Finite Group Databasebeta

← C127×#12697 #44576370

Subgroups — C127×#12697 #44576370

Order 32512.

1182 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 150 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×#535×C127C211yesmaximal, characteristic
2C2×#353×C127C214yesmaximal, complemented
3C127×#1782C212yesmaximal, complemented
4C22×#41×C127C2211yescharacteristic
5C23×D4×C127C411yescharacteristic, complemented
6#535×C127—28no—
7C2×#127×C127C2212yes—
8C2×#124×C127C412yescomplemented
9C2×#121×C127—22no—
10C2×#120×C127—22no—
11C2×#117×C127C411yescharacteristic, complemented
12C2×#99×C127—22no—
13C2×#99×C127C2214yescomplemented
14#381×C127—22no—
15#353×C127—28no—
16C25×C127—21no—
17C2×#41×C127—24no—
18C2×#41×C127—42no—
19C2×#41×C127—42no—
20C22×Q8×C127C2×C411yescharacteristic
21C22×D4×C127—24no—
22C22×D4×C127—24no—
23C22×D4×C127—21no—
24C22×D4×C127—42no—
25C22×D4×C127—42no—
26C22×D4×C127C2×C412yes—
27C22×D4×C127D412yes—
28C22×D4×C127C2311yescharacteristic
29C23×C4×C127C2×C411yescharacteristic
30C23×C4×C127D412yes—
31#127×C127—24no—
32#127×C127D418yes—
33#124×C127—24no—
34#124×C127—24no—
35#121×C127—216no—
36#120×C127—216no—
37#117×C127—24no—
38C2×C4⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—24no—
41C2×C22⋊C4×C127—24no—
42C2×C42×C127C2×C412yes—
43#99×C127—28no—
44#99×C127—44no—
45#95×C127—44no—
46C24×C127—24no—
47C24×C127—22no—
48C24×C127—42no—
49C24×C127—42no—
50C24×C127—84no—
51C24×C127C22⋊C411yescharacteristic
52C24×C127C2×D412yes—
53#41×C127—416no—
54C2×Q8×C127—22no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×D4×C127—24no—
58C2×D4×C127—24no—
59C2×D4×C127—24no—
60C2×D4×C127—216no—
61C2×D4×C127—22no—
62C2×D4×C127—42no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—42no—
66C2×D4×C127—42no—
67C2×D4×C127—416no—
68C2×D4×C127—42no—
69C2×D4×C127—42no—
70C2×D4×C127—88no—
71C2×D4×C127C22⋊C414yes—
72C2×D4×C127C2×D414yes—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—22no—
77C22×C4×C127—24no—
78C22×C4×C127—24no—
79C22×C4×C127—42no—
80C22×C4×C127—44no—
81C22×C4×C127C22⋊C412yes—
82C22×C4×C127C22⋊C411yescharacteristic
83C22×C4×C127C22×C411yescharacteristic
84C4⋊C4×C127—24no—
85C22⋊C4×C127—24no—
86C22⋊C4×C127—216no—
87C22⋊C4×C127—44no—
88C22⋊C4×C127—44no—
89C42×C127—22no—
90C42×C127C22⋊C414yes—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—21no—
94C23×C127—22no—
95C23×C127—24no—
96C23×C127—42no—
97C23×C127—44no—
98C23×C127—48no—
99C23×C127—48no—
100C23×C127—42no—
101C23×C127—42no—
102C23×C127—88no—
103C23×C127—88no—
104C23×C127—88no—
105C23×C127#9912yes—
106C23×C127C2×C22⋊C411yescharacteristic
107C23×C127#12014yes—
108Q8×C127—44no—
109D4×C127—44no—
110D4×C127—44no—
111D4×C127—48no—
112D4×C127—44no—
113D4×C127—48no—
114D4×C127—816no—
115C2×C4×C127—28no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—24no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—22no—
124C2×C4×C127—48no—
125C2×C4×C127—44no—
126C2×C4×C127—84no—
127C2×C4×C127—88no—
128C2×C4×C127#9912yes—
129C2×C4×C127C2×C22⋊C412yes—
130C22×C127—22no—
131C22×C127—22no—
132C22×C127—24no—
133C22×C127—44no—
134C22×C127—42no—
135C22×C127—42no—
136C22×C127—48no—
137C22×C127—44no—
138C22×C127—42no—
139C22×C127—44no—
140C22×C127—82no—
141C22×C127—84no—
142C22×C127—816no—
143C22×C127—84no—
144C22×C127#37911yescharacteristic
145C22×C127C2×#9911yescharacteristic
146C22×C127C2×#9911yescharacteristic
147C4×C127—22no—
148C4×C127—22no—
149C4×C127—44no—
150C4×C127—42no—
151C4×C127—164no—
152#12697C12711yesmaximal, characteristic, direct product factor, Sylow
