Finite Group Databasebeta

← C127×#12694 #44576367

Subgroups — C127×#12694 #44576367

Order 32512.

1206 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×#124×C127C211yesmaximal, characteristic
2C2×#354×C127C214yesmaximal, complemented
3C127×#1781C212yesmaximal
4C23×Q8×C127C411yescharacteristic, complemented
5C23×D4×C127C2211yescharacteristic
6C2×#124×C127—22no—
7C2×#124×C127—28no—
8C2×#124×C127C412yescomplemented
9C2×#124×C127C2212yes—
10C22×C22⋊C4×C127—22no—
11C22×C42×C127C411yescharacteristic, complemented
12C2×#99×C127—22no—
13C2×#99×C127C2214yes—
14#379×C127—22no—
15#354×C127—28no—
16C25×C127D412yes—
17C22×Q8×C127—24no—
18C22×Q8×C127—21no—
19C22×Q8×C127—42no—
20C22×Q8×C127—42no—
21C22×Q8×C127C2×C412yes—
22C22×D4×C127—24no—
23C22×D4×C127—42no—
24C22×D4×C127—42no—
25C22×D4×C127C2×C411yescharacteristic
26C22×D4×C127D412yes—
27C22×D4×C127C2311yescharacteristic
28C23×C4×C127—21no—
29C23×C4×C127—21no—
30C23×C4×C127C2×C411yescharacteristic
31#124×C127—216no—
32#124×C127—24no—
33#124×C127—24no—
34#124×C127—24no—
35#124×C127D418yes—
36C2×C22⋊C4×C127—216no—
37C2×C22⋊C4×C127—24no—
38C2×C22⋊C4×C127—24no—
39C2×C22⋊C4×C127—24no—
40C2×C22⋊C4×C127—44no—
41C2×C42×C127—21no—
42C2×C42×C127—24no—
43C2×C42×C127C2×C412yes—
44#99×C127—28no—
45#99×C127—44no—
46C24×C127—24no—
47C24×C127—24no—
48C24×C127—42no—
49C24×C127—48no—
50C24×C127C22⋊C412yes—
51C24×C127C22⋊C411yescharacteristic
52C24×C127C2×D412yes—
53C2×Q8×C127—24no—
54C2×Q8×C127—22no—
55C2×Q8×C127—416no—
56C2×Q8×C127—42no—
57C2×Q8×C127—42no—
58C2×Q8×C127—42no—
59C2×Q8×C127—42no—
60C2×Q8×C127C22⋊C414yescomplemented
61C2×D4×C127—24no—
62C2×D4×C127—24no—
63C2×D4×C127—22no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127—42no—
67C2×D4×C127—41no—
68C2×D4×C127—41no—
69C2×D4×C127—416no—
70C2×D4×C127C2×D414yes—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—22no—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—22no—
77C22×C4×C127—22no—
78C22×C4×C127—44no—
79C22×C4×C127—42no—
80C22×C4×C127—42no—
81C22×C4×C127—44no—
82C22×C4×C127C22⋊C411yescharacteristic
83C22×C4×C127C22×C411yescharacteristic
84C22⋊C4×C127—216no—
85C22⋊C4×C127—216no—
86C22⋊C4×C127—44no—
87C22⋊C4×C127—44no—
88C22⋊C4×C127—88no—
89C42×C127—24no—
90C42×C127—22no—
91C42×C127C22⋊C414yescomplemented
92C23×C127—24no—
93C23×C127—21no—
94C23×C127—21no—
95C23×C127—22no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127—48no—
101C23×C127—44no—
102C23×C127—44no—
103C23×C127—416no—
104C23×C127—48no—
105C23×C127—84no—
106C23×C127—88no—
107C23×C127#9912yes—
108C23×C127C2×C22⋊C411yescharacteristic
109C23×C127#12014yes—
110Q8×C127—44no—
111Q8×C127—48no—
112Q8×C127—44no—
113D4×C127—44no—
114D4×C127—48no—
115D4×C127—44no—
116C2×C4×C127—28no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—24no—
122C2×C4×C127—24no—
123C2×C4×C127—48no—
124C2×C4×C127—44no—
125C2×C4×C127—48no—
126C2×C4×C127—42no—
127C2×C4×C127—42no—
128C2×C4×C127—84no—
129C2×C4×C127—88no—
130C2×C4×C127#9912yes—
131C2×C4×C127C2×C22⋊C412yes—
132C22×C127—24no—
133C22×C127—24no—
134C22×C127—22no—
135C22×C127—22no—
136C22×C127—22no—
137C22×C127—24no—
138C22×C127—44no—
139C22×C127—44no—
140C22×C127—44no—
141C22×C127—44no—
142C22×C127—42no—
143C22×C127—82no—
144C22×C127—816no—
145C22×C127—82no—
146C22×C127#35414yescomplemented
147C22×C127#37911yescharacteristic
148C22×C127C2×#9911yescharacteristic
149C22×C127C2×#9911yescharacteristic
150C4×C127—22no—
151C4×C127—22no—
152C4×C127—44no—
153C4×C127—44no—
154C4×C127—164no—
155#12694C12711yesmaximal, characteristic, direct product factor, Sylow
156C2×C127—21no—
157C2×C127—24no—
158C2×C127—21no—
159C2×C127—44no—
