Finite Group Databasebeta

← #26120062

Subgroups — #26120062

Order 12032.

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

normal subgroups only

This group has 366 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
1#20253193C211yesmaximal, characteristic
2#20243403C211yesmaximal, characteristic
3#20242163C211yesmaximal, characteristic
4#20242043C211yesmaximal, characteristic
5#20241953C211yesmaximal, characteristic
6#20241247C211yesmaximal, characteristic
7C47×#1461C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9#15689430—22no—
10#15689418—21no—
11#15689408—21no—
12#15689163—21no—
13#15689147C2211yescharacteristic
14C2×C4⋊(C47⋊C8)C2211yescharacteristic
15#15689132—21no—
16C2×#13093948—21no—
17#15688989C2211yescharacteristic
18#15688885—21no—
19#15688876—22no—
20#15688871—21no—
21#15688870—21no—
22#15688861—22no—
23C2×C4×Q8×C47C2211yescharacteristic
24C47×#431C2211yescharacteristic
25C47×#431C2211yescharacteristic
26C47×#324—22no—
27C2×#10625329—21no—
28C22×C47⋊C8—21no—
29#13094024D411yescharacteristic
30#13094024D411yescharacteristic
31#13094016—21no—
32#13094003—21no—
33#13094003—21no—
34C4×Dic94—21no—
35C4×Dic94—21no—
36#13093983—21no—
37C2×C4⋊Dic47—21no—
38#13093957—22no—
39#13093955—22no—
40C4⋊(C47⋊C8)—21no—
41C4⋊(C47⋊C8)—21no—
42C4⋊(C47⋊C8)—21no—
43C4⋊(C47⋊C8)D412yes—
44#13093948—22no—
45#13093948—22no—
46#13093948—42no—
47C47⋊#95—21no—
48C47⋊#95—21no—
49C22×Q8×C47D411yescharacteristic
50C4×Q8×C47—21no—
51C4×Q8×C47—21no—
52C4×Q8×C47—21no—
53C4×Q8×C47—21no—
54C2×C4⋊C4×C47—21no—
55C2×C4⋊C4×C47D411yescharacteristic
56C2×C42×C47—21no—
57C2×C42×C47C2311yescharacteristic
58C4⋊C8×C47—21no—
59C4⋊C8×C47—21no—
60C22⋊C8×C47—21no—
61C22⋊C8×C47D412yes—
62C4×C8×C47—21no—
63C4×C8×C47—21no—
64C2×Q8×C47—21no—
65C2×Q8×C47—21no—
66C2×Q8×C47—21no—
67C2×Q8×C47—21no—
68C2×Q8×C47—21no—
69C2×Q8×C47Q1612yes—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47C2×D411yescharacteristic
73C22×C4×C47C2×D411yescharacteristic
74C22×C4×C47#4111yescharacteristic
75C22×Dic47—41no—
76C2×Dic94—21no—
77C2×Dic94—21no—
78C2×C8×C47—22no—
79C2×C8×C47—41no—
80C4⋊C4×C47—21no—
81C4⋊C4×C47—21no—
82C4⋊C4×C47—41no—
83C4⋊C4×C47—41no—
84C42×C47—41no—
85C42×C47—41no—
86C42×C47C2×D411yescharacteristic
87C42×C47C2×D411yescharacteristic
88C42×C47#4111yescharacteristic
89C42×C47#4111yescharacteristic
90C22⋊Dic47—41no—
91C4⋊Dic47—21no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95C4⋊Dic47—41no—
96C4⋊Dic47—41no—
97Dic47⋊C4—41no—
98Dic47⋊C4—41no—
99C4×Dic47—41no—
100#10625329—41no—
101#10625329—41no—
102C2×C47⋊C8—22no—
103C2×C47⋊C8—41no—
104C2×C47⋊C8—41no—
105C2×C47⋊C8—41no—
106C23×C47#12111yescharacteristic
107Q8×C47—22no—
108Q8×C47—22no—
109Q8×C47—41no—
110Q8×C47—41no—
111C2×C4×C47—22no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—81no—
118C2×C4×C47#12011yescharacteristic
119C2×C4×C47#12111yescharacteristic
120C2×C4×C47#12111yescharacteristic
121C2×C4×C47#12311yescharacteristic
122C2×C4×C47#12311yescharacteristic
123C2×C4×C47#12411yescharacteristic
124C2×C4×C47C2×Q1611yescharacteristic
125C2×C4×C47#13511yescharacteristic
126C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
127C2×C4×C47#13711yescharacteristic
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—81no—
131C2×Dic47—81no—
132C2×Dic47—81no—
133Dic94—81no—
134Dic94—81no—
135C8×C47—42no—
136C8×C47—81no—
137C47⋊C8—42no—
138C47⋊C8—81no—
139#10471—471nomaximal, Sylow
140C22×C47—21no—
141C22×C47—21no—
142C22×C47#39411yescharacteristic
143C22×C47#46711yescharacteristic
144C22×C47#46811yescharacteristic
145C4×C47—21no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—21no—
149C4×C47—42no—
150C4×C47—81no—
151C4×C47#44911yescharacteristic
152C4×C47#45111yescharacteristic
153C4×C47#48411yescharacteristic
154C4×C47#48511yescharacteristic
155Dic47—81no—