153C2×C127—21no—
154C2×C127—21no—
155C2×C127—44no—
156C2×C127—42no—
157C2×C127—84no—
158C2×C127#178111yescharacteristic
159C2×C127#212812yes—
160C2×#535C2×C12711yescharacteristic
161C2×#353C2×C12714yescomplemented
162#1782C2×C12712yescomplemented
163C127#1269711yescharacteristic, direct product factor, Sylow
164C22×#41C22×C12711yescharacteristic
165C23×D4C4×C12711yescharacteristic, complemented
166#535—28no—
167C2×#127C22×C12712yes—
168C2×#124C4×C12712yescomplemented
169C2×#121—22no—
170C2×#120—22no—
171C2×#117C4×C12711yescharacteristic, complemented
172C2×#99—22no—
173C2×#99C22×C12714yescomplemented
174#381—22no—
175#353—28no—
176C25—21no—
177C2×#41—24no—
178C2×#41—42no—
179C2×#41—42no—
180C22×Q8C2×C4×C12711yescharacteristic
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4C2×C4×C12712yes—
187C22×D4D4×C12712yes—
188C22×D4C23×C12711yescharacteristic
189C23×C4C2×C4×C12711yescharacteristic
190C23×C4D4×C12712yes—
191#127—24no—
192#127D4×C12718yes—
193#124—24no—
194#124—24no—
195#121—216no—
196#120—216no—
197#117—24no—
198C2×C4⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C42C2×C4×C12712yes—
203#99—28no—
204#99—44no—
205#95—44no—
206C24—24no—
207C24—22no—
208C24—42no—
209C24—42no—
210C24—84no—
211C24C22⋊C4×C12711yescharacteristic
212C24C2×D4×C12712yes—
213#41—416no—
214C2×Q8—22no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—216no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—416no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—88no—
231C2×D4C22⋊C4×C12714yes—
232C2×D4C2×D4×C12714yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4C22⋊C4×C12712yes—
242C22×C4C22⋊C4×C12711yescharacteristic
243C22×C4C22×C4×C12711yescharacteristic
244C4⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—216no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—22no—
250C42C22⋊C4×C12714yes—
251C23—21no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—24no—
256C23—42no—
257C23—44no—
258C23—48no—
259C23—48no—
260C23—42no—
261C23—42no—
262C23—88no—
263C23—88no—
264C23—88no—
265C23#99×C12712yes—
266C23C2×C22⋊C4×C12711yescharacteristic
267C23#120×C12714yes—
268Q8—44no—
269D4—44no—
270D4—44no—
271D4—48no—
272D4—44no—
273D4—48no—
274D4—816no—
275C2×C4—28no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—48no—
285C2×C4—44no—
286C2×C4—84no—
287C2×C4—88no—
288C2×C4#99×C12712yes—
289C2×C4C2×C22⋊C4×C12712yes—
290C22—22no—
291C22—22no—
292C22—24no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22—48no—
297C22—44no—
298C22—42no—
299C22—44no—
300C22—82no—
301C22—84no—
302C22—816no—
303C22—84no—
304C22#379×C12711yescharacteristic
305C22C2×#99×C12711yescharacteristic
306C22C2×#99×C12711yescharacteristic
307C4—22no—
308C4—22no—
309C4—44no—
310C4—42no—
311C4—164no—
312C2—21no—
313C2—21no—
314C2—44no—
315C2—42no—
316C2—84no—
317C2C127×#178111yescharacteristic
318C2C127×#212812yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC2×#535×C127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#353×C127C2semidirect
3. C127×#1782C2semidirect
5. C23×D4×C127C4semidirect
8. C2×#124×C127C4semidirect
11. C2×#117×C127C4semidirect
13. C2×#99×C127C22semidirect
152. #12697C127direct (N × H)
161. C2×#353C2×C127semidirect
162. #1782C2×C127semidirect
163. C127#12697direct (N × H)
165. C23×D4C4×C127semidirect
168. C2×#124C4×C127semidirect
171. C2×#117C4×C127semidirect
173. C2×#99C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#535×C1272
C2×#353×C1272
C127×#17822
#12697127