Finite Group Databasebeta

← C2×#20242430 #26188513

Subgroups — C2×#20242430 #26188513

Order 12032.

984 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689456C212yesmaximal
2C2×#15689062C211yesmaximal, characteristic
3#20242430C218yesmaximal, direct product factor
4C2×#15688860C212yesmaximal
5C2×#15688849C211yesmaximal, characteristic
6C2×C47×#544C211yesmaximal, characteristic
7C22×C4⋊Dic47C2211yescharacteristic
8#15689456—24no—
9#15689456C2218yes—
10C2×#13093957—22no—
11C2×#13093948—22no—
12C2×C22⋊(C47⋊C8)C2212yes—
13#15689062—22no—
14#15689062C2214yes—
15C2×C47⋊#95—21no—
16C2×C47⋊#95—21no—
17C2×C47⋊#95C2211yescharacteristic
18#15688860C2218yes—
19#15688849C2214yes—
20C47×#544—24no—
21C47×#544—22no—
22C47×#544C2214yes—
23C2×C4⋊Q8×C47—22no—
24C2×#125×C47—22no—
25C2×C22⋊Q8×C47—21no—
26C2×C22⋊Q8×C47C2212yes—
27C2×#117×C47—21no—
28C22×C4⋊C4×C47C2211yescharacteristic
29C23×Dic47—21no—
30C23×Dic47—21no—
31C22×C47⋊C8—22no—
32C2×C4⋊Dic47—22no—
33C2×C4⋊Dic47—22no—
34C2×C4⋊Dic47—21no—
35C2×C4⋊Dic47C2312yes—
36#13093957—28no—
37#13093948—28no—
38C22⋊(C47⋊C8)—22no—
39C22⋊(C47⋊C8)C2314yes—
40C47⋊#95—24no—
41C47⋊#95—24no—
42C47⋊#95—21no—
43C47⋊#95C2312yes—
44C22×Q8×C47—21no—
45C22×Q8×C47D412yes—
46C23×C4×C47D411yescharacteristic
47C23×C4×C47D411yescharacteristic
48C23×C4×C47C2311yescharacteristic
49C4⋊Q8×C47—28no—
50C4⋊Q8×C47—28no—
51#125×C47—28no—
52#125×C47—28no—
53C22⋊Q8×C47—24no—
54C22⋊Q8×C47—24no—
55C22⋊Q8×C47—22no—
56C22⋊Q8×C47—22no—
57C22⋊Q8×C47D414yes—
58C22⋊Q8×C47C2314yes—
59#117×C47—24no—
60#117×C47—24no—
61C2×C4⋊C4×C47—21no—
62C2×C4⋊C4×C47—21no—
63C2×C4⋊C4×C47—21no—
64C2×C4⋊C4×C47—22no—
65C2×C4⋊C4×C47—22no—
66C2×C4⋊C4×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47D412yes—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47C2312yes—
71C2×C22⋊C4×C47—22no—
72C2×C42×C47—22no—
73C24×C47C2×D411yescharacteristic
74C2×Q8×C47—24no—
75C2×Q8×C47—22no—
76C2×Q8×C47—44no—
77C2×Q8×C47—82no—
78C2×Q8×C47C2×D414yes—
79C22×C4×C47—21no—
80C22×C4×C47—21no—
81C22×C4×C47—22no—
82C22×C4×C47—22no—
83C22×C4×C47—22no—
84C22×C4×C47—21no—
85C22×C4×C47—21no—
86C22×C4×C47—42no—
87C22×C4×C47—42no—
88C22×C4×C47Q1614yes—
89C22×C4×C47C2×D412yes—
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2×D412yes—
92C22×C4×C47C2×D411yescharacteristic
93C22×C4×C47C2×D412yes—
94C22×C4×C47C2411yescharacteristic
95C22×Dic47—22no—
96C22×Dic47—22no—
97C22×Dic47—42no—
98C22×Dic47—42no—
99C22×Dic47—44no—
100C22×Dic47—41no—
101C22×Dic47—41no—
102C22×Dic47—42no—
103C4⋊C4×C47—24no—
104C4⋊C4×C47—24no—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—28no—
107C4⋊C4×C47—24no—
108C4⋊C4×C47—24no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47—24no—
111C4⋊C4×C47—24no—
112C4⋊C4×C47C2×D414yes—
113C22⋊C4×C47—24no—
114C22⋊C4×C47—24no—
115C42×C47—28no—
116C4⋊Dic47—24no—
117C4⋊Dic47—42no—
118C2×C47⋊C8—24no—
119C2×C47⋊C8—44no—
120C23×C47—22no—
121C23×C47—22no—
122C23×C47#12011yescharacteristic
123C23×C47#12011yescharacteristic
124C23×C47C2×Q1612yes—
125C23×C47#13712yes—
126C23×C47C22×D411yescharacteristic
127Q8×C47—44no—
128Q8×C47—82no—
129Q8×C47—84no—
130C2×C4×C47—22no—
131C2×C4×C47—22no—
132C2×C4×C47—24no—
133C2×C4×C47—24no—
134C2×C4×C47—24no—
135C2×C4×C47—24no—
136C2×C4×C47—24no—
137C2×C4×C47—22no—
138C2×C4×C47—42no—
139C2×C4×C47—41no—
140C2×C4×C47—41no—
141C2×C4×C47—44no—
142C2×C4×C47—42no—
143C2×C4×C47—44no—
144C2×C4×C47—42no—
145C2×C4×C47—81no—
146C2×C4×C47#12012yes—
147C2×C4×C47C2×Q1614yes—
148C2×C4×C47C22×D412yes—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—44no—
152C2×Dic47—42no—
153C2×Dic47—42no—
154C2×Dic47—84no—
155C2×Dic47—82no—
156C2×Dic47—82no—
157C47⋊C8—44no—
158C2×#1617—471nomaximal, Sylow
159C22×C47—24no—
160C22×C47—24no—
161C22×C47—21no—
162C22×C47—21no—
163C22×C47—21no—
164C22×C47—21no—
165C22×C47#45112yes—
166C22×C47#45112yes—
167C22×C47#45412yes—
168C22×C47#45812yes—
169C22×C47C2×#12011yescharacteristic
170C22×C47C22×Q1611yescharacteristic
