Finite Group Databasebeta

← C2×#20241394 #26126475

Subgroups — C2×#20241394 #26126475

Order 12032.

544 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 131 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×#15688892C211yesmaximal, characteristic
2C2×#15688849C211yesmaximal, characteristic
3#20241394C214yesmaximal, direct product factor
4C2×C47×#398C211yesmaximal, characteristic
5C22×#10625329—21no—
6C2×C22⋊(C47⋊C8)—21no—
7#15688892—21no—
8#15688892C2212yes—
9#15688849—21no—
10#15688849C2212yes—
11C22×C4⋊C4×C47—21no—
12C22×C4⋊C4×C47C411yescharacteristic
13C22×C4⋊C4×C47C2211yescharacteristic
14C47×#398—21no—
15C47×#398—21no—
16C47×#398C412yes—
17C47×#398C2212yes—
18C2×#95×C47—21no—
19C2×#95×C47C411yescharacteristic
20C2×#10625329—24no—
21C2×#10625329—22no—
22C2×#10625329—41no—
23C2×#10625329—42no—
24C22×C47⋊C8—41no—
25C22×C47⋊C8—41no—
26C22⋊(C47⋊C8)—22no—
27C22⋊(C47⋊C8)—41no—
28C23×C4×C47—21no—
29C23×C4×C47—21no—
30C23×C4×C47C2×C411yescharacteristic
31C23×C4×C47D411yescharacteristic
32C23×C4×C47D411yescharacteristic
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47—41no—
37C2×C4⋊C4×C47—41no—
38C2×C4⋊C4×C47—41no—
39C2×C4⋊C4×C47—41no—
40C2×C4⋊C4×C47—42no—
41C2×C4⋊C4×C47—42no—
42C2×C4⋊C4×C47C2×C412yes—
43C2×C4⋊C4×C47C2×C411yescharacteristic
44C2×C4⋊C4×C47D411yescharacteristic
45C2×C4⋊C4×C47D411yescharacteristic
46C2×C4⋊C4×C47C2311yescharacteristic
47#95×C47—22no—
48#95×C47—21no—
49#95×C47—22no—
50#95×C47C2×C412yes—
51C24×C47C22⋊C411yescharacteristic
52C22×C4×C47—21no—
53C22×C4×C47—21no—
54C22×C4×C47—22no—
55C22×C4×C47—24no—
56C22×C4×C47—21no—
57C22×C4×C47—21no—
58C22×C4×C47—21no—
59C22×C4×C47—22no—
60C22×C4×C47—21no—
61C22×C4×C47—22no—
62C22×C4×C47—42no—
63C22×C4×C47—42no—
64C22×C4×C47—42no—
65C22×C4×C47—42no—
66C22×C4×C47—41no—
67C22×C4×C47—41no—
68C22×C4×C47—42no—
69C22×C4×C47—42no—
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C411yescharacteristic
72C22×C4×C47C22⋊C411yescharacteristic
73C22×C4×C47D812yes—
74C22×C4×C47SD1612yes—
75C22×C4×C47C22×C411yescharacteristic
76C22×C4×C47C2×D411yescharacteristic
77C22×C4×C47C2×D411yescharacteristic
78C4⋊C4×C47—41no—
79C4⋊C4×C47—41no—
80C4⋊C4×C47—41no—
81C4⋊C4×C47—41no—
82C4⋊C4×C47—42no—
83C4⋊C4×C47—42no—
84C4⋊C4×C47—44no—
85C4⋊C4×C47—42no—
86C4⋊C4×C47—42no—
87#10625329—24no—
88#10625329—44no—
89#10625329—42no—
90C2×C47⋊C8—44no—
91C2×C47⋊C8—42no—
92C2×C47⋊C8—42no—
93C2×C47⋊C8—82no—
94C23×C47—22no—
95C23×C47—22no—
96C23×C47#9911yescharacteristic
97C23×C47#9911yescharacteristic
98C23×C47#10212yes—
99C23×C47#10412yes—
100C23×C47C2×C22⋊C411yescharacteristic
101C2×C4×C47—22no—
102C2×C4×C47—22no—
103C2×C4×C47—22no—
104C2×C4×C47—22no—
105C2×C4×C47—21no—
106C2×C4×C47—21no—
107C2×C4×C47—21no—
108C2×C4×C47—21no—
109C2×C4×C47—24no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—21no—
113C2×C4×C47—21no—
114C2×C4×C47—48no—
115C2×C4×C47—42no—
116C2×C4×C47—42no—
117C2×C4×C47—42no—
118C2×C4×C47—42no—
119C2×C4×C47—42no—
120C2×C4×C47—41no—
121C2×C4×C47—41no—
122C2×C4×C47—41no—
123C2×C4×C47—41no—
124C2×C4×C47—42no—
125C2×C4×C47—42no—
126C2×C4×C47—42no—
127C2×C4×C47—81no—
128C2×C4×C47—82no—
129C2×C4×C47—82no—
130C2×C4×C47#10212yes—
131C2×C4×C47C2×D811yescharacteristic
132C2×C4×C47C2×SD1611yescharacteristic
133C47⋊C8—44no—
134C47⋊C8—82no—
135C2×#1346—471nomaximal, Sylow
136C22×C47—24no—
137C22×C47—24no—
138C22×C47—21no—
139C22×C47—21no—
140C22×C47—21no—
141C22×C47—21no—
142C22×C47#32712yes—
143C22×C47#32712yes—
144C22×C47#35212yes—
