Finite Group Databasebeta

← C2×#20241413 #26126554

Subgroups — C2×#20241413 #26126554

Order 12032.

712 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 123 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×#15689059C211yesmaximal, characteristic
2C2×#15688890C211yesmaximal, characteristic
3#20241413C214yesmaximal, direct product factor
4C2×C47×#323C211yesmaximal, characteristic
5C22×C4⋊Dic47C411yescharacteristic, complemented
6C22×Dic47⋊C4—21no—
7C22×C22⋊Dic47—21no—
8C22×C22⋊Dic47—21no—
9#15689059—21no—
10#15689059—21no—
11#15689059C412yescomplemented
12#15689059C2212yes—
13C2×C47⋊#95C411yescharacteristic, complemented
14#15688890—21no—
15#15688890C2212yes—
16C22×C22⋊C4×C47C2211yescharacteristic
17C2×C22⋊C8×C47—21no—
18C47×#323—21no—
19C47×#323C2212yes—
20C23×Dic47—21no—
21C23×Dic47—21no—
22C23×Dic47—41no—
23C2×C4⋊Dic47—21no—
24C2×C4⋊Dic47—41no—
25C2×C4⋊Dic47—41no—
26C2×C4⋊Dic47C2×C412yescomplemented
27C2×Dic47⋊C4—22no—
28C2×Dic47⋊C4—22no—
29C2×Dic47⋊C4—44no—
30C2×C22⋊Dic47—24no—
31C2×C22⋊Dic47—22no—
32C2×C22⋊Dic47—22no—
33C2×C22⋊Dic47—22no—
34C2×C22⋊Dic47—44no—
35C2×C22⋊Dic47—41no—
36C2×C22⋊Dic47—42no—
37C47⋊#95—21no—
38C47⋊#95C2×C412yescomplemented
39C25×C47D411yescharacteristic
40C23×C4×C47C2×C411yescharacteristic
41C23×C4×C47D411yescharacteristic
42C22×C8×C47—41no—
43C2×C22⋊C4×C47—41no—
44C2×C22⋊C4×C47—41no—
45C2×C22⋊C4×C47C2×C411yescharacteristic
46C2×C22⋊C4×C47D411yescharacteristic
47C2×C22⋊C4×C47D411yescharacteristic
48C2×C22⋊C4×C47C2311yescharacteristic
49C22⋊C8×C47—22no—
50C22⋊C8×C47—41no—
51C24×C47—21no—
52C24×C47—21no—
53C24×C47—22no—
54C24×C47—42no—
55C24×C47—42no—
56C24×C47C22⋊C411yescharacteristic
57C24×C47C22⋊C411yescharacteristic
58C24×C47SD1612yes—
59C24×C47Q1612yes—
60C24×C47C2×D411yescharacteristic
61C22×C4×C47—21no—
62C22×C4×C47—21no—
63C22×C4×C47—21no—
64C22×C4×C47—21no—
65C22×C4×C47—42no—
66C22×C4×C47—42no—
67C22×C4×C47C22⋊C411yescharacteristic
68C22×C4×C47C22⋊C411yescharacteristic
69C22×C4×C47C22×C411yescharacteristic
70C22×C4×C47C2×D411yescharacteristic
71C22×Dic47—24no—
72C22×Dic47—22no—
73C22×Dic47—22no—
74C22×Dic47—42no—
75C22×Dic47—42no—
76C22×Dic47—44no—
77C22×Dic47—42no—
78C22×Dic47—42no—
79C22×Dic47—42no—
80C22×Dic47—41no—
81C22×Dic47—41no—
82C22×Dic47—82no—
83C22×Dic47—82no—
84C2×C8×C47—42no—
85C2×C8×C47—82no—
86C22⋊C4×C47—41no—
87C22⋊C4×C47—41no—
88C22⋊C4×C47—41no—
89C22⋊C4×C47—41no—
90C22⋊Dic47—24no—
91C22⋊Dic47—44no—
92C22⋊Dic47—44no—
93C22⋊Dic47—44no—
94C22⋊Dic47—42no—
95C4⋊Dic47—42no—
96C4⋊Dic47—42no—
97Dic47⋊C4—44no—
98Dic47⋊C4—44no—
99C23×C47—22no—
100C23×C47—22no—
101C23×C47—22no—
102C23×C47—22no—
103C23×C47—21no—
104C23×C47—21no—
105C23×C47—22no—
106C23×C47—22no—
107C23×C47—21no—
108C23×C47—21no—
109C23×C47—41no—
110C23×C47—41no—
111C23×C47—41no—
112C23×C47—44no—
113C23×C47—44no—
114C23×C47—41no—
115C23×C47—42no—
116C23×C47—42no—
117C23×C47—81no—
118C23×C47—84no—
119C23×C47—81no—
