Finite Group Databasebeta

← #26104812

Subgroups — #26104812

Order 12032.

532 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 107 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#20251719C211yesmaximal, characteristic
2#20243150C211yesmaximal, characteristic, complemented
3#20241210C212yesmaximal, complemented
4#20241208C212yesmaximal, complemented
5C47×#1797C211yesmaximal, characteristic, complemented
6C2×#13094007—21no—
7C2×#13093998C411yescharacteristic
8C2×C22⋊(C47⋊C8)—21no—
9C2×#13093935—21no—
10#15689025C411yescharacteristic
11#15689025C2211yescharacteristic
12#15689021C411yescharacteristic
13#15689021C2211yescharacteristic
14#15689014—21no—
15#15689014—21no—
16#15689002—21no—
17#15688983C411yescharacteristic
18#15688878C2212yes—
19C22×C42×C47C2211yescharacteristic
20C2×C4⋊C8×C47—21no—
21C2×C22⋊C8×C47—21no—
22C47×#336C2212yes—
23C22×D188C2×C411yescharacteristic
24C22×C22⋊D47—21no—
25C22×C47⋊C8—22no—
26#13094007—44no—
27#13093998—42no—
28C2×C4⋊Dic47C2×C411yescharacteristic
29C2×C4⋊Dic47C2×C411yescharacteristic
30C2×C4⋊Dic47C2×C411yescharacteristic
31C2×Dic47⋊C4—21no—
32C2×Dic47⋊C4—21no—
33C2×C22⋊Dic47—21no—
34C2×(C2×D47)⋊C4—21no—
35C2×(C2×D47)⋊C4—21no—
36C2×(C2×D47)⋊C4—21no—
37C2×(C2×D47)⋊C4—21no—
38C2×(C2×D47)⋊C4—21no—
39C22⋊(C47⋊C8)—24no—
40#13093935—42no—
41C47⋊#95—21no—
42C47⋊#95—21no—
43C47⋊#95—21no—
44C23×C4×C47C2×C411yescharacteristic
45C23×C4×C47D411yescharacteristic
46C23×C4×C47D411yescharacteristic
47C22×C8×C47—22no—
48C2×C42×C47—42no—
49C2×C42×C47C2×C411yescharacteristic
50C2×C42×C47D411yescharacteristic
51C2×C42×C47D411yescharacteristic
52C2×C42×C47C2311yescharacteristic
53C4⋊C8×C47—42no—
54C22⋊C8×C47—42no—
55C24×C47C22⋊C411yescharacteristic
56C23×D47—21no—
57C22×C4×C47—22no—
58C22×C4×C47—22no—
59C22×C4×C47—22no—
60C22×C4×C47—22no—
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—41no—
64C22×C4×C47—41no—
65C22×C4×C47—41no—
66C22×C4×C47—41no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22×C411yescharacteristic
71C22×C4×C47C2×D411yescharacteristic
72C22×C4×C47C2×D411yescharacteristic
73C2×C22⋊D47—42no—
74C2×C22⋊D47—42no—
75C2×C22⋊D47—42no—
76C22×Dic47—21no—
77C22×Dic47—21no—
78C22×Dic47—21no—
79C2×D188—22no—
80C2×D188—41no—
81C2×D188—41no—
82C2×C8×C47—42no—
83C2×C8×C47—42no—
84C2×C8×C47—42no—
85C42×C47—22no—
86C42×C47—22no—
87C42×C47—41no—
88C42×C47—41no—
89C22⋊Dic47—42no—
90(C2×D47)⋊C4—42no—
91(C2×D47)⋊C4—42no—
92(C2×D47)⋊C4—42no—
93(C2×D47)⋊C4—41no—
94(C2×D47)⋊C4—42no—
95(C2×D47)⋊C4—41no—
96C4⋊Dic47—22no—
97C4⋊Dic47—41no—
98C4⋊Dic47—41no—
99Dic47⋊C4—42no—
100Dic47⋊C4—42no—
101C2×C47⋊C8—24no—
102C2×C47⋊C8—42no—
103C2×C47⋊C8—42no—
104C23×C47—21no—
105C23×C47—21no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—21no—
110C23×C47#10412yes—
111C23×C47C2×C22⋊C411yescharacteristic
112C22×D47—41no—
