Finite Group Databasebeta

← C2×#20242432 #26188515

Subgroups — C2×#20242432 #26188515

Order 12032.

984 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 203 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×#15689754C211yesmaximal, characteristic
2C2×#15689459C212yesmaximal
3C2×#15689062C211yesmaximal, characteristic
4#20242432C218yesmaximal, direct product factor
5C2×#15688861C212yesmaximal
6C2×C47×#324C211yesmaximal, characteristic
7#15689754—24no—
8#15689754—22no—
9#15689754C2214yes—
10C2×#13094023—22no—
11C2×#13094015—22no—
12C2×#13094003—21no—
13C2×#13094003C2212yes—
14C2×#13093983—21no—
15C22×C4⋊Dic47C2211yescharacteristic
16#15689459—24no—
17#15689459C2218yes—
18C2×#13093958—22no—
19C2×#13093949—22no—
20#15689062—22no—
21#15689062C2214yes—
22C2×C47⋊#95—21no—
23C2×C47⋊#95—21no—
24C2×C47⋊#95C2211yescharacteristic
25#15688861C2218yes—
26C22×C4⋊C4×C47C2211yescharacteristic
27C2×C22⋊C8×C47C2212yes—
28C47×#324C2214yes—
29C22×Dic94—21no—
30C22×Dic94D412yes—
31C23×Dic47—21no—
32C23×Dic47—21no—
33#13094023—28no—
34#13094023—28no—
35#13094015—28no—
36#13094015—28no—
37#13094003—24no—
38#13094003—22no—
39#13094003—22no—
40#13094003—24no—
41#13094003D414yes—
42#13094003C2314yes—
43#13093983—24no—
44#13093983—24no—
45C2×C4⋊Dic47—21no—
46C2×C4⋊Dic47—21no—
47C2×C4⋊Dic47—21no—
48C2×C4⋊Dic47—22no—
49C2×C4⋊Dic47—22no—
50C2×C4⋊Dic47—21no—
51C2×C4⋊Dic47D412yes—
52C2×C4⋊Dic47C2312yes—
53C2×Dic47⋊C4—22no—
54C2×Dic47⋊C4—22no—
55C2×C22⋊Dic47—22no—
56C2×C4×Dic47—22no—
57#13093958—28no—
58#13093949—28no—
59C47⋊#95—24no—
60C47⋊#95—24no—
61C47⋊#95—21no—
62C47⋊#95C2312yes—
63C23×C4×C47D411yescharacteristic
64C23×C4×C47D411yescharacteristic
65C23×C4×C47C2311yescharacteristic
66C22×C8×C47—22no—
67C2×C4⋊C4×C47—22no—
68C2×C4⋊C4×C47—22no—
69C2×C4⋊C4×C47D412yes—
70C2×C4⋊C4×C47C2312yes—
71C22⋊C8×C47—22no—
72C22⋊C8×C47C2314yes—
73C24×C47C2×D411yescharacteristic
74C22×C4×C47—21no—
75C22×C4×C47—21no—
76C22×C4×C47—22no—
77C22×C4×C47—21no—
78C22×C4×C47—21no—
79C22×C4×C47—42no—
80C22×C4×C47—42no—
81C22×C4×C47Q1614yes—
82C22×C4×C47C2×D412yes—
83C22×C4×C47C2×D412yes—
84C22×C4×C47C2×D412yes—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D412yes—
87C22×C4×C47C2411yescharacteristic
88C22×Dic47—22no—
89C22×Dic47—22no—
90C22×Dic47—22no—
91C22×Dic47—22no—
92C22×Dic47—42no—
93C22×Dic47—42no—
94C22×Dic47—44no—
95C22×Dic47—41no—
96C22×Dic47—41no—
97C22×Dic47—42no—
98C2×Dic94—24no—
99C2×Dic94—22no—
100C2×Dic94—44no—
101C2×Dic94—82no—
102C2×Dic94C2×D414yes—
103C2×C8×C47—24no—
104C2×C8×C47—44no—
105C4⋊C4×C47—24no—
106C4⋊C4×C47—24no—
107C22⋊Dic47—24no—
108C22⋊Dic47—24no—
109C4⋊Dic47—24no—
110C4⋊Dic47—24no—
111C4⋊Dic47—24no—
112C4⋊Dic47—22no—
113C4⋊Dic47—24no—
114C4⋊Dic47—42no—
115C4⋊Dic47C2×D414yes—
116Dic47⋊C4—28no—
117Dic47⋊C4—24no—
118Dic47⋊C4—24no—
119C4×Dic47—28no—
120C23×C47—22no—
121C23×C47—22no—
122C23×C47#12011yescharacteristic
123C23×C47#12011yescharacteristic
124C23×C47C2×Q1612yes—
125C23×C47#13712yes—
126C23×C47C22×D411yescharacteristic
127C2×C4×C47—22no—
128C2×C4×C47—22no—
129C2×C4×C47—24no—
130C2×C4×C47—24no—
131C2×C4×C47—24no—
132C2×C4×C47—22no—
133C2×C4×C47—41no—
134C2×C4×C47—42no—
135C2×C4×C47—41no—
136C2×C4×C47—42no—
137C2×C4×C47—42no—
138C2×C4×C47—81no—
139C2×C4×C47#12012yes—
140C2×C4×C47C2×Q1614yes—
141C2×C4×C47C22×D412yes—
142C2×Dic47—24no—
143C2×Dic47—24no—
144C2×Dic47—44no—
145C2×Dic47—44no—
146C2×Dic47—42no—
147C2×Dic47—42no—
148C2×Dic47—44no—
149C2×Dic47—42no—
150C2×Dic47—42no—
151C2×Dic47—84no—
152C2×Dic47—82no—
153C2×Dic47—82no—
154Dic94—44no—
155Dic94—82no—
156Dic94—84no—
157C8×C47—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—42no—
176Dic47—44no—
177Dic47—44no—
178Dic47—82no—
179Dic47—82no—
180C2×#544—471no—
181C2×#484—472no—
182C2×#400—471no—
183#1617—478no—
184C2×#328—472no—
185C2×#324D4711yescharacteristic
