Finite Group Databasebeta

← C2×#20253919 #27085902

Subgroups — C2×#20253919 #27085902

Order 12032.

1854 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C212yesmaximal
2#20253919C2116yesmaximal, direct product factor
3C2×#15689532C214yesmaximal, complemented
4C2×#15689525C212yesmaximal, complemented
5C2×#15689521C212yesmaximal, complemented
6C2×#15689165C212yesmaximal
7C2×#15689161C211yesmaximal, characteristic
8C22×#13093956C211yesmaximal, characteristic
9C2×C47×#405C211yesmaximal, characteristic
10#15689751—28no—
11#15689751C22116yes—
12C2×#13094024C2214yes—
13C2×#13094023—22no—
14C2×#13094016C2214yes—
15C2×#13094015—22no—
16C2×#13094001—24no—
17C2×#13093983C2212yes—
18C22×C4⋊Dic47C2212yes—
19#15689532C22132yescomplemented
20#15689525C22116yescomplemented
21#15689521C22116yescomplemented
22#15689165C22116yes—
23#15689161C2218yes—
24C2×#13093958C2212yes—
25C2×#13093958C2214yes—
26C2×#13093956C2212yes—
27C2×#13093956C2212yes—
28C2×#13093956C2218yes—
29C2×#13093956C2211yescharacteristic
30C2×#13093956C2211yescharacteristic
31C2×#13093956C2214yes—
32C22×M16×C47C2211yescharacteristic
33C23×C8×C47C2211yescharacteristic
34C2×#117×C47C2211yescharacteristic
35C47×#405C2218yes—
36C2×#97×C47C2212yes—
37C2×C4×C8×C47C2212yes—
38C22×Dic94—24no—
39C23×Dic47—22no—
40#13094024—28no—
41#13094024C23116yes—
42#13094023—216no—
43#13094016—28no—
44#13094016C23116yes—
45#13094015—216no—
46#13094001—232no—
47#13093983—24no—
48#13093983C2318yes—
49C2×C4⋊Dic47—24no—
50C2×C4⋊Dic47—22no—
51C2×C4⋊Dic47—24no—
52C2×C4⋊Dic47C2318yes—
53C2×C4⋊Dic47C2314yes—
54C2×C4⋊Dic47C2312yes—
55C2×C4⋊Dic47C2312yes—
56C2×Dic47⋊C4—24no—
57C2×Dic47⋊C4—24no—
58C2×C22⋊Dic47—22no—
59C2×C4×Dic47—22no—
60#13093958C23116yes—
61#13093958C2318yes—
62#13093956C2314yes—
63#13093956C2318yes—
64#13093956C23116yes—
65#13093956C2314yes—
66#13093956C2318yes—
67C23×C4×C47C2311yescharacteristic
68C2×M16×C47Q818yescomplemented
69C2×M16×C47C2314yes—
70C22×C8×C47Q818yescomplemented
71C22×C8×C47C2311yescharacteristic
72C22×C8×C47C2311yescharacteristic
73C22×C8×C47C2312yes—
74C22×C8×C47C2314yes—
75#117×C47D414yes—
76#117×C47C2314yes—
77C2×C4⋊C4×C47D412yes—
78C2×C22⋊C4×C47D412yes—
79C2×C42×C47C2312yes—
80#97×C47C2318yes—
81C4×C8×C47C2318yes—
82C24×C47C2×D411yescharacteristic
83C22×C4×C47C2×D414yes—
84C22×C4×C47C2×D411yescharacteristic
85C22×C4×C47C2×D412yes—
86C22×C4×C47C2×Q814yes—
87C22×C4×C47C2×Q814yes—
88C22×C4×C47C2411yescharacteristic
89C22×C4×C47C2412yes—
90C22×Dic47—22no—
91C22×Dic47—22no—
92C22×Dic47—24no—
93C22×Dic47—28no—
94C22×Dic47—84no—
95C2×Dic94—216no—
96C2×Dic94—88no—
97M16×C47C2×Q8116yescomplemented
98C2×C8×C47C2×Q814yes—
99C2×C8×C47C2×Q814yes—
100C2×C8×C47C2×Q818yes—
101C2×C8×C47C2×Q8116yescomplemented
102C2×C8×C47C2412yes—
103C2×C8×C47C2412yes—
104C2×C8×C47C2414yes—
105C4⋊C4×C47C2×D418yes—
106C22⋊C4×C47C2×D418yes—
107C42×C47C2×D414yes—