120C23×C47#9911yescharacteristic
121C23×C47#9911yescharacteristic
122C23×C47#10312yes—
123C23×C47#10312yes—
124C23×C47#10412yes—
125C23×C47C2×C22⋊C411yescharacteristic
126C23×C47C2×SD1611yescharacteristic
127C23×C47C2×Q1611yescharacteristic
128C2×C4×C47—21no—
129C2×C4×C47—21no—
130C2×C4×C47—21no—
131C2×C4×C47—21no—
132C2×C4×C47—42no—
133C2×C4×C47—42no—
134C2×C4×C47—42no—
135C2×C4×C47—41no—
136C2×C4×C47—41no—
137C2×C4×C47—41no—
138C2×C4×C47—41no—
139C2×C4×C47—81no—
140C2×Dic47—24no—
141C2×Dic47—42no—
142C2×Dic47—48no—
143C2×Dic47—42no—
144C2×Dic47—42no—
145C2×Dic47—44no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47—82no—
149C2×Dic47—82no—
150C2×Dic47—88no—
151C2×Dic47—82no—
152C2×Dic47—82no—
153C8×C47—82no—
154C2×#1352—471nomaximal, Sylow
155C22×C47—21no—
156C22×C47—21no—
157C22×C47—24no—
158C22×C47—24no—
159C22×C47—21no—
160C22×C47—21no—
161C22×C47—21no—
162C22×C47—22no—
163C22×C47—21no—
164C22×C47—42no—
165C22×C47—41no—
166C22×C47—41no—
167C22×C47—41no—
168C22×C47—41no—
169C22×C47—44no—
170C22×C47—44no—
171C22×C47—41no—
172C22×C47—41no—
173C22×C47—81no—
174C22×C47—81no—
175C22×C47—82no—
176C22×C47—82no—
177C22×C47#32812yes—
178C22×C47#32812yes—
179C22×C47#35212yes—
180C22×C47#35312yes—
181C22×C47C2×#9911yescharacteristic
182C22×C47C2×#10311yescharacteristic
183C22×C47C2×#10411yescharacteristic
184C4×C47—41no—
185C4×C47—41no—
186C4×C47—41no—
187C4×C47—41no—
188Dic47—44no—
189Dic47—84no—
190Dic47—82no—
191C2×#399—471no—
192C2×#342—471no—
193C2×#323D4711yescharacteristic
194#1352—474no—
195C2×C47—21no—
196C2×C47—21no—
197C2×C47—22no—
198C2×C47—41no—
199C2×C47—42no—
200C2×C47—41no—
201C2×C47#135214yescomplemented
202C2×C47C2×#32811yescharacteristic
203C2×C47C2×#35211yescharacteristic
204C2×C47C2×#35311yescharacteristic
205C22×C4⋊C4—471no—
206C22×C4⋊C4—941no—
207C22×C22⋊C4—941no—
208C22×C22⋊C4—941no—
209C22×C22⋊C4C2×D4711yescharacteristic
210C2×C22⋊C8—21no—
211#399—472no—
212#399—472no—
213#399—941no—
214#399—941no—
215C2×#95—471no—
216#342—472no—
217#342—941no—
218#323—21no—
219#323C2×D4712yes—
220C47C2×#135211yescharacteristic, complemented, Sylow
221C25D18811yescharacteristic
222C23×C4—941no—
223C23×C4—941no—
224C23×C4—1881no—
225C23×C4C4×D4711yescharacteristic
226C23×C4C22⋊D4711yescharacteristic
227C22×C8—41no—
228C2×C4⋊C4—472no—
229C2×C4⋊C4—941no—
230C2×C4⋊C4—942no—
231C2×C4⋊C4—942no—
232C2×C4⋊C4—1881no—
233C2×C4⋊C4—1881no—
234C2×C4⋊C4—1884no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—944no—
238C2×C22⋊C4—942no—
239C2×C22⋊C4—942no—
240C2×C22⋊C4—942no—
241C2×C22⋊C4—1884no—
242C2×C22⋊C4—1881no—
243C2×C22⋊C4—1882no—
244C2×C22⋊C4C4×D4711yescharacteristic
245C2×C22⋊C4D18811yescharacteristic
246C2×C22⋊C4C22⋊D4711yescharacteristic
247C2×C22⋊C4C22×D4711yescharacteristic
248C22⋊C8—22no—
249C22⋊C8—41no—
250#95—472no—
251#95—941no—
252C24—21no—
253C24—21no—
254C24—22no—
255C24—42no—
256C24—42no—
257C24#1062532512yes—
