Finite Group Databasebeta

← C127×#10561 #44574234

Subgroups — C127×#10561 #44574234

Order 32512.

464 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3193C211yesmaximal, characteristic
2C127×#1901C211yesmaximal, characteristic, complemented
3C127×#1895C211yesmaximal, characteristic
4C127×#1855C211yesmaximal, characteristic, complemented
5C127×#1520C211yesmaximal, characteristic, complemented
6C127×#1471C211yesmaximal, characteristic
7C127×#1305C211yesmaximal, characteristic, complemented
8C22×M16×C127C2211yescharacteristic
9C2×C22⋊Q8×C127C2211yescharacteristic
10C2×#121×C127C2211yescharacteristic
11#484×C127—21no—
12#480×C127—21no—
13#468×C127—21no—
14#468×C127—21no—
15#450×C127—21no—
16#448×C127—21no—
17C2×#108×C127—21no—
18C2×#102×C127—21no—
19C2×#101×C127—21no—
20#407×C127C2211yescharacteristic
21#407×C127C2211yescharacteristic
22C2×C22⋊C8×C127C2211yescharacteristic
23#380×C127C2211yescharacteristic
24#343×C127—21no—
25#341×C127—21no—
26#340×C127—21no—
27#328×C127—22no—
28#323×C127—22no—
29C22×Q8×C127D411yescharacteristic
30C22×D4×C127—21no—
31C22×D4×C127D411yescharacteristic
32C23×C4×C127C2311yescharacteristic
33C2×SD16×C127—21no—
34C2×SD16×C127—21no—
35C2×M16×C127—21no—
36C2×M16×C127—21no—
37C2×M16×C127—21no—
38C2×M16×C127—42no—
39C2×M16×C127D411yescharacteristic
40C2×M16×C127D411yescharacteristic
41C22×C8×C127—21no—
42C22×C8×C127—21no—
43C22⋊Q8×C127—21no—
44C22⋊Q8×C127—21no—
45C22⋊Q8×C127—21no—
46C22⋊Q8×C127—21no—
47#121×C127—21no—
48#121×C127—21no—
49#121×C127—21no—
50#121×C127—21no—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127—21no—
53C2×C4⋊C4×C127D411yescharacteristic
54C2×C4⋊C4×C127D411yescharacteristic
55C2×C22⋊C4×C127—21no—
56C2×C22⋊C4×C127—21no—
57C2×C22⋊C4×C127—21no—
58#108×C127—42no—
59#107×C127—21no—
60#107×C127—21no—
61#103×C127—21no—
62#103×C127—21no—
63#102×C127—21no—
64#102×C127—21no—
65#102×C127—42no—
66#101×C127—42no—
67C22⋊C8×C127—21no—
68C22⋊C8×C127—21no—
69C22⋊C8×C127—21no—
70C22⋊C8×C127—21no—
71C22⋊C8×C127—21no—
72C22⋊C8×C127D412yes—
73#95×C127—21no—
74C24×C127—21no—
75C24×C127#4111yescharacteristic
76C2×Q8×C127—21no—
77C2×Q8×C127—21no—
78C2×Q8×C127—41no—
79C2×Q8×C127—41no—
80C2×D4×C127—21no—
81C2×D4×C127—21no—
82C2×D4×C127—41no—
83C2×D4×C127—41no—
84C2×D4×C127—41no—
85C2×D4×C127—41no—
86C2×D4×C127—82no—
87C22×C4×C127—21no—
88C22×C4×C127—21no—
89C22×C4×C127—21no—
90C22×C4×C127—41no—
91C22×C4×C127—41no—
92C22×C4×C127—41no—
93C22×C4×C127C2×D411yescharacteristic
94C22×C4×C127C2×D411yescharacteristic
95C22×C4×C127C2×D411yescharacteristic
96C22×C4×C127C2×D411yescharacteristic
97C22×C4×C127#4111yescharacteristic
98C22×C4×C127#4111yescharacteristic
99SD16×C127—81no—
100SD16×C127—81no—
101M16×C127—22no—
102M16×C127—22no—
103M16×C127—41no—
104M16×C127—41no—
105M16×C127—41no—
106M16×C127—41no—
107M16×C127—41no—
108M16×C127—41no—
109C2×C8×C127—22no—
110C2×C8×C127—21no—
111C2×C8×C127—21no—
