Finite Group Databasebeta

← #26119896

Subgroups — #26119896

Order 12032.

408 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 95 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×#15689334C211yesmaximal, characteristic
2#20243381C211yesmaximal, characteristic
3#20242158C211yesmaximal, characteristic
4#20242061C211yesmaximal, characteristic
5#20242059C211yesmaximal, characteristic
6#20241784C211yesmaximal, characteristic
7C47×#1301C211yesmaximal, characteristic
8C22×Dic188—21no—
9C2×#13094024C2211yescharacteristic
10C2×C4×Dic94C2211yescharacteristic
11#15689334—22no—
12#15689334—22no—
13#15689176C2211yescharacteristic
14#15689147C2211yescharacteristic
15C2×#13093949—21no—
16C2×#13093942—21no—
17#15688891—21no—
18#15688873—22no—
19#15688870—22no—
20#15688868—22no—
21#15688855—22no—
22C47×#431C2211yescharacteristic
23C47×#423C2211yescharacteristic
24C2×C4⋊C8×C47C2211yescharacteristic
25C22×Dic94—21no—
26C22×Dic94D411yescharacteristic
27C2×Dic188—24no—
28C2×Dic188—42no—
29C2×Dic188—42no—
30C2×#10625329—21no—
31#13094024—21no—
32#13094024—21no—
33#13094024—21no—
34#13094024D412yes—
35C4×Dic94—21no—
36C4×Dic94—21no—
37C4×Dic94—21no—
38C4×Dic94—21no—
39C2×C4⋊Dic47—21no—
40C2×C4⋊Dic47—21no—
41C2×C4⋊Dic47D411yescharacteristic
42C2×Dic47⋊C4—21no—
43C2×C4×Dic47—21no—
44C4⋊(C47⋊C8)—21no—
45C4⋊(C47⋊C8)—21no—
46C4⋊(C47⋊C8)D412yes—
47#13093949—24no—
48#13093942—42no—
49C22⋊(C47⋊C8)—21no—
50C4×C47⋊C8—21no—
51C2×M16×C47—21no—
52C22×C8×C47—21no—
53C2×C42×C47C2311yescharacteristic
54C4⋊C8×C47—21no—
55C4⋊C8×C47—21no—
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47D412yes—
59C22⋊C8×C47—21no—
60C4×C8×C47—21no—
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47#4111yescharacteristic
63C22×C4×C47#4111yescharacteristic
64C22×Dic47—21no—
65C22×Dic47—21no—
66C22×Dic47—41no—
67C2×Dic94—21no—
68C2×Dic94—21no—
69C2×Dic94—21no—
70C2×Dic94—21no—
71C2×Dic94—21no—
72C2×Dic94—22no—
73C2×Dic94—41no—
74C2×Dic94—41no—
75C2×Dic94—41no—
76C2×Dic94—41no—
77C2×Dic94—41no—
78C2×Dic94SD1612yes—
79M16×C47—41no—
80M16×C47—41no—
81C2×C8×C47—22no—
82C2×C8×C47—41no—
83C2×C8×C47—41no—
84C2×C8×C47—41no—
85C2×C8×C47—41no—
86C42×C47C2×D411yescharacteristic
87C42×C47C2×D411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47#4111yescharacteristic
90C4⋊Dic47—21no—
91C4⋊Dic47—22no—
92C4⋊Dic47—21no—
93C4⋊Dic47—22no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96Dic47⋊C4—41no—
97Dic47⋊C4—41no—
98C4×Dic47—41no—
99C4×Dic47—41no—
100#10625329—41no—
101#10625329—41no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—41no—
104Dic188—48no—
105C23×C47#12311yescharacteristic
106C2×C4×C47—22no—
107C2×C4×C47—41no—
108C2×C4×C47#12011yescharacteristic
109C2×C4×C47#12111yescharacteristic
110C2×C4×C47#12111yescharacteristic
111C2×C4×C47#12111yescharacteristic
112C2×C4×C47#12311yescharacteristic
113C2×C4×C47#12411yescharacteristic
114C2×C4×C47C2×SD1611yescharacteristic
115C2×C4×C47#13511yescharacteristic
116C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
117C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
118C2×Dic47—22no—
119C2×Dic47—41no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—42no—
123C2×Dic47—81no—
124C2×Dic47—81no—
125C2×Dic47—81no—
126Dic94—22no—
127Dic94—22no—
128Dic94—41no—
129Dic94—41no—
130Dic94—42no—
131Dic94—42no—
132Dic94—81no—
133Dic94—81no—
134C8×C47—42no—
135C8×C47—81no—
136C8×C47—81no—
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#44911yescharacteristic
149C4×C47#45011yescharacteristic
150C4×C47#46511yescharacteristic
151C4×C47#46911yescharacteristic
152C4×C47#47312yes—
153Dic47—42no—
154Dic47—81no—
155Dic47—82no—
156C2×#462—471no—
