Finite Group Databasebeta

← C4×#15688916 #26131440

Subgroups — C4×#15688916 #26131440

Order 12032.

720 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 139 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×#15688916C212yesmaximal
2C4×#13093939C212yesmaximal, complemented
3#20241513C212yesmaximal, complemented
4C4×#127×C47C211yesmaximal, characteristic
5C2×#13093939C2214yes—
6C4×C22⋊Dic47—22no—
7#15688916C418yesdirect product factor
8#15688890—22no—
9C2×#127×C47C2211yescharacteristic
10C2×C4×D4×C47—22no—
11C2×C4×D4×C47C411yescharacteristic, complemented
12C2×C4×D4×C47C2211yescharacteristic
13C47×#390—22no—
14C47×#390C411yescharacteristic, complemented
15C47×#390C2211yescharacteristic
16C47×#384C411yescharacteristic, complemented
17C43×C47C411yescharacteristic, complemented
18C2×C22⋊Dic47—24no—
19C2×C4×Dic47—42no—
20#13093939—24no—
21#13093939C2×C418yescomplemented
22C47⋊#95—42no—
23C22×D4×C47—22no—
24C22×D4×C47C2×C411yescharacteristic
25C22×D4×C47C2311yescharacteristic
26C23×C4×C47—21no—
27C23×C4×C47D412yes—
28#127×C47—22no—
29#127×C47C2×C414yes—
30C4×D4×C47—41no—
31C4×D4×C47—41no—
32C4×D4×C47—48no—
33C4×D4×C47—41no—
34C4×D4×C47—41no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47C2×C411yescharacteristic
37C2×C4⋊C4×C47C2×C411yescharacteristic
38C2×C22⋊C4×C47—22no—
39C2×C22⋊C4×C47—22no—
40C2×C22⋊C4×C47—21no—
41C2×C22⋊C4×C47D412yes—
42C2×C42×C47—21no—
43C2×C42×C47—21no—
44C2×C42×C47C2×C411yescharacteristic
45C2×C42×C47C2×C411yescharacteristic
46C2×C42×C47C2×C411yescharacteristic
47C24×C47—21no—
48C24×C47C2×D412yes—
49C2×D4×C47—28no—
50C2×D4×C47—21no—
51C2×D4×C47—41no—
52C2×D4×C47—41no—
53C2×D4×C47—44no—
54C2×D4×C47—44no—
55C2×D4×C47—41no—
56C2×D4×C47—41no—
57C2×D4×C47C4212yescomplemented
58C2×D4×C47C22×C412yes—
59C2×D4×C47#4112yes—
60C22×C4×C47—21no—
61C22×C4×C47—21no—
62C22×C4×C47—22no—
63C22×C4×C47—22no—
64C22×C4×C47—21no—
65C22×C4×C47—44no—
66C22×C4×C47—44no—
67C22×C4×C47—41no—
68C22×C4×C47—41no—
69C22×C4×C47C22⋊C411yescharacteristic
70C22×C4×C47C22⋊C411yescharacteristic
71C22×C4×C47C22⋊C412yes—
72C22×C4×C47C22×C411yescharacteristic
73C22×Dic47—44no—
74C4⋊C4×C47—41no—
75C4⋊C4×C47—44no—
76C4⋊C4×C47—41no—
77C22⋊C4×C47—42no—
78C22⋊C4×C47—44no—
79C22⋊C4×C47—44no—
80C22⋊C4×C47—42no—
81C42×C47—24no—
82C42×C47—21no—
83C42×C47—21no—
84C42×C47—21no—
85C42×C47—42no—
86C42×C47—41no—
87C42×C47C4212yescomplemented
88C22⋊Dic47—28no—
89C22⋊Dic47—44no—
90C4×Dic47—84no—
91C23×C47—22no—
92C23×C47—22no—
93C23×C47—22no—
94C23×C47—42no—
95C23×C47—42no—
96C23×C47—42no—
97C23×C47—42no—
98C23×C47—41no—
99C23×C47—41no—
100C23×C47C2×C22⋊C411yescharacteristic
101C23×C47C4×D414yes—
102D4×C47—42no—
103D4×C47—42no—
104D4×C47—42no—
105D4×C47—416no—
106D4×C47—42no—
107C2×C4×C47—22no—
108C2×C4×C47—22no—
109C2×C4×C47—22no—
110C2×C4×C47—21no—
111C2×C4×C47—21no—
112C2×C4×C47—22no—
113C2×C4×C47—22no—
114C2×C4×C47—21no—
115C2×C4×C47—22no—
116C2×C4×C47—41no—
117C2×C4×C47—41no—
118C2×C4×C47—44no—
119C2×C4×C47—41no—
120C2×C4×C47—42no—
121C2×C4×C47—42no—
122C2×C4×C47—41no—
123C2×C4×C47—82no—
124C2×C4×C47—82no—
125C2×C4×C47—82no—
126C2×C4×C47#9911yescharacteristic
127C2×C4×C47#9911yescharacteristic
128C2×C4×C47C2×C4211yescharacteristic
129C2×C4×C47#11711yescharacteristic
130C2×Dic47—48no—
131C2×Dic47—84no—
132C2×Dic47—84no—
133C4×#353—471nomaximal, Sylow
134C22×C47—21no—
135C22×C47—21no—
136C22×C47—24no—
137C22×C47—44no—
138C22×C47—44no—
139C22×C47—42no—
140C22×C47—42no—
141C22×C47—42no—
142C22×C47—82no—
143C22×C47—82no—
144C22×C47—81no—
145C22×C47—81no—
146C22×C47C4×C22⋊C411yescharacteristic
147C22×C47C2×#9911yescharacteristic
148C22×C47#41011yescharacteristic
149C4×C47—24no—
150C4×C47—21no—
151C4×C47—21no—
152C4×C47—21no—
153C4×C47—42no—
154C4×C47—42no—
155C4×C47—41no—
156C4×C47—81no—
157C4×C47#35312yescomplemented
158Dic47—88no—
159C4×#127D4711yescharacteristic
