Finite Group Databasebeta

← C127×#12210 #44575883

Subgroups — C127×#12210 #44575883

Order 32512.

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

normal subgroups only

This group has 414 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×#2728C211yesmaximal, characteristic
2C127×#2028C211yesmaximal, characteristic, complemented
3C127×#1611C211yesmaximal, characteristic, complemented
4C127×#1599C211yesmaximal, characteristic
5C127×#1346C211yesmaximal, characteristic, complemented
6C127×#1345C211yesmaximal, characteristic
7C127×#1319C211yesmaximal, characteristic, complemented
8#544×C127C2211yescharacteristic
9C2×C22⋊Q8×C127—21no—
10#483×C127—21no—
11#450×C127—21no—
12#419×C127—21no—
13#400×C127—21no—
14#398×C127C2211yescharacteristic
15#397×C127—21no—
16#394×C127C2211yescharacteristic
17#391×C127C2211yescharacteristic
18#388×C127—21no—
19#386×C127—21no—
20#385×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—
36C22⋊Q8×C127—42no—
37C22⋊Q8×C127—42no—
38#117×C127—21no—
39C2×C4⋊C4×C127—21no—
40C2×C4⋊C4×C127—21no—
41C2×C4⋊C4×C127—21no—
42C2×C4⋊C4×C127D411yescharacteristic
43C2×C4⋊C4×C127C2311yescharacteristic
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×C127D411yescharacteristic
50#106×C127—41no—
51#103×C127—41no—
52#103×C127—41no—
53#103×C127—41no—
54#102×C127—41no—
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×C127D411yescharacteristic
63#95×C127D411yescharacteristic
64C24×C127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127—21no—
67C2×Q8×C127—21no—
68C2×Q8×C127—41no—
69C2×Q8×C127—41no—
70C2×Q8×C127SD1612yescomplemented
71C2×D4×C127—41no—
72C2×D4×C127—42no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127—21no—
77C22×C4×C127—21no—
78C22×C4×C127—42no—
79C22×C4×C127—81no—
80C22×C4×C127—81no—
81C22×C4×C127C2×D411yescharacteristic
82C22×C4×C127C2×D411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84SD16×C127—82no—
85M16×C127—41no—
86M16×C127—41no—
87C2×C8×C127—41no—
88C2×C8×C127—41no—
89C2×C8×C127—41no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—42no—
97C4⋊C4×C127—42no—
98C4⋊C4×C127—41no—
99C4⋊C4×C127—41no—
100C4⋊C4×C127—41no—
101C4⋊C4×C127—41no—
102C22⋊C4×C127—42no—
103C22⋊C4×C127—42no—
104C22⋊C4×C127—42no—
105C22⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—81no—
109C42×C127—41no—
110C42×C127—41no—
111C23×C127—22no—
112C23×C127—41no—
113C23×C127—41no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—41no—
118C23×C127#12011yescharacteristic
119Q8×C127—42no—
120Q8×C127—81no—
121Q8×C127—81no—
122Q8×C127—81no—
123D4×C127—42no—
124D4×C127—81no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—22no—
128C2×C4×C127—21no—
129C2×C4×C127—21no—
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—41no—
144C2×C4×C127—41no—
145C2×C4×C127—41no—
146C2×C4×C127—41no—
147C2×C4×C127—41no—
148C2×C4×C127—84no—
149C2×C4×C127C2×SD1611yescharacteristic
150C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
151C8×C127—81no—
152C8×C127—81no—
153C8×C127—81no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—42no—
157C22×C127—42no—
158C22×C127—42no—
159C22×C127—81no—
160C22×C127—81no—
161C22×C127—81no—
162C22×C127—81no—
163C22×C127#45011yescharacteristic
164C22×C127#45511yescharacteristic
165C22×C127B(4,𝔽2)11yescharacteristic
166C4×C127—21no—
167C4×C127—21no—
168C4×C127—42no—
169C4×C127—41no—
170C4×C127—41no—
171C4×C127—81no—
172C4×C127—81no—
173C4×C127—81no—
174C4×C127—81no—
175C4×C127—81no—
176C4×C127—81no—
177#12210C12711yesmaximal, characteristic, direct product factor, Sylow
178C2×C127—21no—
179C2×C127—21no—
180C2×C127—82no—
181C2×C127#160111yescharacteristic
182C2×C127#219711yescharacteristic
183C2×C127#220311yescharacteristic
184#2728C2×C12711yescharacteristic
185#2028C2×C12711yescharacteristic, complemented
186#1611C2×C12711yescharacteristic, complemented
187#1599C2×C12711yescharacteristic
188#1346C2×C12711yescharacteristic, complemented
189#1345C2×C12711yescharacteristic
190#1319C2×C12711yescharacteristic, complemented
191C127#1221011yescharacteristic, direct product factor, Sylow
192#544C22×C12711yescharacteristic
193C2×C22⋊Q8—21no—
194#483—21no—
195#450—21no—
196#419—21no—
197#400—21no—
198#398C22×C12711yescharacteristic
199#397—21no—
200#394C22×C12711yescharacteristic
201#391C22×C12711yescharacteristic
202#388—21no—
203#386—21no—
204#385—21no—
205#343C22×C12711yescharacteristic
206#339—21no—
207#328—21no—
208#328—21no—
209#324C22×C12711yescharacteristic
210#324C22×C12711yescharacteristic
211C22×Q8D4×C12711yescharacteristic
212C22×D4D4×C12711yescharacteristic
213C23×C4—21no—
214C2×SD16—41no—
215C2×M16—21no—
216C4⋊Q8—41no—
217#125—41no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—42no—
221C22⋊Q8—42no—
222#117—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4D4×C12711yescharacteristic
227C2×C4⋊C4C23×C12711yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C42D4×C12711yescharacteristic
234#106—41no—
235#103—41no—
236#103—41no—
237#103—41no—
238#102—41no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95D4×C12711yescharacteristic
247#95D4×C12711yescharacteristic
248C24—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8SD16×C12712yescomplemented
255C2×D4—41no—
256C2×D4—42no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4C2×D4×C12711yescharacteristic
266C22×C4C2×D4×C12711yescharacteristic
267C22×C4C2×D4×C12711yescharacteristic
268SD16—82no—
269M16—41no—
270M16—41no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—81no—
293C42—41no—
294C42—41no—
295C23—22no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23#120×C12711yescharacteristic
303Q8—42no—
304Q8—81no—
305Q8—81no—
306Q8—81no—
307D4—42no—
308D4—81no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
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—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—84no—
333C2×C4C2×SD16×C12711yescharacteristic
334C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
335C8—81no—
336C8—81no—
337C8—81no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22#450×C12711yescharacteristic
348C22#455×C12711yescharacteristic
349C22B(4,𝔽2)×C12711yescharacteristic
350C4—21no—
351C4—21no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C2—21no—
362C2—21no—
363C2—82no—
364C2C127×#160111yescharacteristic
365C2C127×#219711yescharacteristic
366C2C127×#220311yescharacteristic
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×#1611C2semidirect
5. C127×#1346C2semidirect
7. C127×#1319C2semidirect
70. C2×Q8×C127SD16semidirect
177. #12210C127direct (N × H)
185. #2028C2×C127semidirect
186. #1611C2×C127semidirect
188. #1346C2×C127semidirect
190. #1319C2×C127semidirect
191. C127#12210direct (N × H)
254. C2×Q8SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#27282
C127×#20282
C127×#16112
C127×#15992
C127×#13462
C127×#13452
C127×#13192
#12210127