Finite Group Databasebeta

← C127×#12134 #44575807

Subgroups — C127×#12134 #44575807

Order 32512.

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

normal subgroups only

This group has 424 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×#2523C211yesmaximal, characteristic, complemented
2C127×#2044C211yesmaximal, characteristic
3C127×#1607C211yesmaximal, characteristic
4C127×#1601C211yesmaximal, characteristic
5C127×#1351C211yesmaximal, characteristic, complemented
6C127×#1344C211yesmaximal, characteristic, complemented
7C127×#1326C211yesmaximal, characteristic, complemented
8#534×C127C2211yescharacteristic
9#480×C127—21no—
10#448×C127—21no—
11C2×#99×C127—21no—
12#397×C127—21no—
13#397×C127C2211yescharacteristic
14#393×C127C2211yescharacteristic
15#391×C127—21no—
16#387×C127—21no—
17#387×C127—21no—
18#386×C127—21no—
19#385×C127C2211yescharacteristic
20#383×C127—21no—
21C4×C22⋊C4×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—41no—
32#121×C127—21no—
33#121×C127—21no—
34#120×C127—41no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127D411yescharacteristic
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—21no—
41C2×C22⋊C4×C127—41no—
42C2×C22⋊C4×C127D411yescharacteristic
43C2×C22⋊C4×C127C2311yescharacteristic
44C2×C42×C127—21no—
45C2×C42×C127—21no—
46#107×C127—41no—
47#103×C127—41no—
48#102×C127—41no—
49#99×C127—42no—
50C22⋊C8×C127—21no—
51C22⋊C8×C127—21no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—21no—
60#95×C127—41no—
61#95×C127D411yescharacteristic
62#95×C127D411yescharacteristic
63C24×C127—21no—
64C24×C127C2×D411yescharacteristic
65C2×Q8×C127—42no—
66C2×Q8×C127—41no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C22×C4×C127—21no—
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—42no—
83C22×C4×C127—41no—
84C22×C4×C127—41no—
85C22×C4×C127C2×D411yescharacteristic
86C22×C4×C127C2×D411yescharacteristic
87SD16×C127—82no—
88C2×C8×C127—41no—
89C2×C8×C127—41no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—42no—
94C4⋊C4×C127—42no—
95C22⋊C4×C127—21no—
96C22⋊C4×C127—21no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—42no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C22⋊C4×C127—41no—
105C22⋊C4×C127—82no—
106C42×C127—42no—
107C42×C127—81no—
108C23×C127—21no—
109C23×C127—21no—
110C23×C127—41no—
111C23×C127—41no—
112C23×C127—41no—
113C23×C127—41no—
114C23×C127—41no—
115C23×C127—81no—
116C23×C127—81no—
117C23×C127#12011yescharacteristic
118C23×C127#13511yescharacteristic
119C23×C127AGL(1,ℤ/8)11yescharacteristic
120Q8×C127—42no—
121Q8×C127—81no—
122D4×C127—81no—
123D4×C127—81no—
124D4×C127—81no—
125D4×C127—81no—
126D4×C127—81no—
127D4×C127—81no—
128C2×C4×C127—22no—
129C2×C4×C127—22no—
130C2×C4×C127—21no—
131C2×C4×C127—21no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—41no—
137C2×C4×C127—41no—
138C2×C4×C127—42no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
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—81no—
155C2×C4×C127—81no—
156C2×C4×C127—81no—
157C2×C4×C127—81no—
158C8×C127—81no—
159C8×C127—81no—
160C22×C127—21no—
161C22×C127—21no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—41no—
165C22×C127—41no—
166C22×C127—41no—
167C22×C127—41no—
168C22×C127—81no—
169C22×C127—82no—
170C22×C127—81no—
171C22×C127—81no—
172C22×C127—81no—
173C22×C127—161no—
174C22×C127#44911yescharacteristic
175C22×C127#45411yescharacteristic
