Finite Group Databasebeta

← #26127296

Subgroups — #26127296

Order 12032.

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

normal subgroups only

This group has 472 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#20247696C211yesmaximal, characteristic
2#20241709C211yesmaximal, characteristic
3#20241387C214yesmaximal, complemented
4C47×#1799C211yesmaximal, characteristic
5C2×#13094005—21no—
6C22×C4⋊Dic47C411yescharacteristic, complemented
7#15689176—21no—
8#15689053C412yescomplemented
9#15689048C2212yes—
10#15689021—21no—
11#15688984—21no—
12#15688983—21no—
13#15688983C411yescharacteristic, complemented
14#15688847—21no—
15#15688847C2212yes—
16C2×C4×D4×C47C2211yescharacteristic
17C2×#99×C47—21no—
18C47×#380—21no—
19C47×#342C2212yes—
20C23×Dic47—21no—
21#13094005—42no—
22#13094005—42no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—21no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47—41no—
27C2×C4⋊Dic47C2×C412yes—
28C2×Dic47⋊C4—22no—
29C2×C22⋊Dic47—21no—
30C2×C22⋊Dic47—22no—
31C4⋊(C47⋊C8)—41no—
32C22⋊(C47⋊C8)—22no—
33C4×C47⋊C8—41no—
34C47⋊#95—22no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C22×D4×C47D411yescharacteristic
38C23×C4×C47C2×C411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C4×D4×C47—41no—
41C4×D4×C47—41no—
42C2×C4⋊C4×C47—41no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47—22no—
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47C2311yescharacteristic
47C2×C42×C47D411yescharacteristic
48#99×C47—42no—
49#95×C47—41no—
50C24×C47C22⋊C411yescharacteristic
51C24×C47C2×D411yescharacteristic
52C2×D4×C47—41no—
53C2×D4×C47—41no—
54C2×D4×C47D812yes—
55C2×D4×C47SD1612yes—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—21no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22×C411yescharacteristic
67C22×C4×C47C2×D411yescharacteristic
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—42no—
71C22×Dic47—42no—
72C22×Dic47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—82no—
75C22⋊C4×C47—41no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—42no—
79C42×C47—21no—
80C42×C47—21no—
81C22⋊Dic47—42no—
82C22⋊Dic47—42no—
83C22⋊Dic47—42no—
84C4⋊Dic47—42no—
85C4⋊Dic47—41no—
86C4⋊Dic47—41no—
87C4⋊Dic47—42no—
88Dic47⋊C4—44no—
89C2×C47⋊C8—42no—
90C23×C47—21no—
91C23×C47—21no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—41no—
95C23×C47—41no—
96C23×C47—41no—
97C23×C47—41no—
98C23×C47#10212yes—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47C2×D811yescharacteristic
101C23×C47C2×SD1611yescharacteristic
102D4×C47—42no—
103D4×C47—42no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—22no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—42no—
111C2×C4×C47—42no—
112C2×C4×C47—42no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—81no—
117C2×C4×C47—82no—
118C2×C4×C47—82no—
119C2×C4×C47—82no—
120C2×C4×C47#9911yescharacteristic
121C2×C4×C47#9911yescharacteristic
122C2×C4×C47#10212yes—
123C2×Dic47—42no—
124C2×Dic47—42no—
125C2×Dic47—42no—
126C2×Dic47—42no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129C2×Dic47—84no—
130C47⋊C8—82no—
131#11874—471nomaximal, Sylow
132C22×C47—21no—
133C22×C47—21no—
134C22×C47—22no—
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—42no—
141C22×C47—41no—
142C22×C47—41no—
143C22×C47—41no—
144C22×C47—41no—
145C22×C47—81no—
146C22×C47C2×#9911yescharacteristic
147C22×C47C2×#10211yescharacteristic
148C22×C47#42111yescharacteristic
