Finite Group Databasebeta

← #26119892

Subgroups — #26119892

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×#15689330C211yesmaximal, characteristic
2#20243381C211yesmaximal, characteristic
3#20242159C211yesmaximal, characteristic
4#20242060C211yesmaximal, characteristic
5#20241782C211yesmaximal, characteristic
6#20241283C211yesmaximal, characteristic
7C47×#1546C211yesmaximal, characteristic
8C22×#10625337—21no—
9C2×C4×Dic94C2211yescharacteristic
10#15689330—22no—
11#15689330—22no—
12#15689176C2211yescharacteristic
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15C2×#13093948—21no—
16C2×#13093942—21no—
17#15688891—21no—
18#15688874—22no—
19#15688869—22no—
20#15688853—22no—
21C2×C4⋊Q8×C47C2211yescharacteristic
22C47×#431C2211yescharacteristic
23C47×#423C2211yescharacteristic
24C47×#332—22no—
25C22×Dic94D411yescharacteristic
26C2×#10625337—24no—
27C2×#10625337—42no—
28C2×#10625337—42no—
29C2×#10625329—21no—
30C22×C47⋊C8—21no—
31C4×Dic94—21no—
32C4×Dic94—21no—
33C4×Dic94—21no—
34C4×Dic94—21no—
35C2×C4⋊Dic47D411yescharacteristic
36C2×Dic47⋊C4—21no—
37C2×C4×Dic47—21no—
38C4⋊(C47⋊C8)—21no—
39C4⋊(C47⋊C8)—21no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)D412yes—
43#13093948—24no—
44#13093942—42no—
45C22⋊(C47⋊C8)—21no—
46C4×C47⋊C8—21no—
47C22×Q8×C47—21no—
48C2×M16×C47—21no—
49C4⋊Q8×C47—21no—
50C4⋊Q8×C47—21no—
51C4⋊Q8×C47—21no—
52C4⋊Q8×C47D412yes—
53C2×C4⋊C4×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C42×C47C2311yescharacteristic
56C4⋊C8×C47—21no—
57C4⋊C8×C47—21no—
58C4⋊C8×C47D412yes—
59C22⋊C8×C47—21no—
60C4×C8×C47—21no—
61C2×Q8×C47—22no—
62C2×Q8×C47—41no—
63C2×Q8×C47—41no—
64C2×Q8×C47—41no—
65C2×Q8×C47—41no—
66C2×Q8×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47C2×D411yescharacteristic
69C22×C4×C47#4111yescharacteristic
70C22×C4×C47#4111yescharacteristic
71C22×Dic47—21no—
72C22×Dic47—21no—
73C2×Dic94—21no—
74C2×Dic94—21no—
75C2×Dic94—21no—
76C2×Dic94—21no—
77C2×Dic94—21no—
78C2×Dic94SD1612yes—
79M16×C47—41no—
80M16×C47—41no—
81C2×C8×C47—22no—
82C2×C8×C47—41no—
83C4⋊C4×C47—22no—
84C4⋊C4×C47—22no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C42×C47C2×D411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91#10625337—48no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94Dic47⋊C4—41no—
95Dic47⋊C4—41no—
96C4×Dic47—41no—
97C4×Dic47—41no—
98#10625329—41no—
99#10625329—41no—
100C2×C47⋊C8—22no—
101C2×C47⋊C8—41no—
102C2×C47⋊C8—41no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C23×C47#12311yescharacteristic
106Q8×C47—42no—
107Q8×C47—42no—
108Q8×C47—81no—
109Q8×C47—81no—
110C2×C4×C47—22no—
111C2×C4×C47—41no—
112C2×C4×C47—42no—
113C2×C4×C47—81no—
114C2×C4×C47—81no—
115C2×C4×C47#12011yescharacteristic
116C2×C4×C47#12111yescharacteristic
117C2×C4×C47#12111yescharacteristic
118C2×C4×C47#12111yescharacteristic
119C2×C4×C47#12311yescharacteristic
120C2×C4×C47#12411yescharacteristic
121C2×C4×C47C2×SD1611yescharacteristic
122C2×C4×C47#13511yescharacteristic
123C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
125C2×Dic47—22no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—81no—
130Dic94—22no—
131Dic94—22no—
132Dic94—41no—
133Dic94—41no—
134C8×C47—42no—
135C8×C47—81no—
136C47⋊C8—42no—
137C47⋊C8—81no—
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—82no—
149C4×C47#44911yescharacteristic
150C4×C47#45011yescharacteristic
151C4×C47#46511yescharacteristic
152C4×C47#46911yescharacteristic
153C4×C47#47312yes—
154Dic47—42no—
155Dic47—81no—