112C2×C8×C127—22no—
113C2×C8×C127—41no—
114C2×C8×C127—41no—
115C2×C8×C127—41no—
116C2×C8×C127—41no—
117C2×C8×C127—41no—
118C4⋊C4×C127—21no—
119C4⋊C4×C127—21no—
120C4⋊C4×C127—21no—
121C4⋊C4×C127—21no—
122C4⋊C4×C127—41no—
123C4⋊C4×C127—41no—
124C4⋊C4×C127—41no—
125C4⋊C4×C127—41no—
126C22⋊C4×C127—41no—
127C22⋊C4×C127—41no—
128C22⋊C4×C127—41no—
129C22⋊C4×C127—41no—
130C22⋊C4×C127—81no—
131C23×C127—21no—
132C23×C127—21no—
133C23×C127—41no—
134C23×C127—41no—
135C23×C127—41no—
136C23×C127—41no—
137C23×C127—41no—
138C23×C127—81no—
139C23×C127—81no—
140C23×C127#12111yescharacteristic
141C23×C127#12311yescharacteristic
142C23×C127#12411yescharacteristic
143Q8×C127—41no—
144Q8×C127—41no—
145Q8×C127—81no—
146D4×C127—42no—
147D4×C127—41no—
148D4×C127—41no—
149D4×C127—82no—
150D4×C127—82no—
151C2×C4×C127—21no—
152C2×C4×C127—21no—
153C2×C4×C127—21no—
154C2×C4×C127—21no—
155C2×C4×C127—21no—
156C2×C4×C127—21no—
157C2×C4×C127—41no—
158C2×C4×C127—41no—
159C2×C4×C127—41no—
160C2×C4×C127—42no—
161C2×C4×C127—41no—
162C2×C4×C127—41no—
163C2×C4×C127—41no—
164C2×C4×C127—81no—
165C2×C4×C127—81no—
166C2×C4×C127—81no—
167C2×C4×C127—81no—
168C2×C4×C127—81no—
169C2×C4×C127#12011yescharacteristic
170C2×C4×C127#12111yescharacteristic
171C2×C4×C127#12111yescharacteristic
172C2×C4×C127#12311yescharacteristic
173C8×C127—42no—
174C8×C127—42no—
175C8×C127—41no—
176C8×C127—41no—
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—21no—
186C22×C127—81no—
187C22×C127—81no—
188C22×C127—81no—
189C22×C127—81no—
190C22×C127—81no—
191C22×C127—161no—
192C22×C127—161no—
193C22×C127#39411yescharacteristic
194C22×C127#47111yescharacteristic
195C22×C127#47311yescharacteristic
196C4×C127—21no—
197C4×C127—21no—
198C4×C127—21no—
199C4×C127—81no—
200C4×C127—81no—
201C4×C127—81no—
202C4×C127—161no—
203C4×C127B(4,𝔽2)11yescharacteristic
204C4×C127#45811yescharacteristic
205#10561C12711yesmaximal, characteristic, direct product factor, Sylow
206C2×C127—21no—
207C2×C127—21no—
208C2×C127—41no—
209C2×C127—41no—
210C2×C127—161no—
211C2×C127#204411yescharacteristic
212C2×C127#204811yescharacteristic
213C2×C127#205211yescharacteristic
214#3193C2×C12711yescharacteristic
215#1901C2×C12711yescharacteristic, complemented
216#1895C2×C12711yescharacteristic
217#1855C2×C12711yescharacteristic, complemented
218#1520C2×C12711yescharacteristic, complemented
219#1471C2×C12711yescharacteristic
220#1305C2×C12711yescharacteristic, complemented
221C127#1056111yescharacteristic, direct product factor, Sylow
222C22×M16C22×C12711yescharacteristic
223C2×C22⋊Q8C22×C12711yescharacteristic
224C2×#121C22×C12711yescharacteristic
225#484—21no—
226#480—21no—
227#468—21no—
228#468—21no—
229#450—21no—
230#448—21no—
231C2×#108—21no—
232C2×#102—21no—
233C2×#101—21no—
234#407C22×C12711yescharacteristic
235#407C22×C12711yescharacteristic
236C2×C22⋊C8C22×C12711yescharacteristic
237#380C22×C12711yescharacteristic
238#343—21no—
