Finite Group Databasebeta

← #26126952

Subgroups — #26126952

Order 12032.

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

normal subgroups only

This group has 502 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
1#20243465C211yesmaximal, characteristic, complemented
2#20241722C211yesmaximal, characteristic, complemented
3#20241385C214yesmaximal, complemented
4C47×#2619C211yesmaximal, characteristic
5#15689180—21no—
6#15688986—21no—
7#15688890—21no—
8#15688890C2212yes—
9#15688849—21no—
10#15688847C2212yes—
11C2×#127×C47C411yescharacteristic, complemented
12C2×#121×C47—21no—
13C2×#117×C47C2211yescharacteristic
14C47×#394—21no—
15C47×#394C412yescomplemented
16C47×#392—21no—
17C47×#392C2212yes—
18C47×#390—21no—
19C47×#382C411yescharacteristic, complemented
20C2×C22⋊Dic47—22no—
21C4⋊(C47⋊C8)—41no—
22C22⋊(C47⋊C8)—22no—
23#13093935—41no—
24C47⋊#95—41no—
25C47⋊#95—41no—
26C22×D4×C47—22no—
27C22×D4×C47—21no—
28C22×D4×C47—21no—
29C22×D4×C47C2×C412yes—
30C23×C4×C47D411yescharacteristic
31#127×C47—41no—
32#127×C47—41no—
33#121×C47—44no—
34#117×C47—41no—
35#117×C47—41no—
36C2×C4⋊C4×C47—21no—
37C2×C4⋊C4×C47C2×C411yescharacteristic
38C2×C4⋊C4×C47D411yescharacteristic
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—22no—
41C2×C22⋊C4×C47—22no—
42C2×C22⋊C4×C47D411yescharacteristic
43C2×C22⋊C4×C47C2311yescharacteristic
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47D411yescharacteristic
46#95×C47—21no—
47#95×C47C2×C412yes—
48C24×C47—22no—
49C24×C47C2×D411yescharacteristic
50C2×D4×C47—42no—
51C2×D4×C47—42no—
52C2×D4×C47—42no—
53C2×D4×C47—42no—
54C2×D4×C47—44no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47—82no—
58C2×D4×C47—82no—
59C2×D4×C47—82no—
60C2×D4×C47—82no—
61C22×C4×C47—22no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—22no—
65C22×C4×C47—41no—
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×Dic47—42no—
73C4⋊C4×C47—41no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—42no—
76C22⋊C4×C47—41no—
77C22⋊C4×C47—41no—
78C22⋊C4×C47—44no—
79C22⋊C4×C47—82no—
80C22⋊C4×C47—82no—
81C42×C47—21no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C22⋊Dic47—42no—
86C22⋊Dic47—42no—
87C2×C47⋊C8—42no—
88C23×C47—21no—
89C23×C47—21no—
90C23×C47—41no—
91C23×C47—41no—
92C23×C47—42no—
93C23×C47—42no—
94C23×C47—42no—
95C23×C47—42no—
96C23×C47—42no—
97C23×C47—42no—
98C23×C47—42no—
99C23×C47C2×C22⋊C411yescharacteristic
100C23×C47AGL(1,ℤ/8)12yes—
101D4×C47—82no—
102D4×C47—82no—
103D4×C47—84no—
104D4×C47—84no—
105D4×C47—88no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—41no—
118C2×C4×C47—41no—
119C2×C4×C47—41no—
120C2×C4×C47—41no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47#9911yescharacteristic
125C2×C4×C47#9911yescharacteristic
126C2×Dic47—82no—
127C2×Dic47—82no—
128C2×Dic47—82no—
129C47⋊C8—82no—
130#11800—471nomaximal, Sylow
131C22×C47—21no—
132C22×C47—21no—
133C22×C47—22no—
134C22×C47—42no—
135C22×C47—41no—
136C22×C47—41no—
137C22×C47—81no—
138C22×C47—82no—
139C22×C47—82no—
140C22×C47—82no—
141C22×C47—82no—
142C22×C47—82no—
143C22×C47—82no—
144C22×C47—82no—
145C22×C47C2×#9911yescharacteristic
146C22×C47#41811yescharacteristic
147C22×C47#42111yescharacteristic
148C4×C47—21no—
149C4×C47—21no—
150C4×C47—41no—
151C4×C47—41no—
152C4×C47—41no—
153C4×C47—41no—
154C4×C47—41no—
155C4×C47—82no—
156Dic47—162no—
157#2619D4711yescharacteristic
