Finite Group Databasebeta

← #26104444

Subgroups — #26104444

Order 12032.

504 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20245852C211yesmaximal, characteristic
2#20243505C211yesmaximal, characteristic, complemented
3#20241212C212yesmaximal, complemented
4#20241208C212yesmaximal
5C47×#1767C211yesmaximal, characteristic, complemented
6C2×C4×D188C411yescharacteristic
7C2×#13093981—21no—
8C2×C4⋊(C47⋊C8)—21no—
9#15689025C411yescharacteristic
10#15689025C2211yescharacteristic
11#15689021C411yescharacteristic
12#15689021C2211yescharacteristic
13#15689014—21no—
14#15689014—21no—
15C4×C4⋊Dic47C411yescharacteristic
16C4×(C2×D47)⋊C4—21no—
17#15688878—21no—
18#15688878C2212yes—
19C2×C4⋊C8×C47—21no—
20C2×#97×C47—21no—
21C43×C47C2211yescharacteristic
22C47×#336C2212yes—
23C22×D188C2×C411yescharacteristic
24C22×C4×D47—21no—
25C22×C47⋊C8—22no—
26C4×D188—22no—
27C4×D188—22no—
28#13093981—42no—
29#13093981—42no—
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×C4⋊Dic47C2×C411yescharacteristic
32C2×C4⋊Dic47C2×C411yescharacteristic
33C2×Dic47⋊C4—21no—
34C2×Dic47⋊C4—21no—
35C2×(C2×D47)⋊C4—21no—
36C2×(C2×D47)⋊C4—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C2×(C2×D47)⋊C4—21no—
40C2×C4×Dic47—21no—
41C4⋊(C47⋊C8)—42no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C47⋊#95—21no—
45C22×C8×C47—22no—
46C2×C42×C47C2×C411yescharacteristic
47C2×C42×C47C2×C411yescharacteristic
48C2×C42×C47D411yescharacteristic
49C2×C42×C47D411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53C4⋊C8×C47—24no—
54#97×C47—42no—
55C23×D47—21no—
56C22×C4×C47C22⋊C411yescharacteristic
57C22×C4×C47C22⋊C411yescharacteristic
58C22×C4×C47C22⋊C411yescharacteristic
59C22×C4×C47C22⋊C411yescharacteristic
60C22×C4×C47C22×C411yescharacteristic
61C22×C4×C47C2×D411yescharacteristic
62C22×C4×C47C2×D411yescharacteristic
63C22×Dic47—21no—
64C22×Dic47—21no—
65C22×Dic47—21no—
66C2×D188—21no—
67C2×D188—41no—
68C2×D188—41no—
69C2×D188M1612yes—
70C2×C4×D47—24no—
71C2×C4×D47—42no—
72C2×C4×D47—42no—
73C2×C8×C47—24no—
74C2×C8×C47—42no—
75C2×C8×C47—42no—
76C42×C47—22no—
77C42×C47—21no—
78C42×C47—21no—
79C42×C47—21no—
80C42×C47—21no—
81C42×C47—41no—
82C42×C47—41no—
83C42×C47—41no—
84C42×C47D812yes—
85C42×C47SD1612yes—
86(C2×D47)⋊C4—24no—
87(C2×D47)⋊C4—42no—
88(C2×D47)⋊C4—42no—
89(C2×D47)⋊C4—41no—
90(C2×D47)⋊C4—41no—
91(C2×D47)⋊C4—41no—
92(C2×D47)⋊C4—41no—
93C4⋊Dic47—21no—
94C4⋊Dic47—41no—
95C4⋊Dic47—41no—
96C4⋊Dic47M1612yes—
97Dic47⋊C4—42no—
98Dic47⋊C4—41no—
99Dic47⋊C4—41no—
100C4×Dic47—42no—
101C2×C47⋊C8—42no—
102C2×C47⋊C8—42no—
103C2×C47⋊C8—42no—
104C23×C47C2×C22⋊C411yescharacteristic
105C22×D47—22no—
106C22×D47—41no—
107C22×D47—41no—
108C22×D47—41no—
109C22×D47—41no—
110C22×D47—41no—
111C22×D47—41no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—21no—
