Finite Group Databasebeta

← #26112269

Subgroups — #26112269

Order 12032.

502 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 99 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#20252417C211yesmaximal, characteristic
2#20249256C211yesmaximal, characteristic
3#20241989C212yesmaximal, complemented
4#20241253C212yesmaximal, complemented
5C4×#100×C47C211yesmaximal, characteristic
6C2×#13094016C2211yescharacteristic
7#15689195—24no—
8#15689180C2212yes—
9#15689087—21no—
10C2×C22⋊(C47⋊C8)—21no—
11#15689059C411yescharacteristic
12#15689059C411yescharacteristic
13#15689054—21no—
14#15689048C411yescharacteristic
15#15689048C411yescharacteristic
16#15689037—21no—
17#15688996—21no—
18#15688996—21no—
19#15688991C2211yescharacteristic
20#15688864—24no—
21C2×C4×D4×C47C2211yescharacteristic
22C2×#100×C47—21no—
23C4×M16×C47C2212yes—
24C47×#343—21no—
25C2×#10625329—22no—
26C22×C47⋊C8—22no—
27#13094016—22no—
28#13094016—22no—
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×C22⋊Dic47—21no—
36C2×C22⋊Dic47—21no—
37C2×C22⋊Dic47—21no—
38C2×C22⋊Dic47—21no—
39C4⋊(C47⋊C8)—22no—
40C22⋊(C47⋊C8)—22no—
41C22⋊(C47⋊C8)—22no—
42C22⋊(C47⋊C8)—22no—
43#13093935D414yes—
44C47⋊#95—21no—
45C47⋊#95—21no—
46C47⋊#95—21no—
47C47⋊#95—21no—
48C47⋊#95C2×C411yescharacteristic
49C47⋊#95C2×C411yescharacteristic
50C22×D4×C47D411yescharacteristic
51C23×C4×C47D411yescharacteristic
52C23×C4×C47D411yescharacteristic
53C2×M16×C47—22no—
54C4×D4×C47—22no—
55C4×D4×C47—22no—
56C2×C4⋊C4×C47D411yescharacteristic
57C2×C22⋊C4×C47C2×C411yescharacteristic
58C2×C22⋊C4×C47C2×C411yescharacteristic
59C2×C42×C47C2311yescharacteristic
60#100×C47—24no—
61#97×C47—22no—
62C4×C8×C47—22no—
63C24×C47C22⋊C411yescharacteristic
64C24×C47C22⋊C411yescharacteristic
65C2×D4×C47—22no—
66C2×D4×C47—22no—
67C2×D4×C47—41no—
68C22×C4×C47—21no—
69C22×C4×C47—21no—
70C22×C4×C47—21no—
71C22×C4×C47—21no—
72C22×C4×C47—81no—
73C22×C4×C47—81no—
74C22×C4×C47C22⋊C411yescharacteristic
75C22×C4×C47C22⋊C411yescharacteristic
76C22×C4×C47C22×C411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C22×C4×C47C2×D411yescharacteristic
79C22×Dic47—21no—
80C22×Dic47—21no—
81C22×Dic47—21no—
82C22×Dic47—21no—
83M16×C47—24no—
84M16×C47—42no—
85C2×C8×C47—42no—
86C4⋊C4×C47—22no—
87C22⋊C4×C47—22no—
88C22⋊C4×C47—22no—
89C42×C47C2×D412yes—
90C42×C47#4112yes—
91C22⋊Dic47—42no—
92C22⋊Dic47—42no—
93C22⋊Dic47—81no—
94C22⋊Dic47—81no—
95C4⋊Dic47—22no—
96C4⋊Dic47—22no—
97Dic47⋊C4—41no—
98Dic47⋊C4—42no—
99Dic47⋊C4—41no—
100Dic47⋊C4—81no—
101Dic47⋊C4—81no—
102#10625329—82no—
103C2×C47⋊C8—24no—
104C2×C47⋊C8—82no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47#10412yes—
115C23×C47C2×C22⋊C411yescharacteristic
116D4×C47—44no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—42no—
124C2×C4×C47—42no—
125C2×C4×C47—41no—
126C2×C4×C47—41no—
127C2×C4×C47—81no—
128C2×C4×C47—81no—
129C2×C4×C47C4×D412yes—
130C2×C4×C47#12011yescharacteristic
131C2×C4×C47#12112yes—
132C2×C4×C47#12311yescharacteristic
133C2×Dic47—41no—
134C2×Dic47—41no—
135C2×Dic47—41no—
136C2×Dic47—41no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C8×C47—44no—
146C47⋊C8—44no—
147#9189—471nomaximal, Sylow
148C22×C47—21no—
149C22×C47—21no—
150C22×C47—22no—
151C22×C47—42no—
152C22×C47—42no—
153C22×C47—41no—
154C22×C47—41no—
155C22×C47—41no—
156C22×C47—41no—
157C22×C47—41no—
158C22×C47—81no—
159C22×C47—81no—
160C22×C47#38611yescharacteristic
161C22×C47C2×#10411yescharacteristic
162C22×C47#42111yescharacteristic
163C4×C47—22no—
164C4×C47—21no—
165C4×C47—21no—
166C4×C47—22no—
167C4×C47—41no—
168C4×C47—41no—
169Dic47—81no—
170Dic47—81no—
171Dic47—81no—
172Dic47—81no—
173#2973—471no—
174#2633—471no—
