Finite Group Databasebeta

← C4×#353×C127 #44576298

Subgroups — C4×#353×C127 #44576298

Order 32512.

720 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C4×#127×C127C211yesmaximal, characteristic
2C2×#353×C127C212yesmaximal
3C4×#99×C127C212yesmaximal, complemented
4C127×#1393C212yesmaximal, complemented
5C2×#127×C127C2211yescharacteristic
6C2×C4×D4×C127—22no—
7C2×C4×D4×C127C411yescharacteristic, complemented
8C2×C4×D4×C127C2211yescharacteristic
9C2×#99×C127C2214yes—
10#390×C127—22no—
11#390×C127C411yescharacteristic, complemented
12#390×C127C2211yescharacteristic
13#384×C127C411yescharacteristic, complemented
14C4×C22⋊C4×C127—22no—
15C43×C127C411yescharacteristic, complemented
16#353×C127C418yesdirect product factor
17#342×C127—22no—
18C22×D4×C127—22no—
19C22×D4×C127C2×C411yescharacteristic
20C22×D4×C127C2311yescharacteristic
21C23×C4×C127—21no—
22C23×C4×C127D412yes—
23#127×C127—22no—
24#127×C127C2×C414yes—
25C4×D4×C127—41no—
26C4×D4×C127—41no—
27C4×D4×C127—48no—
28C4×D4×C127—41no—
29C4×D4×C127—41no—
30C2×C4⋊C4×C127—22no—
31C2×C4⋊C4×C127C2×C411yescharacteristic
32C2×C4⋊C4×C127C2×C411yescharacteristic
33C2×C22⋊C4×C127—24no—
34C2×C22⋊C4×C127—22no—
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—21no—
37C2×C22⋊C4×C127D412yes—
38C2×C42×C127—21no—
39C2×C42×C127—21no—
40C2×C42×C127—42no—
41C2×C42×C127C2×C411yescharacteristic
42C2×C42×C127C2×C411yescharacteristic
43C2×C42×C127C2×C411yescharacteristic
44#99×C127—24no—
45#99×C127C2×C418yescomplemented
46#95×C127—42no—
47C24×C127—21no—
48C24×C127C2×D412yes—
49C2×D4×C127—28no—
50C2×D4×C127—21no—
51C2×D4×C127—41no—
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127—41no—
55C2×D4×C127—44no—
56C2×D4×C127—44no—
57C2×D4×C127C4212yescomplemented
58C2×D4×C127C22×C412yes—
59C2×D4×C127#4112yes—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127—21no—
65C22×C4×C127—44no—
66C22×C4×C127—44no—
67C22×C4×C127—44no—
68C22×C4×C127—41no—
69C22×C4×C127—41no—
70C22×C4×C127C22⋊C411yescharacteristic
71C22×C4×C127C22⋊C412yes—
72C22×C4×C127C22⋊C411yescharacteristic
73C22×C4×C127C22×C411yescharacteristic
74C4⋊C4×C127—41no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127—44no—
77C22⋊C4×C127—28no—
78C22⋊C4×C127—42no—
79C22⋊C4×C127—44no—
80C22⋊C4×C127—44no—
81C22⋊C4×C127—42no—
82C22⋊C4×C127—44no—
83C42×C127—24no—
84C42×C127—21no—
85C42×C127—21no—
86C42×C127—21no—
87C42×C127—41no—
88C42×C127—42no—
89C42×C127—84no—
90C42×C127C4212yescomplemented
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—42no—
95C23×C127—42no—
96C23×C127—42no—
97C23×C127—42no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127C2×C22⋊C411yescharacteristic
101C23×C127C4×D414yes—
102D4×C127—42no—
103D4×C127—42no—
104D4×C127—42no—
105D4×C127—416no—
106D4×C127—42no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—21no—
115C2×C4×C127—22no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—44no—
119C2×C4×C127—41no—
120C2×C4×C127—42no—
121C2×C4×C127—48no—
122C2×C4×C127—42no—
123C2×C4×C127—41no—
124C2×C4×C127—82no—
125C2×C4×C127—82no—
126C2×C4×C127—84no—
127C2×C4×C127—84no—
128C2×C4×C127—82no—
129C2×C4×C127#9911yescharacteristic
130C2×C4×C127#9911yescharacteristic
131C2×C4×C127C2×C4211yescharacteristic
132C2×C4×C127#11711yescharacteristic
133C22×C127—21no—
134C22×C127—21no—
135C22×C127—24no—
136C22×C127—44no—
137C22×C127—44no—
138C22×C127—42no—
139C22×C127—42no—
140C22×C127—42no—
141C22×C127—82no—
142C22×C127—82no—
143C22×C127—81no—
144C22×C127—81no—
145C22×C127C4×C22⋊C411yescharacteristic
146C22×C127C2×#9911yescharacteristic
147C22×C127#41011yescharacteristic
148C4×C127—24no—
149C4×C127—21no—
150C4×C127—21no—
151C4×C127—21no—
152C4×C127—42no—
153C4×C127—42no—
154C4×C127—41no—
155C4×C127—88no—
156C4×C127—81no—
157C4×C127#35312yesdirect product factor
158C4×#353C12711yesmaximal, characteristic, direct product factor, Sylow
159C2×C127—21no—
160C2×C127—21no—
161C2×C127—44no—