115C2×C4×C47—21no—
116C2×C4×C47—21no—
117C2×C4×C47—21no—
118C2×C4×C47—21no—
119C2×C4×C47—21no—
120C2×C4×C47—21no—
121C2×C4×C47—21no—
122C2×C4×C47—21no—
123C2×C4×C47—21no—
124C2×C4×C47—21no—
125C2×C4×C47—21no—
126C2×C4×C47—21no—
127C2×C4×C47#10212yes—
128C2×C4×C47#10212yes—
129C2×C4×C47#10412yes—
130C2×C4×C47C2×M1611yescharacteristic
131C2×C4×C47C2×M1611yescharacteristic
132C2×C4×C47C2×D811yescharacteristic
133C2×C4×C47C2×SD1611yescharacteristic
134C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C47#13711yescharacteristic
136C2×Dic47—22no—
137C2×Dic47—41no—
138C2×Dic47—41no—
139C2×Dic47—41no—
140C2×Dic47—41no—
141C2×Dic47—41no—
142C2×Dic47—41no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145D188—42no—
146D188—42no—
147C4×D47—48no—
148C8×C47—44no—
149C47⋊C8—82no—
150#7614—471nomaximal, Sylow
151C22×C47#40711yescharacteristic
152C22×C47#41011yescharacteristic
153C22×C47#41311yescharacteristic
154C22×C47C2×#10211yescharacteristic
155C22×C47#41711yescharacteristic
156C22×C47C2×#10411yescharacteristic
157C22×C47#42111yescharacteristic
158C2×D47—42no—
159C2×D47—42no—
160C2×D47—42no—
161C2×D47—81no—
162C2×D47—81no—
163C2×D47—81no—
164C2×D47—81no—
165C4×C47—22no—
166C4×C47—21no—
167C4×C47—21no—
168C4×C47—22no—
169C4×C47—21no—
170C4×C47—21no—
171C4×C47—21no—
172C4×C47—21no—
173C4×C47—22no—
174C4×C47—41no—
175Dic47—42no—
176Dic47—81no—
177Dic47—81no—
178#2293—471no—
179#1831—471no—
180#1767D4711yescharacteristic, complemented
181#1271—472no—
182#1270—472no—
183C2×C47#146411yescharacteristic
184C2×C47#148611yescharacteristic
185C2×C47#148911yescharacteristic
186C2×C47#150211yescharacteristic
187C2×C47#151211yescharacteristic
188C2×C47#153211yescharacteristic
189C2×C47#154711yescharacteristic
190D47—82no—
191C2×C4×D4—471no—
192C2×#117—941no—
193C2×C4⋊C8—21no—
194C2×C4⋊C8—941no—
195C2×#97—21no—
196#390—471no—
197#390—471no—
198#389—471no—
199#389—471no—
200#388—941no—
201#388—941no—
202C4×C4⋊C4—471no—
203C4×C22⋊C4—941no—
204C43C2×D4711yescharacteristic
205#336—472no—
206#336—941no—
207#336C2×D4712yes—
208C47#761411yescharacteristic, complemented, Sylow
209C22×D4—471no—
210C23×C4—941no—
211C22×C8—22no—
212C22×C8—942no—
213C4×D4—942no—
214C4×D4—942no—
215#117—1882no—
216#117—1882no—
217C2×C4⋊C4—471no—
218C2×C4⋊C4—471no—
219C2×C4⋊C4—471no—
220C2×C4⋊C4—941no—
221C2×C4⋊C4—941no—
222C2×C22⋊C4—941no—
223C2×C22⋊C4—941no—
224C2×C22⋊C4—941no—
225C2×C22⋊C4—941no—
226C2×C22⋊C4—941no—
227C2×C42—941no—
228C2×C42C4×D4711yescharacteristic
229C2×C42C4×D4711yescharacteristic
230C2×C42D18811yescharacteristic
231C2×C42D18811yescharacteristic
232C2×C42C22⋊D4711yescharacteristic
233C2×C42C22⋊D4711yescharacteristic
234C2×C42C22×D4711yescharacteristic
235C4⋊C8—24no—
236C4⋊C8—1882no—
237#97—42no—
238#95—941no—
239#95—941no—
240#95—941no—
241C24—941no—