156Dic47—81no—
157Dic47—161no—
158#3078—471no—
159#1805—471no—
160#1571—471no—
161#1542—471no—
162#1518—471no—
163#1461D4711yescharacteristic
164#1286—471no—
165C2×C47—21no—
166C2×C47—21no—
167C2×C47#203611yescharacteristic
168C2×C47#204911yescharacteristic
169C2×C47#205311yescharacteristic
170#533—471no—
171C2×C4×Q8C2×D4711yescharacteristic
172#479—942no—
173#477—941no—
174#475—941no—
175#431C2×D4711yescharacteristic
176#431C2×D4711yescharacteristic
177#428—941no—
178#423—471no—
179C2×C4⋊C8—471no—
180#419—941no—
181C2×#103—941no—
182#382—471no—
183#340—941no—
184#335—942no—
185#333—941no—
186#332—941no—
187#328—942no—
188#324—22no—
189C47#1047111yescharacteristic, complemented, Sylow
190C22×Q8D18811yescharacteristic
191C2×M16—941no—
192C22×C8—941no—
193C4⋊Q8—471no—
194C4⋊Q8—471no—
195#125—941no—
196C22⋊Q8—941no—
197C22⋊Q8—941no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—941no—
203C4×Q8—941no—
204#117—941no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—941no—
207C2×C4⋊C4D18811yescharacteristic
208C2×C42—21no—
209C2×C42C22×D4711yescharacteristic
210#107—942no—
211#106—942no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—472no—
215C4⋊C8—941no—
216C4⋊C8—941no—
217C4⋊C8—941no—
218#103—942no—
219#103—942no—
220#103—1882no—
221C22⋊C8—21no—
222C22⋊C8C22⋊D4712yes—
223C4×C8—21no—
224C4×C8—21no—
225#95—941no—
226#95—941no—
227C2×Q8—21no—
228C2×Q8—21no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×Q8—941no—
233C2×Q8—941no—
234C2×Q8Dic18812yes—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—1881no—
238C22×C4C2×D18811yescharacteristic
239C22×C4#1062535011yescharacteristic
240C22×C4C2×C22⋊D4711yescharacteristic
241M16—1881no—
242M16—1881no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—942no—
246C2×C8—1881no—
247C2×C8—1881no—
248C2×C8—1881no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—941no—
254C4⋊C4—941no—
255C4⋊C4—941no—
256C4⋊C4—941no—
257C4⋊C4—1881no—
258C4⋊C4—1881no—
259C4⋊C4—1881no—
260C4⋊C4—1881no—
261C22⋊C4—1881no—
262C42—41no—
263C42—41no—
264C42—1881no—
265C42D4×D4711yescharacteristic
266C42D4×D4711yescharacteristic
267C42#1062535211yescharacteristic
268C42#1062535211yescharacteristic
269C23#1309399811yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—41no—
273Q8—41no—
274Q8—3761no—
275Q8—3761no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—3761no—
284C2×C4—3761no—
285C2×C4—3761no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4#1309399411yescharacteristic
289C2×C4#1309399511yescharacteristic
290C2×C4#1309399611yescharacteristic
291C2×C4#1309400611yescharacteristic
292C2×C4#1309400811yescharacteristic
293C2×C4#1309401011yescharacteristic
294C2×C4C2×Dic18811yescharacteristic
295C2×C4#1309404611yescharacteristic
296C2×C4#1309405211yescharacteristic
297C2×C4#1309405711yescharacteristic
298C8—42no—
299C8—81no—
300C8—3761no—
301C8—1882no—
302C22—21no—
303C22—21no—
304C22#1568903911yescharacteristic
305C22#1568936411yescharacteristic
306C22#1568936911yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—81no—
313C4—3761no—
314C4—3761no—
315C4—7521no—
316C4#1568927011yescharacteristic
317C4#1568928111yescharacteristic
318C4#1568945911yescharacteristic
319C4#1568946511yescharacteristic
320C2—21no—
321C2—21no—
322C2#2024459211yescharacteristic
323C2#2024466711yescharacteristic
324C2#2024469311yescharacteristic
subgroupG/H
centerC22#15689039
commutator, cosocleC2×C42×C47C23
FrattiniC2×C42C22×D47
FittingC47×#1461C2
socleC22×C47#394
prime radicalC23×C47#121
hypercenter#1461D47
hypocenterC47#10471
LCSG ▷ C2×C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
189. C47#10471semidirect

Maximal subgroups

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

subgroupindex
#202531932
#202434032
#202421632
#202420432
#202419532
#202412472
C47×#14612
#1047147