108C42×C47C2414yes—
109C22⋊Dic47—28no—
110C4⋊Dic47—28no—
111C4⋊Dic47—24no—
112C4⋊Dic47—216no—
113C4⋊Dic47—22no—
114C4⋊Dic47—22no—
115C4⋊Dic47C2418yes—
116C4⋊Dic47C2414yes—
117C4⋊Dic47C2414yes—
118Dic47⋊C4—216no—
119Dic47⋊C4—216no—
120C4×Dic47—28no—
121C23×C47—24no—
122C23×C47C4⋊Q814yes—
123C23×C47C22×D412yes—
124C23×C47C22×D411yescharacteristic
125C2×C4×C47—28no—
126C2×C4×C47C4⋊Q818yes—
127C2×C4×C47C4⋊Q814yes—
128C2×C4×C47C22×D418yes—
129C2×C4×C47C22×D412yes—
130C2×C4×C47C22×D411yescharacteristic
131C2×C4×C47C22×Q812yes—
132C2×C4×C47C22×Q812yes—
133C2×C4×C47C22×Q814yes—
134C2×C4×C47C22×Q814yes—
135C2×C4×C47C2511yescharacteristic
136C2×Dic47—24no—
137C2×Dic47—24no—
138C2×Dic47—28no—
139C2×Dic47—44no—
140C2×Dic47—48no—
141C2×Dic47—44no—
142C2×Dic47—88no—
143Dic94—816no—
144C8×C47C22×Q814yes—
145C8×C47C22×Q814yes—
146C8×C47C22×Q818yes—
147C2×#3163—471nomaximal, Sylow
148C22×C47—28no—
149C22×C47—24no—
150C22×C47C2×C4⋊Q814yes—
151C22×C47C2×C4⋊Q812yes—
152C22×C47#57714yes—
153C22×C47C23×D411yescharacteristic
154C4×C47—28no—
155C4×C47C2×C4⋊Q814yes—
156C4×C47C2×C4⋊Q812yes—
157C4×C47C23×Q811yescharacteristic
158C4×C47C23×Q811yescharacteristic
159Dic47—44no—
160Dic47—48no—
161Dic47—44no—
162C2×#544—472no—
163#3163—4716no—
164C2×#501—474no—
165C2×#499—472no—
166C2×#498—472no—
167C2×#428—472no—
168C2×#427—471no—
169C22×#106—471no—
170C2×#405D4711yescharacteristic
171C2×C47—24no—
172C2×C47#316314yescomplemented
173C2×C47C22×C4⋊Q811yescharacteristic
174C2×C47C2×#57712yes—
175C22×M16C2×D4711yescharacteristic
176C23×C8C2×D4711yescharacteristic
177#544—4716no—
178#544—948no—
179C2×C4⋊Q8—474no—
180C2×C4⋊Q8—942no—
181C2×#125—474no—
182C2×#125—942no—
183C2×C22⋊Q8—944no—
184C2×#117—472no—
185C2×#117C2×D4711yescharacteristic
186C22×C4⋊C4—472no—
187#501—4732no—
188#499—4716no—
189#498—4716no—
190#428—4716no—
191#427—478no—
192C2×#107—474no—
193C2×#107—472no—
194C2×#106—471no—
195C2×#106—472no—
196C2×#106—474no—
197C2×#106—471no—
198C2×#106—472no—
199C2×#106—478no—
200#405C2×D4718yes—
201C2×#97C2×D4712yes—
202C2×C4×C8C2×D4712yes—
203C47C2×#316311yescharacteristic, complemented, Sylow
204C22×Q8—944no—
205C23×C4—942no—
206C23×C4C22×D4711yescharacteristic
207C2×M16Dic9418yescomplemented
208C2×M16C22×D4714yes—
209C22×C8Dic9418yescomplemented
210C22×C8C22×D4711yescharacteristic
211C22×C8C22×D4711yescharacteristic
212C22×C8C22×D4712yes—
213C22×C8C22×D4714yes—
214C4⋊Q8—4716no—
215C4⋊Q8—9416no—
216C4⋊Q8—948no—
217#125—4716no—
218#125—9416no—
219#125—948no—
220C22⋊Q8—9432no—
221#117—478no—
222#117—944no—
223#117D18814yes—
224#117C22×D4714yes—
225C2×C4⋊C4—474no—
226C2×C4⋊C4—478no—
227C2×C4⋊C4—472no—
228C2×C4⋊C4—472no—
229C2×C4⋊C4—942no—
230C2×C4⋊C4—944no—
231C2×C4⋊C4—944no—
232C2×C4⋊C4—944no—
233C2×C4⋊C4—944no—
234C2×C4⋊C4D18812yes—
235C2×C22⋊C4—942no—
236C2×C22⋊C4D18812yes—
237C2×C42—942no—
238C2×C42C22×D4712yes—
239#107—478no—