145C22×C47#35612yes—
146C22×C47C2×#9911yescharacteristic
147C22×C47C2×#10211yescharacteristic
148C22×C47C2×#10411yescharacteristic
149C4×C47—21no—
150C4×C47—22no—
151C4×C47—21no—
152C4×C47—44no—
153C4×C47—41no—
154C4×C47—41no—
155C4×C47—41no—
156C4×C47—41no—
157C4×C47—82no—
158C2×#398D4711yescharacteristic
159C2×#343—471no—
160C2×#324—471no—
161#1346—474no—
162C2×C47—21no—
163C2×C47—22no—
164C2×C47—21no—
165C2×C47#134614yescomplemented
166C2×C47C2×#32711yescharacteristic
167C2×C47C2×#35211yescharacteristic
168C2×C47C2×#35611yescharacteristic
169C22×M16—941no—
170C22×C4⋊C4—21no—
171C22×C4⋊C4Dic4711yescharacteristic
172C22×C4⋊C4C2×D4711yescharacteristic
173C2×C22⋊C8—941no—
174#398—21no—
175#398—21no—
176#398Dic4712yes—
177#398C2×D4712yes—
178C2×#95—21no—
179C2×#95Dic4711yescharacteristic
180#343—472no—
181#343—941no—
182#324—472no—
183#324—941no—
184C47C2×#134611yescharacteristic, complemented, Sylow
185C23×C4—21no—
186C23×C4—21no—
187C23×C4C2×Dic4711yescharacteristic
188C23×C4C22⋊D4711yescharacteristic
189C23×C4C22⋊D4711yescharacteristic
190C2×M16—944no—
191C2×M16—942no—
192C2×M16—1881no—
193C2×M16—1882no—
194C22×C8—1881no—
195C22×C8—1881no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4C2×Dic4712yes—
206C2×C4⋊C4C2×Dic4711yescharacteristic
207C2×C4⋊C4C22⋊D4711yescharacteristic
208C2×C4⋊C4C22⋊D4711yescharacteristic
209C2×C4⋊C4C22×D4711yescharacteristic
210C22⋊C8—942no—
211C22⋊C8—1881no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95C2×Dic4712yes—
216C24C22⋊Dic4711yescharacteristic
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4#1062533412yes—
236C22×C4#1062533512yes—
237C22×C4C22⋊Dic4711yescharacteristic
238C22×C4C22⋊Dic4711yescharacteristic
239C22×C4C22⋊Dic4711yescharacteristic
240C22×C4C22×Dic4711yescharacteristic
241C22×C4C2×C22⋊D4711yescharacteristic
242C22×C4C2×C22⋊D4711yescharacteristic
243M16—944no—
244M16—1884no—
245M16—1882no—
246C2×C8—3762no—
247C2×C8—1882no—
248C2×C8—1882no—
249C2×C8—1884no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C23—22no—
260C23—22no—
261C23#1309393911yescharacteristic
262C23#1309393911yescharacteristic
263C23#1309394512yes—
264C23#1309395112yes—
265C23C2×C22⋊Dic4711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—48no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#1309394512yes—
296C2×C4C2×#1062533411yescharacteristic
297C2×C4C2×#1062533511yescharacteristic
298C8—3762no—
299C8—1884no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—24no—
306C22#1568885712yes—
307C22#1568885712yes—
308C22#1568891312yes—
309C22#1568892212yes—
310C22C2×#1309393911yescharacteristic
311C22C2×#1309394511yescharacteristic
312C22C2×#1309395111yescharacteristic
313C4—21no—
314C4—22no—
315C4—21no—
316C4—44no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—82no—
322C2—21no—
323C2—22no—
324C2—21no—
325C2#2024139414yesdirect product factor
326C2C2×#1568885711yescharacteristic
327C2C2×#1568891311yescharacteristic
328C2C2×#1568892211yescharacteristic
subgroupG/H
centerC23#13093939
commutatorC22×C4×C47C22×C4
FrattiniC2×C4⋊C4C22×D47
FittingC2×C47×#398C2
socleC23×C47#99
prime radicalC24×C47C22⋊C4
cosocleC2×C4⋊C4×C47C23
hypercenterC2×#398D47
hypocenterC47C2×#1346
LCSG ▷ C22×C4×C47 ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
3. #20241394C2direct (N × H)
165. C2×C47#1346semidirect
184. C47C2×#1346semidirect
325. C2#20241394direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156888922
C2×#156888492
#202413942
C2×C47×#3982
C2×#134647