Finite Group Databasebeta

← C2×#20241477 #26131147

Subgroups — C2×#20241477 #26131147

Order 12032.

606 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093964C211yesmaximal, characteristic
2C23×C47⋊C16C211yesmaximal, characteristic
3#20241477C214yesmaximal, direct product factor
4C2×C47×#405C211yesmaximal, characteristic
5C2×#13093964—22no—
6C2×#13093964—21no—
7C2×#13093964C414yescomplemented
8C2×#13093964C2212yes—
9C22×C47⋊C16—22no—
10C22×C47⋊C16—21no—
11C22×C47⋊C16—21no—
12C22×C47⋊C16—21no—
13C22×C47⋊C16C414yescomplemented
14C22×C47⋊C16C2212yes—
15C22×M16×C47C411yescharacteristic
16C23×C8×C47C2211yescharacteristic
17C2×#117×C47C411yescharacteristic
18C47×#405—21no—
19C47×#405—21no—
20C47×#405C412yes—
21C47×#405C2212yes—
22C2×#97×C47—21no—
23C2×C4×C8×C47—21no—
24#13093964—24no—
25#13093964—22no—
26#13093964C2×C414yescomplemented
27C2×C47⋊C16—24no—
28C2×C47⋊C16—24no—
29C2×C47⋊C16—24no—
30C2×C47⋊C16—22no—
31C2×C47⋊C16—22no—
32C2×C47⋊C16—22no—
33C2×C47⋊C16C2×C414yescomplemented
34C23×C4×C47C2×C411yescharacteristic
35C2×M16×C47—21no—
36C2×M16×C47—22no—
37C2×M16×C47C814yes—
38C2×M16×C47C2×C412yes—
39C22×C8×C47—21no—
40C22×C8×C47C2×C411yescharacteristic
41C22×C8×C47C2×C412yes—
42C22×C8×C47C2×C412yes—
43C22×C8×C47D412yes—
44C22×C8×C47D411yescharacteristic
45C22×C8×C47D411yescharacteristic
46C22×C8×C47D411yescharacteristic
47C22×C8×C47D411yescharacteristic
48C22×C8×C47Q812yes—
49C22×C8×C47C2311yescharacteristic
50#117×C47—21no—
51#117×C47—21no—
52#117×C47C812yes—
53#117×C47C2×C412yes—
54C2×C4⋊C4×C47C812yes—
55C2×C22⋊C4×C47—21no—
56C2×C42×C47—21no—
57#97×C47—22no—
58#97×C47—22no—
59C4×C8×C47—22no—
60C4×C8×C47—22no—
61C24×C47M1611yescharacteristic
62C22×C4×C47—22no—
63C22×C4×C47C4212yes—
64C22×C4×C47C22⋊C411yescharacteristic
65C22×C4×C47C22⋊C411yescharacteristic
66C22×C4×C47C4⋊C412yes—
67C22×C4×C47C2×C812yes—
68C22×C4×C47C2×C811yescharacteristic
69C22×C4×C47C2×C811yescharacteristic
70C22×C4×C47M1611yescharacteristic
71C22×C4×C47M1612yes—
72C22×C4×C47C22×C411yescharacteristic
73M16×C47—24no—
74M16×C47—22no—
75M16×C47C2×C814yes—
76C2×C8×C47—24no—
77C2×C8×C47—22no—
78C2×C8×C47C4212yes—
79C2×C8×C47C22⋊C411yescharacteristic
80C2×C8×C47C22⋊C411yescharacteristic
81C2×C8×C47C22⋊C412yes—
82C2×C8×C47C22⋊C412yes—
83C2×C8×C47C22⋊C412yes—
84C2×C8×C47C22⋊C412yes—
85C2×C8×C47C4⋊C412yes—
86C2×C8×C47C4⋊C412yes—
87C2×C8×C47C4⋊C412yes—
88C2×C8×C47C4⋊C412yes—
89C2×C8×C47C4⋊C412yes—
90C2×C8×C47C22×C411yescharacteristic
91C2×C8×C47C22×C411yescharacteristic
92C2×C8×C47C2×D411yescharacteristic
93C2×C8×C47C2×D411yescharacteristic
94C2×C8×C47C2×D411yescharacteristic
95C2×C8×C47C2×Q811yescharacteristic
96C4⋊C4×C47—22no—
97C4⋊C4×C47C2×C814yes—
98C22⋊C4×C47—22no—
99C22⋊C4×C47—22no—
100C42×C47—22no—
101C42×C47—22no—
102C47⋊C16—24no—
103C47⋊C16—24no—
104C23×C47—24no—
105C23×C47#9712yes—
106C23×C47C22⋊C811yescharacteristic
107C23×C47C22⋊C811yescharacteristic
108C23×C47C4⋊C812yes—
109C23×C47C2×M1611yescharacteristic
110C2×C4×C47—24no—
111C2×C4×C47—44no—
112C2×C4×C47#9512yes—
