Finite Group Databasebeta

← #26108052

Subgroups — #26108052

Order 12032.

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

normal subgroups only

This group has 802 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×#15689838C211yesmaximal, characteristic
2#20242074C212yesmaximal, complemented
3#20241812C212yesmaximal, complemented
4#20241709C211yesmaximal, characteristic, complemented
5C47×#1800C211yesmaximal, characteristic, complemented
6#15689838—24no—
7#15689838—24no—
8C2×#13094024C2211yescharacteristic
9C2×#13094010—22no—
10C2×#13093998C412yes—
11C2×#13093995—21no—
12C2×C4×D188C2211yescharacteristic
13C2×#13093983C412yes—
14C2×#13093982—21no—
15#15689176C2212yes—
16#15688868—24no—
17#15688850—24no—
18#15688847—22no—
19C2×C4×D4×C47C2211yescharacteristic
20C47×#423C2212yes—
21C2×#100×C47—21no—
22C47×#343—21no—
23C22×Dic94—21no—
24C22×D188C2×C411yescharacteristic
25C22×C22⋊D47—22no—
26C22×C4×D47—21no—
27#13094024—22no—
28#13094024D414yes—
29#13094010—216no—
30#13093998—24no—
31#13093998—24no—
32#13093995—28no—
33C4×D188—22no—
34C4×D188D414yes—
35#13093983—24no—
36#13093983—24no—
37#13093982—28no—
38C2×C4⋊Dic47—21no—
39C2×C4⋊Dic47C2×C411yescharacteristic
40C2×C4⋊Dic47C2×C412yes—
41C2×C22⋊Dic47—22no—
42C2×(C2×D47)⋊C4—22no—
43C2×(C2×D47)⋊C4—21no—
44C2×C4×Dic47—22no—
45C4⋊(C47⋊C8)—22no—
46C22⋊(C47⋊C8)—22no—
47C4×C47⋊C8—22no—
48C22×D4×C47D411yescharacteristic
49C23×C4×C47C2×C412yes—
50C2×M16×C47—22no—
51C4×D4×C47—22no—
52C4×D4×C47—22no—
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C22⋊C4×C47D412yes—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—22no—
57C4⋊C8×C47—22no—
58#100×C47—42no—
59C24×C47C22⋊C412yes—
60C23×D47—21no—
61C2×D4×C47—22no—
62C2×D4×C47—22no—
63C2×D4×C47—21no—
64C2×D4×C47—21no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—44no—
68C22×C4×C47C22⋊C412yes—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C2×C22⋊D47—28no—
73C2×C22⋊D47—84no—
74C22×Dic47—21no—
75C22×Dic47—22no—
76C2×D188—21no—
77C2×D188—42no—
78C2×D188C22⋊C412yes—
79C2×C4×D47—24no—
80C2×C4×D47—42no—
81C2×C4×D47—41no—
82C2×C4×D47—41no—
83C2×Dic94—24no—
84C2×Dic94—82no—
85M16×C47—42no—
86M16×C47—42no—
87C2×C8×C47—42no—
88C4⋊C4×C47—22no—
89C22⋊C4×C47—24no—
90C42×C47C2×D412yes—
91C42×C47C2×D412yes—
92C22⋊Dic47—28no—
93(C2×D47)⋊C4—28no—
94(C2×D47)⋊C4—24no—
95C4⋊Dic47—21no—
96C4⋊Dic47—22no—
97C4⋊Dic47—24no—
98C4⋊Dic47C22⋊C412yes—
99C4⋊Dic47C22⋊C414yes—
100C4×Dic47—28no—
101C2×C47⋊C8—42no—
102C23×C47—22no—
103C23×C47—22no—
104C23×C47—22no—
105C23×C47—42no—
106C23×C47—42no—
107C23×C47C2×C22⋊C411yescharacteristic
108C22×D47—22no—
109C22×D47—42no—
110C22×D47—42no—
111C22×D47—41no—
112C22×D47—41no—
113D4×C47—42no—
114D4×C47—42no—
115C2×C4×C47—22no—
116C2×C4×C47—22no—
117C2×C4×C47—41no—
118C2×C4×C47—44no—
119C2×C4×C47—44no—
120C2×C4×C47—42no—
121C2×C4×C47C2×C22⋊C411yescharacteristic
122C2×C4×C47C2×C22⋊C411yescharacteristic
123C2×C4×C47#12012yes—
124C2×C4×C47#12012yes—
125C2×C4×C47#13512yes—
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C47#13711yescharacteristic
128C22⋊D47—88no—
129C2×Dic47—22no—
130C2×Dic47—24no—
131C2×Dic47—44no—
132C2×Dic47—41no—
133C2×Dic47—41no—
134D188—24no—
135D188—42no—
136C4×D47—44no—
137C4×D47—44no—
138Dic94—84no—
139C8×C47—82no—
140C47⋊C8—82no—
141#8397—471nomaximal, Sylow
142C22×C47—21no—
143C22×C47—21no—
144C22×C47—42no—
145C22×C47—42no—
146C22×C47—42no—
147C22×C47—82no—
148C22×C47—82no—
149C22×C47#37911yescharacteristic
150C22×C47#41611yescharacteristic
151C22×C47#41711yescharacteristic
152C2×D47—42no—
153C2×D47—42no—
154C2×D47—42no—
155C2×D47—82no—
156C2×D47—82no—
157C4×C47—22no—
158C4×C47—21no—
159C4×C47—42no—
160C4×C47#41312yes—
161C4×C47#44912yes—
162C4×C47#45212yes—
163Dic47—44no—
164Dic47—42no—
165C2×#555—471no—
166#1800D4711yescharacteristic, complemented
167#1549—472no—
168#1486—472no—
169#1460—471no—
