Finite Group Databasebeta

← #26104426

Subgroups — #26104426

Order 12032.

612 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20243456C211yesmaximal, characteristic
2#20243082C211yesmaximal, characteristic
3#20241211C214yesmaximal, complemented
4C2×C47×#389C211yesmaximal, characteristic
5C22×#10625329—21no—
6C2×C22⋊(C47⋊C8)—21no—
7#15688878—21no—
8#15688878—21no—
9#15688878C2212yes—
10#15688878C2212yes—
11C22×C4⋊C4×C47—21no—
12C22×C4⋊C4×C47C411yescharacteristic
13C22×C4⋊C4×C47C411yescharacteristic
14C22×C42×C47C2211yescharacteristic
15C47×#389—21no—
16C47×#389—21no—
17C47×#389C412yes—
18C47×#389C2212yes—
19C2×#95×C47—21no—
20C2×#10625329—24no—
21C2×#10625329—42no—
22C2×#10625329—42no—
23C22×C47⋊C8—22no—
24C22×C47⋊C8—22no—
25C22⋊(C47⋊C8)—42no—
26C23×C4×C47—21no—
27C23×C4×C47—21no—
28C23×C4×C47C2×C411yescharacteristic
29C23×C4×C47D411yescharacteristic
30C23×C4×C47D411yescharacteristic
31C2×C4⋊C4×C47—21no—
32C2×C4⋊C4×C47—24no—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—42no—
37C2×C4⋊C4×C47—42no—
38C2×C4⋊C4×C47—42no—
39C2×C4⋊C4×C47C2×C412yes—
40C2×C4⋊C4×C47C2×C412yes—
41C2×C42×C47—42no—
42C2×C42×C47C2×C411yescharacteristic
43C2×C42×C47D411yescharacteristic
44C2×C42×C47D411yescharacteristic
45C2×C42×C47C2311yescharacteristic
46#95×C47—22no—
47#95×C47—22no—
48C24×C47C22⋊C411yescharacteristic
49C22×C4×C47—22no—
50C22×C4×C47—22no—
51C22×C4×C47—22no—
52C22×C4×C47—22no—
53C22×C4×C47—22no—
54C22×C4×C47—24no—
55C22×C4×C47—22no—
56C22×C4×C47—22no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—41no—
60C22×C4×C47—42no—
61C22×C4×C47—42no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C22⋊C411yescharacteristic
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47D812yes—
69C22×C4×C47SD1612yes—
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C4⋊C4×C47—24no—
74C4⋊C4×C47—22no—
75C4⋊C4×C47—42no—
76C4⋊C4×C47—42no—
77C4⋊C4×C47—44no—
78C4⋊C4×C47—42no—
79C4⋊C4×C47—42no—
80C4⋊C4×C47M1614yescomplemented
81C42×C47—41no—
82C42×C47—41no—
83C42×C47—41no—
84C42×C47—41no—
85#10625329—48no—
86C2×C47⋊C8—24no—
87C2×C47⋊C8—42no—
88C2×C47⋊C8—42no—
89C2×C47⋊C8—42no—
90C2×C47⋊C8—42no—
91C2×C47⋊C8—42no—
92C23×C47—21no—
93C23×C47—21no—
94C23×C47—21no—
95C23×C47—21no—
96C23×C47#9912yes—
97C23×C47#10212yes—
98C23×C47#10412yes—
99C23×C47C2×C22⋊C411yescharacteristic
100C2×C4×C47—28no—
101C2×C4×C47—24no—
102C2×C4×C47—24no—
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—21no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—24no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—44no—
118C2×C4×C47—48no—
119C2×C4×C47—42no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—42no—
123C2×C4×C47—42no—
124C2×C4×C47—84no—
125C2×C4×C47#10112yes—
126C2×C4×C47#10212yes—
127C2×C4×C47C2×M1612yes—
128C2×C4×C47C2×D811yescharacteristic
129C2×C4×C47C2×SD1611yescharacteristic
130C2×C4×C47#13712yes—
131C47⋊C8—44no—
132C47⋊C8—82no—
133#7610—471nomaximal, Sylow
134C22×C47—22no—
135C22×C47—22no—
136C22×C47—22no—
137C22×C47—22no—
138C22×C47—22no—
139C22×C47—22no—
140C22×C47#32714yes—
141C22×C47#40711yescharacteristic
142C22×C47C2×#9911yescharacteristic
143C22×C47C2×#10111yescharacteristic
144C22×C47C2×#10211yescharacteristic
145C22×C47#41911yescharacteristic
146C22×C47C2×#10411yescharacteristic
147C22×C47#42111yescharacteristic
148C4×C47—22no—
149C4×C47—22no—
150C4×C47—21no—
151C4×C47—22no—
152C4×C47—21no—
153C4×C47—22no—
154C4×C47—44no—
155C4×C47—41no—
156C4×C47—82no—