113C22×D47—41no—
114C22×D47—41no—
115C22×D47—41no—
116C22×D47—41no—
117C22×D47—41no—
118C22×D47—41no—
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—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—21no—
133C2×C4×C47—44no—
134C2×C4×C47—44no—
135C2×C4×C47—42no—
136C2×C4×C47—42no—
137C2×C4×C47#9912yes—
138C2×C4×C47#13112yes—
139C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C47AGL(1,ℤ/8)12yes—
141C2×C4×C47#13711yescharacteristic
142C22⋊D47—84no—
143C2×Dic47—41no—
144C2×Dic47—41no—
145C2×Dic47—41no—
146C2×Dic47—41no—
147C2×Dic47—41no—
148C2×Dic47—41no—
149C2×Dic47—41no—
150C2×Dic47—41no—
151C2×Dic47—41no—
152D188—82no—
153C8×C47—82no—
154C47⋊C8—44no—
155#7698—471nomaximal, Sylow
156C22×C47—22no—
157C22×C47—22no—
158C22×C47—22no—
159C22×C47—41no—
160C22×C47—41no—
161C22×C47—41no—
162C22×C47—41no—
163C22×C47#40811yescharacteristic
164C22×C47C2×#9911yescharacteristic
165C22×C47#41311yescharacteristic
166C22×C47#41711yescharacteristic
167C22×C47#41811yescharacteristic
168C22×C47C2×#10411yescharacteristic
169C22×C47#42111yescharacteristic
170C2×D47—81no—
171C2×D47—81no—
172C2×D47—81no—
173C2×D47—81no—
174C2×D47—81no—
175C2×D47—81no—
176C2×D47—81no—
177C4×C47—22no—
178C4×C47—22no—
179C4×C47—22no—
180C4×C47—21no—
181C4×C47—21no—
182C4×C47—41no—
183C4×C47—41no—
184Dic47—81no—
185Dic47—81no—
186Dic47—81no—
187#2854—471no—
188#1797D4711yescharacteristic, complemented
189#1749—471no—
190#1271—472no—
191#1270—472no—
192C2×C47—42no—
193C2×C47#146811yescharacteristic
194C2×C47#148311yescharacteristic
195C2×C47#149611yescharacteristic
196C2×C47#150711yescharacteristic
197C2×C47#151311yescharacteristic
198C2×C47#153211yescharacteristic
199C2×C47#155711yescharacteristic
200D47—161no—
201C2×#123—941no—
202C2×#121—471no—
203C22×C42C2×D4711yescharacteristic
204C2×C4⋊C8—21no—
205C2×C22⋊C8—21no—
206C2×C22⋊C8—941no—
207C2×#97—941no—
208#390—471no—
209#390—471no—
210#389—471no—
211#389—471no—
212#388—941no—
213#388—941no—
214#386—941no—
215#380—471no—
216#336—472no—
217#336C2×D4712yes—
218C47#769811yescharacteristic, complemented, Sylow
219C22×D4—471no—
220C22×D4—941no—
221C23×C4C4×D4711yescharacteristic
222C23×C4D18811yescharacteristic
223C23×C4C22⋊D4711yescharacteristic
224C22×C8—22no—
225C22×C8—942no—
226#123—1884no—
227#121—1882no—
228C2×C4⋊C4—471no—
229C2×C4⋊C4—471no—
230C2×C4⋊C4—471no—
231C2×C4⋊C4—941no—
232C2×C4⋊C4—941no—
233C2×C22⋊C4—941no—
234C2×C22⋊C4—941no—
235C2×C22⋊C4—941no—
236C2×C22⋊C4—941no—
237C2×C22⋊C4—941no—
238C2×C22⋊C4—941no—
239C2×C42—42no—
240C2×C42C4×D4711yescharacteristic
241C2×C42D18811yescharacteristic
242C2×C42C22⋊D4711yescharacteristic
243C2×C42C22×D4711yescharacteristic
244C4⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8—944no—
247#97—1882no—
248#95—941no—
249#95—941no—
250#95—941no—
251C24—941no—
