Finite Group Databasebeta

← C127×#10328 #44574001

Subgroups — C127×#10328 #44574001

Order 32512.

698 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3192C211yesmaximal, characteristic
2C127×#2504C211yesmaximal, characteristic, complemented
3C127×#2501C211yesmaximal, characteristic, complemented
4C127×#1821C211yesmaximal, characteristic, complemented
5C127×#1507C211yesmaximal, characteristic, complemented
6C127×#1507C211yesmaximal, characteristic, complemented
7C127×#1297C211yesmaximal, characteristic, complemented
8C2×#123×C127—22no—
9C2×#121×C127C2211yescharacteristic
10C2×#121×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2211yescharacteristic
12#481×C127—21no—
13#481×C127—21no—
14#469×C127—22no—
15#447×C127—21no—
16#447×C127—21no—
17#407×C127C2211yescharacteristic
18#407×C127C2211yescharacteristic
19#399×C127—22no—
20#397×C127—22no—
21#390×C127—21no—
22#390×C127—21no—
23#386×C127—22no—
24#384×C127—21no—
25#382×C127—21no—
26C4×C22⋊C4×C127C2211yescharacteristic
27C4×C22⋊C4×C127C2211yescharacteristic
28#343×C127—22no—
29#327×C127—22no—
30#327×C127—22no—
31C22×D4×C127—21no—
32C22×D4×C127—21no—
33C22×D4×C127D411yescharacteristic
34C22×D4×C127D411yescharacteristic
35C23×C4×C127D412yes—
36C23×C4×C127C2311yescharacteristic
37C2×D8×C127—22no—
38C2×M16×C127—21no—
39C2×M16×C127D412yes—
40#123×C127—48no—
41#121×C127—21no—
42#121×C127—21no—
43#121×C127—21no—
44#121×C127—21no—
45#121×C127—21no—
46#121×C127—21no—
47#121×C127—21no—
48#121×C127—21no—
49C2×C4⋊C4×C127—24no—
50C2×C4⋊C4×C127—21no—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127—21no—
53C2×C4⋊C4×C127—21no—
54C2×C4⋊C4×C127—21no—
55C2×C4⋊C4×C127—21no—
56C2×C4⋊C4×C127D411yescharacteristic
57C2×C4⋊C4×C127D411yescharacteristic
58C2×C22⋊C4×C127—21no—
59C2×C22⋊C4×C127—24no—
60C2×C22⋊C4×C127—21no—
61C2×C22⋊C4×C127—21no—
62C2×C22⋊C4×C127—21no—
63C2×C22⋊C4×C127—24no—
64C2×C42×C127—21no—
65C2×C42×C127—21no—
66#106×C127—22no—
67#102×C127—22no—
68#102×C127—22no—
69C22⋊C8×C127—21no—
70C22⋊C8×C127—21no—
71C22⋊C8×C127—21no—
72C22⋊C8×C127—21no—
73#95×C127—24no—
74#95×C127D412yes—
75#95×C127D412yes—
76C24×C127—21no—
77C24×C127—21no—
78C24×C127C2×D411yescharacteristic
79C2×D4×C127—21no—
80C2×D4×C127—21no—
81C2×D4×C127—21no—
82C2×D4×C127—21no—
83C2×D4×C127—42no—
84C2×D4×C127—42no—
85C2×D4×C127—44no—
86C2×D4×C127—44no—
87C2×D4×C127—41no—
88C2×D4×C127—41no—
89C2×D4×C127—41no—
90C2×D4×C127—41no—
91C22×C4×C127—22no—
92C22×C4×C127—21no—
93C22×C4×C127—22no—
94C22×C4×C127—22no—
95C22×C4×C127—22no—
96C22×C4×C127—21no—
97C22×C4×C127—22no—
98C22×C4×C127—42no—
99C22×C4×C127—44no—
100C22×C4×C127C2×D411yescharacteristic
101C22×C4×C127C2×D411yescharacteristic
102C22×C4×C127C2×D411yescharacteristic
103C22×C4×C127C2×D411yescharacteristic
104C22×C4×C127C2×D411yescharacteristic
105C22×C4×C127#4111yescharacteristic
106D8×C127—82no—
107M16×C127—42no—
108M16×C127—42no—
109C2×C8×C127—22no—
110C2×C8×C127—41no—
111C4⋊C4×C127—21no—
112C4⋊C4×C127—21no—
113C4⋊C4×C127—21no—