157C2×#389D4711yescharacteristic
158#1817—471no—
159#1728—471no—
160#1271—474no—
161C2×C47—24no—
162C2×C47#146311yescharacteristic
163C2×C47#148611yescharacteristic
164C2×C47#149111yescharacteristic
165C2×C47C2×#32711yescharacteristic
166C2×C47#151111yescharacteristic
167C2×C47#152811yescharacteristic
168C2×C47#154611yescharacteristic
169C22×M16—941no—
170C22×C4⋊C4—21no—
171C22×C4⋊C4Dic4711yescharacteristic
172C22×C4⋊C4Dic4711yescharacteristic
173C22×C42C2×D4711yescharacteristic
174C2×C22⋊C8—941no—
175#389—21no—
176#389—21no—
177#389Dic4712yes—
178#389C2×D4712yes—
179C2×#95—21no—
180#336—472no—
181#336—472no—
182#336—941no—
183#336—941no—
184C47#761011yescharacteristic, complemented, Sylow
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×Dic4711yescharacteristic
188C23×C4C22⋊D4711yescharacteristic
189C23×C4C22⋊D4711yescharacteristic
190C2×M16—944no—
191C2×M16—1882no—
192C2×M16—1882no—
193C22×C8—942no—
194C22×C8—942no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4C2×Dic4712yes—
204C2×C4⋊C4C2×Dic4712yes—
205C2×C42—42no—
206C2×C42C2×Dic4711yescharacteristic
207C2×C42C22⋊D4711yescharacteristic
208C2×C42C22⋊D4711yescharacteristic
209C2×C42C22×D4711yescharacteristic
210C22⋊C8—1882no—
211#95—22no—
212#95—22no—
213C24C22⋊Dic4711yescharacteristic
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4#1062533412yes—
231C22×C4#1062533512yes—
232C22×C4C22⋊Dic4711yescharacteristic
233C22×C4C22⋊Dic4711yescharacteristic
234C22×C4C22⋊Dic4711yescharacteristic
235C22×C4C22×Dic4711yescharacteristic
236C22×C4C2×C22⋊D4711yescharacteristic
237C22×C4C2×C22⋊D4711yescharacteristic
238M16—1888no—
239C2×C8—944no—
240C2×C8—1882no—
241C2×C8—1882no—
242C2×C8—1882no—
243C2×C8—1882no—
244C2×C8—1882no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4#1062532914yescomplemented
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23#1309393912yes—
262C23#1309394512yes—
263C23#1309395112yes—
264C23C2×C22⋊Dic4711yescharacteristic
265C2×C4—28no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4#1309394312yes—
291C2×C4#1309394512yes—
292C2×C4C2×#1062532912yes—
293C2×C4C2×#1062533411yescharacteristic
294C2×C4C2×#1062533511yescharacteristic
295C2×C4#1309405512yes—
296C8—3762no—
297C8—1884no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22#1568885714yes—
305C22#1568908711yescharacteristic
306C22C2×#1309393911yescharacteristic
307C22C2×#1309394311yescharacteristic
308C22C2×#1309394511yescharacteristic
309C22#1568913211yescharacteristic
310C22C2×#1309395111yescharacteristic
311C22#1568914011yescharacteristic
312C4—22no—
313C4—22no—
314C4—22no—
315C4—21no—
316C4—22no—
317C4—21no—
318C4—41no—
319C4—44no—
320C4—82no—
321C2—24no—
322C2#2024171811yescharacteristic
323C2#2024181311yescharacteristic
324C2#2024183511yescharacteristic
325C2C2×#1568885711yescharacteristic
326C2#2024192111yescharacteristic
327C2#2024199011yescharacteristic
328C2#2024206011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic47
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC2×C47×#389C2
socleC23×C47C2×C22⋊C4
prime radicalC24×C47C22⋊C4
cosocleC2×C42×C47C23
hypercenterC2×#389D47
hypocenterC47#7610
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
3. #20241211C2semidirect
80. C4⋊C4×C47M16semidirect
184. C47#7610semidirect
252. C4⋊C4#10625329semidirect

Maximal subgroups

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

subgroupindex
#202434562
#202430822
#202412112
C2×C47×#3892
#761047