Finite Group Databasebeta

← C127×#12191 #44575864

Subgroups — C127×#12191 #44575864

Order 32512.

484 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2752C211yesmaximal, characteristic, complemented
2C127×#2022C211yesmaximal, characteristic, complemented
3C127×#1613C211yesmaximal, characteristic, complemented
4C127×#1595C211yesmaximal, characteristic
5C127×#1352C211yesmaximal, characteristic, complemented
6C127×#1343C211yesmaximal, characteristic, complemented
7C127×#1317C211yesmaximal, characteristic, complemented
8#534×C127C2211yescharacteristic
9C2×#123×C127—21no—
10#481×C127—21no—
11#447×C127—21no—
12C2×#99×C127—21no—
13#399×C127C2211yescharacteristic
14#397×C127—21no—
15#394×C127—21no—
16#392×C127C2211yescharacteristic
17#390×C127—21no—
18#388×C127—21no—
19#387×C127—21no—
20#386×C127—21no—
21#381×C127C2211yescharacteristic
22#342×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C22×D4×C127—21no—
28C22×D4×C127—21no—
29C22×D4×C127D411yescharacteristic
30C23×C4×C127D411yescharacteristic
31C2×D8×C127—41no—
32#123×C127—42no—
33#123×C127—41no—
34#123×C127—42no—
35#121×C127—21no—
36#121×C127—21no—
37#120×C127—41no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127—41no—
41C2×C4⋊C4×C127D411yescharacteristic
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C22⋊C4×C127—21no—
48C2×C22⋊C4×C127—21no—
49C2×C22⋊C4×C127—41no—
50C2×C22⋊C4×C127D411yescharacteristic
51C2×C22⋊C4×C127C2311yescharacteristic
52C2×C42×C127—21no—
53#106×C127—41no—
54#102×C127—41no—
55#102×C127—41no—
56#99×C127—42no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59#95×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127—21no—
63#95×C127D411yescharacteristic
64#95×C127D411yescharacteristic
65C24×C127—21no—
66C24×C127—21no—
67C24×C127C2×D411yescharacteristic
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—41no—
71C2×D4×C127—42no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127—41no—
77C2×D4×C127—41no—
78C2×D4×C127—41no—
79C2×D4×C127—41no—
80C2×D4×C127—81no—
81C2×D4×C127—81no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—21no—
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
95D8×C127—82no—
96C2×C8×C127—41no—
97C2×C8×C127—41no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127—21no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—42no—
102C4⋊C4×C127—44no—
103C4⋊C4×C127—81no—
104C22⋊C4×C127—22no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—42no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—44no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—81no—
113C22⋊C4×C127—81no—
114C22⋊C4×C127—81no—
115C22⋊C4×C127—81no—
116C22⋊C4×C127—81no—
117C42×C127—81no—
118C23×C127—21no—
119C23×C127—21no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127—41no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—81no—
133C23×C127—81no—
134C23×C127#12011yescharacteristic
135C23×C127#13511yescharacteristic
136C23×C127AGL(1,ℤ/8)11yescharacteristic
137D4×C127—42no—
138D4×C127—82no—
139D4×C127—82no—
140D4×C127—81no—
141D4×C127—81no—
142D4×C127—81no—
143D4×C127—81no—
144D4×C127—81no—
145D4×C127—81no—
146D4×C127—81no—
147C2×C4×C127—22no—
148C2×C4×C127—21no—
149C2×C4×C127—21no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—41no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—42no—
157C2×C4×C127—42no—
158C2×C4×C127—41no—
159C2×C4×C127—41no—
160C2×C4×C127—41no—
161C2×C4×C127—41no—
162C2×C4×C127—41no—
163C2×C4×C127—41no—
164C2×C4×C127—41no—
165C2×C4×C127—41no—
166C2×C4×C127—41no—
167C2×C4×C127—41no—
168C2×C4×C127—41no—
169C2×C4×C127—41no—
170C2×C4×C127—41no—
171C2×C4×C127—81no—
172C2×C4×C127—81no—
173C2×C4×C127—81no—
174C8×C127—81no—
175C8×C127—81no—
176C22×C127—21no—
177C22×C127—21no—
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—41no—
183C22×C127—41no—
184C22×C127—81no—
185C22×C127—81no—
186C22×C127—81no—
187C22×C127—81no—
188C22×C127—81no—
189C22×C127—82no—
190C22×C127—81no—
191C22×C127—81no—
192C22×C127—81no—
193C22×C127—81no—
194C22×C127—81no—
