Finite Group Databasebeta

← C127×#12202 #44575875

Subgroups — C127×#12202 #44575875

Order 32512.

976 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2403C211yesmaximal, characteristic
2C127×#2029C211yesmaximal, characteristic, complemented
3C127×#1596C212yesmaximal, complemented
4C127×#1343C212yesmaximal, complemented
5C127×#1316C211yesmaximal, characteristic, complemented
6C23×D4×C127—21no—
7#543×C127C2211yescharacteristic, complemented
8C2×#120×C127—22no—
9#450×C127—22no—
10#392×C127—21no—
11#392×C127C2212yes—
12#386×C127—22no—
13#379×C127—21no—
14#379×C127C2211yescharacteristic
15#342×C127—21no—
16#342×C127C2211yescharacteristic, complemented
17#328×C127—22no—
18#323×C127C2212yescomplemented
19C25×C127—21no—
20C25×C127D411yescharacteristic
21C22×D4×C127—22no—
22C22×D4×C127—22no—
23C22×D4×C127—21no—
24C22×D4×C127—48no—
25C22×D4×C127—42no—
26C22×D4×C127—42no—
27C22×D4×C127D412yes—
28C23×C4×C127—21no—
29C2×SD16×C127—42no—
30C22⋊Q8×C127—22no—
31C22⋊Q8×C127—42no—
32#120×C127—48no—
33C2×C22⋊C4×C127—22no—
34C2×C22⋊C4×C127—21no—
35C2×C22⋊C4×C127—22no—
36C2×C22⋊C4×C127—22no—
37C2×C22⋊C4×C127—21no—
38C2×C22⋊C4×C127—42no—
39C2×C22⋊C4×C127D411yescharacteristic
40C2×C22⋊C4×C127C2311yescharacteristic
41#102×C127—42no—
42C22⋊C8×C127—22no—
43#95×C127—21no—
44#95×C127D412yes—
45C24×C127—21no—
46C24×C127—22no—
47C24×C127—21no—
48C24×C127—41no—
49C24×C127—44no—
50C24×C127—42no—
51C24×C127—44no—
52C24×C127—41no—
53C24×C127—44no—
54C24×C127—81no—
55C24×C127—81no—
56C24×C127SD1612yescomplemented
57C24×C127C2×D411yescharacteristic
58C2×Q8×C127—22no—
59C2×D4×C127—28no—
60C2×D4×C127—44no—
61C2×D4×C127—42no—
62C2×D4×C127—44no—
63C2×D4×C127—44no—
64C2×D4×C127—44no—
65C2×D4×C127—416no—
66C2×D4×C127—416no—
67C2×D4×C127—42no—
68C2×D4×C127—42no—
69C2×D4×C127—44no—
70C2×D4×C127—44no—
71C2×D4×C127—81no—
72C22×C4×C127—22no—
73C22×C4×C127—21no—
74C22×C4×C127—22no—
75C22×C4×C127—44no—
76C22×C4×C127—42no—
77C22×C4×C127—41no—
78C22×C4×C127C2×D412yes—
79SD16×C127—84no—
80C2×C8×C127—42no—
81C4⋊C4×C127—22no—
82C4⋊C4×C127—42no—
83C22⋊C4×C127—41no—
84C22⋊C4×C127—44no—
85C22⋊C4×C127—44no—
86C22⋊C4×C127—44no—
87C22⋊C4×C127—41no—
88C22⋊C4×C127—41no—
89C22⋊C4×C127—41no—
90C22⋊C4×C127—41no—
91C22⋊C4×C127—41no—
92C22⋊C4×C127—84no—
93C23×C127—24no—
94C23×C127—22no—
95C23×C127—21no—
96C23×C127—21no—
97C23×C127—24no—
98C23×C127—42no—
99C23×C127—44no—
100C23×C127—42no—
101C23×C127—42no—
102C23×C127—42no—
103C23×C127—42no—
104C23×C127—42no—
105C23×C127—44no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127—44no—
109C23×C127—48no—
110C23×C127—42no—
111C23×C127—42no—
112C23×C127—42no—
113C23×C127—81no—
114C23×C127—84no—
115C23×C127—84no—
116C23×C127—82no—
117C23×C127—88no—
118C23×C127—88no—
119C23×C127—84no—
120C23×C127#12011yescharacteristic
121C23×C127C2×SD1611yescharacteristic
122C23×C127AGL(1,ℤ/8)11yescharacteristic
123Q8×C127—82no—
124D4×C127—416no—
125D4×C127—44no—
126D4×C127—48no—
127D4×C127—48no—
128D4×C127—82no—
129D4×C127—84no—
130C2×C4×C127—24no—
131C2×C4×C127—22no—
132C2×C4×C127—42no—
133C2×C4×C127—48no—
134C2×C4×C127—42no—
135C2×C4×C127—42no—
136C2×C4×C127—42no—
137C2×C4×C127—41no—
138C2×C4×C127—42no—
139C2×C4×C127—82no—
140C2×C4×C127—81no—
141C2×C4×C127—82no—
142C8×C127—82no—
143C22×C127—22no—
144C22×C127—24no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—41no—
148C22×C127—41no—
149C22×C127—42no—
150C22×C127—44no—
151C22×C127—44no—
152C22×C127—44no—
153C22×C127—44no—
154C22×C127—42no—
155C22×C127—42no—
156C22×C127—44no—
157C22×C127—44no—
158C22×C127—81no—
159C22×C127—82no—
160C22×C127—82no—
161C22×C127—84no—
162C22×C127—82no—
163C22×C127—816no—
164C22×C127—82no—
165C22×C127—82no—
166C22×C127#45011yescharacteristic
167C22×C127#45511yescharacteristic
168C22×C127B(4,𝔽2)11yescharacteristic
169C4×C127—44no—
170C4×C127—42no—
171C4×C127—81no—
172C4×C127—82no—
