Finite Group Databasebeta

← #26111463

Subgroups — #26111463

Order 12032.

424 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 109 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#20253428C211yesmaximal, characteristic
2#20252328C211yesmaximal, characteristic
3#20244328C211yesmaximal, characteristic
4#20242043C211yesmaximal, characteristic
5#20241818C211yesmaximal, characteristic
6#20241284C211yesmaximal, characteristic
7C47×#1281C211yesmaximal, characteristic
8#15689680C2211yescharacteristic
9C2×C4×Dic94C2211yescharacteristic
10#15689459C412yes—
11#15689419—21no—
12#15689334—21no—
13#15689281C412yes—
14#15689251—22no—
15#15689251—22no—
16C4×Dic188—22no—
17#15689149—21no—
18#15689147C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20#15689078—21no—
21#15688877—21no—
22#15688877—21no—
23#15688871—21no—
24#15688870—21no—
25#15688855—22no—
26C47×#431C2211yescharacteristic
27C47×#431C2211yescharacteristic
28C2×#97×C47C2211yescharacteristic
29C22×Dic94C2×C411yescharacteristic
30C2×Dic188—22no—
31C2×#10625329—21no—
32C22×C47⋊C8—21no—
33#13094024D411yescharacteristic
34#13094024D411yescharacteristic
35#13094016—21no—
36#13094003—21no—
37#13094003C2×C412yes—
38C4×Dic94—21no—
39C4×Dic94—21no—
40C4×Dic94—21no—
41C4×Dic94—21no—
42C4×Dic94—21no—
43C4×Dic94D412yes—
44#13093983—21no—
45C2×C4⋊Dic47C2×C411yescharacteristic
46C2×Dic47⋊C4—21no—
47C2×C4×Dic47—21no—
48#13093958—22no—
49C4⋊(C47⋊C8)—21no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)—21no—
52C4⋊(C47⋊C8)D412yes—
53Dic47⋊C8—24no—
54#13093949—22no—
55#13093949—22no—
56#13093934—22no—
57C8×Dic47—22no—
58C22×C8×C47D411yescharacteristic
59C22×C8×C47D411yescharacteristic
60C2×C42×C47C2311yescharacteristic
61C4⋊C8×C47—21no—
62C4⋊C8×C47—21no—
63C22⋊C8×C47—21no—
64C22⋊C8×C47C2×C412yes—
65#97×C47—21no—
66#97×C47—21no—
67C4×C8×C47—21no—
68C4×C8×C47—21no—
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×C4×C47#4111yescharacteristic
72C22×Dic47—21no—
73C22×Dic47—21no—
74C2×Dic94—21no—
75C2×Dic94—21no—
76C2×Dic94—21no—
77C2×Dic94—42no—
78C2×Dic94C22⋊C412yes—
79C2×C8×C47—21no—
80C2×C8×C47—21no—
81C2×C8×C47—22no—
82C2×C8×C47—21no—
83C2×C8×C47—21no—
84C2×C8×C47—21no—
85C2×C8×C47—41no—
86C2×C8×C47Q1612yescomplemented
87C42×C47C2×D411yescharacteristic
88C42×C47C2×D411yescharacteristic
89C42×C47C2×D411yescharacteristic
90C42×C47#4111yescharacteristic
91C22⋊Dic47—22no—
92C4⋊Dic47—21no—
93C4⋊Dic47—21no—
94C4⋊Dic47—21no—
95C4⋊Dic47—21no—
96C4⋊Dic47—21no—
97C4⋊Dic47C22⋊C412yes—
98Dic47⋊C4—22no—
99Dic47⋊C4—22no—
100Dic47⋊C4—41no—
101Dic47⋊C4—41no—
102C4×Dic47—22no—
103C4×Dic47—41no—
104C4×Dic47—41no—
105#10625329—41no—
106#10625329—41no—
107C2×C47⋊C8—22no—
108C2×C47⋊C8—41no—
109C2×C47⋊C8—41no—
110C2×C47⋊C8—41no—
111Dic188—44no—
112C23×C47C4×D411yescharacteristic
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47C2×C22⋊C411yescharacteristic
116C2×C4×C47C4×D411yescharacteristic
117C2×C4×C47#12011yescharacteristic
118C2×C4×C47#12111yescharacteristic
119C2×C4×C47#12111yescharacteristic
120C2×C4×C47#12311yescharacteristic
121C2×C4×C47C2×Q1611yescharacteristic
122C2×C4×C47#13511yescharacteristic
123C2×C4×C47#13711yescharacteristic
124C2×C4×C47#13711yescharacteristic
125C2×Dic47—22no—
126C2×Dic47—22no—
127C2×Dic47—22no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130C2×Dic47—42no—
131C2×Dic47—41no—
132Dic94—22no—
133Dic94—22no—
134Dic94—41no—
135Dic94—41no—
136Dic94—42no—
137Dic94—81no—
138C8×C47—22no—
139C8×C47—42no—
140C8×C47—41no—
141C8×C47—81no—
142C47⋊C8—42no—
143C47⋊C8—81no—
144#9048—471nomaximal, Sylow
145C22×C47—21no—
146C22×C47—21no—
147C22×C47#38611yescharacteristic
148C22×C47C4×Q1611yescharacteristic
149C22×C47#44111yescharacteristic
150C4×C47—21no—
151C4×C47—21no—
152C4×C47—21no—
153C4×C47—21no—
154C4×C47#45111yescharacteristic
155C4×C47#45211yescharacteristic
156C4×C47#46211yescharacteristic
157C4×C47#46411yescharacteristic
158Dic47—42no—
159Dic47—42no—
160Dic47—42no—
161Dic47—81no—
162#3104—471no—
