Finite Group Databasebeta

← C127×#12708 #44576381

Subgroups — C127×#12708 #44576381

Order 32512.

1002 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1002 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C127×#3487C211yesmaximal, characteristic
2C127×#2127C214yesmaximal, complemented
3C127×#1782C212yesmaximal, complemented
4C2×2−1+4×C127C411yescharacteristic, complemented
5C22×#41×C127C2211yescharacteristic
6#550×C127—24no—
7#548×C127—24no—
8#535×C127—22no—
9#535×C127C2212yes—
10#525×C127—22no—
11#525×C127C412yescomplemented
12#519×C127C411yescharacteristic, complemented
13C2×#99×C127—22no—
14C2×#99×C127C2214yes—
15#381×C127—22no—
16#354×C127—24no—
17#353×C127—24no—
182−1+4×C127—44no—
19C2×#41×C127—22no—
20C2×#41×C127—24no—
21C2×#41×C127—22no—
22C2×#41×C127—44no—
23C2×#41×C127C2×C412yes—
24C22×Q8×C127—22no—
25C22×Q8×C127C2×C411yescharacteristic
26C22×D4×C127D412yes—
27C22×D4×C127C2311yescharacteristic
28C23×C4×C127C2×C411yescharacteristic
29C23×C4×C127D412yes—
30#127×C127—24no—
31#127×C127D414yes—
32#124×C127—24no—
33#124×C127D414yes—
34#123×C127—22no—
35C22⋊Q8×C127—24no—
36C22⋊Q8×C127—22no—
37#121×C127—24no—
38#121×C127—28no—
39#121×C127—22no—
40#121×C127—24no—
41#120×C127—22no—
42#120×C127—24no—
43C4×Q8×C127—24no—
44C4×D4×C127—28no—
45C4×D4×C127—24no—
46#117×C127—22no—
47#117×C127C2×C412yes—
48C2×C22⋊C4×C127—24no—
49#99×C127—28no—
50#99×C127—44no—
51#95×C127—44no—
52C24×C127C22⋊C411yescharacteristic
53C24×C127C2×D412yes—
54#41×C127—44no—
55#41×C127—44no—
56#41×C127—416no—
57#41×C127—48no—
58#41×C127—44no—
59C2×Q8×C127—22no—
60C2×Q8×C127—24no—
61C2×Q8×C127—22no—
62C2×Q8×C127—48no—
63C2×Q8×C127C22⋊C412yes—
64C2×D4×C127—22no—
65C2×D4×C127—22no—
66C2×D4×C127—24no—
67C2×D4×C127—24no—
68C2×D4×C127—28no—
69C2×D4×C127—24no—
70C2×D4×C127—42no—
71C2×D4×C127—42no—
72C2×D4×C127C22⋊C412yes—
73C2×D4×C127C2×D414yes—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—22no—
78C22×C4×C127—22no—
79C22×C4×C127—42no—
80C22×C4×C127—42no—
81C22×C4×C127—44no—
82C22×C4×C127C22⋊C411yescharacteristic
83C22×C4×C127C22⋊C412yes—
84C22×C4×C127C22×C411yescharacteristic
85C4⋊C4×C127—24no—
86C4⋊C4×C127—24no—
87C4⋊C4×C127—22no—
88C22⋊C4×C127—24no—
89C22⋊C4×C127—24no—
90C22⋊C4×C127—22no—
91C22⋊C4×C127—28no—
92C22⋊C4×C127—44no—
93C22⋊C4×C127—44no—
94C42×C127—24no—
95C42×C127C22⋊C414yes—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—21no—
99C23×C127—22no—
100C23×C127—22no—
101C23×C127—82no—
102C23×C127—81no—
103C23×C127C2×C22⋊C411yescharacteristic
104C23×C127#12014yes—
105Q8×C127—28no—
106Q8×C127—42no—
107Q8×C127—44no—
108D4×C127—44no—
109D4×C127—48no—
110D4×C127—42no—
111D4×C127—44no—
112D4×C127—44no—
113D4×C127—88no—
114D4×C127—84no—
115C2×C4×C127—22no—
116C2×C4×C127—22no—
117C2×C4×C127—22no—
118C2×C4×C127—24no—
119C2×C4×C127—22no—
120C2×C4×C127—21no—
121C2×C4×C127—22no—
122C2×C4×C127—48no—
123C2×C4×C127—44no—
124C2×C4×C127—48no—
125C2×C4×C127—44no—
126C2×C4×C127—84no—
127C2×C4×C127—88no—
128C2×C4×C127#9914yes—
129C2×C4×C127C2×C22⋊C412yes—
130C22×C127—24no—
131C22×C127—41no—
132C22×C127—44no—
133C22×C127—44no—
134C22×C127—44no—
135C22×C127—42no—
136C22×C127—42no—
137C22×C127—41no—
138C22×C127—82no—
139C22×C127—82no—
140C22×C127#37911yescharacteristic
141C22×C127C2×#9912yes—
142C4×C127—24no—
143C4×C127—22no—
144C4×C127—44no—
145C4×C127—41no—
146C4×C127—42no—
147C4×C127—164no—
148#12708C12711yesmaximal, characteristic, direct product factor, Sylow
