Finite Group Databasebeta

← #26114070

Subgroups — #26114070

Order 12032.

576 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 115 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#20252417C211yesmaximal, characteristic
2#20247696C211yesmaximal, characteristic
3#20243472C212yesmaximal, complemented
4#20241255C212yesmaximal, complemented
5C47×#1799C211yesmaximal, characteristic
6C2×#13094005—21no—
7C22×C4⋊Dic47C2211yescharacteristic
8#15689195—24no—
9#15689180—21no—
10#15689087—21no—
11C2×C22⋊(C47⋊C8)C2212yes—
12#15689053C412yescomplemented
13#15689048C412yescomplemented
14#15689021—21no—
15#15688984—21no—
16#15688983—21no—
17#15688983C2211yescharacteristic
18#15688885—24no—
19C2×C4×D4×C47C2211yescharacteristic
20C2×#99×C47—21no—
21C47×#380C2212yes—
22C47×#342—21no—
23C23×Dic47—21no—
24C2×#10625329—22no—
25C22×C47⋊C8D414yes—
26#13094005—44no—
27C2×C4⋊Dic47—22no—
28C2×C4⋊Dic47—22no—
29C2×C4⋊Dic47—21no—
30C2×C4⋊Dic47—21no—
31C2×C4⋊Dic47C2×C412yes—
32C2×Dic47⋊C4—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—21no—
35C4⋊(C47⋊C8)—22no—
36C22⋊(C47⋊C8)—22no—
37C22⋊(C47⋊C8)—22no—
38C22⋊(C47⋊C8)—22no—
39#13093935—22no—
40C47⋊#95—22no—
41C47⋊#95—22no—
42C47⋊#95C2×C412yes—
43C22×D4×C47D411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C23×C4×C47C2311yescharacteristic
46C4×D4×C47—22no—
47C4×D4×C47—22no—
48C2×C4⋊C4×C47—22no—
49C2×C4⋊C4×C47C2×C411yescharacteristic
50C2×C22⋊C4×C47—22no—
51C2×C22⋊C4×C47C2×C411yescharacteristic
52C2×C22⋊C4×C47D411yescharacteristic
53C2×C42×C47D411yescharacteristic
54#99×C47—42no—
55#95×C47—22no—
56C24×C47C22⋊C411yescharacteristic
57C24×C47C2×D411yescharacteristic
58C2×D4×C47—22no—
59C2×D4×C47—22no—
60C2×D4×C47—21no—
61C2×D4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D412yes—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47#4112yes—
74C22×Dic47—22no—
75C22×Dic47—22no—
76C22×Dic47—42no—
77C22×Dic47—42no—
78C22×Dic47—41no—
79C22×Dic47—41no—
80C4⋊C4×C47—22no—
81C4⋊C4×C47—44no—
82C22⋊C4×C47—22no—
83C22⋊C4×C47—22no—
84C22⋊C4×C47—42no—
85C22⋊C4×C47—42no—
86C42×C47—21no—
87C42×C47—21no—
88C22⋊Dic47—44no—
89C22⋊Dic47—42no—
90C4⋊Dic47—24no—
91C4⋊Dic47—42no—
92C4⋊Dic47—41no—
93C4⋊Dic47—41no—
94Dic47⋊C4—44no—
95#10625329—82no—
96C2×C47⋊C8—24no—
97C2×C47⋊C8—44no—
98C23×C47—21no—
99C23×C47—21no—
100C23×C47—21no—
101C23×C47—41no—
102C23×C47—41no—
103C23×C47—41no—
104C23×C47—41no—
105C23×C47C2×C22⋊C411yescharacteristic
106C23×C47#12112yes—
107C23×C47AGL(1,ℤ/8)12yes—
108C23×C47AGL(1,ℤ/8)11yescharacteristic
109C23×C47#13711yescharacteristic
110D4×C47—42no—
111D4×C47—42no—
112C2×C4×C47—24no—
113C2×C4×C47—22no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—41no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—81no—
125C2×C4×C47—82no—
126C2×C4×C47—82no—
127C2×C4×C47—82no—
128C2×C4×C47C4×D412yes—
129C2×C4×C47#12011yescharacteristic
130C2×C4×C47#12311yescharacteristic
131C2×Dic47—42no—
132C2×Dic47—42no—
133C2×Dic47—42no—
134C2×Dic47—42no—
135C2×Dic47—42no—
136C2×Dic47—42no—
137C2×Dic47—82no—
138C2×Dic47—82no—
139C47⋊C8—44no—
140#9474—471nomaximal, Sylow
141C22×C47—21no—
142C22×C47—22no—
143C22×C47—21no—
144C22×C47—22no—
145C22×C47—42no—
146C22×C47—41no—
147C22×C47—41no—
148C22×C47—42no—
149C22×C47—41no—
150C22×C47—41no—
151C22×C47—41no—
152C22×C47—81no—
153C22×C47#38611yescharacteristic
154C22×C47#41711yescharacteristic
155C22×C47#41811yescharacteristic
156C22×C47#46812yes—
157C22×C47#46912yes—
158C4×C47—21no—
159C4×C47—22no—
160C4×C47—41no—