160C2×C127—82no—
161C2×C127#178111yescharacteristic
162C2×C127C2×#35412yes—
163C22×#124C2×C12711yescharacteristic
164C2×#354C2×C12714yescomplemented
165#1781C2×C12712yes—
166C127#1269411yescharacteristic, direct product factor, Sylow
167C23×Q8C4×C12711yescharacteristic, complemented
168C23×D4C22×C12711yescharacteristic
169C2×#124—22no—
170C2×#124—28no—
171C2×#124C4×C12712yescomplemented
172C2×#124C22×C12712yes—
173C22×C22⋊C4—22no—
174C22×C42C4×C12711yescharacteristic, complemented
175C2×#99—22no—
176C2×#99C22×C12714yes—
177#379—22no—
178#354—28no—
179C25D4×C12712yes—
180C22×Q8—24no—
181C22×Q8—21no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8C2×C4×C12712yes—
185C22×D4—24no—
186C22×D4—42no—
187C22×D4—42no—
188C22×D4C2×C4×C12711yescharacteristic
189C22×D4D4×C12712yes—
190C22×D4C23×C12711yescharacteristic
191C23×C4—21no—
192C23×C4—21no—
193C23×C4C2×C4×C12711yescharacteristic
194#124—216no—
195#124—24no—
196#124—24no—
197#124—24no—
198#124D4×C12718yes—
199C2×C22⋊C4—216no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—44no—
204C2×C42—21no—
205C2×C42—24no—
206C2×C42C2×C4×C12712yes—
207#99—28no—
208#99—44no—
209C24—24no—
210C24—24no—
211C24—42no—
212C24—48no—
213C24C22⋊C4×C12711yescharacteristic
214C24C22⋊C4×C12712yes—
215C24C2×D4×C12712yes—
216C2×Q8—24no—
217C2×Q8—22no—
218C2×Q8—416no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8C22⋊C4×C12714yescomplemented
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—22no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—416no—
233C2×D4C2×D4×C12714yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—44no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4C22⋊C4×C12711yescharacteristic
246C22×C4C22×C4×C12711yescharacteristic
247C22⋊C4—216no—
248C22⋊C4—216no—
249C22⋊C4—44no—
250C22⋊C4—44no—
251C22⋊C4—88no—
252C42—24no—
253C42—22no—
254C42C22⋊C4×C12714yescomplemented
255C23—24no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—48no—
264C23—44no—
265C23—44no—
266C23—416no—
267C23—48no—
268C23—84no—
269C23—88no—
270C23#99×C12712yes—
271C23C2×C22⋊C4×C12711yescharacteristic
272C23#120×C12714yes—
273Q8—44no—
274Q8—48no—
275Q8—44no—
276D4—44no—
277D4—48no—
278D4—44no—
279C2×C4—28no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—48no—
287C2×C4—48no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—84no—
292C2×C4—88no—
293C2×C4#99×C12712yes—
294C2×C4C2×C22⋊C4×C12712yes—
295C22—24no—
296C22—24no—
297C22—22no—
298C22—22no—
299C22—22no—
300C22—24no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—82no—
307C22—816no—
308C22—82no—
309C22#354×C12714yescomplemented
310C22#379×C12711yescharacteristic
311C22C2×#99×C12711yescharacteristic
312C22C2×#99×C12711yescharacteristic
313C4—22no—
314C4—22no—
315C4—44no—
316C4—44no—
317C4—164no—
318C2—21no—
319C2—24no—
320C2—21no—
321C2—44no—
322C2—82no—
323C2C127×#178111yescharacteristic
324C2C2×#354×C12712yes—
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC23×D4×C127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C23×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#354×C127C2semidirect
4. C23×Q8×C127C4semidirect
8. C2×#124×C127C4semidirect
11. C22×C42×C127C4semidirect
60. C2×Q8×C127C22⋊C4semidirect
91. C42×C127C22⋊C4semidirect
146. C22×C127#354semidirect
155. #12694C127direct (N × H)
164. C2×#354C2×C127semidirect
166. C127#12694direct (N × H)
167. C23×Q8C4×C127semidirect
171. C2×#124C4×C127semidirect
174. C22×C42C4×C127semidirect
223. C2×Q8C22⋊C4×C127semidirect
254. C42C22⋊C4×C127semidirect
309. C22#354×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#124×C1272
C2×#354×C1272
C127×#17812
#12694127