163#2961—471no—
164#1985—471no—
165#1542—471no—
166#1487—471no—
167#1301—471no—
168#1281D4711yescharacteristic
169C2×C47—21no—
170C2×C47—21no—
171C2×C47#187811yescharacteristic
172C2×C47#188111yescharacteristic
173C2×C47#189111yescharacteristic
174#533—471no—
175C2×C4×Q8—471no—
176#484—472no—
177#477—941no—
178#462—941no—
179#451—472no—
180#446—942no—
181#446—942no—
182C4×Q16—942no—
183#431C2×D4711yescharacteristic
184#431C2×D4711yescharacteristic
185#424—941no—
186#423—471no—
187C2×C4⋊C8—471no—
188#405—941no—
189C2×#97C2×D4711yescharacteristic
190#336—941no—
191#336—941no—
192#333—941no—
193#332—941no—
194Q8⋊C8—942no—
195C47#904811yescharacteristic, complemented, Sylow
196C22×Q8—471no—
197C2×Q16—942no—
198C2×M16—941no—
199C22×C8—941no—
200C22×C8D18811yescharacteristic
201C22×C8D18811yescharacteristic
202C4⋊Q8—471no—
203C4⋊Q8—471no—
204#125—941no—
205C22⋊Q8—472no—
206C22⋊Q8—941no—
207C4×Q8—472no—
208C4×Q8—941no—
209C4×Q8—941no—
210C4×Q8—941no—
211C4×Q8—941no—
212C4×Q8—941no—
213#117—941no—
214C2×C4⋊C4—471no—
215C2×C4⋊C4—941no—
216C2×C42—941no—
217C2×C42C22×D4711yescharacteristic
218#107—942no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—472no—
222C4⋊C8—944no—
223C4⋊C8—941no—
224C4⋊C8—941no—
225C4⋊C8—941no—
226#103—942no—
227#103—942no—
228C22⋊C8—21no—
229C22⋊C8C4×D4712yes—
230#97—21no—
231#97—21no—
232#97—942no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—942no—
236C2×Q8—472no—
237C2×Q8—941no—
238C2×Q8—941no—
239C2×Q8—941no—
240C2×Q8—1882no—
241C22×C4—941no—
242C22×C4—941no—
243C22×C4C2×C4×D4711yescharacteristic
244C22×C4C2×D18811yescharacteristic
245C22×C4#1062535011yescharacteristic
246Q16—1884no—
247M16—1881no—
248M16—1881no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—22no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—942no—
257C2×C8—1881no—
258C2×C8—1881no—
259C2×C8—1881no—
260C2×C8Dic18812yescomplemented
261C4⋊C4—472no—
262C4⋊C4—942no—
263C4⋊C4—941no—
264C4⋊C4—941no—
265C4⋊C4—941no—
266C4⋊C4—942no—
267C4⋊C4—941no—
268C4⋊C4—941no—
269C4⋊C4—1881no—
270C4⋊C4—1881no—
271C22⋊C4—942no—
272C42—942no—
273C42—1881no—
274C42—1881no—
275C42D4×D4711yescharacteristic
276C42D4×D4711yescharacteristic
277C42D4×D4711yescharacteristic
278C42#1062535411yescharacteristic
279C23C4×D18811yescharacteristic
280Q8—942no—
281Q8—942no—
282Q8—3761no—
283Q8—1882no—
284Q8—1881no—
285Q8—1881no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—942no—
289C2×C4—942no—
290C2×C4—942no—
291C2×C4—1882no—
292C2×C4—1881no—
293C2×C4—1881no—
294C2×C4—1881no—
295C2×C4C22⋊C4×D4711yescharacteristic
296C2×C4#1309398611yescharacteristic
297C2×C4#1309399411yescharacteristic
298C2×C4#1309399711yescharacteristic
299C2×C4#1309399911yescharacteristic
300C2×C4#1309400911yescharacteristic
301C2×C4C2×Dic18811yescharacteristic
302C2×C4#1309404611yescharacteristic
303C2×C4#1309405711yescharacteristic
304C2×C4#1309405711yescharacteristic
305C8—22no—
306C8—42no—
307C8—41no—
308C8—81no—
309C8—3761no—
310C8—1882no—
311C22—21no—
312C22—21no—
313C22#1568900111yescharacteristic
314C22C4×Dic18811yescharacteristic
315C22#1568923111yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—3761no—
321C4—1882no—
322C4—1882no—
323C4—1882no—
324C4#1568928111yescharacteristic
325C4#1568928711yescharacteristic
326C4#1568933411yescharacteristic
327C4#1568934611yescharacteristic
328C2—21no—
329C2—21no—
330C2#2024372811yescharacteristic
331C2#2024374911yescharacteristic
332C2#2024380511yescharacteristic
subgroupG/H
centerC2×C4#13093994
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1281C2
socleC22×C47#386
prime radicalC23×C47C4×D4
cosocleC2×C42×C47C23
hypercenter#1281D47
hypocenterC47#9048
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
86. C2×C8×C47Q16semidirect
195. C47#9048semidirect
260. C2×C8Dic188semidirect

Maximal subgroups

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

subgroupindex
#202534282
#202523282
#202443282
#202420432
#202418182
#202412842
C47×#12812
#904847