149C2×C127—21no—
150C2×C127—22no—
151C2×C127—44no—
152C2×C127—82no—
153C2×C127—81no—
154C2×C127#178111yescharacteristic
155#3487C2×C12711yescharacteristic
156#2127C2×C12714yescomplemented
157#1782C2×C12712yescomplemented
158C127#1270811yescharacteristic, direct product factor, Sylow
159C2×2−1+4C4×C12711yescharacteristic, complemented
160C22×#41C22×C12711yescharacteristic
161#550—24no—
162#548—24no—
163#535—22no—
164#535C22×C12712yes—
165#525—22no—
166#525C4×C12712yescomplemented
167#519C4×C12711yescharacteristic, complemented
168C2×#99—22no—
169C2×#99C22×C12714yes—
170#381—22no—
171#354—24no—
172#353—24no—
1732−1+4—44no—
174C2×#41—22no—
175C2×#41—24no—
176C2×#41—22no—
177C2×#41—44no—
178C2×#41C2×C4×C12712yes—
179C22×Q8—22no—
180C22×Q8C2×C4×C12711yescharacteristic
181C22×D4D4×C12712yes—
182C22×D4C23×C12711yescharacteristic
183C23×C4C2×C4×C12711yescharacteristic
184C23×C4D4×C12712yes—
185#127—24no—
186#127D4×C12714yes—
187#124—24no—
188#124D4×C12714yes—
189#123—22no—
190C22⋊Q8—24no—
191C22⋊Q8—22no—
192#121—24no—
193#121—28no—
194#121—22no—
195#121—24no—
196#120—22no—
197#120—24no—
198C4×Q8—24no—
199C4×D4—28no—
200C4×D4—24no—
201#117—22no—
202#117C2×C4×C12712yes—
203C2×C22⋊C4—24no—
204#99—28no—
205#99—44no—
206#95—44no—
207C24C22⋊C4×C12711yescharacteristic
208C24C2×D4×C12712yes—
209#41—44no—
210#41—44no—
211#41—416no—
212#41—48no—
213#41—44no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—22no—
217C2×Q8—48no—
218C2×Q8C22⋊C4×C12712yes—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—28no—
224C2×D4—24no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4C22⋊C4×C12712yes—
228C2×D4C2×D4×C12714yes—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—44no—
237C22×C4C22⋊C4×C12711yescharacteristic
238C22×C4C22⋊C4×C12712yes—
239C22×C4C22×C4×C12711yescharacteristic
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C22⋊C4—24no—
244C22⋊C4—24no—
245C22⋊C4—22no—
246C22⋊C4—28no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—24no—
250C42C22⋊C4×C12714yes—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—22no—
256C23—82no—
257C23—81no—
258C23C2×C22⋊C4×C12711yescharacteristic
259C23#120×C12714yes—
260Q8—28no—
261Q8—42no—
262Q8—44no—
263D4—44no—
264D4—48no—
265D4—42no—
266D4—44no—
267D4—44no—
268D4—88no—
269D4—84no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—48no—
280C2×C4—44no—
281C2×C4—84no—
282C2×C4—88no—
283C2×C4#99×C12714yes—
284C2×C4C2×C22⋊C4×C12712yes—
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—44no—
290C22—42no—
291C22—42no—
292C22—41no—
293C22—82no—
294C22—82no—
295C22#379×C12711yescharacteristic
296C22C2×#99×C12712yes—
297C4—24no—
298C4—22no—
299C4—44no—
300C4—41no—
301C4—42no—
302C4—164no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2C127×#178111yescharacteristic
subgroupG/H
center, socleC2×C127#1781
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC127×#3487C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C23×C127 ◁ C22×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2127C2semidirect
3. C127×#1782C2semidirect
4. C2×2−1+4×C127C4semidirect
11. #525×C127C4semidirect
12. #519×C127C4semidirect
148. #12708C127direct (N × H)
156. #2127C2×C127semidirect
157. #1782C2×C127semidirect
158. C127#12708direct (N × H)
159. C2×2−1+4C4×C127semidirect
166. #525C4×C127semidirect
167. #519C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34872
C127×#21272
C127×#17822
#12708127