Finite Group Databasebeta

← C127×#12276 #44575949

Subgroups — C127×#12276 #44575949

Order 32512.

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

normal subgroups only

This group has 362 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×#2830C211yesmaximal, characteristic
2C127×#2027C211yesmaximal, characteristic
3C127×#1611C211yesmaximal, characteristic, complemented
4C127×#1605C211yesmaximal, characteristic, complemented
5C127×#1354C211yesmaximal, characteristic, complemented
6C127×#1346C211yesmaximal, characteristic, complemented
7C127×#1318C211yesmaximal, characteristic
8#544×C127C2211yescharacteristic
9#483×C127—21no—
10#451×C127—21no—
11#418×C127—21no—
12#398×C127C2211yescharacteristic
13#397×C127—21no—
14#397×C127—21no—
15#396×C127—21no—
16#395×C127C2211yescharacteristic
17#394×C127—21no—
18#393×C127—21no—
19#390×C127C2211yescharacteristic
20#387×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
29C2×Q16×C127—41no—
30C2×M16×C127—21no—
31C4⋊Q8×C127—41no—
32#125×C127—41no—
33C22⋊Q8×C127—21no—
34C22⋊Q8×C127—21no—
35#117×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C4⋊C4×C127D411yescharacteristic
40C2×C4⋊C4×C127C2311yescharacteristic
41C2×C22⋊C4×C127—21no—
42C2×C22⋊C4×C127—21no—
43C2×C22⋊C4×C127—21no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—21no—
47C2×C42×C127D411yescharacteristic
48#106×C127—41no—
49#103×C127—41no—
50#103×C127—41no—
51#102×C127—41no—
52#102×C127—41no—
53C22⋊C8×C127—21no—
54C22⋊C8×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—41no—
66C2×Q8×C127—81no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—21no—
74C22×C4×C127—21no—
75C22×C4×C127—21no—
76C22×C4×C127C2×D411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C22×C4×C127C2×D411yescharacteristic
79Q16×C127—82no—
80M16×C127—41no—
81M16×C127—41no—
82C2×C8×C127—41no—
83C2×C8×C127—41no—
84C2×C8×C127—41no—
85C4⋊C4×C127—21no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—21no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—81no—
96C4⋊C4×C127—81no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—81no—
99C22⋊C4×C127—81no—
100C22⋊C4×C127—81no—
101C22⋊C4×C127—81no—
102C22⋊C4×C127—81no—
103C22⋊C4×C127—81no—
104C42×C127—41no—
105C42×C127—41no—
106C23×C127—41no—
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—21no—
123C2×C4×C127—21no—
124C2×C4×C127—41no—
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#13511yescharacteristic
144C2×C4×C127#13711yescharacteristic
145C8×C127—81no—
146C8×C127—81no—
147C8×C127—81no—
148C22×C127—21no—
149C22×C127—21no—
150C22×C127—81no—
151C22×C127—81no—
152C22×C127—81no—
153C22×C127—81no—
154C22×C127—81no—
155C22×C127—81no—
156C22×C127—81no—
157C22×C127#45211yescharacteristic
158C22×C127#45511yescharacteristic
159C22×C127B(4,𝔽2)11yescharacteristic
160C4×C127—21no—
161C4×C127—21no—
162C4×C127—41no—
163C4×C127—41no—
164C4×C127—81no—
165C4×C127—81no—
166C4×C127—81no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127—81no—
170C4×C127—81no—
171#12276C12711yesmaximal, characteristic, direct product factor, Sylow
172C2×C127—21no—
173C2×C127—21no—
174C2×C127—161no—
175C2×C127#160611yescharacteristic
176C2×C127#220011yescharacteristic
177C2×C127#220311yescharacteristic
178#2830C2×C12711yescharacteristic
179#2027C2×C12711yescharacteristic
180#1611C2×C12711yescharacteristic, complemented
181#1605C2×C12711yescharacteristic, complemented
182#1354C2×C12711yescharacteristic, complemented
183#1346C2×C12711yescharacteristic, complemented
184#1318C2×C12711yescharacteristic
185C127#1227611yescharacteristic, direct product factor, Sylow
186#544C22×C12711yescharacteristic
187#483—21no—
188#451—21no—
189#418—21no—
190#398C22×C12711yescharacteristic
191#397—21no—
192#397—21no—
193#396—21no—
194#395C22×C12711yescharacteristic
195#394—21no—
196#393—21no—
197#390C22×C12711yescharacteristic
198#387—21no—
199#343C22×C12711yescharacteristic
200#339—21no—
201#328—21no—
202#328—21no—
203#324C22×C12711yescharacteristic
204#324C22×C12711yescharacteristic
205C22×Q8D4×C12711yescharacteristic
206C22×D4D4×C12711yescharacteristic
207C2×Q16—41no—
208C2×M16—21no—
209C4⋊Q8—41no—
210#125—41no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213#117—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4D4×C12711yescharacteristic
218C2×C4⋊C4C23×C12711yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42D4×C12711yescharacteristic
226#106—41no—
227#103—41no—
228#103—41no—
229#102—41no—
230#102—41no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95D4×C12711yescharacteristic
239#95D4×C12711yescharacteristic
240C24—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—81no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
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
257Q16—82no—
258M16—41no—
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—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C42—41no—
283C42—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#120×C12711yescharacteristic
292Q8—42no—
293Q8—81no—
294Q8—81no—
295Q8—81no—
296D4—42no—
297D4—81no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
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#135×C12711yescharacteristic
322C2×C4#137×C12711yescharacteristic
323C8—81no—
324C8—81no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#452×C12711yescharacteristic
336C22#455×C12711yescharacteristic
337C22B(4,𝔽2)×C12711yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—21no—
350C2—21no—
351C2—161no—
352C2C127×#160611yescharacteristic
353C2C127×#220011yescharacteristic
354C2C127×#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
3. C127×#1611C2semidirect
4. C127×#1605C2semidirect
5. C127×#1354C2semidirect
6. C127×#1346C2semidirect
171. #12276C127direct (N × H)
180. #1611C2×C127semidirect
181. #1605C2×C127semidirect
182. #1354C2×C127semidirect
183. #1346C2×C127semidirect
185. C127#12276direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#28302
C127×#20272
C127×#16112
C127×#16052
C127×#13542
C127×#13462
C127×#13182
#12276127