239#341—21no—
240#340—21no—
241#328—22no—
242#323—22no—
243C22×Q8D4×C12711yescharacteristic
244C22×D4—21no—
245C22×D4D4×C12711yescharacteristic
246C23×C4C23×C12711yescharacteristic
247C2×SD16—21no—
248C2×SD16—21no—
249C2×M16—21no—
250C2×M16—21no—
251C2×M16—21no—
252C2×M16—42no—
253C2×M16D4×C12711yescharacteristic
254C2×M16D4×C12711yescharacteristic
255C22×C8—21no—
256C22×C8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264#121—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4D4×C12711yescharacteristic
268C2×C4⋊C4D4×C12711yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272#108—42no—
273#107—21no—
274#107—21no—
275#103—21no—
276#103—21no—
277#102—21no—
278#102—21no—
279#102—42no—
280#101—42no—
281C22⋊C8—21no—
282C22⋊C8—21no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286C22⋊C8D4×C12712yes—
287#95—21no—
288C24—21no—
289C24#41×C12711yescharacteristic
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4C2×D4×C12711yescharacteristic
308C22×C4C2×D4×C12711yescharacteristic
309C22×C4C2×D4×C12711yescharacteristic
310C22×C4C2×D4×C12711yescharacteristic
311C22×C4#41×C12711yescharacteristic
312C22×C4#41×C12711yescharacteristic
313SD16—81no—
314SD16—81no—
315M16—22no—
316M16—22no—
317M16—41no—
318M16—41no—
319M16—41no—
320M16—41no—
321M16—41no—
322M16—41no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—81no—
345C23—21no—
346C23—21no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—81no—
353C23—81no—
354C23#121×C12711yescharacteristic
355C23#123×C12711yescharacteristic
356C23#124×C12711yescharacteristic
357Q8—41no—
358Q8—41no—
359Q8—81no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—82no—
364D4—82no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4#120×C12711yescharacteristic
384C2×C4#121×C12711yescharacteristic
385C2×C4#121×C12711yescharacteristic
386C2×C4#123×C12711yescharacteristic
387C8—42no—
388C8—42no—
389C8—41no—
390C8—41no—
391C8—81no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—161no—
406C22—161no—
407C22#394×C12711yescharacteristic
408C22#471×C12711yescharacteristic
409C22#473×C12711yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—161no—
417C4B(4,𝔽2)×C12711yescharacteristic
418C4#458×C12711yescharacteristic
419C2—21no—
420C2—21no—
421C2—41no—
422C2—41no—
423C2—161no—
424C2C127×#204411yescharacteristic
425C2C127×#204811yescharacteristic
426C2C127×#205211yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC2×#121×C127C22
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×#1901C2semidirect
4. C127×#1855C2semidirect
5. C127×#1520C2semidirect
7. C127×#1305C2semidirect
205. #10561C127direct (N × H)
215. #1901C2×C127semidirect
217. #1855C2×C127semidirect
218. #1520C2×C127semidirect
220. #1305C2×C127semidirect
221. C127#10561direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31932
C127×#19012
C127×#18952
C127×#18552
C127×#15202
C127×#14712
C127×#13052
#10561127