Finite Group Databasebeta

← C127×#12214 #44575887

Subgroups — C127×#12214 #44575887

Order 32512.

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

normal subgroups only

This group has 578 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×#2437C211yesmaximal, characteristic
2C127×#2029C211yesmaximal, characteristic
3C127×#1614C211yesmaximal, characteristic, complemented
4C127×#1602C211yesmaximal, characteristic, complemented
5C127×#1352C211yesmaximal, characteristic, complemented
6C127×#1351C211yesmaximal, characteristic, complemented
7C127×#1316C211yesmaximal, characteristic
8#543×C127C2211yescharacteristic
9C2×C22⋊Q8×C127—21no—
10C22×C22⋊C4×C127—21no—
11#484×C127—21no—
12#451×C127—21no—
13#399×C127C2211yescharacteristic
14#396×C127—21no—
15#393×C127C2211yescharacteristic
16#385×C127—21no—
17#385×C127—21no—
18#380×C127—21no—
19#379×C127—21no—
20#379×C127C2211yescharacteristic
21#342×C127—21no—
22#342×C127C2211yescharacteristic
23#328×C127—21no—
24#328×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C25×C127D411yescharacteristic
28C22×Q8×C127D411yescharacteristic
29C23×C4×C127—21no—
30C23×C4×C127—21no—
31C2×Q16×C127—41no—
32C22⋊Q8×C127—21no—
33C22⋊Q8×C127—21no—
34C22⋊Q8×C127—44no—
35C22⋊Q8×C127—41no—
36C22⋊Q8×C127—41no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127—21no—
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—21no—
50C2×C22⋊C4×C127—44no—
51C2×C22⋊C4×C127—41no—
52C2×C22⋊C4×C127—41no—
53C2×C22⋊C4×C127D411yescharacteristic
54C2×C22⋊C4×C127C2311yescharacteristic
55#107×C127—41no—
56#103×C127—41no—
57#103×C127—41no—
58C22⋊C8×C127—21no—
59C22⋊C8×C127—21no—
60#95×C127—21no—
61#95×C127—21no—
62#95×C127D411yescharacteristic
63#95×C127D411yescharacteristic
64C24×C127—21no—
65C24×C127—21no—
66C24×C127—42no—
67C24×C127—41no—
68C24×C127—42no—
69C24×C127—41no—
70C24×C127Q1612yescomplemented
71C24×C127C2×D411yescharacteristic
72C2×Q8×C127—21no—
73C2×Q8×C127—21no—
74C2×Q8×C127—42no—
75C2×Q8×C127—41no—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—21no—
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×C127—41no—
88C22×C4×C127—81no—
89C22×C4×C127—81no—
90C22×C4×C127C2×D411yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92Q16×C127—82no—
93C2×C8×C127—41no—
94C2×C8×C127—41no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—41no—
98C4⋊C4×C127—42no—
99C4⋊C4×C127—42no—
100C4⋊C4×C127—42no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—81no—
104C22⋊C4×C127—24no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—42no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—42no—
109C22⋊C4×C127—42no—
110C22⋊C4×C127—42no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C22⋊C4×C127—41no—
115C22⋊C4×C127—41no—
116C22⋊C4×C127—82no—
117C22⋊C4×C127—82no—
118C23×C127—22no—
119C23×C127—22no—
120C23×C127—22no—
121C23×C127—21no—
122C23×C127—21no—
123C23×C127—41no—
124C23×C127—41no—
125C23×C127—41no—
126C23×C127—44no—
127C23×C127—42no—
128C23×C127—41no—
129C23×C127—44no—
130C23×C127—41no—
131C23×C127—41no—
132C23×C127—42no—
133C23×C127—42no—
134C23×C127—81no—
135C23×C127—84no—
136C23×C127—81no—
137C23×C127—81no—
138C23×C127#12011yescharacteristic
139C23×C127C2×Q1611yescharacteristic
140C23×C127#13711yescharacteristic
141Q8×C127—42no—
142Q8×C127—81no—
143Q8×C127—81no—
144Q8×C127—81no—
145C2×C4×C127—22no—
146C2×C4×C127—22no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—41no—
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—41no—
157C2×C4×C127—41no—
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—84no—
167C2×C4×C127—84no—
168C2×C4×C127—81no—
169C2×C4×C127—81no—
170C2×C4×C127—81no—
171C2×C4×C127—81no—
172C2×C4×C127—81no—
173C8×C127—81no—
174C8×C127—81no—
175C22×C127—21no—
176C22×C127—21no—
177C22×C127—24no—
178C22×C127—21no—
179C22×C127—21no—
180C22×C127—41no—
181C22×C127—41no—
182C22×C127—42no—
183C22×C127—42no—
184C22×C127—42no—
185C22×C127—44no—
186C22×C127—42no—
187C22×C127—41no—
188C22×C127—42no—
189C22×C127—41no—
190C22×C127—42no—
191C22×C127—81no—
192C22×C127—82no—
193C22×C127—82no—
