Finite Group Databasebeta

← C2×#20244186 #26219862

Subgroups — C2×#20244186 #26219862

Order 12032.

2094 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D188C211yesmaximal, characteristic
2C22×C4⋊(C47⋊C8)C211yesmaximal, characteristic, complemented
3C22×(C2×D47)⋊C8C214yesmaximal, complemented
4#20244186C218yesmaximal, direct product factor
5C22×#97×C47C211yesmaximal, characteristic, complemented
6C23×D188C411yescharacteristic
7C23×C4×D47C2212yes—
8C23×C47⋊C8C2212yes—
9C2×C4×D188—24no—
10C2×C4×D188—24no—
11C2×C4×D188C414yes—
12C2×C4×D188C2214yes—
13C22×C4⋊Dic47C411yescharacteristic
14C22×(C2×D47)⋊C4C412yes—
15C2×C4⋊(C47⋊C8)—24no—
16C2×C4⋊(C47⋊C8)C2214yescomplemented
17C2×(C2×D47)⋊C8—216no—
18C2×(C2×D47)⋊C8C22116yescomplemented
19C23×C8×C47C2212yes—
20C22×C42×C47C2211yescharacteristic
21C2×#97×C47—24no—
22C2×#97×C47C2214yescomplemented
23C24×D47C2×C412yes—
24C22×D188—28no—
25C22×D188—24no—
26C22×D188C2×C414yes—
27C23×Dic47C2×C412yes—
28C22×C4×D47—24no—
29C22×C4×D47—24no—
30C22×C4×D47—28no—
31C22×C4×D47—28no—
32C22×C4×D47C2×C414yes—
33C22×C4×D47C2314yes—
34C22×C47⋊C8—28no—
35C22×C47⋊C8—24no—
36C22×C47⋊C8C2314yes—
37C4×D188—216no—
38C4×D188—28no—
39C4×D188—28no—
40C2×C4⋊Dic47—24no—
41C2×C4⋊Dic47C2×C414yes—
42C2×(C2×D47)⋊C4—28no—
43C2×(C2×D47)⋊C4C2×C418yes—
44C4⋊(C47⋊C8)—28no—
45(C2×D47)⋊C8—232no—
46C23×C4×C47C2×C411yescharacteristic
47C23×C4×C47C2×C411yescharacteristic
48C23×C4×C47C2311yescharacteristic
49C22×C8×C47—24no—
50C22×C8×C47—24no—
51C22×C8×C47D418yescomplemented
52C22×C8×C47C2314yes—
53C2×C42×C47C2×C412yes—
54C2×C42×C47D414yes—
55C2×C42×C47D414yes—
56C2×C42×C47C2312yes—
57#97×C47—28no—
58C24×C47C22×C411yescharacteristic
59C23×D47—24no—
60C23×D47—28no—
61C23×D47—24no—
62C23×D47—28no—
63C23×D47M1618yes—
64C23×D47C22×C414yes—
65C22×C4×C47—24no—
66C22×C4×C47—24no—
67C22×C4×C47C22⋊C414yes—
68C22×C4×C47C22⋊C414yes—
69C22×C4×C47C22×C412yes—
70C22×C4×C47C22×C412yes—
71C22×C4×C47C22×C411yescharacteristic
72C22×C4×C47C2×D412yes—
73C22×C4×C47C2×D412yes—
74C22×C4×C47C2×D414yes—
75C22×C4×C47#4114yes—
76C22×C4×C47C2411yescharacteristic
77C22×Dic47—24no—
78C22×Dic47—28no—
79C22×Dic47C22×C414yes—
80C2×D188—232no—
81C2×D188—216no—
82C2×D188—28no—
83C2×C4×D47—28no—
84C2×C4×D47—216no—
85C2×C4×D47—28no—
86C2×C4×D47—24no—
87C2×C4×D47—24no—
88C2×C4×D47—28no—
89C2×C4×D47—416no—
90C2×C4×D47—48no—
91C2×C4×D47—48no—
92C2×C8×C47—216no—
93C2×C8×C47—24no—
94C2×C8×C47—24no—
95C2×C8×C47C2×D418yescomplemented
96C42×C47C22⋊C418yes—
97C42×C47C2×D414yes—
98C42×C47C2×D414yes—
99(C2×D47)⋊C4—216no—
100C4⋊Dic47—28no—
101C2×C47⋊C8—28no—
102C2×C47⋊C8—24no—
103C2×C47⋊C8—48no—
104C23×C47C2×C22⋊C412yes—
105C23×C47C4×D414yes—
106C23×C47C2×M1614yes—
107C23×C47#13114yes—
108C23×C47C23×C411yescharacteristic
109C22×D47—24no—
110C22×D47—28no—
111C22×D47—24no—
112C22×D47—28no—
113C22×D47—216no—
114C22×D47—232no—
115C22×D47—48no—
116C22×D47—48no—