113C2×C4×C47#9512yes—
114C2×C4×C47C4×C812yes—
115C2×C4×C47C4×C812yes—
116C2×C4×C47#9712yes—
117C2×C4×C47C22⋊C811yescharacteristic
118C2×C4×C47C22⋊C811yescharacteristic
119C2×C4×C47C22⋊C812yes—
120C2×C4×C47C22⋊C812yes—
121C2×C4×C47C4⋊C812yes—
122C2×C4×C47C4⋊C812yes—
123C2×C4×C47C4⋊C812yes—
124C2×C4×C47C2×C4211yescharacteristic
125C2×C4×C47C2×C22⋊C411yescharacteristic
126C2×C4×C47C2×C4⋊C411yescharacteristic
127C2×C4×C47C22×C811yescharacteristic
128C2×C4×C47C22×C811yescharacteristic
129C2×C4×C47C2×M1611yescharacteristic
130C8×C47—24no—
131C8×C47#9512yes—
132C8×C47#9512yes—
133C8×C47C2×C22⋊C411yescharacteristic
134C8×C47C2×C22⋊C411yescharacteristic
135C8×C47C2×C4⋊C411yescharacteristic
136C8×C47C2×C4⋊C411yescharacteristic
137C2×#1379—471nomaximal, Sylow
138C22×C47—22no—
139C22×C47—22no—
140C22×C47—24no—
141C22×C47—24no—
142C22×C47#33612yes—
143C22×C47#33612yes—
144C22×C47#35014yes—
145C22×C47C2×#9711yescharacteristic
146C22×C47C2×C22⋊C811yescharacteristic
147C22×C47C2×C4⋊C811yescharacteristic
148C4×C47—44no—
149C4×C47#33612yes—
150C4×C47#33612yes—
151C4×C47C2×#9511yescharacteristic
152C4×C47C2×C4×C811yescharacteristic
153C4×C47C2×C22⋊C811yescharacteristic
154C4×C47C2×C4⋊C811yescharacteristic
155C22×M32—471no—
156C23×C16—471no—
157C2×#405D4711yescharacteristic
158#1379—474no—
159C2×C47—22no—
160C2×C47—21no—
161C2×C47—21no—
162C2×C47#137914yescomplemented
163C2×C47C2×#33611yescharacteristic
164C2×C47C2×#35012yes—
165C22×M16Dic4711yescharacteristic
166C23×C8C2×D4711yescharacteristic
167C2×#117Dic4711yescharacteristic
168C2×M32—474no—
169C2×M32—472no—
170C2×M32—942no—
171C2×M32—941no—
172C22×C16—474no—
173C22×C16—472no—
174C22×C16—942no—
175C22×C16—941no—
176C22×C16—941no—
177C22×C16—941no—
178#405—21no—
179#405—21no—
180#405Dic4712yes—
181#405C2×D4712yes—
182C2×#97—21no—
183C2×C4×C8—21no—
184C47C2×#137911yescharacteristic, complemented, Sylow
185C23×C4C2×Dic4711yescharacteristic
186C2×M16—22no—
187C2×M16—21no—
188C2×M16C47⋊C814yes—
189C2×M16C2×Dic4712yes—
190C22×C8—21no—
191C22×C8Dic9412yes—
192C22×C8C4×D4712yes—
193C22×C8C4×D4712yes—
194C22×C8D18812yes—
195C22×C8C2×Dic4711yescharacteristic
196C22×C8C22⋊D4711yescharacteristic
197C22×C8C22⋊D4711yescharacteristic
198C22×C8C22⋊D4711yescharacteristic
199C22×C8C22⋊D4711yescharacteristic
200C22×C8C22×D4711yescharacteristic
201#117—21no—
202#117—21no—
203#117C47⋊C812yes—
204#117C2×Dic4712yes—
205C2×C4⋊C4C47⋊C812yes—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208M32—474no—
209M32—944no—
210M32—942no—
211C2×C16—474no—
212C2×C16—944no—
213C2×C16—944no—
214C2×C16—944no—
215C2×C16—942no—
216C2×C16—942no—
217C2×C16—942no—
218#97—22no—
219#97—22no—
220C4×C8—22no—
221C4×C8—22no—
222C24#1062532911yescharacteristic
223C22×C4—22no—
224C22×C4C2×C47⋊C812yes—
225C22×C4C2×C47⋊C811yescharacteristic
226C22×C4C2×C47⋊C811yescharacteristic
227C22×C4#1062532911yescharacteristic
228C22×C4#1062532912yes—
229C22×C4C4×Dic4712yes—
230C22×C4C4⋊Dic4712yes—
231C22×C4C22⋊Dic4711yescharacteristic
232C22×C4C22⋊Dic4711yescharacteristic