252C24(C2×D47)⋊C411yescharacteristic
253C2×D4—942no—
254C2×D4—1881no—
255C2×D4—1881no—
256C2×D4—1882no—
257C2×D4—1882no—
258C2×D4—1882no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—941no—
270C22×C4—941no—
271C22×C4—941no—
272C22×C4(C2×D47)⋊C411yescharacteristic
273C22×C4(C2×D47)⋊C411yescharacteristic
274C22×C4(C2×D47)⋊C411yescharacteristic
275C22×C4C2×C4×D4711yescharacteristic
276C22×C4C2×D18811yescharacteristic
277C22×C4C2×C22⋊D4711yescharacteristic
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—944no—
282C2×C8—1882no—
283C2×C8—1882no—
284C4⋊C4—942no—
285C4⋊C4—1881no—
286C4⋊C4—1881no—
287C4⋊C4—1882no—
288C4⋊C4—1882no—
289C22⋊C4—1882no—
290C22⋊C4—1882no—
291C22⋊C4—1882no—
292C22⋊C4—1882no—
293C22⋊C4—1881no—
294C22⋊C4—1882no—
295C22⋊C4—1881no—
296C42—22no—
297C42—22no—
298C42—41no—
299C42—41no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—1881no—
307C23—1881no—
308C23—1881no—
309C23—1881no—
310C23—1881no—
311C23—1881no—
312C23—1881no—
313C23#1309395012yes—
314C23C2×(C2×D47)⋊C411yescharacteristic
315D4—3762no—
316D4—3764no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—1881no—
336C2×C4—1881no—
337C2×C4—1881no—
338C2×C4—1881no—
339C2×C4—1881no—
340C2×C4—1881no—
341C2×C4—1881no—
342C2×C4—1881no—
343C2×C4—1881no—
344C2×C4#1309393812yes—
345C2×C4#1309403012yes—
346C2×C4#1309405011yescharacteristic
347C2×C4#1309405212yes—
348C2×C4#1309405511yescharacteristic
349C8—82no—
350C8—1884no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—3761no—
359C22—3761no—
360C22—3761no—
361C22—3761no—
362C22—3761no—
363C22—3761no—
364C22—3761no—
365C22#1568908911yescharacteristic
366C22C2×#1309393811yescharacteristic
367C22#1568910511yescharacteristic
368C22#1568912211yescharacteristic
369C22#1568912911yescharacteristic
370C22C2×#1309395011yescharacteristic
371C22#1568914111yescharacteristic
372C4—21no—
373C4—21no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—3761no—
380C4—3761no—
381C4—3761no—
382C2—42no—
383C2—7521no—
384C2#2024173411yescharacteristic
385C2#2024179911yescharacteristic
386C2#2024185811yescharacteristic
387C2#2024190511yescharacteristic
388C2#2024192911yescharacteristic
389C2#2024200911yescharacteristic
390C2#2024210811yescharacteristic
subgroupG/H
centerC23C2×(C2×D47)⋊C4
commutatorC22×C4×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1797C2
socleC23×C47C2×C22⋊C4
prime radicalC2×#13093998C4
cosocleC2×C42×C47C23
hypercenter#1797D47
hypocenterC47#7698
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #20243150C2semidirect
3. #20241210C2semidirect
4. #20241208C2semidirect
5. C47×#1797C2semidirect
188. #1797D47semidirect
218. C47#7698semidirect

Maximal subgroups

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

subgroupindex
#202517192
#202431502
#202412102
#202412082
C47×#17972
#769847