157#1801—471no—
158#1570—471no—
159#1546—471no—
160#1546—471no—
161#1479—471no—
162#1301D4711yescharacteristic
163C2×C47—21no—
164C2×C47—21no—
165C2×C47#203411yescharacteristic
166C2×C47#204711yescharacteristic
167C2×C47#204911yescharacteristic
168C22×Q16—941no—
169C2×C4⋊Q8—471no—
170C2×C4×Q8—471no—
171#462—942no—
172#462—942no—
173#431—471no—
174#431C2×D4711yescharacteristic
175#423—471no—
176#423C2×D4711yescharacteristic
177C2×C4⋊C8C2×D4711yescharacteristic
178C2×#103—941no—
179C2×#101—941no—
180#343—941no—
181#334—942no—
182#332—942no—
183#332—942no—
184Q8⋊C8—942no—
185C47#1044711yescharacteristic, complemented, Sylow
186C22×Q8—471no—
187C22×Q8—941no—
188C2×Q16—944no—
189C2×Q16—1882no—
190C2×Q16—1882no—
191C2×M16—21no—
192C2×M16—941no—
193C22×C8—21no—
194C4⋊Q8—472no—
195C4⋊Q8—941no—
196C4⋊Q8—941no—
197C4⋊Q8—941no—
198C4×Q8—941no—
199C4×Q8—941no—
200C4×Q8—941no—
201C4×Q8—941no—
202C2×C4⋊C4—471no—
203C2×C4⋊C4—941no—
204C2×C4⋊C4—941no—
205C2×C4⋊C4—941no—
206C2×C42—941no—
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—472no—
213C4⋊C8—941no—
214C4⋊C8—941no—
215C4⋊C8D18812yes—
216#103—944no—
217#101—1882no—
218C22⋊C8—21no—
219C22⋊C8—941no—
220C4×C8—21no—
221C4×C8—941no—
222C2×Q8—472no—
223C2×Q8—941no—
224C2×Q8—941no—
225C2×Q8—942no—
226C2×Q8—941no—
227C2×Q8—941no—
228C2×Q8—941no—
229C2×Q8—1881no—
230C2×Q8—1881no—
231C2×Q8—1881no—
232C2×Q8—1881no—
233C2×Q8—1881no—
234C22×C4—941no—
235C22×C4—941no—
236C22×C4—1881no—
237C22×C4#1062535011yescharacteristic
238C22×C4D4×D4711yescharacteristic
239C22×C4#1062535411yescharacteristic
240Q16—1888no—
241M16—41no—
242M16—41no—
243M16—1881no—
244M16—1881no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—942no—
251C2×C8—1881no—
252C4⋊C4—941no—
253C4⋊C4—941no—
254C4⋊C4—942no—
255C4⋊C4—942no—
256C4⋊C4—1881no—
257C4⋊C4—1881no—
258C4⋊C4—1881no—
259C4⋊C4—1881no—
260C42—1881no—
261C42—1881no—
262C42C2×D18811yescharacteristic
263C42#1062535011yescharacteristic
264C42D4×D4711yescharacteristic
265C42D4×D4711yescharacteristic
266C23#1309400911yescharacteristic
267Q8—942no—
268Q8—942no—
269Q8—3761no—
270Q8—3761no—
271Q8—1882no—
272Q8—1882no—
273Q8—1881no—
274Q8—1881no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—942no—
278C2×C4—3761no—
279C2×C4—3761no—
280C2×C4—3761no—
281C2×C4—1881no—
282C2×C4—1882no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4#1309399411yescharacteristic
286C2×C4#1309399711yescharacteristic
287C2×C4#1309399711yescharacteristic
288C2×C4#1309399911yescharacteristic
289C2×C4#1309400911yescharacteristic
290C2×C4#1309401211yescharacteristic
291C2×C4SD16×D4711yescharacteristic
292C2×C4#1309404711yescharacteristic
293C2×C4#1309404811yescharacteristic
294C2×C4#1309405111yescharacteristic
295C8—42no—
296C8—81no—
297C8—81no—
298C8—3761no—
299C8—1882no—
300C22—21no—
301C22—21no—
302C22#1568904111yescharacteristic
303C22#1568944311yescharacteristic
304C22#1568945011yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—3762no—
309C4—3761no—
310C4—1882no—
311C4#1568927111yescharacteristic
312C4#1568927511yescharacteristic
313C4#1568935511yescharacteristic
314C4#1568937511yescharacteristic
315C4#1568939912yes—
316C2—21no—
317C2—21no—
318C2#2024458011yescharacteristic
319C2#2024465711yescharacteristic
320C2#2024467111yescharacteristic
subgroupG/H
centerC22#15689041
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1301C2
socleC22×C47#394
prime radicalC23×C47#123
hypercenter#1301D47
hypocenterC47#10447
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1301

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×#156893342
#202433812
#202421582
#202420612
#202420592
#202417842
C47×#13012
#1044747