194C22×C127—82no—
195C22×C127#45111yescharacteristic
196C22×C127#45511yescharacteristic
197C22×C127B(4,𝔽2)11yescharacteristic
198C4×C127—42no—
199C4×C127—41no—
200C4×C127—42no—
201C4×C127—41no—
202C4×C127—81no—
203C4×C127—81no—
204C4×C127—81no—
205C4×C127—81no—
206C4×C127—81no—
207C4×C127—161no—
208#12214C12711yesmaximal, characteristic, direct product factor, Sylow
209C2×C127—21no—
210C2×C127—21no—
211C2×C127—41no—
212C2×C127—44no—
213C2×C127—41no—
214C2×C127#160211yescharacteristic
215C2×C127#219611yescharacteristic
216C2×C127#219711yescharacteristic
217#2437C2×C12711yescharacteristic
218#2029C2×C12711yescharacteristic
219#1614C2×C12711yescharacteristic, complemented
220#1602C2×C12711yescharacteristic, complemented
221#1352C2×C12711yescharacteristic, complemented
222#1351C2×C12711yescharacteristic, complemented
223#1316C2×C12711yescharacteristic
224C127#1221411yescharacteristic, direct product factor, Sylow
225#543C22×C12711yescharacteristic
226C2×C22⋊Q8—21no—
227C22×C22⋊C4—21no—
228#484—21no—
229#451—21no—
230#399C22×C12711yescharacteristic
231#396—21no—
232#393C22×C12711yescharacteristic
233#385—21no—
234#385—21no—
235#380—21no—
236#379—21no—
237#379C22×C12711yescharacteristic
238#342—21no—
239#342C22×C12711yescharacteristic
240#328—21no—
241#328—21no—
242#323C22×C12711yescharacteristic
243#323C22×C12711yescharacteristic
244C25D4×C12711yescharacteristic
245C22×Q8D4×C12711yescharacteristic
246C23×C4—21no—
247C23×C4—21no—
248C2×Q16—41no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—44no—
252C22⋊Q8—41no—
253C22⋊Q8—41no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4D4×C12711yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
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—44no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4D4×C12711yescharacteristic
271C2×C22⋊C4C23×C12711yescharacteristic
272#107—41no—
273#103—41no—
274#103—41no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277#95—21no—
278#95—21no—
279#95D4×C12711yescharacteristic
280#95D4×C12711yescharacteristic
281C24—21no—
282C24—21no—
283C24—42no—
284C24—41no—
285C24—42no—
286C24—41no—
287C24Q16×C12712yescomplemented
288C24C2×D4×C12711yescharacteristic
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—42no—
292C2×Q8—41no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4C2×D4×C12711yescharacteristic
308C22×C4C2×D4×C12711yescharacteristic
309Q16—82no—
310C2×C8—41no—
311C2×C8—41no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C22⋊C4—24no—
322C22⋊C4—41no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—44no—
344C23—42no—
345C23—41no—
346C23—44no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—84no—
353C23—81no—
354C23—81no—
355C23#120×C12711yescharacteristic
356C23C2×Q16×C12711yescharacteristic
357C23#137×C12711yescharacteristic
358Q8—42no—
359Q8—81no—
360Q8—81no—
361Q8—81no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—81no—
390C8—81no—
391C8—81no—
392C22—21no—
393C22—21no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—42no—
404C22—41no—
405C22—42no—
406C22—41no—
407C22—42no—
408C22—81no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22#451×C12711yescharacteristic
413C22#455×C12711yescharacteristic
414C22B(4,𝔽2)×C12711yescharacteristic
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—161no—
425C2—21no—
426C2—21no—
427C2—41no—
428C2—44no—
429C2—41no—
430C2C127×#160211yescharacteristic
431C2C127×#219611yescharacteristic
432C2C127×#219711yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC25×C127D4
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
3. C127×#1614C2semidirect
4. C127×#1602C2semidirect
5. C127×#1352C2semidirect
6. C127×#1351C2semidirect
70. C24×C127Q16semidirect
208. #12214C127direct (N × H)
219. #1614C2×C127semidirect
220. #1602C2×C127semidirect
221. #1352C2×C127semidirect
222. #1351C2×C127semidirect
224. C127#12214direct (N × H)
287. C24Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24372
C127×#20292
C127×#16142
C127×#16022
C127×#13522
C127×#13512
C127×#13162
#12214127