Finite Group Databasebeta

← C127×#12249 #44575922

Subgroups — C127×#12249 #44575922

Order 32512.

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

normal subgroups only

This group has 386 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×#2823C211yesmaximal, characteristic
2C127×#2060C211yesmaximal, characteristic, complemented
3C127×#1611C211yesmaximal, characteristic, complemented
4C127×#1599C211yesmaximal, characteristic
5C127×#1346C211yesmaximal, characteristic, complemented
6C127×#1345C211yesmaximal, characteristic
7C127×#1325C211yesmaximal, characteristic, complemented
8#544×C127C2211yescharacteristic
9C2×#126×C127—21no—
10#483×C127—21no—
11#450×C127—21no—
12#428×C127—21no—
13#398×C127—21no—
14#398×C127C2211yescharacteristic
15#396×C127—21no—
16#396×C127—21no—
17#396×C127—21no—
18#394×C127—21no—
19#394×C127C2211yescharacteristic
20#384×C127C2211yescharacteristic
21#343×C127C2211yescharacteristic
22#339×C127—21no—
23#328×C127—21no—
24#328×C127—21no—
25#324×C127C2211yescharacteristic
26#324×C127C2211yescharacteristic
27C22×D4×C127D411yescharacteristic
28C2×SD16×C127—41no—
29C2×M16×C127—21no—
30C4⋊Q8×C127—41no—
31#126×C127—42no—
32#126×C127—42no—
33#125×C127—41no—
34C22⋊Q8×C127—21no—
35C22⋊Q8×C127—21no—
36#117×C127—21no—
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×C127D411yescharacteristic
44C2×C4⋊C4×C127D411yescharacteristic
45C2×C4⋊C4×C127C2311yescharacteristic
46C2×C22⋊C4×C127—21no—
47C2×C22⋊C4×C127—21no—
48C2×C22⋊C4×C127—21no—
49C2×C22⋊C4×C127—21no—
50C2×C22⋊C4×C127—21no—
51C2×C22⋊C4×C127—21no—
52C2×C42×C127D411yescharacteristic
53#107×C127—41no—
54#106×C127—41no—
55#106×C127—41no—
56#103×C127—41no—
57#102×C127—41no—
58C22⋊C8×C127—21no—
59C22⋊C8×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×D4×C127—41no—
68C2×D4×C127—81no—
69C22×C4×C127—21no—
70C22×C4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127C2×D411yescharacteristic
75C22×C4×C127C2×D411yescharacteristic
76C22×C4×C127C2×D411yescharacteristic
77SD16×C127—82no—
78M16×C127—22no—
79M16×C127—41no—
80C2×C8×C127—41no—
81C2×C8×C127—41no—
82C2×C8×C127—41no—
83C4⋊C4×C127—21no—
84C4⋊C4×C127—21no—
85C4⋊C4×C127—21no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—42no—
91C4⋊C4×C127—42no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—81no—
97C4⋊C4×C127—81no—
98C4⋊C4×C127—81no—
99C22⋊C4×C127—42no—
100C22⋊C4×C127—42no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C22⋊C4×C127—81no—
105C22⋊C4×C127—81no—
106C22⋊C4×C127—81no—
107C42×C127—21no—
108C42×C127—41no—
109C42×C127SD1612yescomplemented
110C23×C127—41no—
111C23×C127—41no—
112C23×C127—41no—
113C23×C127—41no—
114C23×C127—41no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127#12011yescharacteristic
118Q8×C127—81no—
119Q8×C127—81no—
120D4×C127—42no—
121D4×C127—81no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—22no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
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—41no—
144C2×C4×C127—41no—
145C2×C4×C127C2×SD1611yescharacteristic
146C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
147C8×C127—42no—
148C8×C127—81no—
149C8×C127—81no—
150C22×C127—21no—
151C22×C127—21no—
152C22×C127—81no—
153C22×C127—81no—
154C22×C127—81no—
155C22×C127—81no—
156C22×C127—81no—
157C22×C127—81no—
158C22×C127—81no—
159C22×C127#45011yescharacteristic
160C22×C127#45611yescharacteristic
161C22×C127B(4,𝔽2)11yescharacteristic
162C4×C127—21no—
163C4×C127—21no—
164C4×C127—42no—
165C4×C127—41no—
166C4×C127—41no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127—81no—
