Finite Group Databasebeta

← #26104956

Subgroups — #26104956

Order 12032.

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

normal subgroups only

This group has 520 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#20251689C211yesmaximal, characteristic
2#20243368C211yesmaximal, characteristic, complemented
3#20241213C212yesmaximal, complemented
4#20241208C212yesmaximal, complemented
5C47×#1817C211yesmaximal, characteristic, complemented
6C2×#13094012C411yescharacteristic
7C2×#13094003C411yescharacteristic
8C2×#13093998C411yescharacteristic
9C2×C4⋊(C47⋊C8)—21no—
10C2×C22⋊(C47⋊C8)—21no—
11#15689029C2211yescharacteristic
12#15689025C2211yescharacteristic
13#15689014—21no—
14#15689014—21no—
15#15689002—21no—
16#15689002—21no—
17#15688989C411yescharacteristic
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C22⋊C8×C47—21no—
21C47×#336—21no—
22C47×#336C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C22×D188C2×C411yescharacteristic
25C22×C22⋊D47—21no—
26C22×C47⋊C8—22no—
27#13094012—42no—
28#13094003—42no—
29#13093998—42no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×(C2×D47)⋊C4—21no—
35C2×(C2×D47)⋊C4—21no—
36C2×(C2×D47)⋊C4—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C4⋊(C47⋊C8)—42no—
40C22⋊(C47⋊C8)—42no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C23×C4×C47C2×C411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C22×C8×C47—22no—
49C2×C42×C47—42no—
50C2×C42×C47C2×C411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47D411yescharacteristic
53C2×C42×C47C2311yescharacteristic
54C22⋊C8×C47—42no—
55C24×C47C22⋊C411yescharacteristic
56C23×D47—21no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C2×C22⋊D47—42no—
74C2×C22⋊D47—42no—
75C2×C22⋊D47—42no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C2×D188—22no—
80C2×D188—41no—
81C2×D188—41no—
82C2×Dic94—22no—
83C2×Dic94—41no—
84C2×Dic94—41no—
85C2×C8×C47—42no—
86C2×C8×C47—42no—
87C2×C8×C47—42no—
88C42×C47—22no—
89C42×C47—22no—
90C42×C47—41no—
91C42×C47—41no—
92C22⋊Dic47—42no—
93(C2×D47)⋊C4—42no—
94(C2×D47)⋊C4—42no—
95(C2×D47)⋊C4—42no—
96(C2×D47)⋊C4—42no—
97(C2×D47)⋊C4—41no—
98(C2×D47)⋊C4—41no—
99C4⋊Dic47—41no—
100C4⋊Dic47—41no—
101Dic47⋊C4—42no—
102C2×C47⋊C8—42no—
103C2×C47⋊C8—42no—
104C2×C47⋊C8—42no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—21no—
111C23×C47#10012yes—
112C23×C47C2×C22⋊C411yescharacteristic
113C22×D47—41no—
114C22×D47—41no—
115C22×D47—41no—
116C22×D47—41no—
117C22×D47—41no—
118C22×D47—41no—
119C22×D47—41no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—21no—
134C2×C4×C47—44no—
135C2×C4×C47—44no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47#9912yes—
139C2×C4×C47#13112yes—
140C2×C4×C47#13512yes—
141C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C47#13711yescharacteristic
143C22⋊D47—84no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152C2×Dic47—41no—
153D188—82no—
154Dic94—82no—
155C8×C47—82no—
156C47⋊C8—82no—
157#7730—471nomaximal, Sylow
158C22×C47—22no—
159C22×C47—22no—
160C22×C47—22no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47—41no—
164C22×C47—41no—
165C22×C47#40811yescharacteristic
166C22×C47C2×#9911yescharacteristic
167C22×C47C2×#10011yescharacteristic
168C22×C47#41611yescharacteristic
169C22×C47#41711yescharacteristic
170C22×C47#42111yescharacteristic
171C22×C47#42111yescharacteristic
172C2×D47—81no—
173C2×D47—81no—
174C2×D47—81no—
175C2×D47—81no—
176C2×D47—81no—
177C2×D47—81no—
178C2×D47—81no—
179C4×C47—22no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—22no—
183C4×C47—22no—
184C4×C47—41no—
185C4×C47—41no—
186Dic47—81no—
187Dic47—81no—
188Dic47—81no—
189#2850—471no—
190#1817D4711yescharacteristic, complemented
191#1797—471no—
192#1272—472no—
193#1270—472no—
194C2×C47—42no—
195C2×C47#147111yescharacteristic
196C2×C47#148211yescharacteristic
197C2×C47#149511yescharacteristic
