Finite Group Databasebeta

← C127×#13223 #44576896

Subgroups — C127×#13223 #44576896

Order 32512.

724 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 126 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×#2541C211yesmaximal, characteristic
2C2×#353×C127C214yesmaximal, complemented
3C127×#1799C211yesmaximal, characteristic, complemented
4C4×#99×C127C211yesmaximal, characteristic, complemented
5C2×#127×C127C411yescharacteristic, complemented
6C2×#127×C127C2212yes—
7C2×#121×C127—21no—
8C2×#121×C127—21no—
9C2×C4×D4×C127C2211yescharacteristic
10C2×#99×C127C2212yes—
11C2×#99×C127C2212yes—
12#392×C127—21no—
13#392×C127—21no—
14#390×C127C412yescomplemented
15#380×C127—21no—
16C4×C4⋊C4×C127C411yescharacteristic, complemented
17C4×C22⋊C4×C127—21no—
18#353×C127—28no—
19#342×C127—21no—
20#342×C127—21no—
21C22×D4×C127—22no—
22C22×D4×C127—22no—
23C22×D4×C127—21no—
24C22×D4×C127—21no—
25C22×D4×C127C2×C412yes—
26C22×D4×C127C2311yescharacteristic
27C23×C4×C127D411yescharacteristic
28C23×C4×C127D411yescharacteristic
29#127×C127—24no—
30#127×C127—24no—
31#127×C127—41no—
32#127×C127—41no—
33#121×C127—44no—
34#121×C127—44no—
35C4×D4×C127—41no—
36C4×D4×C127—41no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—41no—
39C2×C4⋊C4×C127C2×C411yescharacteristic
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127—22no—
43C2×C22⋊C4×C127—22no—
44C2×C22⋊C4×C127D411yescharacteristic
45C2×C22⋊C4×C127D411yescharacteristic
46C2×C42×C127—41no—
47C2×C42×C127C2×C412yes—
48C2×C42×C127C2×C411yescharacteristic
49#99×C127—22no—
50#99×C127—24no—
51#99×C127D414yescomplemented
52#95×C127—21no—
53#95×C127—41no—
54#95×C127—41no—
55C24×C127—22no—
56C24×C127C2×D411yescharacteristic
57C24×C127C2×D411yescharacteristic
58C2×D4×C127—24no—
59C2×D4×C127—41no—
60C2×D4×C127—41no—
61C2×D4×C127—42no—
62C2×D4×C127—42no—
63C2×D4×C127—42no—
64C2×D4×C127—42no—
65C2×D4×C127—44no—
66C2×D4×C127—44no—
67C2×D4×C127—42no—
68C2×D4×C127—44no—
69C2×D4×C127—42no—
70C2×D4×C127—42no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—82no—
74C2×D4×C127—82no—
75C2×D4×C127—82no—
76C2×D4×C127C2×D412yes—
77C2×D4×C127#4112yes—
78C22×C4×C127—22no—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—42no—
84C22×C4×C127—42no—
85C22×C4×C127—42no—
86C22×C4×C127—42no—
87C22×C4×C127C22⋊C411yescharacteristic
88C22×C4×C127C22⋊C411yescharacteristic
89C22×C4×C127C22⋊C411yescharacteristic
90C22×C4×C127C22⋊C411yescharacteristic
91C22×C4×C127C22×C411yescharacteristic
92C4⋊C4×C127—24no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—82no—
95C22⋊C4×C127—24no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—44no—
99C22⋊C4×C127—44no—
100C22⋊C4×C127—42no—
101C22⋊C4×C127—42no—
102C22⋊C4×C127—42no—
103C42×C127—21no—
104C42×C127—21no—
105C42×C127—24no—
106C42×C127—82no—
107C23×C127—21no—
108C23×C127—21no—
109C23×C127—21no—
110C23×C127—21no—
111C23×C127—41no—
112C23×C127—41no—
113C23×C127—41no—
114C23×C127—41no—
115C23×C127—42no—
116C23×C127—42no—
117C23×C127—42no—
118C23×C127—42no—
119C23×C127—42no—
120C23×C127—42no—
121C23×C127—42no—
122C23×C127C2×C22⋊C411yescharacteristic
123C23×C127#12112yes—
124C23×C127#12411yescharacteristic
125C23×C127#12711yescharacteristic
126D4×C127—42no—
127D4×C127—42no—
128D4×C127—84no—
129D4×C127—82no—
130D4×C127—88no—
131D4×C127—84no—
132D4×C127—84no—
133D4×C127—84no—
134D4×C127—82no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—22no—
138C2×C4×C127—24no—
139C2×C4×C127—21no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—42no—
143C2×C4×C127—42no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—41no—
149C2×C4×C127—44no—
150C2×C4×C127—81no—
151C2×C4×C127—81no—
152C2×C4×C127—82no—
153C2×C4×C127—82no—
154C2×C4×C127—82no—
155C2×C4×C127—82no—
156C2×C4×C127—82no—
157C2×C4×C127#9911yescharacteristic
158C2×C4×C127#9911yescharacteristic
159C2×C4×C127C4×D412yes—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—22no—
163C22×C127—21no—
164C22×C127—21no—
165C22×C127—42no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127—82no—
175C22×C127—82no—
176C22×C127—82no—
177C22×C127—82no—
178C22×C127—82no—
179C22×C127—82no—
180C22×C127—82no—
181C22×C127#39011yescharacteristic
182C22×C127C2×#9911yescharacteristic
183C22×C127#41011yescharacteristic
184C4×C127—21no—
