Finite Group Databasebeta

← #26127015

Subgroups — #26127015

Order 12032.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 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
1C2×#15689043C211yesmaximal, characteristic
2#20241718C211yesmaximal, characteristic
3#20241417C214yesmaximal, complemented
4C47×#1821C211yesmaximal, characteristic
5C23×Dic94C411yescharacteristic
6C22×Dic47⋊C4—21no—
7#15689087—21no—
8#15689043—21no—
9#15689043—21no—
10#15689043C412yes—
11#15689043C2212yes—
12C2×C47⋊#95—21no—
13C2×C47⋊#95C411yescharacteristic
14#15688849—21no—
15#15688849C2212yes—
16C22×C4⋊C4×C47C2211yescharacteristic
17C47×#407—21no—
18C47×#343—21no—
19C47×#343C2212yes—
20C22×Dic94—21no—
21C22×Dic94—41no—
22C22×Dic94—41no—
23C22×Dic94—44no—
24C22×Dic94C2×C412yes—
25C23×Dic47—21no—
26C23×Dic47—21no—
27C2×#10625329—41no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—22no—
30C2×Dic47⋊C4—44no—
31C22⋊(C47⋊C8)—22no—
32C22⋊(C47⋊C8)—41no—
33C47⋊#95—22no—
34C47⋊#95—21no—
35C47⋊#95—22no—
36C47⋊#95C2×C412yes—
37C23×C4×C47C2×C411yescharacteristic
38C23×C4×C47D411yescharacteristic
39C23×C4×C47D411yescharacteristic
40C2×M16×C47—22no—
41C2×C4⋊C4×C47—41no—
42C2×C4⋊C4×C47—41no—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47C2311yescharacteristic
47C22⋊C8×C47—41no—
48C22⋊C8×C47—41no—
49C24×C47C22⋊C411yescharacteristic
50C22×C4×C47—21no—
51C22×C4×C47—21no—
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—21no—
55C22×C4×C47—21no—
56C22×C4×C47—22no—
57C22×C4×C47—41no—
58C22×C4×C47—42no—
59C22×C4×C47—42no—
60C22×C4×C47C22⋊C411yescharacteristic
61C22×C4×C47C22⋊C411yescharacteristic
62C22×C4×C47C22⋊C411yescharacteristic
63C22×C4×C47C22×C411yescharacteristic
64C22×C4×C47C2×D411yescharacteristic
65C22×C4×C47C2×D411yescharacteristic
66C22×Dic47—24no—
67C22×Dic47—24no—
68C22×Dic47—22no—
69C22×Dic47—22no—
70C22×Dic47—42no—
71C22×Dic47—42no—
72C22×Dic47—42no—
73C22×Dic47—41no—
74C22×Dic47—41no—
75C2×Dic94—44no—
76C2×Dic94—42no—
77C2×Dic94—44no—
78C2×Dic94—48no—
79C2×Dic94—42no—
80C2×Dic94—44no—
81C2×Dic94—44no—
82M16×C47—42no—
83M16×C47—42no—
84C2×C8×C47—42no—
85C4⋊C4×C47—41no—
86C4⋊C4×C47—41no—
87C4⋊C4×C47—41no—
88C4⋊C4×C47—41no—
89Dic47⋊C4—44no—
90Dic47⋊C4—44no—
91#10625329—82no—
92C2×C47⋊C8—42no—
93C23×C47—22no—
94C23×C47—22no—
95C23×C47—21no—
96C23×C47—21no—
97C23×C47#9911yescharacteristic
98C23×C47#9911yescharacteristic
99C23×C47C2×C22⋊C411yescharacteristic
100C2×C4×C47—21no—
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—42no—
112C2×C4×C47—42no—
113C2×C4×C47—42no—
114C2×C4×C47—41no—
115C2×C4×C47—41no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—81no—
119C2×C4×C47#13712yes—
120C2×Dic47—24no—
121C2×Dic47—24no—
122C2×Dic47—42no—
123C2×Dic47—42no—
124C2×Dic47—44no—
125C2×Dic47—48no—
126C2×Dic47—44no—
127C2×Dic47—42no—
128C2×Dic47—42no—
129Dic94—48no—
130Dic94—44no—
131Dic94—44no—
132C8×C47—82no—
133C47⋊C8—82no—
134#11815—471nomaximal, Sylow
135C22×C47—24no—
136C22×C47—24no—
137C22×C47—21no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47—21no—
143C22×C47AΣL(1,𝔽16)12yes—
144C22×C47#35612yes—
145C22×C47C2×#9911yescharacteristic
146C22×C47#41911yescharacteristic
147C22×C47#42111yescharacteristic
148C4×C47—22no—
149C4×C47—41no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154Dic47—44no—
155Dic47—44no—
156C2×#395—471no—
