Finite Group Databasebeta

← C127×#12196 #44575869

Subgroups — C127×#12196 #44575869

Order 32512.

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

normal subgroups only

This group has 434 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×#2593C211yesmaximal, characteristic, complemented
2C127×#2006C211yesmaximal, characteristic
3C127×#1616C211yesmaximal, characteristic, complemented
4C127×#1610C211yesmaximal, characteristic, complemented
5C127×#1353C211yesmaximal, characteristic, complemented
6C127×#1346C211yesmaximal, characteristic, complemented
7C127×#1318C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9#537×C127C2211yescharacteristic
10#481×C127—21no—
11#480×C127—21no—
12C2×#104×C127—21no—
13#400×C127C2211yescharacteristic
14#398×C127C2211yescharacteristic
15#394×C127—21no—
16#390×C127C2211yescharacteristic
17#388×C127—21no—
18#388×C127—21no—
19#388×C127—21no—
20#388×C127—21no—
21C4×C4⋊C4×C127—21no—
22#343×C127C2211yescharacteristic
23#327×C127—21no—
24#327×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C22×D4×C127D411yescharacteristic
29AGL(1,ℤ/8)×C127—42no—
30AGL(1,ℤ/8)×C127—42no—
31C2×SD16×C127—41no—
32C2×D8×C127—41no—
33C2×M16×C127—21no—
34C22⋊Q8×C127—41no—
35#121×C127—21no—
36#121×C127—21no—
37#121×C127—41no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127D411yescharacteristic
41C2×C4⋊C4×C127D411yescharacteristic
42C2×C4⋊C4×C127C2311yescharacteristic
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×C42×C127—21no—
50C2×C42×C127—21no—
51C2×C42×C127D411yescharacteristic
52#107×C127—41no—
53#106×C127—41no—
54#104×C127—42no—
55#102×C127—41no—
56#102×C127—41no—
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—
66#41×C127—41no—
67#41×C127—41no—
68#41×C127—41no—
69#41×C127—41no—
70C2×Q8×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—21no—
73C2×D4×C127—21no—
74C2×D4×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127—41no—
77C2×D4×C127—41no—
78C2×D4×C127—41no—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—21no—
82C22×C4×C127—21no—
83C22×C4×C127—21no—
84C22×C4×C127—41no—
85C22×C4×C127C2×D411yescharacteristic
86C22×C4×C127C2×D411yescharacteristic
87C22×C4×C127C2×D411yescharacteristic
88SD16×C127—44no—
89D8×C127—44no—
90M16×C127—22no—
91M16×C127—41no—
92C2×C8×C127—41no—
93C2×C8×C127—41no—
94C2×C8×C127—41no—
95C4⋊C4×C127—21no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—41no—
98C4⋊C4×C127—41no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—42no—
101C4⋊C4×C127—41no—
102C4⋊C4×C127—41no—
103C22⋊C4×C127—42no—
104C22⋊C4×C127—42no—
105C22⋊C4×C127—42no—
106C22⋊C4×C127—42no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—81no—
110C22⋊C4×C127—81no—
111C42×C127—41no—
112C42×C127—42no—
113C42×C127—81no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127—41no—
119C23×C127—41no—
120C23×C127—41no—
121C23×C127—41no—
122C23×C127#12011yescharacteristic
123Q8×C127—41no—
124Q8×C127—41no—
125D4×C127—41no—
126D4×C127—42no—
127D4×C127—41no—
128D4×C127—81no—
129D4×C127—81no—
130D4×C127—81no—
131D4×C127—81no—
132D4×C127—81no—
133C2×C4×C127—21no—
134C2×C4×C127—21no—
135C2×C4×C127—22no—
136C2×C4×C127—22no—
137C2×C4×C127—21no—
138C2×C4×C127—21no—
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—41no—
155C2×C4×C127—81no—
156C2×C4×C127—81no—
157C2×C4×C127#13511yescharacteristic
158C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
159C8×C127—42no—
160C8×C127—81no—
161C8×C127—81no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—81no—
165C22×C127—81no—
166C22×C127—81no—
167C22×C127—81no—
168C22×C127—81no—
169C22×C127—81no—
170C22×C127—81no—
171C22×C127—81no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127#44911yescharacteristic
175C22×C127#45311yescharacteristic
