Finite Group Databasebeta

← #26127019

Subgroups — #26127019

Order 12032.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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#20246656C211yesmaximal, characteristic
2#20241722C211yesmaximal, characteristic
3#20241417C212yesmaximal, complemented
4#20241414C212yesmaximal, complemented
5C47×#1822C211yesmaximal, characteristic
6C2×C4×Dic94C411yescharacteristic
7#15689180—21no—
8#15689062C2211yescharacteristic
9#15689059C411yescharacteristic
10#15689056—21no—
11#15689046—21no—
12#15689043C2211yescharacteristic
13#15689035C411yescharacteristic
14C4×Dic47⋊C4—21no—
15C4×C4⋊Dic47C411yescharacteristic
16C4×C22⋊Dic47—21no—
17#15688849C2212yes—
18#15688847—21no—
19C2×#117×C47C2211yescharacteristic
20C47×#407—21no—
21C47×#343—21no—
22C47×#343C2212yes—
23C22×Dic94C2×C411yescharacteristic
24C4×Dic94—41no—
25C4×Dic94—41no—
26C2×C4⋊Dic47—21no—
27C2×C4⋊Dic47C2×C411yescharacteristic
28C2×Dic47⋊C4—21no—
29C2×Dic47⋊C4—21no—
30C2×Dic47⋊C4—21no—
31C2×C22⋊Dic47—21no—
32C2×C22⋊Dic47—21no—
33C2×C4×Dic47—21no—
34C2×C4×Dic47—21no—
35C4⋊(C47⋊C8)—41no—
36C22⋊(C47⋊C8)—22no—
37#13093935—41no—
38C47⋊#95—21no—
39C47⋊#95—21no—
40C47⋊#95—21no—
41C47⋊#95—21no—
42C47⋊#95C2×C411yescharacteristic
43C47⋊#95C2×C411yescharacteristic
44C23×C4×C47D411yescharacteristic
45C2×M16×C47—22no—
46#117×C47—41no—
47#117×C47—41no—
48C2×C4⋊C4×C47D411yescharacteristic
49C2×C4⋊C4×C47C2311yescharacteristic
50C2×C22⋊C4×C47C2×C411yescharacteristic
51C2×C22⋊C4×C47D411yescharacteristic
52C2×C42×C47C2×C411yescharacteristic
53C2×C42×C47D411yescharacteristic
54C22⋊C8×C47—41no—
55C22⋊C8×C47—41no—
56C24×C47C22⋊C411yescharacteristic
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—41no—
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22⋊C411yescharacteristic
64C22×C4×C47C22×C411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×C4×C47C2×D411yescharacteristic
67C22×Dic47—21no—
68C22×Dic47—21no—
69C22×Dic47—21no—
70C22×Dic47—21no—
71C2×Dic94—41no—
72C2×Dic94—42no—
73M16×C47—42no—
74M16×C47—42no—
75C2×C8×C47—42no—
76C4⋊C4×C47—41no—
77C4⋊C4×C47—41no—
78C22⋊C4×C47—41no—
79C22⋊C4×C47—41no—
80C42×C47—21no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C22⋊Dic47—42no—
85C22⋊Dic47—42no—
86C4⋊Dic47—42no—
87C4⋊Dic47—41no—
88Dic47⋊C4—42no—
89Dic47⋊C4—42no—
90Dic47⋊C4—42no—
91C4×Dic47—42no—
92C4×Dic47—41no—
93C4×Dic47—41no—
94C2×C47⋊C8—42no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47—21no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47C2×C22⋊C411yescharacteristic
101C2×C4×C47—21no—
102C2×C4×C47—21no—
103C2×C4×C47—21no—
104C2×C4×C47—21no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—42no—
110C2×C4×C47—42no—
111C2×C4×C47—41no—
112C2×C4×C47—41no—
113C2×C4×C47—41no—
114C2×C4×C47—41no—
115C2×C4×C47#9911yescharacteristic
116C2×C4×C47#9911yescharacteristic
117C2×C4×C47#13712yes—
118C2×Dic47—22no—
119C2×Dic47—22no—
120C2×Dic47—41no—
121C2×Dic47—41no—
122C2×Dic47—41no—
123C2×Dic47—41no—
124C2×Dic47—41no—
125C2×Dic47—41no—
126C2×Dic47—41no—
127C2×Dic47—41no—
128C2×Dic47—41no—
129C2×Dic47—41no—
130Dic94—42no—
131Dic94—42no—
132C8×C47—82no—
133C47⋊C8—82no—
134#11816—471nomaximal, Sylow
135C22×C47—21no—
136C22×C47—21no—
137C22×C47—42no—
138C22×C47—41no—
139C22×C47—41no—
140C22×C47—41no—
141C22×C47—81no—
142C22×C47C2×#9911yescharacteristic
143C22×C47#41911yescharacteristic
144C22×C47#42111yescharacteristic
145C4×C47—22no—
146C4×C47—21no—
147C4×C47—21no—
