Finite Group Databasebeta

← #26119893

Subgroups — #26119893

Order 12032.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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×#15689333C211yesmaximal, characteristic
2#20242160C211yesmaximal, characteristic
3#20242061C211yesmaximal, characteristic
4#20242061C211yesmaximal, characteristic
5#20241783C211yesmaximal, characteristic
6#20241283C211yesmaximal, characteristic
7C47×#1801C211yesmaximal, characteristic
8C22×#10625337—21no—
9C2×#13094024C2211yescharacteristic
10#15689333—22no—
11#15689333—22no—
12#15689176C2211yescharacteristic
13#15689176C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093947—21no—
16#15688873—22no—
17#15688870—22no—
18#15688870—22no—
19Q8⋊(C47⋊C8)—22no—
20C2×C4×Q8×C47C2211yescharacteristic
21C47×#423C2211yescharacteristic
22C47×#423C2211yescharacteristic
23C2×#101×C47—21no—
24C47×#343—21no—
25C22×Dic94—21no—
26C2×#10625337—24no—
27C2×#10625337—42no—
28C2×#10625337—42no—
29C22×C47⋊C8—21no—
30#13094024—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024D412yes—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47—21no—
36C4⋊(C47⋊C8)—21no—
37C4⋊(C47⋊C8)—21no—
38C4⋊(C47⋊C8)D412yes—
39C4⋊(C47⋊C8)D412yes—
40#13093947—24no—
41C22⋊(C47⋊C8)—21no—
42C22⋊(C47⋊C8)—21no—
43C4×C47⋊C8—21no—
44C4×C47⋊C8—21no—
45C22×Q8×C47D411yescharacteristic
46C2×M16×C47—21no—
47C2×M16×C47—21no—
48C4×Q8×C47—21no—
49C4×Q8×C47—21no—
50C4×Q8×C47—21no—
51C4×Q8×C47—21no—
52C2×C4⋊C4×C47—21no—
53C2×C4⋊C4×C47D411yescharacteristic
54C2×C42×C47—21no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60#101×C47—42no—
61C2×Q8×C47—21no—
62C2×Q8×C47—21no—
63C2×Q8×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—21no—
66C2×Q8×C47SD1612yes—
67C22×C4×C47—21no—
68C22×C4×C47—21no—
69C22×C4×C47C2×D411yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—41no—
73C2×Dic94—22no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94—41no—
79M16×C47—41no—
80M16×C47—41no—
81M16×C47—41no—
82M16×C47—41no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C4⋊C4×C47—21no—
86C4⋊C4×C47—21no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89C42×C47—41no—
90C42×C47—41no—
91C42×C47C2×D411yescharacteristic
92C42×C47C2×D411yescharacteristic
93C42×C47C2×D411yescharacteristic
94C42×C47#4111yescharacteristic
95#10625337—48no—
96C4⋊Dic47—22no—
97C4⋊Dic47—22no—
98C4⋊Dic47—41no—
99C4⋊Dic47—41no—
100C2×C47⋊C8—22no—
101C2×C47⋊C8—22no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C23×C47#12311yescharacteristic
106Q8×C47—22no—
107Q8×C47—22no—
108Q8×C47—41no—
109Q8×C47—41no—
110C2×C4×C47—22no—
111C2×C4×C47—22no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—81no—
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12111yescharacteristic
119C2×C4×C47#12111yescharacteristic
120C2×C4×C47#12111yescharacteristic
121C2×C4×C47#12311yescharacteristic
122C2×C4×C47#12411yescharacteristic
123C2×C4×C47C2×SD1611yescharacteristic
124C2×C4×C47#13511yescharacteristic
125C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×Dic47—42no—
128C2×Dic47—81no—
129C2×Dic47—81no—
130Dic94—42no—
131Dic94—42no—
132Dic94—81no—
133Dic94—81no—
134C8×C47—81no—
135C8×C47—81no—
136C47⋊C8—42no—
137C47⋊C8—42no—
138C47⋊C8—81no—
139#10447—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47#39411yescharacteristic
143C22×C47#48111yescharacteristic
144C22×C47#48211yescharacteristic
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—42no—
149C4×C47—81no—
150C4×C47#44911yescharacteristic
151C4×C47#45011yescharacteristic
152C4×C47#46511yescharacteristic
153C4×C47#46911yescharacteristic