156C2×#462—471no—
157#1801—471no—
158#1570—471no—
159#1546—471no—
160#1546D4711yescharacteristic
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⋊Q8C2×D4711yescharacteristic
170C2×C4×Q8—471no—
171#462—942no—
172#462—942no—
173#431—471no—
174#431C2×D4711yescharacteristic
175#423—471no—
176#423C2×D4711yescharacteristic
177C2×C4⋊C8—471no—
178C2×#103—941no—
179C2×#101—941no—
180#343—941no—
181#334—942no—
182#332—22no—
183#332—942no—
184Q8⋊C8—942no—
185C47#1044711yescharacteristic, complemented, Sylow
186C22×Q8—21no—
187C22×Q8—471no—
188C2×Q16—944no—
189C2×Q16—1882no—
190C2×Q16—1882no—
191C2×M16—21no—
192C2×M16—941no—
193C22×C8—941no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8D18812yes—
198C4×Q8—941no—
199C4×Q8—941no—
200C4×Q8—941no—
201C4×Q8—941no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—471no—
205C2×C4⋊C4—941no—
206C2×C42—941no—
207C2×C42C22×D4711yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—472no—
211C4⋊C8—941no—
212C4⋊C8—941no—
213C4⋊C8—941no—
214C4⋊C8—941no—
215C4⋊C8C22⋊D4712yes—
216#103—944no—
217#101—1882no—
218C22⋊C8—21no—
219C22⋊C8—941no—
220C4×C8—21no—
221C4×C8—941no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—472no—
229C2×Q8—941no—
230C2×Q8—941no—
231C2×Q8—941no—
232C2×Q8—941no—
233C2×Q8—941no—
234C22×C4—41no—
235C22×C4—941no—
236C22×C4—941no—
237C22×C4#1062535011yescharacteristic
238C22×C4D4×D4711yescharacteristic
239C22×C4#1062535211yescharacteristic
240Q16—1888no—
241M16—41no—
242M16—41no—
243M16—1881no—
244M16—1881no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—942no—
248C2×C8—1881no—
249C2×C8—1881no—
250C2×C8—1881no—
251C2×C8—1881no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—941no—
257C4⋊C4—941no—
258C4⋊C4—1881no—
259C4⋊C4—1881no—
260C42—1881no—
261C42—1881no—
262C42C2×D18811yescharacteristic
263C42D4×D4711yescharacteristic
264C42#1062535211yescharacteristic
265C42C2×C22⋊D4711yescharacteristic
266C23#1309400611yescharacteristic
267Q8—42no—
268Q8—42no—
269Q8—81no—
270Q8—81no—
271Q8—942no—
272Q8—942no—
273Q8—1881no—
274Q8—1881no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—942no—
281C2×C4—3761no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4#1309399411yescharacteristic
286C2×C4#1309399511yescharacteristic
287C2×C4#1309399611yescharacteristic
288C2×C4#1309399811yescharacteristic
289C2×C4#1309400811yescharacteristic
290C2×C4#1309401011yescharacteristic
291C2×C4SD16×D4711yescharacteristic
292C2×C4#1309404511yescharacteristic
293C2×C4#1309404811yescharacteristic
294C2×C4#1309404811yescharacteristic
295C8—42no—
296C8—81no—
297C8—3761no—
298C8—3761no—
299C8—1882no—
300C22—21no—
301C22—21no—
302C22#1568903911yescharacteristic
303C22#1568944011yescharacteristic
304C22#1568944611yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—82no—
309C4—3761no—
310C4—1882no—
311C4#1568926911yescharacteristic
312C4#1568927511yescharacteristic
313C4#1568934911yescharacteristic
314C4#1568937311yescharacteristic
315C4#1568939612yes—
316C2—21no—
317C2—21no—
318C2#2024457611yescharacteristic
319C2#2024465511yescharacteristic
320C2#2024466711yescharacteristic
subgroupG/H
centerC22#15689039
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1546C2
socleC22×C47#394
prime radicalC23×C47#123
hypercenter#1546D47
hypocenterC47#10447
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

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×#156893302
#202433812
#202421592
#202420602
#202417822
#202412832
C47×#15462
#1044747