157#1821D4711yescharacteristic
158#1463—471no—
159#1354—474no—
160C2×C47—21no—
161C2×C47—22no—
162C2×C47—22no—
163C2×C47—21no—
164C2×C47#151111yescharacteristic
165C2×C47C2×AΣL(1,𝔽16)11yescharacteristic
166C2×C47C2×#35611yescharacteristic
167C23×Q8—471no—
168C22×C4⋊C4—941no—
169C22×C4⋊C4C2×D4711yescharacteristic
170#407—21no—
171#407—941no—
172#395—472no—
173#395—472no—
174#395—941no—
175#395—941no—
176C2×#95—471no—
177C2×#95—941no—
178#343—21no—
179#343C2×D4712yes—
180#324—472no—
181#324—941no—
182C47#1181511yescharacteristic, complemented, Sylow
183C22×Q8—472no—
184C22×Q8—941no—
185C22×Q8—1881no—
186C22×Q8—1881no—
187C22×Q8—1884no—
188C23×C4—941no—
189C23×C4—941no—
190C23×C4C4×D4711yescharacteristic
191C23×C4D18811yescharacteristic
192C23×C4C22⋊D4711yescharacteristic
193C2×M16—22no—
194C2×M16—1881no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—942no—
198C2×C4⋊C4—942no—
199C2×C4⋊C4—1884no—
200C2×C4⋊C4C4×D4711yescharacteristic
201C2×C4⋊C4D18811yescharacteristic
202C2×C4⋊C4C22⋊D4711yescharacteristic
203C2×C4⋊C4C22×D4711yescharacteristic
204C22⋊C8—41no—
205C22⋊C8—41no—
206C22⋊C8—942no—
207C22⋊C8—1881no—
208#95—472no—
209#95—941no—
210#95—942no—
211#95—942no—
212C24(C2×D47)⋊C411yescharacteristic
213C2×Q8—1882no—
214C2×Q8—1882no—
215C2×Q8—1884no—
216C2×Q8—1884no—
217C2×Q8—1888no—
218C2×Q8—1884no—
219C2×Q8—1884no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—41no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—944no—
231C22×C4—944no—
232C22×C4—942no—
233C22×C4—942no—
234C22×C4—1882no—
235C22×C4—1882no—
236C22×C4—1882no—
237C22×C4—1881no—
238C22×C4—1881no—
239C22×C4(C2×D47)⋊C411yescharacteristic
240C22×C4(C2×D47)⋊C411yescharacteristic
241C22×C4(C2×D47)⋊C411yescharacteristic
242C22×C4C2×C4×D4711yescharacteristic
243C22×C4C2×D18811yescharacteristic
244C22×C4C2×C22⋊D4711yescharacteristic
245M16—42no—
246M16—42no—
247M16—3762no—
248C2×C8—42no—
249C2×C8—1882no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—1884no—
255C4⋊C4—1884no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23#1309393811yescharacteristic
261C23#1309393811yescharacteristic
262C23C2×(C2×D47)⋊C411yescharacteristic
263Q8—1884no—
264Q8—1888no—
265Q8—1884no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—944no—
286C2×C4—944no—
287C2×C4—1882no—
288C2×C4—1888no—
289C2×C4—1882no—
290C2×C4—1884no—
291C2×C4—1882no—
292C2×C4—1882no—
293C2×C4—1884no—
294C2×C4#1309405412yes—
295C8—82no—
296C8—3762no—
297C22—24no—
298C22—24no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#1568891112yes—
306C22#1568892112yes—
307C22C2×#1309393811yescharacteristic
308C22#1568913111yescharacteristic
309C22#1568914111yescharacteristic
310C4—22no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—1884no—
317C4—1884no—
318C2—21no—
319C2—21no—
320C2—22no—
321C2—22no—
322C2#2024192011yescharacteristic
323C2C2×#1568891111yescharacteristic
324C2C2×#1568892111yescharacteristic
subgroupG/H
centerC22C2×#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC47×#1821C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenter#1821D47
hypocenterC47#11815
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
3. #20241417C2semidirect
182. C47#11815semidirect

Maximal subgroups

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

subgroupindex
C2×#156890432
#202417182
#202414172
C47×#18212
#1181547