160C2×#353—472no—
161C4×#99—472no—
162#1393—472no—
163C2×C47—21no—
164C2×C47—21no—
165C2×C47—44no—
166C2×C47—82no—
167C2×C47C4×#9911yescharacteristic
168C2×C47C2×#35311yescharacteristic
169C2×C47#212611yescharacteristic
170C2×#127C2×D4711yescharacteristic
171C2×C4×D4—22no—
172C2×C4×D4Dic4711yescharacteristic, complemented
173C2×C4×D4C2×D4711yescharacteristic
174C2×#99—474no—
175#390—22no—
176#390Dic4711yescharacteristic, complemented
177#390C2×D4711yescharacteristic
178#384Dic4711yescharacteristic, complemented
179C4×C22⋊C4—942no—
180C43Dic4711yescharacteristic, complemented
181#353—478no—
182#342—942no—
183C47C4×#35311yescharacteristic, complemented, Sylow
184C22×D4—22no—
185C22×D4C2×Dic4711yescharacteristic
186C22×D4C22×D4711yescharacteristic
187C23×C4—21no—
188C23×C4C22⋊D4712yes—
189#127—22no—
190#127C4×D4714yes—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—48no—
194C4×D4—41no—
195C4×D4—41no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×Dic4711yescharacteristic
198C2×C4⋊C4C2×Dic4711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—944no—
203C2×C22⋊C4C22⋊D4712yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—1882no—
207C2×C42C2×Dic4711yescharacteristic
208C2×C42C2×Dic4711yescharacteristic
209C2×C42C2×Dic4711yescharacteristic
210#99—478no—
211#99—944no—
212#95—1882no—
213C24—21no—
214C24C2×C22⋊D4712yes—
215C2×D4—28no—
216C2×D4—21no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4C4×Dic4712yescomplemented
224C2×D4C2×C4×D4712yes—
225C2×D4#1062535012yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—1884no—
236C22×C4C22⋊Dic4711yescharacteristic
237C22×C4C22⋊Dic4712yes—
238C22×C4C22⋊Dic4711yescharacteristic
239C22×C4C22×Dic4711yescharacteristic
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C22⋊C4—948no—
248C22⋊C4—1884no—
249C42—24no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—42no—
255C42—3764no—
256C42C4×Dic4712yescomplemented
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic4711yescharacteristic
267C23C4×C22⋊D4714yes—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—416no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—3764no—
293C2×C4—3764no—
294C2×C4—1888no—
295C2×C4#1309393911yescharacteristic
296C2×C4#1309393911yescharacteristic
297C2×C4C2×C4×Dic4711yescharacteristic
298C2×C4#1309398311yescharacteristic
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—44no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22C4×C22⋊Dic4711yescharacteristic
312C22C2×#1309393911yescharacteristic
313C22#1568909711yescharacteristic
314C4—24no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—42no—
320C4—41no—
321C4—81no—
322C4—3768no—
323C4#1568891612yesdirect product factor
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2C4×#1309393911yescharacteristic
329C2C2×#1568891611yescharacteristic
330C2#2024505311yescharacteristic
subgroupG/H
centerC2×C4#13093939
commutatorC2×C4×C47C2×C42
FrattiniC22×D4C22×D47
FittingC4×#127×C47C2
socleC22×C47C2×#99
prime radicalC2×#127×C47C22
cosocleC22×D4×C47C23
hypercenterC4×#127D47
hypocenterC47C4×#353
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#127

Complements

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

NH ≅ G/Ntype
2. C4×#13093939C2semidirect
3. #20241513C2semidirect
7. #15688916C4direct (N × H)
11. C2×C4×D4×C47C4semidirect
14. C47×#390C4semidirect
16. C47×#384C4semidirect
17. C43×C47C4semidirect
21. #13093939C2×C4semidirect
57. C2×D4×C47C42semidirect
87. C42×C47C42semidirect
157. C4×C47#353semidirect
172. C2×C4×D4Dic47semidirect
176. #390Dic47semidirect
178. #384Dic47semidirect
180. C43Dic47semidirect
183. C47C4×#353semidirect
223. C2×D4C4×Dic47semidirect
256. C42C4×Dic47semidirect
323. C4#15688916direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156889162
C4×#130939392
#202415132
C4×#127×C472
C4×#35347