173C4×C127—161no—
174#12202C12711yesmaximal, characteristic, direct product factor, Sylow
175C2×C127—21no—
176C2×C127—21no—
177C2×C127—41no—
178C2×C127—41no—
179C2×C127—44no—
180C2×C127—84no—
181C2×C127#160111yescharacteristic
182C2×C127#219612yes—
183#2403C2×C12711yescharacteristic
184#2029C2×C12711yescharacteristic, complemented
185#1596C2×C12712yescomplemented
186#1343C2×C12712yescomplemented
187#1316C2×C12711yescharacteristic, complemented
188C127#1220211yescharacteristic, direct product factor, Sylow
189C23×D4—21no—
190#543C22×C12711yescharacteristic, complemented
191C2×#120—22no—
192#450—22no—
193#392—21no—
194#392C22×C12712yes—
195#386—22no—
196#379—21no—
197#379C22×C12711yescharacteristic
198#342—21no—
199#342C22×C12711yescharacteristic, complemented
200#328—22no—
201#323C22×C12712yescomplemented
202C25—21no—
203C25D4×C12711yescharacteristic
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—48no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4D4×C12712yes—
211C23×C4—21no—
212C2×SD16—42no—
213C22⋊Q8—22no—
214C22⋊Q8—42no—
215#120—48no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4D4×C12711yescharacteristic
223C2×C22⋊C4C23×C12711yescharacteristic
224#102—42no—
225C22⋊C8—22no—
226#95—21no—
227#95D4×C12712yes—
228C24—21no—
229C24—22no—
230C24—21no—
231C24—41no—
232C24—44no—
233C24—42no—
234C24—44no—
235C24—41no—
236C24—44no—
237C24—81no—
238C24—81no—
239C24SD16×C12712yescomplemented
240C24C2×D4×C12711yescharacteristic
241C2×Q8—22no—
242C2×D4—28no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—416no—
249C2×D4—416no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—81no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4C2×D4×C12712yes—
262SD16—84no—
263C2×C8—42no—
264C4⋊C4—22no—
265C4⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—84no—
276C23—24no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—24no—
281C23—42no—
282C23—44no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—44no—
289C23—41no—
290C23—41no—
291C23—44no—
292C23—48no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—81no—
297C23—84no—
298C23—84no—
299C23—82no—
300C23—88no—
301C23—88no—
302C23—84no—
303C23#120×C12711yescharacteristic
304C23C2×SD16×C12711yescharacteristic
305C23AGL(1,ℤ/8)×C12711yescharacteristic
306Q8—82no—
307D4—416no—
308D4—44no—
309D4—48no—
310D4—48no—
311D4—82no—
312D4—84no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—42no—
316C2×C4—48no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—82no—
323C2×C4—81no—
324C2×C4—82no—
325C8—82no—
326C22—22no—
327C22—24no—
328C22—21no—
329C22—21no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—44no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—44no—
340C22—44no—
341C22—81no—
342C22—82no—
343C22—82no—
344C22—84no—
345C22—82no—
346C22—816no—
347C22—82no—
348C22—82no—
349C22#450×C12711yescharacteristic
350C22#455×C12711yescharacteristic
351C22B(4,𝔽2)×C12711yescharacteristic
352C4—44no—
353C4—42no—
354C4—81no—
355C4—82no—
356C4—161no—
357C2—21no—
358C2—21no—
359C2—41no—
360C2—44no—
361C2—41no—
362C2—84no—
363C2C127×#160111yescharacteristic
364C2C127×#219612yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#2403C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2029C2semidirect
3. C127×#1596C2semidirect
4. C127×#1343C2semidirect
5. C127×#1316C2semidirect
7. #543×C127C22semidirect
16. #342×C127C22semidirect
18. #323×C127C22semidirect
56. C24×C127SD16semidirect
174. #12202C127direct (N × H)
184. #2029C2×C127semidirect
185. #1596C2×C127semidirect
186. #1343C2×C127semidirect
187. #1316C2×C127semidirect
188. C127#12202direct (N × H)
190. #543C22×C127semidirect
199. #342C22×C127semidirect
201. #323C22×C127semidirect
239. C24SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24032
C127×#20292
C127×#15962
C127×#13432
C127×#13162
#12202127