258C24Dic18812yes—
259C24(C2×D47)⋊C411yescharacteristic
260C24(C2×D47)⋊C411yescharacteristic
261C24C2×D18811yescharacteristic
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—944no—
269C22×C4—942no—
270C22×C4—942no—
271C22×C4—3762no—
272C22×C4—3762no—
273C22×C4—1882no—
274C22×C4—1884no—
275C22×C4—1882no—
276C22×C4—1882no—
277C22×C4—1881no—
278C22×C4—1881no—
279C22×C4—1882no—
280C22×C4—1882no—
281C22×C4(C2×D47)⋊C411yescharacteristic
282C22×C4(C2×D47)⋊C411yescharacteristic
283C22×C4C2×C4×D4711yescharacteristic
284C22×C4C2×C22⋊D4711yescharacteristic
285C2×C8—42no—
286C2×C8—82no—
287C4⋊C4—1882no—
288C4⋊C4—1882no—
289C4⋊C4—1884no—
290C4⋊C4—1884no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—944no—
296C22⋊C4—1884no—
297C22⋊C4—1884no—
298C22⋊C4—1884no—
299C22⋊C4—1882no—
300C23—22no—
301C23—22no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—44no—
313C23—44no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—84no—
319C23—81no—
320C23—81no—
321C23#1309393811yescharacteristic
322C23#1309393811yescharacteristic
323C23#1309394912yes—
324C23#1309394912yes—
325C23#1309395012yes—
326C23C2×(C2×D47)⋊C411yescharacteristic
327C23C2×#1062532511yescharacteristic
328C23C2×Dic18811yescharacteristic
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—81no—
341C2×C4—944no—
342C2×C4—3762no—
343C2×C4—3768no—
344C2×C4—3762no—
345C2×C4—3762no—
346C2×C4—3762no—
347C2×C4—1884no—
348C2×C4—1882no—
349C2×C4—1882no—
350C2×C4—1882no—
351C2×C4—1888no—
352C2×C4—1882no—
353C2×C4—1882no—
354C8—82no—
355C22—24no—
356C22—24no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—22no—
363C22—21no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—44no—
368C22—44no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—81no—
375C22—82no—
376C22—82no—
377C22#1568886112yes—
378C22#1568886112yes—
379C22#1568891412yes—
380C22#1568891512yes—
381C22C2×#1309393811yescharacteristic
382C22C2×#1309394911yescharacteristic
383C22C2×#1309395011yescharacteristic
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—3764no—
389C4—3762no—
390C4—1884no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—42no—
396C2—41no—
397C2#2024141314yesdirect product factor
398C2C2×#1568886111yescharacteristic
399C2C2×#1568891411yescharacteristic
400C2C2×#1568891511yescharacteristic
subgroupG/H
centerC23#13093938
commutatorC22×C4×C47C22×C4
FrattiniC2×C22⋊C4C22×D47
FittingC2×C47×#323C2
socleC23×C47#99
prime radicalC25×C47D4
cosocleC2×C22⋊C4×C47C23
hypercenterC2×#323D47
hypocenterC47C2×#1352
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
3. #20241413C2direct (N × H)
5. C22×C4⋊Dic47C4semidirect
11. #15689059C4semidirect
13. C2×C47⋊#95C4semidirect
26. C2×C4⋊Dic47C2×C4semidirect
38. C47⋊#95C2×C4semidirect
201. C2×C47#1352semidirect
220. C47C2×#1352semidirect
397. C2#20241413direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156890592
C2×#156888902
#202414132
C2×C47×#3232
C2×#135247