186C2×C47—21no—
187C2×C47—21no—
188C2×C47—22no—
189C2×C47#161714yescomplemented
190C2×C47C2×#45111yescharacteristic
191C2×C47C2×#45411yescharacteristic
192C2×C47C2×#45811yescharacteristic
193#544—474no—
194#544—944no—
195#544—942no—
196C2×C4⋊Q8—942no—
197C2×#125—942no—
198C2×C22⋊Q8—472no—
199C2×C22⋊Q8—941no—
200C2×#117—941no—
201C22×C4⋊C4—471no—
202C22×C4⋊C4C2×D4711yescharacteristic
203#484—478no—
204#484—944no—
205C2×#107—942no—
206C2×#103—942no—
207C2×C22⋊C8C2×D4712yes—
208#400—474no—
209#400—942no—
210C2×#95—471no—
211C2×#95—941no—
212C2×#95—941no—
213#328—478no—
214#324C2×D4714yes—
215C47C2×#161711yescharacteristic, complemented, Sylow
216C22×Q8—472no—
217C22×Q8—941no—
218C23×C4—941no—
219C23×C4—941no—
220C23×C4D18811yescharacteristic
221C23×C4D18811yescharacteristic
222C23×C4C22×D4711yescharacteristic
223C22×C8—22no—
224C4⋊Q8—948no—
225C4⋊Q8—948no—
226#125—948no—
227#125—948no—
228C22⋊Q8—474no—
229C22⋊Q8—474no—
230C22⋊Q8—944no—
231C22⋊Q8—942no—
232C22⋊Q8—944no—
233C22⋊Q8—942no—
234#117—944no—
235#117—944no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—472no—
239C2×C4⋊C4—472no—
240C2×C4⋊C4—941no—
241C2×C4⋊C4—941no—
242C2×C4⋊C4—941no—
243C2×C4⋊C4—942no—
244C2×C4⋊C4—942no—
245C2×C4⋊C4—942no—
246C2×C4⋊C4—942no—
247C2×C4⋊C4—941no—
248C2×C4⋊C4D18812yes—
249C2×C4⋊C4C22×D4712yes—
250C2×C22⋊C4—942no—
251C2×C42—942no—
252#107—948no—
253#103—948no—
254C22⋊C8—22no—
255C22⋊C8C22×D4714yes—
256#95—472no—
257#95—941no—
258#95—944no—
259#95—944no—
260C24C2×D18811yescharacteristic
261C2×Q8—474no—
262C2×Q8—944no—
263C2×Q8—942no—
264C2×Q8—3762no—
265C2×Q8—1884no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—942no—
274C22×C4—942no—
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×C4Dic18814yes—
284C22×C4C2×D18812yes—
285C22×C4C2×D18812yes—
286C22×C4C2×D18811yescharacteristic
287C22×C4D4×D4712yes—
288C22×C4D4×D4712yes—
289C22×C4C23×D4711yescharacteristic
290C2×C8—24no—
291C2×C8—44no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—474no—
295C4⋊C4—948no—
296C4⋊C4—944no—
297C4⋊C4—944no—
298C4⋊C4—944no—
299C4⋊C4—944no—
300C4⋊C4—942no—
301C4⋊C4—944no—
302C4⋊C4—944no—
303C4⋊C4—1882no—
304C22⋊C4—944no—
305C22⋊C4—944no—
306C42—948no—
307C23—22no—
308C23—22no—
309C23#1309399411yescharacteristic
310C23#1309399411yescharacteristic
311C23C2×Dic18812yes—
312C23#1309405712yes—
313C23C22×D18811yescharacteristic
314Q8—3762no—
315Q8—3764no—
316Q8—1884no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—81no—
329C2×C4—944no—
330C2×C4—944no—
331C2×C4—3764no—
332C2×C4—3762no—
333C2×C4—3762no—
334C2×C4—1882no—
335C2×C4—1882no—
336C2×C4—1884no—
337C2×C4—1882no—
338C2×C4—1882no—
339C2×C4—1884no—
340C2×C4—1884no—
341C2×C4#1309399412yes—
342C2×C4C2×Dic18814yes—
343C2×C4C2×D4×D4712yes—
344C8—44no—
345C22—24no—
346C22—24no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22#1568928112yes—
352C22#1568928112yes—
353C22#1568929812yes—
354C22#1568931412yes—
355C22C2×#1309399411yescharacteristic
356C22C22×Dic18811yescharacteristic
357C22C2×#1309405711yescharacteristic
358C4—22no—
359C4—22no—
360C4—24no—
361C4—42no—
362C4—3762no—
363C4—3762no—
364C4—1884no—
365C4—1884no—
366C2—21no—
367C2—22no—
368C2—21no—
369C2#2024243214yesdirect product factor
370C2C2×#1568928111yescharacteristic
371C2C2×#1568929811yescharacteristic
372C2C2×#1568931411yescharacteristic
subgroupG/H
centerC23#13093994
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC2×C47×#324C2
socleC23×C47#120
prime radicalC24×C47C2×D4
hypercenterC2×#324D47
hypocenterC47C2×#1617
LCSG ▷ C22×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
4. #20242432C2direct (N × H)
189. C2×C47#1617semidirect
215. C47C2×#1617semidirect
369. C2#20242432direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897542
C2×#156894592
C2×#156890622
#202424322
C2×#156888612
C2×C47×#3242
C2×#161747