114C4⋊C4×C127—21no—
115C4⋊C4×C127—48no—
116C4⋊C4×C127—41no—
117C4⋊C4×C127—41no—
118C4⋊C4×C127—41no—
119C4⋊C4×C127—41no—
120C22⋊C4×C127—24no—
121C22⋊C4×C127—48no—
122C22⋊C4×C127—42no—
123C22⋊C4×C127—41no—
124C22⋊C4×C127—41no—
125C22⋊C4×C127—41no—
126C22⋊C4×C127—41no—
127C22⋊C4×C127—84no—
128C42×C127—42no—
129C42×C127—42no—
130C23×C127—22no—
131C23×C127—22no—
132C23×C127—41no—
133C23×C127—42no—
134C23×C127—42no—
135C23×C127—42no—
136C23×C127—42no—
137C23×C127—41no—
138C23×C127—41no—
139C23×C127—42no—
140C23×C127—42no—
141C23×C127#12011yescharacteristic
142C23×C127#12011yescharacteristic
143C23×C127#12111yescharacteristic
144D4×C127—42no—
145D4×C127—42no—
146D4×C127—81no—
147D4×C127—81no—
148D4×C127—82no—
149D4×C127—82no—
150D4×C127—82no—
151D4×C127—82no—
152C2×C4×C127—22no—
153C2×C4×C127—22no—
154C2×C4×C127—22no—
155C2×C4×C127—24no—
156C2×C4×C127—41no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—42no—
160C2×C4×C127—42no—
161C2×C4×C127—42no—
162C2×C4×C127—44no—
163C2×C4×C127—42no—
164C2×C4×C127—42no—
165C2×C4×C127—42no—
166C2×C4×C127—41no—
167C2×C4×C127—41no—
168C2×C4×C127—42no—
169C2×C4×C127—42no—
170C2×C4×C127—44no—
171C2×C4×C127—82no—
172C2×C4×C127#12011yescharacteristic
173C2×C4×C127#12111yescharacteristic
174C2×C4×C127#12111yescharacteristic
175C2×C4×C127#12711yescharacteristic
176C8×C127—42no—
177C8×C127—81no—
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—21no—
181C22×C127—21no—
182C22×C127—21no—
183C22×C127—21no—
184C22×C127—21no—
185C22×C127—44no—
186C22×C127—84no—
187C22×C127—84no—
188C22×C127—81no—
189C22×C127—81no—
190C22×C127—81no—
191C22×C127—81no—
192C22×C127—81no—
193C22×C127—81no—
194C22×C127#39211yescharacteristic
195C22×C127#45312yes—
196C22×C127#45412yes—
197C4×C127—21no—
198C4×C127—21no—
199C4×C127—21no—
200C4×C127—44no—
201C4×C127—82no—
202C4×C127—82no—
203C4×C127—81no—
204C4×C127—81no—
205C4×C127#45812yes—
206#10328C12711yesmaximal, characteristic, direct product factor, Sylow
207C2×C127—22no—
208C2×C127—22no—
209C2×C127—41no—
210C2×C127—161no—
211C2×C127—161no—
212C2×C127#200211yescharacteristic
213C2×C127#200311yescharacteristic
214C2×C127#202211yescharacteristic
215#3192C2×C12711yescharacteristic
216#2504C2×C12711yescharacteristic, complemented
217#2501C2×C12711yescharacteristic, complemented
218#1821C2×C12711yescharacteristic, complemented
219#1507C2×C12711yescharacteristic, complemented
220#1507C2×C12711yescharacteristic, complemented
221#1297C2×C12711yescharacteristic, complemented
222C127#1032811yescharacteristic, direct product factor, Sylow
223C2×#123—22no—
224C2×#121C22×C12711yescharacteristic
225C2×#121C22×C12711yescharacteristic
226C22×C4⋊C4C22×C12711yescharacteristic
227#481—21no—
228#481—21no—
229#469—22no—
230#447—21no—
231#447—21no—
232#407C22×C12711yescharacteristic
233#407C22×C12711yescharacteristic
234#399—22no—
235#397—22no—
236#390—21no—
237#390—21no—
238#386—22no—
239#384—21no—
240#382—21no—
241C4×C22⋊C4C22×C12711yescharacteristic