171C22×C47C2×#13711yescharacteristic
172C4×C47—24no—
173C4×C47—22no—
174C4×C47—22no—
175C4×C47—44no—
176C4×C47—42no—
177C4×C47—44no—
178Dic47—82no—
179Dic47—82no—
180C2×#544D4711yescharacteristic
181C2×#484—472no—
182C2×#400—471no—
183#1617—478no—
184C2×#328—472no—
185C2×#324—471no—
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—22no—
189C2×C47#161714yescomplemented
190C2×C47C2×#45111yescharacteristic
191C2×C47C2×#45411yescharacteristic
192C2×C47C2×#45811yescharacteristic
193#544—24no—
194#544—22no—
195#544C2×D4714yes—
196C2×C4⋊Q8—22no—
197C2×#125—22no—
198C2×C22⋊Q8—21no—
199C2×C22⋊Q8C2×D4712yes—
200C2×#117—21no—
201C22×C4⋊C4—471no—
202C22×C4⋊C4C2×D4711yescharacteristic
203#484—478no—
204#484—944no—
205C2×#107—942no—
206C2×#103—942no—
207C2×C22⋊C8—472no—
208#400—474no—
209#400—942no—
210C2×#95—471no—
211C2×#95—941no—
212C2×#95—941no—
213#328—478no—
214#324—474no—
215C47C2×#161711yescharacteristic, complemented, Sylow
216C22×Q8—21no—
217C22×Q8C22⋊D4712yes—
218C23×C4—941no—
219C23×C4—941no—
220C23×C4C22⋊D4711yescharacteristic
221C23×C4C22⋊D4711yescharacteristic
222C23×C4C22×D4711yescharacteristic
223C22×C8—942no—
224C4⋊Q8—28no—
225C4⋊Q8—28no—
226#125—28no—
227#125—28no—
228C22⋊Q8—24no—
229C22⋊Q8—24no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8C22⋊D4714yes—
233C22⋊Q8C22×D4714yes—
234#117—24no—
235#117—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—472no—
244C2×C4⋊C4—942no—
245C2×C4⋊C4—942no—
246C2×C4⋊C4—941no—
247C2×C4⋊C4C22⋊D4712yes—
248C2×C4⋊C4C22⋊D4712yes—
249C2×C4⋊C4C22×D4712yes—
250C2×C22⋊C4—22no—
251C2×C42—22no—
252#107—948no—
253#103—948no—
254C22⋊C8—474no—
255C22⋊C8—942no—
256#95—472no—
257#95—941no—
258#95—944no—
259#95—944no—
260C24C2×C22⋊D4711yescharacteristic
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—44no—
264C2×Q8—82no—
265C2×Q8C2×C22⋊D4714yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—942no—
276C22×C4—942no—
277C22×C4—1882no—
278C22×C4—1882no—
279C22×C4—1884no—
280C22×C4—1881no—
281C22×C4—1881no—
282C22×C4—1882no—
283C22×C4#1062533714yes—
284C22×C4C2×C22⋊D4712yes—
285C22×C4C2×C22⋊D4712yes—
286C22×C4C2×C22⋊D4712yes—
287C22×C4C2×C22⋊D4711yescharacteristic
288C22×C4C2×C22⋊D4712yes—
289C22×C4C23×D4711yescharacteristic
290C2×C8—944no—
291C2×C8—1884no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—944no—
302C4⋊C4—1882no—
303C4⋊C4C2×C22⋊D4714yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C42—28no—
307C23—22no—
308C23—22no—
309C23#1309399311yescharacteristic
310C23#1309399311yescharacteristic
311C23C2×#1062533712yes—
312C23#1309405512yes—
313C23C22×C22⋊D4711yescharacteristic
314Q8—44no—
315Q8—82no—
316Q8—84no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—3764no—
334C2×C4—3762no—
335C2×C4—3762no—
336C2×C4—1882no—
337C2×C4—1882no—
338C2×C4—1884no—
339C2×C4—1882no—
340C2×C4—1882no—
341C2×C4#1309399312yes—
342C2×C4C2×#1062533714yes—
343C2×C4C22×C22⋊D4712yes—
344C8—1884no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—24no—
350C22—24no—
351C22#1568927912yes—
352C22#1568927912yes—
353C22#1568929612yes—
354C22#1568931312yes—
355C22C2×#1309399311yescharacteristic
356C22C22×#1062533711yescharacteristic
357C22C2×#1309405511yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—44no—
362C4—42no—
363C4—44no—
364C4—3762no—
365C4—3762no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2#2024243014yesdirect product factor
370C2C2×#1568927911yescharacteristic
371C2C2×#1568929611yescharacteristic
372C2C2×#1568931311yescharacteristic
subgroupG/H
centerC23#13093993
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#544C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenterC2×#544D47
hypocenterC47C2×#1617
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #20242430C2direct (N × H)
189. C2×C47#1617semidirect
215. C47C2×#1617semidirect
369. C2#20242430direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156894562
C2×#156890622
#202424302
C2×#156888602
C2×#156888492
C2×C47×#5442
C2×#161747