148C4×C47—41no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152Dic47—42no—
153Dic47—42no—
154Dic47—81no—
155Dic47—81no—
156#2396—471no—
157#1822D4711yescharacteristic
158#1464—471no—
159#1354—472no—
160#1353—472no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—81no—
164C2×C47#151111yescharacteristic
165C2×C47#212011yescharacteristic
166C2×C47#213211yescharacteristic
167C2×C4×Q8—471no—
168C2×#117C2×D4711yescharacteristic
169#432—941no—
170#407—21no—
171#400—471no—
172#399—471no—
173#398—941no—
174#396—941no—
175#395—471no—
176#393—471no—
177C4×C4⋊C4—471no—
178C4×C4⋊C4—941no—
179C4×C22⋊C4—941no—
180#343—21no—
181#343C2×D4712yes—
182#324—472no—
183#323—941no—
184C47#1181611yescharacteristic, complemented, Sylow
185C22×Q8—471no—
186C23×C4C22⋊D4711yescharacteristic
187C2×M16—22no—
188C4×Q8—1881no—
189C4×Q8—1881no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—471no—
193C2×C4⋊C4—941no—
194C2×C4⋊C4—941no—
195C2×C4⋊C4—941no—
196C2×C4⋊C4—941no—
197C2×C4⋊C4C22⋊D4711yescharacteristic
198C2×C4⋊C4C22×D4711yescharacteristic
199C2×C22⋊C4—941no—
200C2×C22⋊C4—941no—
201C2×C22⋊C4C4×D4711yescharacteristic
202C2×C22⋊C4D18811yescharacteristic
203C2×C42—941no—
204C2×C42—941no—
205C2×C42C4×D4711yescharacteristic
206C2×C42D18811yescharacteristic
207C4⋊C8—1881no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210C22⋊C8—942no—
211#97—1881no—
212#95—471no—
213#95—471no—
214#95—941no—
215#95—941no—
216#95—941no—
217#95—941no—
218C24(C2×D47)⋊C411yescharacteristic
219C2×Q8—1881no—
220C2×Q8—1882no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—941no—
226C22×C4—941no—
227C22×C4—941no—
228C22×C4—941no—
229C22×C4(C2×D47)⋊C411yescharacteristic
230C22×C4(C2×D47)⋊C411yescharacteristic
231C22×C4(C2×D47)⋊C411yescharacteristic
232C22×C4C2×C4×D4711yescharacteristic
233C22×C4C2×D18811yescharacteristic
234C22×C4C2×C22⋊D4711yescharacteristic
235M16—42no—
236M16—42no—
237C2×C8—42no—
238C2×C8—1882no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—1881no—
242C4⋊C4—1882no—
243C4⋊C4—1882no—
244C4⋊C4—1882no—
245C4⋊C4—1882no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—1882no—
249C22⋊C4—1882no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—1882no—
255C42—1881no—
256C42—1881no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×(C2×D47)⋊C411yescharacteristic
263Q8—1882no—
264Q8—1882no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—942no—
280C2×C4—942no—
281C2×C4—1881no—
282C2×C4—1881no—
283C2×C4—1881no—
284C2×C4—1881no—
285C2×C4—1881no—
286C2×C4—1881no—
287C2×C4—1881no—
288C2×C4—1881no—
289C2×C4—1881no—
290C2×C4—1881no—
291C2×C4#1309393811yescharacteristic
292C2×C4#1309393811yescharacteristic
293C2×C4#1309405412yes—
294C8—82no—
295C8—3762no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#1309393811yescharacteristic
304C22#1568913111yescharacteristic
305C22#1568914111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3761no—
314C4—3761no—
315C4—1882no—
316C4—1882no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2024192011yescharacteristic
321C2#2024503011yescharacteristic
322C2#2024507111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1822C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1822D47
hypocenterC47#11816
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #20241417C2semidirect
4. #20241414C2semidirect
184. C47#11816semidirect

Maximal subgroups

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

subgroupindex
#202466562
#202417222
#202414172
#202414142
C47×#18222
#1181647