Finite Group Databasebeta

← #26104500

Subgroups — #26104500

Order 12032.

874 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 127 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×#15689025C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic, complemented
3#20241208C214yesmaximal, complemented
4C47×#1749C211yesmaximal, characteristic, complemented
5C23×D188C411yescharacteristic
6C22×C4⋊Dic47C411yescharacteristic
7C22×(C2×D47)⋊C4—21no—
8C22×(C2×D47)⋊C4—21no—
9C2×C22⋊(C47⋊C8)—21no—
10C2×#13093935—21no—
11#15689025—21no—
12#15689025—21no—
13#15689025C412yes—
14#15689025C2212yes—
15#15688878C2212yescomplemented
16C22×C42×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C2×#97×C47—21no—
19C47×#336C2212yescomplemented
20C24×D47—21no—
21C22×D188—21no—
22C22×D188—24no—
23C22×D188—42no—
24C22×D188—42no—
25C22×D188C2×C412yes—
26C23×Dic47—21no—
27C22×C47⋊C8—22no—
28C2×C4⋊Dic47—21no—
29C2×C4⋊Dic47—42no—
30C2×C4⋊Dic47C2×C412yes—
31C2×(C2×D47)⋊C4—22no—
32C2×(C2×D47)⋊C4—22no—
33C2×(C2×D47)⋊C4—22no—
34C2×(C2×D47)⋊C4—22no—
35C2×(C2×D47)⋊C4—44no—
36C2×(C2×D47)⋊C4—44no—
37C22⋊(C47⋊C8)—24no—
38#13093935—42no—
39C23×C4×C47C2×C411yescharacteristic
40C23×C4×C47D411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C22×C8×C47—22no—
43C2×C42×C47—42no—
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46C2×C42×C47D411yescharacteristic
47C2×C42×C47C2311yescharacteristic
48C22⋊C8×C47—24no—
49#97×C47—42no—
50C24×C47C22⋊C411yescharacteristic
51C23×D47—24no—
52C23×D47—22no—
53C23×D47—42no—
54C23×D47—42no—
55C23×D47—42no—
56C23×D47—42no—
57C23×D47—42no—
58C23×D47—42no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—22no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—22no—
65C22×C4×C47—22no—
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×Dic47—22no—
75C22×Dic47—42no—
76C22×Dic47—42no—
77C22×Dic47—42no—
78C2×D188—24no—
79C2×D188—22no—
80C2×D188—416no—
81C2×D188—42no—
82C2×D188—42no—
83C2×D188—42no—
84C2×D188—42no—
85C2×D188M1614yes—
86C2×C8×C47—24no—
87C2×C8×C47—42no—
88C2×C8×C47—42no—
89C42×C47—22no—
90C42×C47—22no—
91C42×C47—41no—
92C42×C47—41no—
93(C2×D47)⋊C4—44no—
94(C2×D47)⋊C4—44no—
95(C2×D47)⋊C4—44no—
96(C2×D47)⋊C4—44no—
97C4⋊Dic47—42no—
98C4⋊Dic47—42no—
99C2×C47⋊C8—24no—
100C2×C47⋊C8—42no—
101C2×C47⋊C8—42no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47#10012yes—
107C23×C47#10412yes—
108C23×C47#10412yes—
109C23×C47C2×C22⋊C411yescharacteristic
110C22×D47—24no—
111C22×D47—48no—
112C22×D47—48no—
113C22×D47—48no—
114C22×D47—42no—
115C22×D47—42no—
116C22×D47—42no—
117C22×D47—42no—
118C22×D47—42no—
119C22×D47—42no—
120C22×D47—84no—
121C22×D47—84no—
122C22×D47—84no—
123C22×D47—84no—
124C2×C4×C47—28no—
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—21no—
135C2×C4×C47—21no—
136C2×C4×C47—21no—
137C2×C4×C47—21no—
138C2×C4×C47—21no—
139C2×C4×C47—44no—
140C2×C4×C47—44no—
141C2×C4×C47#9912yes—
142C2×C4×C47C2×M1612yes—
143C2×C4×C47AGL(1,ℤ/8)12yes—
144C2×C4×C47AGL(1,ℤ/8)12yes—
145C2×Dic47—42no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47—84no—
149D188—44no—
150D188—48no—
151D188—44no—
152C8×C47—44no—
153C47⋊C8—44no—
154#7626—471nomaximal, Sylow
155C22×C47—22no—
156C22×C47—22no—
157C22×C47—22no—
158C22×C47—22no—
159C22×C47—22no—
160C22×C47—22no—
161C22×C47#32914yes—
162C22×C47#40711yescharacteristic
163C22×C47C2×#9911yescharacteristic
164C22×C47C2×#10011yescharacteristic
165C22×C47#41811yescharacteristic
166C22×C47#41811yescharacteristic
167C22×C47C2×#10411yescharacteristic
168C22×C47C2×#10411yescharacteristic
169C2×D47—44no—
170C2×D47—44no—
171C2×D47—44no—
172C2×D47—816no—
173C2×D47—82no—
174C2×D47—82no—
175C2×D47—82no—
176C2×D47—82no—
177C4×C47—22no—
178C4×C47—22no—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—41no—
182C4×C47—41no—
183Dic47—82no—
184C2×#390—471no—
185#1749—471no—
186#1749D4711yescharacteristic, complemented
187#1270—474no—
188C2×C47—24no—
189C2×C47#146211yescharacteristic
190C2×C47#149011yescharacteristic
191C2×C47#149011yescharacteristic
192C2×C47#150711yescharacteristic