117C22×D47—416no—
118C22×D47C2×M1618yes—
119C2×C4×C47—24no—
120C2×C4×C47—24no—
121C2×C4×C47—28no—
122C2×C4×C47—28no—
123C2×C4×C47C2×C22⋊C414yes—
124C2×C4×C47C2×C22⋊C414yes—
125C2×C4×C47C2×C22⋊C412yes—
126C2×C4×C47C4×D414yes—
127C2×C4×C47#12118yes—
128C2×C4×C47#12414yes—
129C2×C4×C47#12714yes—
130C2×C4×C47C22×D411yescharacteristic
131C2×C4×C47C22×D412yes—
132C2×C4×C47C22×D411yescharacteristic
133C2×C4×C47C2×#4112yes—
134C2×Dic47—24no—
135C2×Dic47—28no—
136C2×Dic47—48no—
137D188—232no—
138C4×D47—48no—
139C4×D47—44no—
140C4×D47—44no—
141C4×D47—416no—
142C8×C47—28no—
143C47⋊C8—44no—
144C2×#1955—471nomaximal, Sylow
145C22×C47#39014yes—
146C22×C47#40714yes—
147C22×C47#40814yes—
148C22×C47#435116yes—
149C22×C47C22×C22⋊C411yescharacteristic
150C22×C47C2×C4×D412yes—
151C22×C47C22×M1612yes—
152C22×C47C2×#13112yes—
153C2×D47—28no—
154C2×D47—216no—
155C2×D47—44no—
156C2×D47—48no—
157C2×D47—432no—
158C2×D47—44no—
159C4×C47—24no—
160C4×C47—24no—
161C4×C47#39014yes—
162C4×C47C2×#12112yes—
163C4×C47C2×#12411yescharacteristic
164C4×C47C2×#12711yescharacteristic
165Dic47—44no—
166C22×C4×D4—471no—
167C22×C4⋊C8—471no—
168C22×C22⋊C8—474no—
169C22×#97D4711yescharacteristic, complemented
170#1955—478no—
171C2×C47#195518yescomplemented
172C2×C47C2×#39011yescharacteristic
173C2×C47C2×#40711yescharacteristic
174C2×C47C2×#40811yescharacteristic
175C2×C47C2×#43514yes—
176D47—48no—
177C23×D4—471no—
178C24×C4—472no—
179C23×C8—472no—
180C23×C8C2×D4712yes—
181C2×C4×D4—474no—
182C2×C4×D4—474no—
183C2×C4×D4—944no—
184C2×C4×D4—944no—
185C22×C4⋊C4—471no—
186C22×C22⋊C4—472no—
187C22×C42C2×D4711yescharacteristic
188C2×C4⋊C8—474no—
189C2×C4⋊C8—944no—
190C2×C22⋊C8—4716no—
191C2×C22⋊C8—9416no—
192C2×#97—24no—
193C2×#97C2×D4714yescomplemented
194C47C2×#195511yescharacteristic, complemented, Sylow
195C25—472no—
196C22×D4—474no—
197C22×D4—948no—
198C22×D4—944no—
199C23×C4—472no—
200C23×C4—474no—
201C23×C4—474no—
202C23×C4—944no—
203C23×C4—948no—
204C23×C4—948no—
205C23×C4—944no—
206C23×C4C4×D4711yescharacteristic
207C23×C4C4×D4711yescharacteristic
208C23×C4C22×D4711yescharacteristic
209C22×C8—24no—
210C22×C8—24no—
211C22×C8—474no—
212C22×C8—948no—
213C22×C8—944no—
214C22×C8D18818yescomplemented
215C22×C8C22×D4714yes—
216C4×D4—9416no—
217C4×D4—948no—
218C4×D4—948no—
219C2×C4⋊C4—474no—
220C2×C4⋊C4—944no—
221C2×C22⋊C4—478no—
222C2×C22⋊C4—948no—
223C2×C42C4×D4712yes—
224C2×C42D18814yes—
225C2×C42C22⋊D4714yes—
226C2×C42C22×D4712yes—
227C4⋊C8—948no—
228C22⋊C8—9432no—
229#97—28no—
230C24—478no—
231C24—474no—
232C24—944no—
233C24—948no—
234C24—944no—
235C24—948no—
236C24C2×C4×D4711yescharacteristic
237C2×D4—9416no—
238C2×D4—9432no—
239C2×D4—948no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—474no—
243C22×C4—944no—
244C22×C4—948no—
245C22×C4—948no—
246C22×C4—9416no—
247C22×C4—944no—
248C22×C4—948no—
249C22×C4—944no—
250C22×C4—948no—