162C2×C127—82no—
163C2×C127C4×#9911yescharacteristic
164C2×C127C2×#35311yescharacteristic
165C2×C127#212611yescharacteristic
166C4×#127C2×C12711yescharacteristic
167C2×#353C2×C12712yes—
168C4×#99C2×C12712yescomplemented
169#1393C2×C12712yescomplemented
170C127C4×#35311yescharacteristic, direct product factor, Sylow
171C2×#127C22×C12711yescharacteristic
172C2×C4×D4—22no—
173C2×C4×D4C4×C12711yescharacteristic, complemented
174C2×C4×D4C22×C12711yescharacteristic
175C2×#99C22×C12714yes—
176#390—22no—
177#390C4×C12711yescharacteristic, complemented
178#390C22×C12711yescharacteristic
179#384C4×C12711yescharacteristic, complemented
180C4×C22⋊C4—22no—
181C43C4×C12711yescharacteristic, complemented
182#353C4×C12718yesdirect product factor
183#342—22no—
184C22×D4—22no—
185C22×D4C2×C4×C12711yescharacteristic
186C22×D4C23×C12711yescharacteristic
187C23×C4—21no—
188C23×C4D4×C12712yes—
189#127—22no—
190#127C2×C4×C12714yes—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—48no—
194C4×D4—41no—
195C4×D4—41no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×C4×C12711yescharacteristic
198C2×C4⋊C4C2×C4×C12711yescharacteristic
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4D4×C12712yes—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—42no—
207C2×C42C2×C4×C12711yescharacteristic
208C2×C42C2×C4×C12711yescharacteristic
209C2×C42C2×C4×C12711yescharacteristic
210#99—24no—
211#99C2×C4×C12718yescomplemented
212#95—42no—
213C24—21no—
214C24C2×D4×C12712yes—
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×D4C42×C12712yescomplemented
224C2×D4C22×C4×C12712yes—
225C2×D4#41×C12712yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4C22⋊C4×C12711yescharacteristic
237C22×C4C22⋊C4×C12711yescharacteristic
238C22×C4C22⋊C4×C12712yes—
239C22×C4C22×C4×C12711yescharacteristic
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—44no—
243C22⋊C4—28no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—44no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C42—24no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—42no—
255C42—84no—
256C42C42×C12712yescomplemented
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊C4×C12711yescharacteristic
267C23C4×D4×C12714yes—
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—48no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—82no—
295C2×C4#99×C12711yescharacteristic
296C2×C4#99×C12711yescharacteristic
297C2×C4C2×C42×C12711yescharacteristic
298C2×C4#117×C12711yescharacteristic
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⋊C4×C12711yescharacteristic
312C22C2×#99×C12711yescharacteristic
313C22#410×C12711yescharacteristic
314C4—24no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—42no—
320C4—41no—
321C4—88no—
322C4—81no—
323C4#353×C12712yesdirect product factor
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2C4×#99×C12711yescharacteristic
329C2C2×#353×C12711yescharacteristic
330C2C127×#212611yescharacteristic
subgroupG/H
centerC2×C4×C127#99
commutatorC2×C4C2×C42×C127
Frattini, cosocleC22×D4C23×C127
socleC22×C127C2×#99
prime radicalC2×#127×C127C22
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#99×C127C2semidirect
4. C127×#1393C2semidirect
7. C2×C4×D4×C127C4semidirect
11. #390×C127C4semidirect
13. #384×C127C4semidirect
15. C43×C127C4semidirect
16. #353×C127C4direct (N × H)
45. #99×C127C2×C4semidirect
57. C2×D4×C127C42semidirect
90. C42×C127C42semidirect
157. C4×C127#353direct (N × H)
158. C4×#353C127direct (N × H)
168. C4×#99C2×C127semidirect
169. #1393C2×C127semidirect
170. C127C4×#353direct (N × H)
173. C2×C4×D4C4×C127semidirect
177. #390C4×C127semidirect
179. #384C4×C127semidirect
181. C43C4×C127semidirect
182. #353C4×C127direct (N × H)
211. #99C2×C4×C127semidirect
223. C2×D4C42×C127semidirect
256. C42C42×C127semidirect
323. C4#353×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#127×C1272
C2×#353×C1272
C4×#99×C1272
C127×#13932
C4×#353127