185C4×C127—22no—
186C4×C127—44no—
187C4×C127—41no—
188C4×C127—41no—
189C4×C127—41no—
190C4×C127—84no—
191C4×C127—162no—
192C4×C127#35312yescomplemented
193#13223C12711yesmaximal, characteristic, direct product factor, Sylow
194C2×C127—21no—
195C2×C127—21no—
196C2×C127—42no—
197C2×C127—41no—
198C2×C127—41no—
199C2×C127—162no—
200C2×C127#196411yescharacteristic
201C2×C127C2×#35311yescharacteristic
202C2×C127#212811yescharacteristic
203#2541C2×C12711yescharacteristic
204C2×#353C2×C12714yescomplemented
205#1799C2×C12711yescharacteristic, complemented
206C4×#99C2×C12711yescharacteristic, complemented
207C127#1322311yescharacteristic, direct product factor, Sylow
208C2×#127C4×C12711yescharacteristic, complemented
209C2×#127C22×C12712yes—
210C2×#121—21no—
211C2×#121—21no—
212C2×C4×D4C22×C12711yescharacteristic
213C2×#99C22×C12712yes—
214C2×#99C22×C12712yes—
215#392—21no—
216#392—21no—
217#390C4×C12712yescomplemented
218#380—21no—
219C4×C4⋊C4C4×C12711yescharacteristic, complemented
220C4×C22⋊C4—21no—
221#353—28no—
222#342—21no—
223#342—21no—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—21no—
227C22×D4—21no—
228C22×D4C2×C4×C12712yes—
229C22×D4C23×C12711yescharacteristic
230C23×C4D4×C12711yescharacteristic
231C23×C4D4×C12711yescharacteristic
232#127—24no—
233#127—24no—
234#127—41no—
235#127—41no—
236#121—44no—
237#121—44no—
238C4×D4—41no—
239C4×D4—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4C2×C4×C12711yescharacteristic
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4D4×C12711yescharacteristic
248C2×C22⋊C4D4×C12711yescharacteristic
249C2×C42—41no—
250C2×C42C2×C4×C12712yes—
251C2×C42C2×C4×C12711yescharacteristic
252#99—22no—
253#99—24no—
254#99D4×C12714yescomplemented
255#95—21no—
256#95—41no—
257#95—41no—
258C24—22no—
259C24C2×D4×C12711yescharacteristic
260C24C2×D4×C12711yescharacteristic
261C2×D4—24no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—42no—
271C2×D4—44no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4C2×D4×C12712yes—
280C2×D4#41×C12712yes—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4C22⋊C4×C12711yescharacteristic
291C22×C4C22⋊C4×C12711yescharacteristic
292C22×C4C22⋊C4×C12711yescharacteristic
293C22×C4C22⋊C4×C12711yescharacteristic
294C22×C4C22×C4×C12711yescharacteristic
295C4⋊C4—24no—
296C4⋊C4—41no—
297C4⋊C4—82no—
298C22⋊C4—24no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C42—21no—
307C42—21no—
308C42—24no—
309C42—82no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23C2×C22⋊C4×C12711yescharacteristic
326C23#121×C12712yes—
327C23#124×C12711yescharacteristic
328C23#127×C12711yescharacteristic
329D4—42no—
330D4—42no—
331D4—84no—
332D4—82no—
333D4—88no—
334D4—84no—
335D4—84no—
336D4—84no—
337D4—82no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—44no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4#99×C12711yescharacteristic
361C2×C4#99×C12711yescharacteristic
362C2×C4C4×D4×C12712yes—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—21no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—81no—
376C22—81no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22#390×C12711yescharacteristic
385C22C2×#99×C12711yescharacteristic
386C22#410×C12711yescharacteristic
387C4—21no—
388C4—22no—
389C4—44no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—84no—
394C4—162no—
395C4#353×C12712yescomplemented
396C2—21no—
397C2—21no—
398C2—42no—
399C2—41no—
400C2—41no—
401C2—162no—
402C2C127×#196411yescharacteristic
403C2C2×#353×C12711yescharacteristic
404C2C127×#212811yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC127×#2541C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#353×C127C2semidirect
3. C127×#1799C2semidirect
4. C4×#99×C127C2semidirect
5. C2×#127×C127C4semidirect
14. #390×C127C4semidirect
16. C4×C4⋊C4×C127C4semidirect
51. #99×C127D4semidirect
192. C4×C127#353semidirect
193. #13223C127direct (N × H)
204. C2×#353C2×C127semidirect
205. #1799C2×C127semidirect
206. C4×#99C2×C127semidirect
207. C127#13223direct (N × H)
208. C2×#127C4×C127semidirect
217. #390C4×C127semidirect
219. C4×C4⋊C4C4×C127semidirect
254. #99D4×C127semidirect
395. C4#353×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#25412
C2×#353×C1272
C127×#17992
C4×#99×C1272
#13223127