193C2×C47#150711yescharacteristic
194C2×C47C2×#32911yescharacteristic
195C2×C47#155511yescharacteristic
196D47—84no—
197C23×D4—471no—
198C22×C4⋊C4—471no—
199C22×C22⋊C4—941no—
200C22×C22⋊C4—941no—
201C22×C42C2×D4711yescharacteristic
202C2×C22⋊C8—21no—
203C2×C22⋊C8—941no—
204C2×#97—21no—
205C2×#97—941no—
206#390—472no—
207#390—472no—
208#390—941no—
209#390—941no—
210#336—472no—
211#336C2×D4712yescomplemented
212C47#762611yescharacteristic, complemented, Sylow
213C25—941no—
214C22×D4—472no—
215C22×D4—944no—
216C22×D4—941no—
217C22×D4—1882no—
218C22×D4—1882no—
219C23×C4—941no—
220C23×C4C4×D4711yescharacteristic
221C23×C4D18811yescharacteristic
222C23×C4C22⋊D4711yescharacteristic
223C22×C8—22no—
224C22×C8—942no—
225C2×C4⋊C4—472no—
226C2×C4⋊C4—941no—
227C2×C4⋊C4—1882no—
228C2×C22⋊C4—942no—
229C2×C22⋊C4—942no—
230C2×C22⋊C4—942no—
231C2×C22⋊C4—942no—
232C2×C22⋊C4—1884no—
233C2×C22⋊C4—1884no—
234C2×C42—42no—
235C2×C42C4×D4711yescharacteristic
236C2×C42D18811yescharacteristic
237C2×C42C22⋊D4711yescharacteristic
238C2×C42C22×D4711yescharacteristic
239C22⋊C8—24no—
240C22⋊C8—944no—
241#97—42no—
242#97—1882no—
243C24—944no—
244C24—942no—
245C24—1882no—
246C24—1882no—
247C24—1882no—
248C24—1882no—
249C24—1882no—
250C24—1882no—
251C24(C2×D47)⋊C411yescharacteristic
252C2×D4—474no—
253C2×D4—944no—
254C2×D4—942no—
255C2×D4—1882no—
256C2×D4—1882no—
257C2×D4—18816no—
258C2×D4—1882no—
259C2×D4—1882no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—942no—
270C22×C4—1882no—
271C22×C4—1882no—
272C22×C4—1882no—
273C22×C4(C2×D47)⋊C411yescharacteristic
274C22×C4(C2×D47)⋊C411yescharacteristic
275C22×C4(C2×D47)⋊C411yescharacteristic
276C22×C4C2×C4×D4711yescharacteristic
277C22×C4C2×D18811yescharacteristic
278C22×C4C2×C22⋊D4711yescharacteristic
279C2×C8—24no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—944no—
283C2×C8—1882no—
284C2×C8—1882no—
285C4⋊C4—1882no—
286C4⋊C4—1882no—
287C22⋊C4—1884no—
288C22⋊C4—1884no—
289C22⋊C4—1884no—
290C22⋊C4—1884no—
291C42—22no—
292C42—22no—
293C42—41no—
294C42—41no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—944no—
300C23—3764no—
301C23—3764no—
302C23—3764no—
303C23—3764no—
304C23—1888no—
305C23—1888no—
306C23—1888no—
307C23—1882no—
308C23—1882no—
309C23—1882no—
310C23—1882no—
311C23—1882no—
312C23—1882no—
313C23#1309394012yes—
314C23#1309395012yes—
315C23#1309395212yes—
316C23C2×(C2×D47)⋊C411yescharacteristic
317D4—1884no—
318D4—1888no—
319D4—1884no—
320C2×C4—28no—
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—3764no—
338C2×C4—1882no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4#1309393812yes—
342C2×C4M16×D4712yes—
343C2×C4#1309404912yes—
344C2×C4#1309405212yes—
345C8—44no—
346C8—1884no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—37616no—
354C22—3762no—
355C22—3762no—
356C22—3762no—
357C22—3762no—
358C22—1884no—
359C22—1884no—
360C22—1884no—
361C22#1568886214yes—
362C22#1568908611yescharacteristic
363C22C2×#1309393811yescharacteristic
364C22C2×#1309394011yescharacteristic
365C22#1568912711yescharacteristic
366C22#1568912911yescharacteristic
367C22C2×#1309395011yescharacteristic
368C22C2×#1309395211yescharacteristic
369C4—22no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—3762no—
376C2—24no—
377C2—3764no—
378C2#2024171411yescharacteristic
379C2#2024183011yescharacteristic
380C2#2024183211yescharacteristic
381C2#2024190311yescharacteristic
382C2#2024190511yescharacteristic
383C2C2×#1568886211yescharacteristic
384C2#2024210111yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1749C2
socleC23×C47C2×C22⋊C4
prime radicalC23×D188C4
cosocleC2×C42×C47C23
hypercenter#1749D47
hypocenterC47#7626
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
2. #20243150C2semidirect
3. #20241208C2semidirect
4. C47×#1749C2semidirect
15. #15688878C22semidirect
19. C47×#336C22semidirect
186. #1749D47semidirect
211. #336C2×D47semidirect
212. C47#7626semidirect

Maximal subgroups

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

subgroupindex
C2×#156890252
#202431502
#202412082
C47×#17492
#762647