Finite Group Databasebeta

← C2×#20241388 #26126449

Subgroups — C2×#20241388 #26126449

Order 12032.

668 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15688890C211yesmaximal, characteristic
2C2×#15688847C211yesmaximal, characteristic
3#20241388C214yesmaximal, direct product factor
4C2×C47×#397C211yesmaximal, characteristic
5C22×C22⋊Dic47—21no—
6C2×C22⋊(C47⋊C8)—21no—
7#15688890—21no—
8#15688890C2212yes—
9#15688847—21no—
10#15688847C2212yes—
11C22×C4⋊C4×C47C411yescharacteristic, complemented
12C22×C22⋊C4×C47—21no—
13C22×C22⋊C4×C47C2211yescharacteristic
14C47×#397—21no—
15C47×#397—21no—
16C47×#397C412yescomplemented
17C47×#397C2212yes—
18C2×#95×C47—21no—
19C2×#95×C47C411yescharacteristic, complemented
20C23×Dic47—41no—
21C22×C47⋊C8—41no—
22C2×C22⋊Dic47—24no—
23C2×C22⋊Dic47—22no—
24C2×C22⋊Dic47—41no—
25C2×C22⋊Dic47—42no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—41no—
28C25×C47D411yescharacteristic
29C23×C4×C47—21no—
30C23×C4×C47—21no—
31C23×C4×C47C2×C411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C2×C4⋊C4×C47—21no—
34C2×C4⋊C4×C47—41no—
35C2×C4⋊C4×C47—41no—
36C2×C4⋊C4×C47C2×C412yescomplemented
37C2×C22⋊C4×C47—22no—
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—41no—
40C2×C22⋊C4×C47—41no—
41C2×C22⋊C4×C47—42no—
42C2×C22⋊C4×C47—42no—
43C2×C22⋊C4×C47C2×C411yescharacteristic
44C2×C22⋊C4×C47D411yescharacteristic
45C2×C22⋊C4×C47D411yescharacteristic
46C2×C22⋊C4×C47C2311yescharacteristic
47#95×C47—22no—
48#95×C47—21no—
49#95×C47—22no—
50#95×C47C2×C412yescomplemented
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—22no—
54C24×C47—42no—
55C24×C47—42no—
56C24×C47C22⋊C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47D812yes—
59C24×C47SD1612yes—
60C24×C47C2×D411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—22no—
66C22×C4×C47—22no—
67C22×C4×C47—42no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47—42no—
71C22×C4×C47—41no—
72C22×C4×C47—41no—
73C22×C4×C47—42no—
74C22×C4×C47—42no—
75C22×C4×C47—42no—
76C22×C4×C47C22⋊C411yescharacteristic
77C22×C4×C47C22⋊C411yescharacteristic
78C22×C4×C47C22×C411yescharacteristic
79C22×C4×C47C2×D411yescharacteristic
80C22×Dic47—44no—
81C22×Dic47—42no—
82C22×Dic47—82no—
83C22×Dic47—82no—
84C4⋊C4×C47—42no—
85C4⋊C4×C47—42no—
86C22⋊C4×C47—41no—
87C22⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C22⋊C4×C47—44no—
91C22⋊C4×C47—42no—
92C22⋊C4×C47—42no—
93C22⋊Dic47—24no—
94C22⋊Dic47—44no—
95C22⋊Dic47—42no—
96C2×C47⋊C8—42no—
97C2×C47⋊C8—82no—
98C23×C47—22no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—21no—
103C23×C47—21no—
104C23×C47—22no—
105C23×C47—22no—
106C23×C47—21no—
107C23×C47—21no—
108C23×C47—41no—
109C23×C47—41no—
110C23×C47—44no—
111C23×C47—44no—
112C23×C47—41no—
113C23×C47—41no—
114C23×C47—42no—
115C23×C47—42no—
116C23×C47—84no—
117C23×C47—81no—
118C23×C47—81no—
119C23×C47#9911yescharacteristic
120C23×C47#9911yescharacteristic
121C23×C47#10212yes—
122C23×C47#10212yes—
123C23×C47#10412yes—
124C23×C47C2×C22⋊C411yescharacteristic
125C23×C47C2×D811yescharacteristic
126C23×C47C2×SD1611yescharacteristic
127C2×C4×C47—21no—
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—42no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—41no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—42no—
139C2×C4×C47—42no—
140C2×C4×C47—42no—
141C2×C4×C47—42no—
142C2×C4×C47—42no—
143C2×C4×C47—42no—
144C2×C4×C47—81no—
145C2×C4×C47—84no—
146C2×C4×C47—82no—
147C2×C4×C47—82no—
148C2×Dic47—44no—
149C2×Dic47—88no—
150C2×Dic47—82no—
151C2×Dic47—82no—
152C47⋊C8—82no—
153C2×#1344—471nomaximal, Sylow
154C22×C47—24no—
155C22×C47—24no—
156C22×C47—21no—
157C22×C47—21no—
158C22×C47—21no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—22no—
162C22×C47—21no—
163C22×C47—42no—
164C22×C47—41no—
165C22×C47—41no—
166C22×C47—44no—
167C22×C47—44no—
168C22×C47—41no—
169C22×C47—41no—
170C22×C47—41no—
171C22×C47—41no—
172C22×C47—82no—
173C22×C47—82no—
174C22×C47—81no—
175C22×C47—81no—
176C22×C47#32712yes—
177C22×C47#32712yes—
178C22×C47#35212yes—
179C22×C47#35412yes—
180C22×C47C2×#9911yescharacteristic
181C22×C47C2×#10211yescharacteristic
182C22×C47C2×#10411yescharacteristic
183C4×C47—41no—
184C4×C47—41no—
185C4×C47—41no—
186C4×C47—41no—
187C4×C47—82no—
188C4×C47—82no—