175C4×#100D4711yescharacteristic
176#1528—472no—
177#1288—472no—
178C2×C47—21no—
179C2×C47—21no—
180C2×C47—42no—
181C2×C47—81no—
182C2×C47#187111yescharacteristic
183C2×C47#191111yescharacteristic
184C2×C47#191511yescharacteristic
185C2×#125—471no—
186C2×C4×D4C2×D4711yescharacteristic
187#435—944no—
188#432—472no—
189C2×#100—21no—
190#407—941no—
191C2×C22⋊C8—941no—
192C4×M16C2×D4712yes—
193#399—471no—
194#399—471no—
195#398—941no—
196#397—471no—
197#397—471no—
198#394—941no—
199#385—941no—
200#385—941no—
201#383—471no—
202#343—21no—
203#330—944no—
204C47#918911yescharacteristic, complemented, Sylow
205C22×D4C22⋊D4711yescharacteristic
206C23×C4D18811yescharacteristic
207C23×C4D18811yescharacteristic
208C2×M16—22no—
209C2×M16—942no—
210C22×C8—942no—
211#125—942no—
212#125—942no—
213C4×D4—22no—
214C4×D4—22no—
215C2×C4⋊C4—471no—
216C2×C4⋊C4—471no—
217C2×C4⋊C4—941no—
218C2×C4⋊C4—941no—
219C2×C4⋊C4—941no—
220C2×C4⋊C4—941no—
221C2×C4⋊C4C22⋊D4711yescharacteristic
222C2×C22⋊C4—941no—
223C2×C22⋊C4—941no—
224C2×C22⋊C4—941no—
225C2×C22⋊C4—941no—
226C2×C22⋊C4C4×D4711yescharacteristic
227C2×C22⋊C4C4×D4711yescharacteristic
228C2×C42C22×D4711yescharacteristic
229C4⋊C8—942no—
230#100—24no—
231C22⋊C8—942no—
232C22⋊C8—942no—
233C22⋊C8—942no—
234#97—22no—
235#97—474no—
236C4×C8—22no—
237#95—471no—
238#95—471no—
239#95—941no—
240#95—941no—
241#95—941no—
242#95—941no—
243C24(C2×D47)⋊C411yescharacteristic
244C24(C2×D47)⋊C411yescharacteristic
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—81no—
253C22×C4—81no—
254C22×C4—941no—
255C22×C4—941no—
256C22×C4—941no—
257C22×C4—941no—
258C22×C4(C2×D47)⋊C411yescharacteristic
259C22×C4(C2×D47)⋊C411yescharacteristic
260C22×C4C2×C4×D4711yescharacteristic
261C22×C4C2×D18811yescharacteristic
262C22×C4C2×C22⋊D4711yescharacteristic
263M16—24no—
264M16—42no—
265M16—3762no—
266C2×C8—42no—
267C2×C8—944no—
268C2×C8—3762no—
269C4⋊C4—22no—
270C4⋊C4—942no—
271C4⋊C4—942no—
272C4⋊C4—3761no—
273C4⋊C4—3761no—
274C4⋊C4—1881no—
275C4⋊C4—1882no—
276C4⋊C4—1881no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—3761no—
280C22⋊C4—3761no—
281C22⋊C4—1882no—
282C22⋊C4—1882no—
283C42D4×D4712yes—
284C42#1062535212yes—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23#1309395212yes—
295C23C2×(C2×D47)⋊C411yescharacteristic
296D4—44no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—1881no—
310C2×C4—1881no—
311C2×C4—1881no—
312C2×C4—1881no—
313C2×C4—1881no—
314C2×C4—1881no—
315C2×C4—1881no—
316C2×C4—1881no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4(C22⋊D47)⋊C412yes—
322C2×C4#1309399411yescharacteristic
323C2×C4#1309399612yes—
324C2×C4#1309400811yescharacteristic
325C8—44no—
326C8—1884no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22#1568900011yescharacteristic
340C22C2×#1309395211yescharacteristic
341C22#1568914111yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—22no—
346C4—41no—
347C4—41no—
348C4—3761no—
349C4—3761no—
350C4—3761no—
351C4—3761no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2#2024368811yescharacteristic
357C2#2024391811yescharacteristic
358C2#2024393811yescharacteristic
subgroupG/H
centerC22#15689000
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC4×#100×C47C2
socleC22×C47#386
prime radicalC22×D4×C47D4
cosocleC2×C42×C47C23
hypercenterC4×#100D47
hypocenterC47#9189
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×#100

Complements

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

NH ≅ G/Ntype
3. #20241989C2semidirect
4. #20241253C2semidirect
204. C47#9189semidirect

Maximal subgroups

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

subgroupindex
#202524172
#202492562
#202419892
#202412532
C4×#100×C472
#918947