161C4×C47—41no—
162C4×C47—41no—
163C4×C47—162no—
164C4×C47#45812yes—
165Dic47—82no—
166Dic47—82no—
167#2973—471no—
168#2486—471no—
169#1823—472no—
170#1799D4711yescharacteristic
171#1289—472no—
172C2×C47—21no—
173C2×C47—22no—
174C2×C47—21no—
175C2×C47—22no—
176C2×C47—81no—
177C2×C47#189011yescharacteristic
178C2×C47#189511yescharacteristic
179C2×C47#190911yescharacteristic
180C2×#123—941no—
181C2×C4×D4C2×D4711yescharacteristic
182C22×C4⋊C4—471no—
183#435—944no—
184#432—941no—
185C2×#99—21no—
186#407—941no—
187C2×C22⋊C8—472no—
188#398—472no—
189#397—472no—
190#389—941no—
191#381—941no—
192#380—471no—
193#380—941no—
194#380C2×D4712yes—
195#342—21no—
196#340—944no—
197C47#947411yescharacteristic, complemented, Sylow
198C22×D4C22⋊D4711yescharacteristic
199C23×C4—941no—
200C23×C4D18811yescharacteristic
201C23×C4C22×D4711yescharacteristic
202C2×M16—942no—
203C22×C8—474no—
204#123—1884no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—472no—
209C2×C4⋊C4—942no—
210C2×C4⋊C4—942no—
211C2×C4⋊C4—942no—
212C2×C4⋊C4—941no—
213C2×C4⋊C4—941no—
214C2×C4⋊C4C4×D4711yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—942no—
217C2×C22⋊C4—941no—
218C2×C22⋊C4C4×D4711yescharacteristic
219C2×C22⋊C4C22⋊D4711yescharacteristic
220C2×C42D18811yescharacteristic
221C4⋊C8—942no—
222#99—42no—
223C22⋊C8—942no—
224C22⋊C8—942no—
225C22⋊C8—942no—
226#97—942no—
227#95—22no—
228#95—472no—
229#95—942no—
230#95—942no—
231C24(C2×D47)⋊C411yescharacteristic
232C24C2×C22⋊D4711yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—942no—
243C22×C4—942no—
244C22×C4—1882no—
245C22×C4—1882no—
246C22×C4—1881no—
247C22×C4—1881no—
248C22×C4(C2×D47)⋊C411yescharacteristic
249C22×C4(C2×D47)⋊C411yescharacteristic
250C22×C4(C2×D47)⋊C411yescharacteristic
251C22×C4C2×C4×D4711yescharacteristic
252C22×C4C2×D18811yescharacteristic
253C22×C4D4×D4712yes—
254C22×C4#1062535212yes—
255M16—3762no—
256C2×C8—944no—
257C2×C8—1884no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—944no—
261C4⋊C4—1884no—
262C4⋊C4—1882no—
263C4⋊C4—1881no—
264C4⋊C4—1881no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—1884no—
270C22⋊C4—1882no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D47)⋊C411yescharacteristic
281C23#1309399612yes—
282C23#1309404912yes—
283C23#1309405011yescharacteristic
284C23#1309405511yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—3762no—
304C2×C4—3762no—
305C2×C4—1882no—
306C2×C4—1882no—
307C2×C4—1882no—
308C2×C4—1882no—
309C2×C4—1882no—
310C2×C4—1882no—
311C2×C4(C22⋊D47)⋊C412yes—
312C2×C4#1309399411yescharacteristic
313C2×C4#1309400811yescharacteristic
314C8—1884no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1568900011yescharacteristic
328C22#1568912211yescharacteristic
329C22#1568912711yescharacteristic
330C22#1568937112yes—
331C22#1568937412yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—162no—
338C4—3762no—
339C4—3762no—
340C4#1568931412yes—
341C2—22no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2024379911yescharacteristic
347C2#2024382411yescharacteristic
348C2#2024390811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#1799C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC23×C4×C47C23
hypercenter#1799D47
hypocenterC47#9474
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20243472C2semidirect
4. #20241255C2semidirect
12. #15689053C4semidirect
13. #15689048C4semidirect
197. C47#9474semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202476962
#202434722
#202412552
C47×#17992
#947447