158#1820—471no—
159#1464—471no—
160#1343—474no—
161C2×C47—21no—
162C2×C47—21no—
163C2×C47—42no—
164C2×C47—162no—
165C2×C47#151011yescharacteristic
166C2×C47#212211yescharacteristic
167C2×C47#212811yescharacteristic
168C2×#127Dic4711yescharacteristic, complemented
169C2×#121—21no—
170C2×#117C2×D4711yescharacteristic
171#432—941no—
172#394—21no—
173#394Dic4712yescomplemented
174#392—21no—
175#392C2×D4712yes—
176#390—21no—
177#382Dic4711yescharacteristic, complemented
178#381—941no—
179#342—472no—
180#342—941no—
181#324—941no—
182#323—472no—
183C47#1180011yescharacteristic, complemented, Sylow
184C22×D4—22no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4C2×Dic4712yes—
188C23×C4C22⋊D4711yescharacteristic
189#127—41no—
190#127—41no—
191#121—44no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic4711yescharacteristic
196C2×C4⋊C4C22⋊D4711yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—942no—
201C2×C22⋊C4C22⋊D4711yescharacteristic
202C2×C22⋊C4C22×D4711yescharacteristic
203C2×C42C2×Dic4711yescharacteristic
204C2×C42C22⋊D4711yescharacteristic
205C4⋊C8—1881no—
206C22⋊C8—942no—
207#97—1881no—
208#95—21no—
209#95—1881no—
210#95—1881no—
211#95C2×Dic4712yes—
212C24—22no—
213C24C2×C22⋊D4711yescharacteristic
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—44no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4—82no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—1882no—
231C22×C4C22⋊Dic4711yescharacteristic
232C22×C4C22⋊Dic4711yescharacteristic
233C22×C4C22⋊Dic4711yescharacteristic
234C22×C4C22⋊Dic4711yescharacteristic
235C22×C4C22×Dic4711yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237C2×C8—1882no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C22⋊C4—41no—
242C22⋊C4—41no—
243C22⋊C4—44no—
244C22⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—1882no—
247C22⋊C4—1882no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—42no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23C2×C22⋊Dic4711yescharacteristic
264C23#1309405012yes—
265D4—82no—
266D4—82no—
267D4—84no—
268D4—84no—
269D4—88no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—3762no—
289C2×C4—3762no—
290C2×C4—3762no—
291C2×C4#1309393911yescharacteristic
292C2×C4#1309393911yescharacteristic
293C8—3762no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—41no—
299C22—41no—
300C22—81no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22C2×#1309393911yescharacteristic
309C22#1568912811yescharacteristic
310C22#1568914011yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—82no—
319C4—7522no—
320C2—21no—
321C2—21no—
322C2—42no—
323C2—162no—
324C2#2024191711yescharacteristic
325C2#2024503811yescharacteristic
326C2#2024506011yescharacteristic
subgroupG/H
centerC22C2×#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
Fitting, prime radicalC47×#2619C2
socleC22×C47C2×#99
cosocleC2×C22⋊C4×C47C23
hypercenter#2619D47
hypocenterC47#11800
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #20243465C2semidirect
2. #20241722C2semidirect
3. #20241385C2semidirect
11. C2×#127×C47C4semidirect
15. C47×#394C4semidirect
19. C47×#382C4semidirect
168. C2×#127Dic47semidirect
173. #394Dic47semidirect
177. #382Dic47semidirect
183. C47#11800semidirect

Maximal subgroups

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

subgroupindex
#202434652
#202417222
#202413852
C47×#26192
#1180047