198C2×C47#151311yescharacteristic
199C2×C47#151611yescharacteristic
200C2×C47#153311yescharacteristic
201C2×C47#155111yescharacteristic
202D47—161no—
203C2×#124—471no—
204C2×C22⋊Q8—471no—
205C2×#121—471no—
206C22×C42C2×D4711yescharacteristic
207C2×C4⋊C8—941no—
208C2×C22⋊C8—21no—
209C2×C22⋊C8—941no—
210#391—471no—
211#390—471no—
212#388—941no—
213#388—941no—
214#386—941no—
215#386—941no—
216#382—471no—
217#336—21no—
218#336—472no—
219#336C2×D4712yes—
220C47#773011yescharacteristic, complemented, Sylow
221C22×Q8—471no—
222C22×D4—471no—
223C22×D4—941no—
224C23×C4C4×D4711yescharacteristic
225C23×C4D18811yescharacteristic
226C23×C4C22⋊D4711yescharacteristic
227C22×C8—22no—
228C22×C8—942no—
229#124—1882no—
230C22⋊Q8—1882no—
231#121—1882no—
232C2×C4⋊C4—471no—
233C2×C4⋊C4—471no—
234C2×C4⋊C4—941no—
235C2×C22⋊C4—941no—
236C2×C22⋊C4—941no—
237C2×C22⋊C4—941no—
238C2×C22⋊C4—941no—
239C2×C22⋊C4—941no—
240C2×C22⋊C4—941no—
241C2×C42—42no—
242C2×C42C4×D4711yescharacteristic
243C2×C42D18811yescharacteristic
244C2×C42C22⋊D4711yescharacteristic
245C2×C42C22×D4711yescharacteristic
246C4⋊C8—1882no—
247C22⋊C8—42no—
248C22⋊C8—1882no—
249#95—941no—
250#95—941no—
251#95—941no—
252#95—941no—
253C24—941no—
254C24(C2×D47)⋊C411yescharacteristic
255C2×Q8—942no—
256C2×Q8—1881no—
257C2×Q8—1881no—
258C2×D4—942no—
259C2×D4—1881no—
260C2×D4—1881no—
261C2×D4—1882no—
262C2×D4—1882no—
263C2×D4—1882no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—941no—
275C22×C4—941no—
276C22×C4—941no—
277C22×C4(C2×D47)⋊C411yescharacteristic
278C22×C4(C2×D47)⋊C411yescharacteristic
279C22×C4(C2×D47)⋊C411yescharacteristic
280C22×C4C2×C4×D4711yescharacteristic
281C22×C4C2×D18811yescharacteristic
282C22×C4C2×C22⋊D4711yescharacteristic
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—1882no—
287C2×C8—1882no—
288C2×C8—1882no—
289C4⋊C4—1881no—
290C4⋊C4—1881no—
291C4⋊C4—1882no—
292C22⋊C4—1882no—
293C22⋊C4—1882no—
294C22⋊C4—1882no—
295C22⋊C4—1882no—
296C22⋊C4—1882no—
297C22⋊C4—1881no—
298C22⋊C4—1881no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—1881no—
310C23—1881no—
311C23—1881no—
312C23—1881no—
313C23—1881no—
314C23—1881no—
315C23—1881no—
316C23#1309394012yes—
317C23C2×(C2×D47)⋊C411yescharacteristic
318Q8—3762no—
319D4—3762no—
320D4—3764no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4—1881no—
345C2×C4—1881no—
346C2×C4—1881no—
347C2×C4—1881no—
348C2×C4#1309393812yes—
349C2×C4#1309403012yes—
350C2×C4#1309404412yes—
351C2×C4#1309405211yescharacteristic
352C2×C4#1309405711yescharacteristic
353C8—82no—
354C8—3762no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—3761no—
363C22—3761no—
364C22—3761no—
365C22—3761no—
366C22—3761no—
367C22—3761no—
368C22—3761no—
369C22#1568908911yescharacteristic
370C22C2×#1309393811yescharacteristic
371C22C2×#1309394011yescharacteristic
372C22#1568911711yescharacteristic
373C22#1568912511yescharacteristic
374C22#1568913911yescharacteristic
375C22#1568914111yescharacteristic
376C4—21no—
377C4—21no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—3761no—
384C4—3761no—
385C4—3761no—
386C2—42no—
387C2—7521no—
388C2#2024174711yescharacteristic
389C2#2024179411yescharacteristic
390C2#2024185411yescharacteristic
391C2#2024193211yescharacteristic
392C2#2024194211yescharacteristic
393C2#2024201111yescharacteristic
394C2#2024208411yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1817C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C4
cosocleC2×C42×C47C23
hypercenter#1817D47
hypocenterC47#7730
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. #20243368C2semidirect
3. #20241213C2semidirect
4. #20241208C2semidirect
5. C47×#1817C2semidirect
190. #1817D47semidirect
220. C47#7730semidirect

Maximal subgroups

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

subgroupindex
#202516892
#202433682
#202412132
#202412082
C47×#18172
#773047