251C22×C4—18816no—
252C22×C4—1888no—
253C22×C4—1888no—
254C22×C4(C2×D47)⋊C414yes—
255C22×C4(C2×D47)⋊C414yes—
256C22×C4C2×C4×D4712yes—
257C22×C4C2×C4×D4712yes—
258C22×C4C2×C4×D4711yescharacteristic
259C22×C4C2×D18812yes—
260C22×C4C2×D18814yes—
261C22×C4#1062535014yes—
262C22×C4C2×C22⋊D4712yes—
263C22×C4C23×D4711yescharacteristic
264C2×C8—216no—
265C2×C8—24no—
266C2×C8—24no—
267C2×C8—948no—
268C2×C8—944no—
269C2×C8—1888no—
270C2×C8C2×D18818yescomplemented
271C4⋊C4—948no—
272C22⋊C4—9416no—
273C42(C2×D47)⋊C418yes—
274C42C2×D18814yes—
275C42C2×C22⋊D4714yes—
276C23—478no—
277C23—944no—
278C23—948no—
279C23—9416no—
280C23—944no—
281C23—948no—
282C23—9432no—
283C23—1888no—
284C23—1888no—
285C23—18816no—
286C23C2×(C2×D47)⋊C412yes—
287C23C4×D18814yes—
288C23M16×D4714yes—
289C23#1309403214yes—
290C23C22×C4×D4711yescharacteristic
291D4—9432no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—944no—
297C2×C4—948no—
298C2×C4—1888no—
299C2×C4—18816no—
300C2×C4—1888no—
301C2×C4—1884no—
302C2×C4—1884no—
303C2×C4C2×(C2×D47)⋊C414yes—
304C2×C4C2×(C2×D47)⋊C414yes—
305C2×C4C2×(C2×D47)⋊C412yes—
306C2×C4C4×D18814yes—
307C2×C4#1309399818yes—
308C2×C4#1309401214yes—
309C2×C4#1309402114yes—
310C2×C4C22×C22⋊D4711yescharacteristic
311C2×C4C22×D18812yes—
312C2×C4C22×D18811yescharacteristic
313C2×C4C2×#1062535012yes—
314C8—28no—
315C8—1884no—
316C22—9416no—
317C22—948no—
318C22—1884no—
319C22—1888no—
320C22—1884no—
321C22—18832no—
322C22#1568902514yes—
323C22#1568908614yes—
324C22#1568909114yes—
325C22#15689196116yes—
326C22C22×(C2×D47)⋊C411yescharacteristic
327C22C2×C4×D18812yes—
328C22C2×M16×D4712yes—
329C22C2×#1309403212yes—
330C4—24no—
331C4—24no—
332C4—1884no—
333C4#1568902514yes—
334C4C2×#1309399812yes—
335C4C2×#1309401211yescharacteristic
336C4C2×#1309402111yescharacteristic
337C2—1888no—
338C2#2024418618yesdirect product factor
339C2C2×#1568902511yescharacteristic
340C2C2×#1568908611yescharacteristic
341C2C2×#1568909111yescharacteristic
342C2C2×#1568919614yes—
subgroupG/H
centerC23×C4C22×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC22×#97×C47C2
socleC24×C47C22×C4
prime radicalC23×D188C4
cosocleC22×C4×C47C24
hypercenterC22×#97D47
hypocenterC47C2×#1955
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C23×C4 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊(C47⋊C8)C2semidirect
3. C22×(C2×D47)⋊C8C2semidirect
4. #20244186C2direct (N × H)
5. C22×#97×C47C2semidirect
16. C2×C4⋊(C47⋊C8)C22semidirect
18. C2×(C2×D47)⋊C8C22semidirect
22. C2×#97×C47C22semidirect
51. C22×C8×C47D4semidirect
95. C2×C8×C47C2×D4semidirect
169. C22×#97D47semidirect
171. C2×C47#1955semidirect
193. C2×#97C2×D47semidirect
194. C47C2×#1955semidirect
214. C22×C8D188semidirect
270. C2×C8C2×D188semidirect
338. C2#20244186direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D1882
C22×C4⋊(C47⋊C8)2
C22×(C2×D47)⋊C82
#202441862
C22×#97×C472
C2×#195547