195C22×C127—81no—
196C22×C127—161no—
197C22×C127#44911yescharacteristic
198C22×C127#45311yescharacteristic
199C22×C127B(4,𝔽2)11yescharacteristic
200C4×C127—42no—
201C4×C127—41no—
202C4×C127—41no—
203C4×C127—81no—
204C4×C127—81no—
205C4×C127—82no—
206C4×C127—81no—
207C4×C127—81no—
208C4×C127—81no—
209#12191C12711yesmaximal, characteristic, direct product factor, Sylow
210C2×C127—21no—
211C2×C127—21no—
212C2×C127—41no—
213C2×C127—41no—
214C2×C127—161no—
215C2×C127—161no—
216C2×C127#160011yescharacteristic
217C2×C127#219811yescharacteristic
218C2×C127#220011yescharacteristic
219#2752C2×C12711yescharacteristic, complemented
220#2022C2×C12711yescharacteristic, complemented
221#1613C2×C12711yescharacteristic, complemented
222#1595C2×C12711yescharacteristic
223#1352C2×C12711yescharacteristic, complemented
224#1343C2×C12711yescharacteristic, complemented
225#1317C2×C12711yescharacteristic, complemented
226C127#1219111yescharacteristic, direct product factor, Sylow
227#534C22×C12711yescharacteristic
228C2×#123—21no—
229#481—21no—
230#447—21no—
231C2×#99—21no—
232#399C22×C12711yescharacteristic
233#397—21no—
234#394—21no—
235#392C22×C12711yescharacteristic
236#390—21no—
237#388—21no—
238#387—21no—
239#386—21no—
240#381C22×C12711yescharacteristic
241#342C22×C12711yescharacteristic
242#327—21no—
243#327—21no—
244#323C22×C12711yescharacteristic
245#323C22×C12711yescharacteristic
246C22×D4—21no—
247C22×D4—21no—
248C22×D4D4×C12711yescharacteristic
249C23×C4D4×C12711yescharacteristic
250C2×D8—41no—
251#123—42no—
252#123—41no—
253#123—42no—
254#121—21no—
255#121—21no—
256#120—41no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4D4×C12711yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4D4×C12711yescharacteristic
270C2×C22⋊C4C23×C12711yescharacteristic
271C2×C42—21no—
272#106—41no—
273#102—41no—
274#102—41no—
275#99—42no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282#95D4×C12711yescharacteristic
283#95D4×C12711yescharacteristic
284C24—21no—
285C24—21no—
286C24C2×D4×C12711yescharacteristic
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4C2×D4×C12711yescharacteristic
313C22×C4C2×D4×C12711yescharacteristic
314D8—82no—
315C2×C8—41no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—81no—
323C22⋊C4—22no—
324C22⋊C4—41no—
325C22⋊C4—42no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—44no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—81no—
332C22⋊C4—81no—
333C22⋊C4—81no—
334C22⋊C4—81no—
335C22⋊C4—81no—
336C42—81no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—81no—
352C23—81no—
353C23#120×C12711yescharacteristic
354C23#135×C12711yescharacteristic
355C23AGL(1,ℤ/8)×C12711yescharacteristic
356D4—42no—
357D4—82no—
358D4—82no—
359D4—81no—
360D4—81no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C8—81no—
394C8—81no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—82no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22—81no—
414C22—81no—
415C22—161no—
416C22#449×C12711yescharacteristic
417C22#453×C12711yescharacteristic
418C22B(4,𝔽2)×C12711yescharacteristic
419C4—42no—
420C4—41no—
421C4—41no—
422C4—81no—
423C4—81no—
424C4—82no—
425C4—81no—
426C4—81no—
427C4—81no—
428C2—21no—
429C2—21no—
430C2—41no—
431C2—41no—
432C2—161no—
433C2—161no—
434C2C127×#160011yescharacteristic
435C2C127×#219811yescharacteristic
436C2C127×#220011yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC127×#1595C2
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
1. C127×#2752C2semidirect
2. C127×#2022C2semidirect
3. C127×#1613C2semidirect
5. C127×#1352C2semidirect
6. C127×#1343C2semidirect
7. C127×#1317C2semidirect
209. #12191C127direct (N × H)
219. #2752C2×C127semidirect
220. #2022C2×C127semidirect
221. #1613C2×C127semidirect
223. #1352C2×C127semidirect
224. #1343C2×C127semidirect
225. #1317C2×C127semidirect
226. C127#12191direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#27522
C127×#20222
C127×#16132
C127×#15952
C127×#13522
C127×#13432
C127×#13172
#12191127