189Dic47—84no—
190C2×#397D4711yescharacteristic
191C2×#342—471no—
192C2×#323—471no—
193#1344—474no—
194C2×C47—21no—
195C2×C47—22no—
196C2×C47—21no—
197C2×C47—41no—
198C2×C47—42no—
199C2×C47—41no—
200C2×C47#134414yescomplemented
201C2×C47C2×#32711yescharacteristic
202C2×C47C2×#35211yescharacteristic
203C2×C47C2×#35411yescharacteristic
204C22×C4⋊C4Dic4711yescharacteristic, complemented
205C22×C22⋊C4—21no—
206C22×C22⋊C4—941no—
207C22×C22⋊C4C2×D4711yescharacteristic
208C2×C22⋊C8—941no—
209#397—21no—
210#397—21no—
211#397Dic4712yescomplemented
212#397C2×D4712yes—
213C2×#95—21no—
214C2×#95Dic4711yescharacteristic, complemented
215#342—472no—
216#342—941no—
217#323—472no—
218#323—941no—
219C47C2×#134411yescharacteristic, complemented, Sylow
220C25C22⋊D4711yescharacteristic
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—1881no—
224C23×C4C2×Dic4711yescharacteristic
225C23×C4C22⋊D4711yescharacteristic
226C22×C8—1881no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4C2×Dic4712yescomplemented
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—944no—
238C2×C22⋊C4—942no—
239C2×C22⋊C4—1881no—
240C2×C22⋊C4—1882no—
241C2×C22⋊C4C2×Dic4711yescharacteristic
242C2×C22⋊C4C22⋊D4711yescharacteristic
243C2×C22⋊C4C22⋊D4711yescharacteristic
244C2×C22⋊C4C22×D4711yescharacteristic
245C22⋊C8—942no—
246C22⋊C8—1881no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95C2×Dic4712yescomplemented
251C24—21no—
252C24—21no—
253C24—22no—
254C24—42no—
255C24—42no—
256C24#1062533412yes—
257C24#1062533512yes—
258C24C22⋊Dic4711yescharacteristic
259C24C22⋊Dic4711yescharacteristic
260C24C2×C22⋊D4711yescharacteristic
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—3762no—
277C22×C4—3762no—
278C22×C4—1882no—
279C22×C4—1884no—
280C22×C4C22⋊Dic4711yescharacteristic
281C22×C4C22⋊Dic4711yescharacteristic
282C22×C4C22×Dic4711yescharacteristic
283C22×C4C2×C22⋊D4711yescharacteristic
284C2×C8—3762no—
285C2×C8—1882no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—44no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—944no—
296C22⋊C4—1884no—
297C22⋊C4—1882no—
298C23—22no—
299C23—22no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—44no—
309C23—44no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—42no—
315C23—42no—
316C23—81no—
317C23—81no—
318C23—84no—
319C23#1309393911yescharacteristic
320C23#1309393911yescharacteristic
321C23#1309394512yes—
322C23#1309394512yes—
323C23#1309395112yes—
324C23C2×C22⋊Dic4711yescharacteristic
325C23C2×#1062533411yescharacteristic
326C23C2×#1062533511yescharacteristic
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4—84no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—3762no—
349C2×C4—3762no—
350C2×C4—3768no—
351C2×C4—1884no—
352C8—3762no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—24no—
358C22—24no—
359C22—21no—
360C22—22no—
361C22—21no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—44no—
370C22—44no—
371C22—82no—
372C22—82no—
373C22—81no—
374C22—81no—
375C22#1568885712yes—
376C22#1568885712yes—
377C22#1568891312yes—
378C22#1568891812yes—
379C22C2×#1309393911yescharacteristic
380C22C2×#1309394511yescharacteristic
381C22C2×#1309395111yescharacteristic
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—82no—
387C4—82no—
388C4—3764no—
389C2—21no—
390C2—22no—
391C2—21no—
392C2—41no—
393C2—42no—
394C2—41no—
395C2#2024138814yesdirect product factor
396C2C2×#1568885711yescharacteristic
397C2C2×#1568891311yescharacteristic
398C2C2×#1568891811yescharacteristic
subgroupG/H
centerC23#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC2×C47×#397C2
socleC23×C47#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenterC2×#397D47
hypocenterC47C2×#1344
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
3. #20241388C2direct (N × H)
11. C22×C4⋊C4×C47C4semidirect
16. C47×#397C4semidirect
19. C2×#95×C47C4semidirect
36. C2×C4⋊C4×C47C2×C4semidirect
50. #95×C47C2×C4semidirect
200. C2×C47#1344semidirect
204. C22×C4⋊C4Dic47semidirect
211. #397Dic47semidirect
214. C2×#95Dic47semidirect
219. C47C2×#1344semidirect
230. C2×C4⋊C4C2×Dic47semidirect
250. #95C2×Dic47semidirect
395. C2#20241388direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156888902
C2×#156888472
#202413882
C2×C47×#3972
C2×#134447