176C22×C127B(4,𝔽2)11yescharacteristic
177C4×C127—41no—
178C4×C127—41no—
179C4×C127—42no—
180C4×C127—42no—
181C4×C127—81no—
182C4×C127—81no—
183C4×C127—81no—
184C4×C127—81no—
185C4×C127—81no—
186C4×C127—161no—
187#12134C12711yesmaximal, characteristic, direct product factor, Sylow
188C2×C127—21no—
189C2×C127—21no—
190C2×C127—41no—
191C2×C127—41no—
192C2×C127—161no—
193C2×C127#159711yescharacteristic
194C2×C127#219711yescharacteristic
195C2×C127#219711yescharacteristic
196#2523C2×C12711yescharacteristic, complemented
197#2044C2×C12711yescharacteristic
198#1607C2×C12711yescharacteristic
199#1601C2×C12711yescharacteristic
200#1351C2×C12711yescharacteristic, complemented
201#1344C2×C12711yescharacteristic, complemented
202#1326C2×C12711yescharacteristic, complemented
203C127#1213411yescharacteristic, direct product factor, Sylow
204#534C22×C12711yescharacteristic
205#480—21no—
206#448—21no—
207C2×#99—21no—
208#397—21no—
209#397C22×C12711yescharacteristic
210#393C22×C12711yescharacteristic
211#391—21no—
212#387—21no—
213#387—21no—
214#386—21no—
215#385C22×C12711yescharacteristic
216#383—21no—
217C4×C22⋊C4—21no—
218#342C22×C12711yescharacteristic
219#327—21no—
220#327—21no—
221#323C22×C12711yescharacteristic
222#323C22×C12711yescharacteristic
223C22×Q8D4×C12711yescharacteristic
224C22×D4—21no—
225C23×C4D4×C12711yescharacteristic
226C2×SD16—41no—
227#123—41no—
228#121—21no—
229#121—21no—
230#120—41no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4D4×C12711yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4D4×C12711yescharacteristic
239C2×C22⋊C4C23×C12711yescharacteristic
240C2×C42—21no—
241C2×C42—21no—
242#107—41no—
243#103—41no—
244#102—41no—
245#99—42no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95D4×C12711yescharacteristic
258#95D4×C12711yescharacteristic
259C24—21no—
260C24C2×D4×C12711yescharacteristic
261C2×Q8—42no—
262C2×Q8—41no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4C2×D4×C12711yescharacteristic
282C22×C4C2×D4×C12711yescharacteristic
283SD16—82no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—82no—
302C42—42no—
303C42—81no—
304C23—21no—
305C23—21no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—81no—
312C23—81no—
313C23#120×C12711yescharacteristic
314C23#135×C12711yescharacteristic
315C23AGL(1,ℤ/8)×C12711yescharacteristic
316Q8—42no—
317Q8—81no—
318D4—81no—
319D4—81no—
320D4—81no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C8—81no—
355C8—81no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—82no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—161no—
370C22#449×C12711yescharacteristic
371C22#454×C12711yescharacteristic
372C22B(4,𝔽2)×C12711yescharacteristic
373C4—41no—
374C4—41no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—161no—
383C2—21no—
384C2—21no—
385C2—41no—
386C2—41no—
387C2—161no—
388C2C127×#159711yescharacteristic
389C2C127×#219711yescharacteristic
390C2C127×#219711yescharacteristic
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×#2523C2semidirect
5. C127×#1351C2semidirect
6. C127×#1344C2semidirect
7. C127×#1326C2semidirect
187. #12134C127direct (N × H)
196. #2523C2×C127semidirect
200. #1351C2×C127semidirect
201. #1344C2×C127semidirect
202. #1326C2×C127semidirect
203. C127#12134direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25232
C127×#20442
C127×#16072
C127×#16012
C127×#13512
C127×#13442
C127×#13262
#12134127