240#107—4716no—
241#106—478no—
242#106—478no—
243#106—474no—
244#106—474no—
245#106—4716no—
246#97C22×D4718yes—
247C4×C8C22×D4718yes—
248C24C2×D18811yescharacteristic
249C2×Q8—9416no—
250C2×Q8—3768no—
251C22×C4—948no—
252C22×C4—942no—
253C22×C4—942no—
254C22×C4—944no—
255C22×C4—3764no—
256C22×C4C2×Dic9414yes—
257C22×C4C2×Dic9414yes—
258C22×C4C2×D18814yes—
259C22×C4C2×D18812yes—
260C22×C4C2×D18811yescharacteristic
261C22×C4C23×D4712yes—
262C22×C4C23×D4711yescharacteristic
263M16C2×Dic94116yescomplemented
264C2×C8C2×Dic9414yes—
265C2×C8C2×Dic9414yes—
266C2×C8C2×Dic9418yes—
267C2×C8C2×Dic94116yescomplemented
268C2×C8C23×D4712yes—
269C2×C8C23×D4714yes—
270C2×C8C23×D4712yes—
271C4⋊C4—478no—
272C4⋊C4—474no—
273C4⋊C4—474no—
274C4⋊C4—9416no—
275C4⋊C4—9416no—
276C4⋊C4—948no—
277C4⋊C4—944no—
278C4⋊C4—942no—
279C4⋊C4—942no—
280C4⋊C4—9416no—
281C4⋊C4C2×D18818yes—
282C22⋊C4—948no—
283C22⋊C4C2×D18818yes—
284C42—948no—
285C42C2×D18814yes—
286C42C23×D4714yes—
287C23—24no—
288C23#1309402414yes—
289C23C22×D18812yes—
290C23C22×D18811yescharacteristic
291Q8—37616no—
292C2×C4—28no—
293C2×C4—944no—
294C2×C4—944no—
295C2×C4—948no—
296C2×C4—3768no—
297C2×C4—1884no—
298C2×C4—1888no—
299C2×C4—1884no—
300C2×C4#1309402418yes—
301C2×C4#1309402414yes—
302C2×C4C22×D18818yes—
303C2×C4C22×D18812yes—
304C2×C4C22×D18811yescharacteristic
305C2×C4C22×Dic9412yes—
306C2×C4C22×Dic9412yes—
307C2×C4C22×Dic9414yes—
308C2×C4C22×Dic9414yes—
309C2×C4C24×D4711yescharacteristic
310C8C22×Dic9414yes—
311C8C22×Dic9414yes—
312C8C22×Dic9418yes—
313C22—24no—
314C22—28no—
315C22C2×#1309402414yes—
316C22C2×#1309402412yes—
317C22#1568993514yes—
318C22C23×D18811yescharacteristic
319C4—28no—
320C4—1888no—
321C4—1884no—
322C4—1884no—
323C4C2×#1309402414yes—
324C4C2×#1309402412yes—
325C4C23×Dic9411yescharacteristic
326C4C23×Dic9411yescharacteristic
327C2—24no—
328C2#2025391914yesdirect product factor
329C2C22×#1309402411yescharacteristic
330C2C2×#1568993512yes—
subgroupG/H
centerC23C22×D188
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#405C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenterC2×#405D47
hypocenterC47C2×#3163
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. #20253919C2direct (N × H)
3. C2×#15689532C2semidirect
4. C2×#15689525C2semidirect
5. C2×#15689521C2semidirect
19. #15689532C22semidirect
20. #15689525C22semidirect
21. #15689521C22semidirect
68. C2×M16×C47Q8semidirect
70. C22×C8×C47Q8semidirect
97. M16×C47C2×Q8semidirect
101. C2×C8×C47C2×Q8semidirect
172. C2×C47#3163semidirect
203. C47C2×#3163semidirect
207. C2×M16Dic94semidirect
209. C22×C8Dic94semidirect
263. M16C2×Dic94semidirect
267. C2×C8C2×Dic94semidirect
328. C2#20253919direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156897512
#202539192
C2×#156895322
C2×#156895252
C2×#156895212
C2×#156891652
C2×#156891612
C22×#130939562
C2×C47×#4052
C2×#316347