170C4×C127—81no—
171C4×C127—81no—
172C4×C127—81no—
173#12249C12711yesmaximal, characteristic, direct product factor, Sylow
174C2×C127—21no—
175C2×C127—21no—
176C2×C127—161no—
177C2×C127#160411yescharacteristic
178C2×C127#220011yescharacteristic
179C2×C127#220311yescharacteristic
180#2823C2×C12711yescharacteristic
181#2060C2×C12711yescharacteristic, complemented
182#1611C2×C12711yescharacteristic, complemented
183#1599C2×C12711yescharacteristic
184#1346C2×C12711yescharacteristic, complemented
185#1345C2×C12711yescharacteristic
186#1325C2×C12711yescharacteristic, complemented
187C127#1224911yescharacteristic, direct product factor, Sylow
188#544C22×C12711yescharacteristic
189C2×#126—21no—
190#483—21no—
191#450—21no—
192#428—21no—
193#398—21no—
194#398C22×C12711yescharacteristic
195#396—21no—
196#396—21no—
197#396—21no—
198#394—21no—
199#394C22×C12711yescharacteristic
200#384C22×C12711yescharacteristic
201#343C22×C12711yescharacteristic
202#339—21no—
203#328—21no—
204#328—21no—
205#324C22×C12711yescharacteristic
206#324C22×C12711yescharacteristic
207C22×D4D4×C12711yescharacteristic
208C2×SD16—41no—
209C2×M16—21no—
210C4⋊Q8—41no—
211#126—42no—
212#126—42no—
213#125—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#117—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4D4×C12711yescharacteristic
224C2×C4⋊C4D4×C12711yescharacteristic
225C2×C4⋊C4C23×C12711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42D4×C12711yescharacteristic
233#107—41no—
234#106—41no—
235#106—41no—
236#103—41no—
237#102—41no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240#95—21no—
241#95—21no—
242#95D4×C12711yescharacteristic
243#95D4×C12711yescharacteristic
244C24—21no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×D4—41no—
248C2×D4—81no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C2×D4×C12711yescharacteristic
255C22×C4C2×D4×C12711yescharacteristic
256C22×C4C2×D4×C12711yescharacteristic
257SD16—82no—
258M16—22no—
259M16—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C4⋊C4—81no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—81no—
287C42—21no—
288C42—41no—
289C42SD16×C12712yescomplemented
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#120×C12711yescharacteristic
298Q8—81no—
299Q8—81no—
300D4—42no—
301D4—81no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
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—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4C2×SD16×C12711yescharacteristic
326C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
327C8—42no—
328C8—81no—
329C8—81no—
330C22—21no—
331C22—21no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#450×C12711yescharacteristic
340C22#456×C12711yescharacteristic
341C22B(4,𝔽2)×C12711yescharacteristic
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—81no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C2—21no—
354C2—21no—
355C2—161no—
356C2C127×#160411yescharacteristic
357C2C127×#220011yescharacteristic
358C2C127×#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×#2060C2semidirect
3. C127×#1611C2semidirect
5. C127×#1346C2semidirect
7. C127×#1325C2semidirect
109. C42×C127SD16semidirect
173. #12249C127direct (N × H)
181. #2060C2×C127semidirect
182. #1611C2×C127semidirect
184. #1346C2×C127semidirect
186. #1325C2×C127semidirect
187. C127#12249direct (N × H)
289. C42SD16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#28232
C127×#20602
C127×#16112
C127×#15992
C127×#13462
C127×#13452
C127×#13252
#12249127