176C22×C127B(4,𝔽2)11yescharacteristic
177C4×C127—21no—
178C4×C127—21no—
179C4×C127—42no—
180C4×C127—42no—
181C4×C127—41no—
182C4×C127—41no—
183C4×C127—81no—
184C4×C127—81no—
185C4×C127—81no—
186C4×C127—81no—
187#12196C12711yesmaximal, characteristic, direct product factor, Sylow
188C2×C127—21no—
189C2×C127—21no—
190C2×C127—161no—
191C2×C127—161no—
192C2×C127#160011yescharacteristic
193C2×C127#219811yescharacteristic
194C2×C127#220411yescharacteristic
195#2593C2×C12711yescharacteristic, complemented
196#2006C2×C12711yescharacteristic
197#1616C2×C12711yescharacteristic, complemented
198#1610C2×C12711yescharacteristic, complemented
199#1353C2×C12711yescharacteristic, complemented
200#1346C2×C12711yescharacteristic, complemented
201#1318C2×C12711yescharacteristic, complemented
202C127#1219611yescharacteristic, direct product factor, Sylow
203C2×AGL(1,ℤ/8)—21no—
204#537C22×C12711yescharacteristic
205#481—21no—
206#480—21no—
207C2×#104—21no—
208#400C22×C12711yescharacteristic
209#398C22×C12711yescharacteristic
210#394—21no—
211#390C22×C12711yescharacteristic
212#388—21no—
213#388—21no—
214#388—21no—
215#388—21no—
216C4×C4⋊C4—21no—
217#343C22×C12711yescharacteristic
218#327—21no—
219#327—21no—
220#324C22×C12711yescharacteristic
221#324C22×C12711yescharacteristic
222C2×#41—21no—
223C22×D4D4×C12711yescharacteristic
224AGL(1,ℤ/8)—42no—
225AGL(1,ℤ/8)—42no—
226C2×SD16—41no—
227C2×D8—41no—
228C2×M16—21no—
229C22⋊Q8—41no—
230#121—21no—
231#121—21no—
232#121—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4D4×C12711yescharacteristic
236C2×C4⋊C4D4×C12711yescharacteristic
237C2×C4⋊C4C23×C12711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42D4×C12711yescharacteristic
247#107—41no—
248#106—41no—
249#104—42no—
250#102—41no—
251#102—41no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95D4×C12711yescharacteristic
259#95D4×C12711yescharacteristic
260C24—21no—
261#41—41no—
262#41—41no—
263#41—41no—
264#41—41no—
265C2×Q8—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4C2×D4×C12711yescharacteristic
281C22×C4C2×D4×C12711yescharacteristic
282C22×C4C2×D4×C12711yescharacteristic
283SD16—44no—
284D8—44no—
285M16—22no—
286M16—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—81no—
305C22⋊C4—81no—
306C42—41no—
307C42—42no—
308C42—81no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23#120×C12711yescharacteristic
318Q8—41no—
319Q8—41no—
320D4—41no—
321D4—42no—
322D4—41no—
323D4—81no—
324D4—81no—
325D4—81no—
326D4—81no—
327D4—81no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
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#135×C12711yescharacteristic
353C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
354C8—42no—
355C8—81no—
356C8—81no—
357C22—21no—
358C22—21no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#449×C12711yescharacteristic
370C22#453×C12711yescharacteristic
371C22B(4,𝔽2)×C12711yescharacteristic
372C4—21no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C2—21no—
383C2—21no—
384C2—161no—
385C2—161no—
386C2C127×#160011yescharacteristic
387C2C127×#219811yescharacteristic
388C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC127×#2006C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C127×#2593C2semidirect
3. C127×#1616C2semidirect
4. C127×#1610C2semidirect
5. C127×#1353C2semidirect
6. C127×#1346C2semidirect
7. C127×#1318C2semidirect
187. #12196C127direct (N × H)
195. #2593C2×C127semidirect
197. #1616C2×C127semidirect
198. #1610C2×C127semidirect
199. #1353C2×C127semidirect
200. #1346C2×C127semidirect
201. #1318C2×C127semidirect
202. C127#12196direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25932
C127×#20062
C127×#16162
C127×#16102
C127×#13532
C127×#13462
C127×#13182
#12196127