149C4×C47—21no—
150C4×C47—22no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—162no—
155C4×C47#35212yes—
156Dic47—82no—
157Dic47—82no—
158#2486—471no—
159#1799D4711yescharacteristic
160#1460—471no—
161#1344—474no—
162C2×C47—21no—
163C2×C47—21no—
164C2×C47—41no—
165C2×C47—42no—
166C2×C47—41no—
167C2×C47#151911yescharacteristic
168C2×C47C2×#35211yescharacteristic
169C2×C47#212911yescharacteristic
170C2×#123—941no—
171C2×C4×D4C2×D4711yescharacteristic
172C22×C4⋊C4—471no—
173#431—941no—
174C2×#99—21no—
175#398—472no—
176#397—472no—
177#389—941no—
178#381—941no—
179#380—21no—
180#380—471no—
181#380—941no—
182#342C2×D4712yes—
183#323—472no—
184#323—941no—
185C47#1187411yescharacteristic, complemented, Sylow
186C22×D4C22⋊D4711yescharacteristic
187C23×C4—941no—
188C23×C4C4×D4711yescharacteristic
189C23×C4C22⋊D4711yescharacteristic
190#123—1882no—
191#123—1882no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—472no—
196C2×C4⋊C4—942no—
197C2×C4⋊C4—941no—
198C2×C4⋊C4—941no—
199C2×C4⋊C4—1881no—
200C2×C4⋊C4—1881no—
201C2×C4⋊C4C4×D4711yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—941no—
204C2×C22⋊C4—942no—
205C2×C22⋊C4D18811yescharacteristic
206C2×C22⋊C4C22×D4711yescharacteristic
207C2×C42D18811yescharacteristic
208C4⋊C8—1881no—
209#99—42no—
210C22⋊C8—942no—
211C4×C8—1881no—
212#95—41no—
213#95—472no—
214#95—942no—
215#95—942no—
216C24(C2×D47)⋊C411yescharacteristic
217C24C2×C22⋊D4711yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1062533412yes—
221C2×D4#1062533612yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—942no—
230C22×C4—942no—
231C22×C4—1882no—
232C22×C4—1882no—
233C22×C4—1882no—
234C22×C4(C2×D47)⋊C411yescharacteristic
235C22×C4(C2×D47)⋊C411yescharacteristic
236C22×C4(C2×D47)⋊C411yescharacteristic
237C22×C4C2×C4×D4711yescharacteristic
238C22×C4C2×D18811yescharacteristic
239C2×C8—1882no—
240C4⋊C4—41no—
241C4⋊C4—82no—
242C4⋊C4—1882no—
243C4⋊C4—1882no—
244C4⋊C4—1881no—
245C4⋊C4—1881no—
246C4⋊C4—1884no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—1882no—
252C22⋊C4—1882no—
253C22⋊C4—1882no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#1309394412yes—
265C23C2×(C2×D47)⋊C411yescharacteristic
266C23C2×#1062533411yescharacteristic
267C23C2×#1062533611yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—3764no—
287C2×C4—1882no—
288C2×C4—1882no—
289C2×C4—1882no—
290C2×C4—1882no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4#1309393811yescharacteristic
294C2×C4#1309393811yescharacteristic
295C2×C4#1309394412yes—
296C8—3762no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1309393811yescharacteristic
312C22C2×#1309394411yescharacteristic
313C22#1568914111yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—162no—
320C4—3762no—
321C4—3762no—
322C4#1568891212yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2024195511yescharacteristic
329C2C2×#1568891211yescharacteristic
330C2#2024506211yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC47×#1799C2
socleC22×C47C2×#99
prime radicalC22×D4×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenter#1799D47
hypocenterC47#11874
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20241387C2semidirect
6. C22×C4⋊Dic47C4semidirect
8. #15689053C4semidirect
13. #15688983C4semidirect
185. C47#11874semidirect

Maximal subgroups

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

subgroupindex
#202476962
#202417092
#202413872
C47×#17992
#1187447