Finite Group Databasebeta

← C127×#12125 #44575798

Subgroups — C127×#12125 #44575798

Order 32512.

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

normal subgroups only

This group has 440 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×#2841C211yesmaximal, characteristic, complemented
2C127×#2022C211yesmaximal, characteristic
3C127×#1613C211yesmaximal, characteristic
4C127×#1601C211yesmaximal, characteristic
5C127×#1352C211yesmaximal, characteristic, complemented
6C127×#1351C211yesmaximal, characteristic, complemented
7C127×#1317C211yesmaximal, characteristic, complemented
8#534×C127C2211yescharacteristic
9C2×#123×C127—21no—
10#481×C127—21no—
11#448×C127—21no—
12C2×#99×C127—21no—
13#400×C127—21no—
14#399×C127C2211yescharacteristic
15#398×C127—21no—
16#398×C127—21no—
17#396×C127—21no—
18#395×C127—21no—
19#393×C127C2211yescharacteristic
20#381×C127C2211yescharacteristic
21#380×C127—21no—
22#342×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#323×C127C2211yescharacteristic
26#323×C127C2211yescharacteristic
27C22×Q8×C127D411yescharacteristic
28C22×D4×C127—21no—
29C23×C4×C127D411yescharacteristic
30C2×SD16×C127—41no—
31#123×C127—42no—
32#123×C127—41no—
33#123×C127—42no—
34#121×C127—21no—
35#121×C127—21no—
36#120×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×C127—21no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—41no—
44C2×C4⋊C4×C127D411yescharacteristic
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C22⋊C4×C127—21no—
48C2×C22⋊C4×C127—41no—
49C2×C22⋊C4×C127D411yescharacteristic
50C2×C22⋊C4×C127C2311yescharacteristic
51#106×C127—41no—
52#103×C127—41no—
53#102×C127—41no—
54#99×C127—42no—
55C22⋊C8×C127—21no—
56C22⋊C8×C127—21no—
57#95×C127—21no—
58#95×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×C127C2×D411yescharacteristic
67C2×Q8×C127—41no—
68C2×Q8×C127—81no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127—41no—
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—21no—
84C22×C4×C127—21no—
85C22×C4×C127—41no—
86C22×C4×C127—41no—
87C22×C4×C127—41no—
88C22×C4×C127SD1612yescomplemented
89C22×C4×C127C2×D411yescharacteristic
90C22×C4×C127C2×D411yescharacteristic
91SD16×C127—82no—
92C2×C8×C127—41no—
93C2×C8×C127—41no—
94C4⋊C4×C127—21no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—44no—
98C4⋊C4×C127—81no—
99C4⋊C4×C127—81no—
100C4⋊C4×C127—81no—
101C4⋊C4×C127—81no—
102C4⋊C4×C127—81no—
103C4⋊C4×C127—81no—
104C22⋊C4×C127—22no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—44no—
109C22⋊C4×C127—41no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—81no—
112C22⋊C4×C127—81no—
113C23×C127—21no—
114C23×C127—21no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—81no—
121C23×C127—81no—
122C23×C127#12011yescharacteristic
123C23×C127C2×SD1611yescharacteristic
124C23×C127#13711yescharacteristic
125Q8×C127—42no—
126Q8×C127—81no—
127D4×C127—81no—
128D4×C127—81no—
129D4×C127—81no—
130D4×C127—81no—
131D4×C127—81no—
132D4×C127—81no—
133C2×C4×C127—22no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—42no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—42no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—41no—
145C2×C4×C127—41no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—41no—
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—81no—
161C2×C4×C127—81no—
162C2×C4×C127—81no—
163C8×C127—81no—
164C8×C127—81no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—21no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—41no—
172C22×C127—41no—
173C22×C127—81no—
174C22×C127—82no—
175C22×C127—81no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127—161no—
179C22×C127#44811yescharacteristic
180C22×C127#45311yescharacteristic
181C22×C127B(4,𝔽2)11yescharacteristic
182C4×C127—42no—
183C4×C127—41no—
184C4×C127—41no—
185C4×C127—81no—
186C4×C127—81no—
187C4×C127—81no—
188C4×C127—82no—
189C4×C127—81no—
190C4×C127—81no—
191C4×C127—81no—
192#12125C12711yesmaximal, characteristic, direct product factor, Sylow
193C2×C127—21no—
194C2×C127—21no—
195C2×C127—41no—
196C2×C127—41no—
197C2×C127—161no—
198C2×C127#159611yescharacteristic
199C2×C127#220011yescharacteristic
200C2×C127#220111yescharacteristic
201#2841C2×C12711yescharacteristic, complemented
202#2022C2×C12711yescharacteristic
203#1613C2×C12711yescharacteristic
204#1601C2×C12711yescharacteristic
205#1352C2×C12711yescharacteristic, complemented
206#1351C2×C12711yescharacteristic, complemented
207#1317C2×C12711yescharacteristic, complemented
208C127#1212511yescharacteristic, direct product factor, Sylow
209#534C22×C12711yescharacteristic
210C2×#123—21no—
211#481—21no—
212#448—21no—
213C2×#99—21no—
214#400—21no—
215#399C22×C12711yescharacteristic
216#398—21no—
217#398—21no—
218#396—21no—
219#395—21no—
220#393C22×C12711yescharacteristic
221#381C22×C12711yescharacteristic
222#380—21no—
223#342C22×C12711yescharacteristic
224#327—21no—
225#327—21no—
226#323C22×C12711yescharacteristic
227#323C22×C12711yescharacteristic
228C22×Q8D4×C12711yescharacteristic
229C22×D4—21no—
230C23×C4D4×C12711yescharacteristic
231C2×SD16—41no—
232#123—42no—
233#123—41no—
234#123—42no—
235#121—21no—
236#121—21no—
237#120—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4D4×C12711yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4D4×C12711yescharacteristic
251C2×C22⋊C4C23×C12711yescharacteristic
252#106—41no—
253#103—41no—
254#102—41no—
255#99—42no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95D4×C12711yescharacteristic
265#95D4×C12711yescharacteristic
266C24—21no—
267C24C2×D4×C12711yescharacteristic
268C2×Q8—41no—
269C2×Q8—81no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4SD16×C12712yescomplemented
290C22×C4C2×D4×C12711yescharacteristic
291C22×C4C2×D4×C12711yescharacteristic
292SD16—82no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—44no—
299C4⋊C4—81no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C22⋊C4—22no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—44no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—81no—
322C23—81no—
323C23#120×C12711yescharacteristic
324C23C2×SD16×C12711yescharacteristic
325C23#137×C12711yescharacteristic
326Q8—42no—
327Q8—81no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C8—81no—
365C8—81no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—82no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—161no—
380C22#448×C12711yescharacteristic
381C22#453×C12711yescharacteristic
382C22B(4,𝔽2)×C12711yescharacteristic
383C4—42no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—82no—
390C4—81no—
391C4—81no—
392C4—81no—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—161no—
398C2C127×#159611yescharacteristic
399C2C127×#220011yescharacteristic
400C2C127×#220111yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radical#534×C127C22
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×#2841C2semidirect
5. C127×#1352C2semidirect
6. C127×#1351C2semidirect
7. C127×#1317C2semidirect
88. C22×C4×C127SD16semidirect
192. #12125C127direct (N × H)
201. #2841C2×C127semidirect
205. #1352C2×C127semidirect
206. #1351C2×C127semidirect
207. #1317C2×C127semidirect
208. C127#12125direct (N × H)
289. C22×C4SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#28412
C127×#20222
C127×#16132
C127×#16012
C127×#13522
C127×#13512
C127×#13172
#12125127