170C2×C47—21no—
171C2×C47—21no—
172C2×C47—82no—
173C2×C47#178511yescharacteristic
174C2×C47#178811yescharacteristic
175C2×C47#179511yescharacteristic
176D47—82no—
177#555—944no—
178#555—944no—
179C2×C4⋊Q8—471no—
180C2×#124—942no—
181C2×#121—472no—
182C2×#121—941no—
183C2×C4×D4—471no—
184C2×C4×D4C2×D4711yescharacteristic
185C2×#117—472no—
186C2×#117—941no—
187#431—472no—
188#423C2×D4712yes—
189C2×#100—21no—
190#343—21no—
191#332—944no—
192D4⋊C8—944no—
193#323—942no—
194C47#839711yescharacteristic, complemented, Sylow
195C22×Q8—941no—
196C22×D4—471no—
197C22×D4—942no—
198C22×D4C22⋊D4711yescharacteristic
199C23×C4—941no—
200C23×C4C4×D4712yes—
201C2×M16—22no—
202C4⋊Q8—474no—
203C4⋊Q8—942no—
204#124—9416no—
205#121—944no—
206#121—944no—
207#121—948no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—474no—
211C4×D4—942no—
212#117—948no—
213#117—944no—
214#117—944no—
215C2×C4⋊C4—471no—
216C2×C4⋊C4—472no—
217C2×C4⋊C4—941no—
218C2×C4⋊C4C22⋊D4711yescharacteristic
219C2×C22⋊C4—942no—
220C2×C22⋊C4—942no—
221C2×C22⋊C4—941no—
222C2×C22⋊C4D18812yes—
223C2×C42—942no—
224C2×C42C22×D4711yescharacteristic
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—942no—
228#100—42no—
229C22⋊C8—942no—
230C4×C8—942no—
231C24—941no—
232C24(C2×D47)⋊C412yes—
233C2×Q8—944no—
234C2×Q8—3762no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—472no—
240C2×D4—948no—
241C2×D4—941no—
242C2×D4—3764no—
243C2×D4—1882no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—44no—
247C22×C4—941no—
248C22×C4—942no—
249C22×C4—944no—
250C22×C4—1882no—
251C22×C4—1881no—
252C22×C4—1881no—
253C22×C4(C2×D47)⋊C412yes—
254C22×C4C2×C4×D4711yescharacteristic
255C22×C4C2×D18811yescharacteristic
256C22×C4C2×C22⋊D4711yescharacteristic
257M16—42no—
258M16—42no—
259C2×C8—42no—
260C2×C8—1882no—
261C4⋊C4—22no—
262C4⋊C4—472no—
263C4⋊C4—474no—
264C4⋊C4—941no—
265C4⋊C4—942no—
266C4⋊C4—944no—
267C22⋊C4—24no—
268C22⋊C4—948no—
269C22⋊C4—948no—
270C22⋊C4—944no—
271C42—948no—
272C42D4×D4712yes—
273C42D4×D4712yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—42no—
278C23—42no—
279C23—942no—
280C23—1882no—
281C23—1882no—
282C23—1881no—
283C23—1881no—
284C23C2×(C2×D47)⋊C411yescharacteristic
285Q8—3764no—
286D4—42no—
287D4—42no—
288D4—944no—
289D4—3768no—
290D4—1882no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—44no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—942no—
298C2×C4—944no—
299C2×C4—1884no—
300C2×C4—1881no—
301C2×C4—1881no—
302C2×C4—1884no—
303C2×C4—1884no—
304C2×C4C22⋊C4×D4711yescharacteristic
305C2×C4C22⋊C4×D4711yescharacteristic
306C2×C4#1309399212yes—
307C2×C4#1309399412yes—
308C2×C4#1309404412yes—
309C2×C4#1309405011yescharacteristic
310C2×C4#1309405511yescharacteristic
311C8—82no—
312C8—3762no—
313C22—21no—
314C22—21no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—82no—
320C22—3762no—
321C22—3762no—
322C22—1882no—
323C22—1882no—
324C22—1882no—
325C22#1568897811yescharacteristic
326C22#1568911711yescharacteristic
327C22#1568912211yescharacteristic
328C4—21no—
329C4—22no—
330C4—42no—
331C4—1882no—
332C4—1884no—
333C4#1568910412yes—
334C4#1568926612yes—
335C4#1568928312yes—
336C2—21no—
337C2—21no—
338C2—82no—
339C2—3762no—
340C2#2024331311yescharacteristic
341C2#2024332611yescharacteristic
342C2#2024335911yescharacteristic
subgroupG/H
centerC22#15688978
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1800C2
socleC22×C47#379
prime radicalC2×#15689838C2
cosocleC2×C42×C47C23
hypercenter#1800D47
hypocenterC47#8397
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
2. #20242074C2semidirect
3. #20241812C2semidirect
4. #20241709C2semidirect
5. C47×#1800C2semidirect
166. #1800D47semidirect
194. C47#8397semidirect

Maximal subgroups

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

subgroupindex
C2×#156898382
#202420742
#202418122
#202417092
C47×#18002
#839747