242C4×C22⋊C4C22×C12711yescharacteristic
243#343—22no—
244#327—22no—
245#327—22no—
246C22×D4—21no—
247C22×D4—21no—
248C22×D4D4×C12711yescharacteristic
249C22×D4D4×C12711yescharacteristic
250C23×C4D4×C12712yes—
251C23×C4C23×C12711yescharacteristic
252C2×D8—22no—
253C2×M16—21no—
254C2×M16D4×C12712yes—
255#123—48no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4D4×C12711yescharacteristic
272C2×C4⋊C4D4×C12711yescharacteristic
273C2×C22⋊C4—21no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—24no—
279C2×C42—21no—
280C2×C42—21no—
281#106—22no—
282#102—22no—
283#102—22no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288#95—24no—
289#95D4×C12712yes—
290#95D4×C12712yes—
291C24—21no—
292C24—21no—
293C24C2×D4×C12711yescharacteristic
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4C2×D4×C12711yescharacteristic
316C22×C4C2×D4×C12711yescharacteristic
317C22×C4C2×D4×C12711yescharacteristic
318C22×C4C2×D4×C12711yescharacteristic
319C22×C4C2×D4×C12711yescharacteristic
320C22×C4#41×C12711yescharacteristic
321D8—82no—
322M16—42no—
323M16—42no—
324C2×C8—22no—
325C2×C8—41no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—48no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C22⋊C4—24no—
336C22⋊C4—48no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—84no—
343C42—42no—
344C42—42no—
345C23—22no—
346C23—22no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—41no—
354C23—42no—
355C23—42no—
356C23#120×C12711yescharacteristic
357C23#120×C12711yescharacteristic
358C23#121×C12711yescharacteristic
359D4—42no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—82no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—24no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—82no—
387C2×C4#120×C12711yescharacteristic
388C2×C4#121×C12711yescharacteristic
389C2×C4#121×C12711yescharacteristic
390C2×C4#127×C12711yescharacteristic
391C8—42no—
392C8—81no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—44no—
401C22—84no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#392×C12711yescharacteristic
410C22#453×C12712yes—
411C22#454×C12712yes—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—44no—
416C4—82no—
417C4—82no—
418C4—81no—
419C4—81no—
420C4#458×C12712yes—
421C2—22no—
422C2—22no—
423C2—41no—
424C2—161no—
425C2—161no—
426C2C127×#200211yescharacteristic
427C2C127×#200311yescharacteristic
428C2C127×#202211yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC127×#3192C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2504C2semidirect
3. C127×#2501C2semidirect
4. C127×#1821C2semidirect
5. C127×#1507C2semidirect
6. C127×#1507C2semidirect
7. C127×#1297C2semidirect
206. #10328C127direct (N × H)
216. #2504C2×C127semidirect
217. #2501C2×C127semidirect
218. #1821C2×C127semidirect
219. #1507C2×C127semidirect
220. #1507C2×C127semidirect
221. #1297C2×C127semidirect
222. C127#10328direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31922
C127×#25042
C127×#25012
C127×#18212
C127×#15072
C127×#12972
#10328127