242C2×D4—472no—
243C2×D4—941no—
244C2×D4—1881no—
245C2×D4—1881no—
246C22×C4—944no—
247C22×C4—941no—
248C22×C4—941no—
249C22×C4—941no—
250C22×C4—1882no—
251C22×C4—1882no—
252C22×C4(C2×D47)⋊C411yescharacteristic
253C22×C4(C2×D47)⋊C411yescharacteristic
254C22×C4(C2×D47)⋊C411yescharacteristic
255C22×C4(C2×D47)⋊C411yescharacteristic
256C22×C4C2×C4×D4711yescharacteristic
257C22×C4C2×D18811yescharacteristic
258C22×C4C2×C22⋊D4711yescharacteristic
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—1882no—
263C2×C8—1882no—
264C2×C8—1882no—
265C4⋊C4—472no—
266C4⋊C4—941no—
267C4⋊C4—1881no—
268C4⋊C4—1881no—
269C4⋊C4—1882no—
270C4⋊C4—1881no—
271C4⋊C4—1881no—
272C22⋊C4—944no—
273C22⋊C4—1882no—
274C22⋊C4—1882no—
275C22⋊C4—1881no—
276C22⋊C4—1881no—
277C22⋊C4—1881no—
278C22⋊C4—1881no—
279C42—22no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—1882no—
288C42#1062533412yes—
289C42#1062533612yes—
290C23—942no—
291C23—1881no—
292C23—1881no—
293C23—1881no—
294C23—1881no—
295C23—1881no—
296C23—1881no—
297C23C2×(C2×D47)⋊C411yescharacteristic
298D4—1882no—
299D4—1882no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—942no—
316C2×C4—1888no—
317C2×C4—1881no—
318C2×C4—1881no—
319C2×C4—1881no—
320C2×C4—1881no—
321C2×C4—1881no—
322C2×C4—1881no—
323C2×C4—1881no—
324C2×C4—1881no—
325C2×C4#1309394412yes—
326C2×C4#1309394412yes—
327C2×C4#1309395212yes—
328C2×C4M16×D4711yescharacteristic
329C2×C4M16×D4711yescharacteristic
330C2×C4C2×#1062533411yescharacteristic
331C2×C4C2×#1062533611yescharacteristic
332C2×C4#1309405211yescharacteristic
333C2×C4#1309405711yescharacteristic
334C8—44no—
335C8—3762no—
336C22—3761no—
337C22—3761no—
338C22—3761no—
339C22—3761no—
340C22—1882no—
341C22—1882no—
342C22—1882no—
343C22#1568908611yescharacteristic
344C22#1568909611yescharacteristic
345C22#1568910511yescharacteristic
346C22C2×#1309394411yescharacteristic
347C22#1568912511yescharacteristic
348C22C2×#1309395211yescharacteristic
349C22#1568913911yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—3761no—
361C4—3761no—
362C4—1882no—
363C2—3762no—
364C2#2024172011yescharacteristic
365C2#2024181211yescharacteristic
366C2#2024182811yescharacteristic
367C2#2024188211yescharacteristic
368C2#2024192711yescharacteristic
369C2#2024200611yescharacteristic
370C2#2024206611yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1767C2
socleC23×C47C2×C22⋊C4
prime radicalC22×D188C2×C4
cosocleC2×C42×C47C23
hypercenter#1767D47
hypocenterC47#7614
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
2. #20243505C2semidirect
3. #20241212C2semidirect
5. C47×#1767C2semidirect
180. #1767D47semidirect
208. C47#7614semidirect

Maximal subgroups

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

subgroupindex
#202458522
#202435052
#202412122
#202412082
C47×#17672
#761447