Finite Group Databasebeta

← C127×#12271 #44575944

Subgroups — C127×#12271 #44575944

Order 32512.

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

normal subgroups only

This group has 396 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×#2572C211yesmaximal, characteristic
2C127×#2028C211yesmaximal, characteristic, complemented
3C127×#1617C211yesmaximal, characteristic, complemented
4C127×#1599C211yesmaximal, characteristic
5C127×#1353C211yesmaximal, characteristic, complemented
6C127×#1345C211yesmaximal, characteristic
7C127×#1319C211yesmaximal, characteristic, complemented
8#544×C127C2211yescharacteristic
9#484×C127—21no—
10#450×C127—21no—
11#419×C127—21no—
12#400×C127C2211yescharacteristic
13#394×C127C2211yescharacteristic
14#393×C127—21no—
15#391×C127C2211yescharacteristic
16#388×C127—21no—
17#386×C127—21no—
18#383×C127—21no—
19#381×C127—21no—
20C4×C22⋊C4×C127—21no—
21#343×C127C2211yescharacteristic
22#339×C127—21no—
23#328×C127—21no—
24#328×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C22×Q8×C127D411yescharacteristic
28C22×D4×C127D411yescharacteristic
29C23×C4×C127—21no—
30C2×SD16×C127—41no—
31C2×M16×C127—21no—
32C4⋊Q8×C127—41no—
33#125×C127—41no—
34C22⋊Q8×C127—21no—
35C22⋊Q8×C127—21no—
36#117×C127—21no—
37C2×C4⋊C4×C127D411yescharacteristic
38C2×C4⋊C4×C127C2311yescharacteristic
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—21no—
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C42×C127—21no—
45C2×C42×C127D411yescharacteristic
46#107×C127—41no—
47#103×C127—41no—
48#103×C127—41no—
49#103×C127—41no—
50#102×C127—41no—
51C22⋊C8×C127—21no—
52C22⋊C8×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×C127D411yescharacteristic
61#95×C127D411yescharacteristic
62C24×C127—21no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127—21no—
67C2×Q8×C127—41no—
68C2×Q8×C127—41no—
69C2×D4×C127—41no—
70C2×D4×C127—42no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—42no—
77C22×C4×C127—42no—
78C22×C4×C127—81no—
79C22×C4×C127C2×D411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81C22×C4×C127C2×D411yescharacteristic
82SD16×C127—82no—
83M16×C127—41no—
84M16×C127—41no—
85C2×C8×C127—41no—
86C2×C8×C127—41no—
87C2×C8×C127—41no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C22⋊C4×C127—42no—
97C22⋊C4×C127—42no—
98C22⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—81no—
102C22⋊C4×C127—81no—
103C42×C127—41no—
104C42×C127—42no—
105C42×C127—41no—
106C23×C127—22no—
107C23×C127—41no—
108C23×C127—41no—
109C23×C127—41no—
110C23×C127—41no—
111C23×C127—41no—
112C23×C127—41no—
113C23×C127#12011yescharacteristic
114Q8×C127—42no—
115Q8×C127—81no—
116Q8×C127—81no—
117Q8×C127—81no—
118D4×C127—42no—
119D4×C127—81no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—22no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
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—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—84no—
144C2×C4×C127#13511yescharacteristic
145C2×C4×C127#13711yescharacteristic
146C8×C127—81no—
147C8×C127—81no—
148C8×C127—81no—
149C22×C127—21no—
150C22×C127—21no—
151C22×C127—42no—
152C22×C127—42no—
153C22×C127—42no—
154C22×C127—81no—
155C22×C127—81no—
156C22×C127—81no—
157C22×C127—81no—
158C22×C127#45211yescharacteristic
159C22×C127#45511yescharacteristic
160C22×C127B(4,𝔽2)11yescharacteristic
161C4×C127—21no—
162C4×C127—21no—
163C4×C127—42no—
164C4×C127—41no—
165C4×C127—41no—
166C4×C127—81no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127—81no—
170C4×C127—81no—
171C4×C127—81no—
172#12271C12711yesmaximal, characteristic, direct product factor, Sylow
173C2×C127—21no—
174C2×C127—21no—
175C2×C127—82no—
176C2×C127#160611yescharacteristic
177C2×C127#219711yescharacteristic
178C2×C127#220411yescharacteristic
179#2572C2×C12711yescharacteristic
180#2028C2×C12711yescharacteristic, complemented
181#1617C2×C12711yescharacteristic, complemented
182#1599C2×C12711yescharacteristic
183#1353C2×C12711yescharacteristic, complemented
184#1345C2×C12711yescharacteristic
185#1319C2×C12711yescharacteristic, complemented
186C127#1227111yescharacteristic, direct product factor, Sylow
187#544C22×C12711yescharacteristic
188#484—21no—
189#450—21no—
190#419—21no—
191#400C22×C12711yescharacteristic
192#394C22×C12711yescharacteristic
193#393—21no—
194#391C22×C12711yescharacteristic
195#388—21no—
196#386—21no—
197#383—21no—
198#381—21no—
199C4×C22⋊C4—21no—
200#343C22×C12711yescharacteristic
201#339—21no—
202#328—21no—
203#328—21no—
204#324C22×C12711yescharacteristic
205#324C22×C12711yescharacteristic
206C22×Q8D4×C12711yescharacteristic
207C22×D4D4×C12711yescharacteristic
208C23×C4—21no—
209C2×SD16—41no—
210C2×M16—21no—
211C4⋊Q8—41no—
212#125—41no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215#117—21no—
216C2×C4⋊C4D4×C12711yescharacteristic
217C2×C4⋊C4C23×C12711yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42D4×C12711yescharacteristic
225#107—41no—
226#103—41no—
227#103—41no—
228#103—41no—
229#102—41no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95D4×C12711yescharacteristic
240#95D4×C12711yescharacteristic
241C24—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×Q8—21no—
245C2×Q8—21no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—81no—
258C22×C4C2×D4×C12711yescharacteristic
259C22×C4C2×D4×C12711yescharacteristic
260C22×C4C2×D4×C12711yescharacteristic
261SD16—82no—
262M16—41no—
263M16—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C42—41no—
283C42—42no—
284C42—41no—
285C23—22no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23#120×C12711yescharacteristic
293Q8—42no—
294Q8—81no—
295Q8—81no—
296Q8—81no—
297D4—42no—
298D4—81no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—84no—
323C2×C4#135×C12711yescharacteristic
324C2×C4#137×C12711yescharacteristic
325C8—81no—
326C8—81no—
327C8—81no—
328C22—21no—
329C22—21no—
330C22—42no—
331C22—42no—
332C22—42no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22#452×C12711yescharacteristic
338C22#455×C12711yescharacteristic
339C22B(4,𝔽2)×C12711yescharacteristic
340C4—21no—
341C4—21no—
342C4—42no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C2—21no—
352C2—21no—
353C2—82no—
354C2C127×#160611yescharacteristic
355C2C127×#219711yescharacteristic
356C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC22×D4×C127D4
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
2. C127×#2028C2semidirect
3. C127×#1617C2semidirect
5. C127×#1353C2semidirect
7. C127×#1319C2semidirect
172. #12271C127direct (N × H)
180. #2028C2×C127semidirect
181. #1617C2×C127semidirect
183. #1353C2×C127semidirect
185. #1319C2×C127semidirect
186. C127#12271direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25722
C127×#20282
C127×#16172
C127×#15992
C127×#13532
C127×#13452
C127×#13192
#12271127