C22×C47—21no—
188C22×C47—21no—
189C22×C47—21no—
190C22×C47—81no—
191C22×C47—81no—
192C22×C47#38611yescharacteristic
193C22×C47#44311yescharacteristic
194C22×C47#44411yescharacteristic
195C4×C47—21no—
196C4×C47—21no—
197C4×C47—21no—
198C4×C47—41no—
199C4×C47—41no—
200C4×C47—41no—
201C4×C47—41no—
202C4×C47—82no—
203C4×C47#41011yescharacteristic
204C4×C47#41011yescharacteristic
205Dic47—81no—
206Dic47—81no—
207C2×#485—471no—
208#2924—471no—
209#1826D4711yescharacteristic
210#1508—471no—
211#1472—471no—
212#1293—471no—
213#1282—471no—
214C2×C47—21no—
215C2×C47—21no—
216C2×C47—41no—
217C2×C47—41no—
218C2×C47#187611yescharacteristic
219C2×C47#189911yescharacteristic
220C2×C47#190211yescharacteristic
221C2×C22⋊Q8—471no—
222C2×#117C2×D4711yescharacteristic
223#485—941no—
224#485—941no—
225#485—941no—
226#485—941no—
227#433—942no—
228#432—941no—
229#431—941no—
230C2×#107—471no—
231C2×#106—471no—
232#424—941no—
233C2×#103—471no—
234C2×#103—471no—
235#407—471no—
236#407C2×D4711yescharacteristic
237C2×C22⋊C8—471no—
238C2×C22⋊C8—471no—
239#405—941no—
240C4×C22⋊C4C2×D4711yescharacteristic
241#339—21no—
242#339—21no—
243#328—942no—
244#324—941no—
245#324—941no—
246C47#924011yescharacteristic, complemented, Sylow
247C22×Q8—471no—
248C23×C4C22×D4711yescharacteristic
249C2×M16—21no—
250C2×M16—941no—
251C22×C8—471no—
252C22×C8—471no—
253C22×C8—941no—
254C22⋊Q8—941no—
255C22⋊Q8—941no—
256C22⋊Q8—941no—
257C22⋊Q8—941no—
258#117—21no—
259#117—21no—
260#117D18811yescharacteristic
261#117D18811yescharacteristic
262#117C22⋊D4711yescharacteristic
263#117C22⋊D4711yescharacteristic
264C2×C4⋊C4—471no—
265C2×C4⋊C4—941no—
266C2×C4⋊C4C4×D4711yescharacteristic
267C2×C4⋊C4C4×D4711yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—941no—
271C2×C42—21no—
272C2×C42—21no—
273#107—941no—
274#107—941no—
275#106—941no—
276#106—941no—
277C4⋊C8—942no—
278C4⋊C8—942no—
279#103—941no—
280#103—941no—
281#103—941no—
282#103—941no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—471no—
286C22⋊C8—471no—
287C22⋊C8—472no—
288C22⋊C8—941no—
289C22⋊C8—941no—
290C22⋊C8—941no—
291C22⋊C8—941no—
292#97—942no—
293C4×C8—942no—
294#95—21no—
295C24#1062535211yescharacteristic
296C2×Q8—941no—
297C2×Q8—941no—
298C2×Q8—1881no—
299C2×Q8—1881no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—941no—
306C22×C4—941no—
307C22×C4C2×C4×D4711yescharacteristic
308C22×C4C2×D18811yescharacteristic
309C22×C4D4×D4711yescharacteristic
310C22×C4D4×D4711yescharacteristic
311C22×C4#1062535211yescharacteristic
312C22×C4C2×C22⋊D4711yescharacteristic
313M16—41no—
314M16—41no—
315M16—1881no—
316M16—1881no—
317C2×C8—41no—
318C2×C8—941no—
319C2×C8—941no—
320C2×C8—941no—
321C2×C8—942no—
322C2×C8—941no—
323C2×C8—1881no—
324C2×C8—1881no—
325C2×C8—1881no—
326C2×C8—1881no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—941no—
330C4⋊C4—941no—
331C4⋊C4—1881no—
332C4⋊C4—1881no—
333C4⋊C4(C2×D47)⋊C411yescharacteristic
334C4⋊C4(C2×D47)⋊C411yescharacteristic
335C4⋊C4(C2×D47)⋊C411yescharacteristic
336C4⋊C4(C2×D47)⋊C411yescharacteristic
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—22no—
342C22⋊C4—41no—
343C22⋊C4—1881no—
344C22⋊C4—1881no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—41no—
350C42—41no—
351C23—21no—
352C23—21no—
353C23—41no—
354C23—41no—
355C23(C22⋊D47)⋊C411yescharacteristic
356C23#1309399611yescharacteristic
357C23#1309400811yescharacteristic
358Q8—3761no—
359Q8—1881no—
360Q8—1881no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—3761no—
381C2×C4—1881no—
382C2×C4—1881no—
383C2×C4—1881no—
384C2×C4—1881no—
385C2×C4C2×(C2×D47)⋊C411yescharacteristic
386C2×C4(C22⋊D47)⋊C411yescharacteristic
387C2×C4#1309399411yescharacteristic
388C2×C4#1309399611yescharacteristic
389C8—81no—
390C8—1881no—
391C8—1881no—
392C8—1882no—
393C8—1881no—
394C8—1881no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—81no—
404C22—81no—
405C22#1568900011yescharacteristic
406C22#1568923711yescharacteristic
407C22#1568924211yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—82no—
416C4—3761no—
417C4—3761no—
418C4#1568909611yescharacteristic
419C4#1568909611yescharacteristic
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2#2024371311yescharacteristic
425C2#2024384411yescharacteristic
426C2#2024385911yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1826C2
socleC22×C47#386
prime radicalC24×C47#41
cosocleC23×C4×C47C23
hypercenter#1826D47
hypocenterC47#9240
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
16. C2×#13093957C4semidirect
17. C2×#13093955C4semidirect
19. C2×#13093947C4semidirect
20. C2×#13093947C4semidirect
246. C47#9240semidirect

Maximal subgroups

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

subgroupindex
C2×#156894642
#202520462
#202419072
#202417532
#202412642
#202412362
C47×#18262
#924047