233C22×C4C22×Dic4711yescharacteristic
234M16—24no—
235M16—22no—
236M16C2×C47⋊C814yes—
237C2×C8—24no—
238C2×C8—22no—
239C2×C8C4×Dic4712yes—
240C2×C8Dic47⋊C412yes—
241C2×C8Dic47⋊C412yes—
242C2×C8Dic47⋊C412yes—
243C2×C8Dic47⋊C412yes—
244C2×C8C4⋊Dic4712yes—
245C2×C8(C2×D47)⋊C412yes—
246C2×C8(C2×D47)⋊C412yes—
247C2×C8(C2×D47)⋊C412yes—
248C2×C8(C2×D47)⋊C412yes—
249C2×C8C22⋊Dic4711yescharacteristic
250C2×C8C22⋊Dic4711yescharacteristic
251C2×C8C2×Dic9411yescharacteristic
252C2×C8C2×C4×D4711yescharacteristic
253C2×C8C2×C4×D4711yescharacteristic
254C2×C8C2×D18811yescharacteristic
255C2×C8C2×C22⋊D4711yescharacteristic
256C2×C8C2×C22⋊D4711yescharacteristic
257C4⋊C4—22no—
258C4⋊C4C2×C47⋊C814yes—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C42—22no—
262C42—22no—
263C16—944no—
264C16—944no—
265C23—24no—
266C23#1309393512yes—
267C23C22⋊(C47⋊C8)11yescharacteristic
268C23C22⋊(C47⋊C8)11yescharacteristic
269C23C4⋊(C47⋊C8)12yes—
270C23C2×#1062532911yescharacteristic
271C2×C4—24no—
272C2×C4—44no—
273C2×C4C47⋊#9512yes—
274C2×C4C47⋊#9512yes—
275C2×C4C4×C47⋊C812yes—
276C2×C4C4×C47⋊C812yes—
277C2×C4#1309393512yes—
278C2×C4C22⋊(C47⋊C8)12yes—
279C2×C4C22⋊(C47⋊C8)12yes—
280C2×C4C22⋊(C47⋊C8)11yescharacteristic
281C2×C4C22⋊(C47⋊C8)11yescharacteristic
282C2×C4C4⋊(C47⋊C8)12yes—
283C2×C4C4⋊(C47⋊C8)12yes—
284C2×C4C4⋊(C47⋊C8)12yes—
285C2×C4C2×C4×Dic4711yescharacteristic
286C2×C4C2×C22⋊Dic4711yescharacteristic
287C2×C4C2×C4⋊Dic4711yescharacteristic
288C2×C4C22×C47⋊C811yescharacteristic
289C2×C4C22×C47⋊C811yescharacteristic
290C2×C4C2×#1062532911yescharacteristic
291C8—24no—
292C8C47⋊#9512yes—
293C8C47⋊#9512yes—
294C8C2×(C2×D47)⋊C411yescharacteristic
295C8C2×(C2×D47)⋊C411yescharacteristic
296C8C2×Dic47⋊C411yescharacteristic
297C8C2×Dic47⋊C411yescharacteristic
298C22—22no—
299C22—22no—
300C22—24no—
301C22—24no—
302C22#1568887812yes—
303C22#1568887812yes—
304C22#1568890714yes—
305C22C2×#1309393511yescharacteristic
306C22C2×C22⋊(C47⋊C8)11yescharacteristic
307C22C2×C4⋊(C47⋊C8)11yescharacteristic
308C4—44no—
309C4#1568887812yes—
310C4#1568887812yes—
311C4C2×C47⋊#9511yescharacteristic
312C4C2×C4×C47⋊C811yescharacteristic
313C4C2×C22⋊(C47⋊C8)11yescharacteristic
314C4C2×C4⋊(C47⋊C8)11yescharacteristic
315C2—21no—
316C2—22no—
317C2—21no—
318C2#2024147714yesdirect product factor
319C2C2×#1568887811yescharacteristic
320C2C2×#1568890712yes—
subgroupG/H
centerC22×C8C22⋊D47
commutatorC4×C47C2×C4×C8
FrattiniC22×C8C22×D47
FittingC2×C47×#405C2
socleC23×C47C22⋊C8
prime radicalC24×C47M16
cosocleC22×C8×C47C23
hypercenterC2×#405D47
hypocenterC47C2×#1379
LCSG ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22×C8 ◁ C23×C8 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
3. #20241477C2direct (N × H)
7. C2×#13093964C4semidirect
13. C22×C47⋊C16C4semidirect
26. #13093964C2×C4semidirect
33. C2×C47⋊C16C2×C4semidirect
162. C2×C47#1379semidirect
184. C47C2×#1379semidirect
318. C2#20241477direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130939642
C23×C47⋊C162
#202414772
C2×C47×#4052
C2×#137947