154C4×C47#47312yes—
155Dic47—82no—
156C2×#462—471no—
157#1801D4711yescharacteristic
158#1570—471no—
159#1546—471no—
160#1546—471no—
161#1479—471no—
162#1301—471no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47#203411yescharacteristic
166C2×C47#204711yescharacteristic
167C2×C47#204911yescharacteristic
168C22×Q16—941no—
169C2×C4⋊Q8—471no—
170C2×C4×Q8C2×D4711yescharacteristic
171#462—942no—
172#462—942no—
173#431—471no—
174#431—471no—
175#423C2×D4711yescharacteristic
176#423C2×D4711yescharacteristic
177C2×C4⋊C8—471no—
178C2×#103—941no—
179C2×#101—21no—
180#343—21no—
181#334—942no—
182#332—942no—
183#332—942no—
184Q8⋊C8—942no—
185C47#1044711yescharacteristic, complemented, Sylow
186C22×Q8—941no—
187C22×Q8C22⋊D4711yescharacteristic
188C2×Q16—944no—
189C2×Q16—1882no—
190C2×Q16—1882no—
191C2×M16—21no—
192C2×M16—21no—
193C22×C8—941no—
194C4⋊Q8—472no—
195C4⋊Q8—941no—
196C4⋊Q8—941no—
197C4⋊Q8—941no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4C22⋊D4711yescharacteristic
206C2×C42—21no—
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—472no—
213C4⋊C8—472no—
214C4⋊C8—941no—
215C4⋊C8—941no—
216#103—944no—
217#101—42no—
218C22⋊C8—941no—
219C22⋊C8—941no—
220C4×C8—941no—
221C4×C8—941no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—942no—
228C2×Q8—1881no—
229C2×Q8—1881no—
230C2×Q8—1881no—
231C2×Q8—1881no—
232C2×Q8—1881no—
233C2×Q8#1062533612yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—1881no—
237C22×C4#1062535011yescharacteristic
238C22×C4#1062535011yescharacteristic
239C22×C4C2×C22⋊D4711yescharacteristic
240Q16—1888no—
241M16—41no—
242M16—41no—
243M16—41no—
244M16—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—942no—
248C2×C8—942no—
249C2×C8—1881no—
250C2×C8—1881no—
251C2×C8—1881no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—942no—
257C4⋊C4—942no—
258C4⋊C4—1881no—
259C4⋊C4—1881no—
260C42—41no—
261C42—41no—
262C42D4×D4711yescharacteristic
263C42D4×D4711yescharacteristic
264C42D4×D4711yescharacteristic
265C42#1062535211yescharacteristic
266C23#1309400711yescharacteristic
267Q8—22no—
268Q8—22no—
269Q8—41no—
270Q8—41no—
271Q8—3761no—
272Q8—3761no—
273Q8—1882no—
274Q8—1882no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—3761no—
283C2×C4—3761no—
284C2×C4—1882no—
285C2×C4#1309399211yescharacteristic
286C2×C4#1309399611yescharacteristic
287C2×C4#1309399711yescharacteristic
288C2×C4#1309399711yescharacteristic
289C2×C4#1309400611yescharacteristic
290C2×C4#1309401011yescharacteristic
291C2×C4C2×#1062533611yescharacteristic
292C2×C4#1309404411yescharacteristic
293C2×C4#1309404911yescharacteristic
294C2×C4#1309405011yescharacteristic
295C8—81no—
296C8—81no—
297C8—3761no—
298C8—1882no—
299C8—1882no—
300C22—21no—
301C22—21no—
302C22#1568903811yescharacteristic
303C22#1568944111yescharacteristic
304C22#1568944711yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—81no—
310C4—3762no—
311C4#1568926611yescharacteristic
312C4#1568927311yescharacteristic
313C4#1568935411yescharacteristic
314C4#1568937411yescharacteristic
315C4#1568939712yes—
316C2—21no—
317C2—21no—
318C2#2024457711yescharacteristic
319C2#2024465411yescharacteristic
320C2#2024466811yescharacteristic
subgroupG/H
centerC22#15689038
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1801C2
socleC22×C47#394
prime radicalC23×C47#123
hypercenter#1801D47
hypocenterC47#10447
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
185. C47#10447semidirect

Maximal subgroups

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

subgroupindex
C2×#156893332
#202421602
#202420612
#202417832
#202412832
C47×#18012
#1044747