Finite Group Databasebeta

← C127×#12280 #44575953

Subgroups — C127×#12280 #44575953

Order 32512.

348 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3191C211yesmaximal, characteristic
2C127×#2235C211yesmaximal, characteristic, complemented
3C127×#2232C211yesmaximal, characteristic, complemented
4C127×#2075C211yesmaximal, characteristic, complemented
5C127×#1348C211yesmaximal, characteristic, complemented
6C127×#1347C211yesmaximal, characteristic, complemented
7C127×#1314C211yesmaximal, characteristic, complemented
8#525×C127—21no—
9#485×C127—21no—
10#482×C127—21no—
11#467×C127C2211yescharacteristic
12#466×C127C2211yescharacteristic
13#465×C127—21no—
14#452×C127—21no—
15#449×C127—21no—
16C2×#107×C127C2211yescharacteristic
17C2×C22⋊C8×C127C2211yescharacteristic
18#380×C127—21no—
19#367×C127—21no—
20#366×C127—21no—
21#358×C127—21no—
22#357×C127—21no—
23C22⋊C16×C127C2211yescharacteristic
24C22⋊C16×C127C2211yescharacteristic
25#340×C127—21no—
26#340×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C23×C4×C127D411yescharacteristic
29C2×Q16×C127D411yescharacteristic
30C2×SD16×C127—21no—
31C2×D8×C127D411yescharacteristic
32C22×C8×C127D411yescharacteristic
33C22×C8×C127C2311yescharacteristic
34#123×C127—41no—
35C22⋊Q8×C127—21no—
36C22⋊Q8×C127—21no—
37#121×C127—21no—
38#121×C127—21no—
39#120×C127—41no—
40C4×D4×C127—41no—
41C4×D4×C127—41no—
42#117×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127—21no—
45C2×C22⋊C4×C127—41no—
46C2×C16×C127—21no—
47C2×C16×C127—21no—
48#107×C127—21no—
49#107×C127—21no—
50#107×C127D411yescharacteristic
51#107×C127D411yescharacteristic
52#106×C127—21no—
53#103×C127—21no—
54#103×C127—21no—
55#102×C127—21no—
56#102×C127—21no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59#95×C127—41no—
60C24×C127D811yescharacteristic
61#41×C127—81no—
62#41×C127—81no—
63C2×Q8×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—41no—
66C2×D4×C127—41no—
67C2×D4×C127—41no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—41no—
71C22×C4×C127—41no—
72C22×C4×C127—41no—
73C22×C4×C127—41no—
74C22×C4×C127—41no—
75C22×C4×C127D811yescharacteristic
76C22×C4×C127C2×D411yescharacteristic
77Q16×C127—41no—
78SD16×C127—81no—
79D8×C127—41no—
80C2×C8×C127—21no—
81C2×C8×C127—41no—
82C2×C8×C127—41no—
83C2×C8×C127C2×D411yescharacteristic
84C2×C8×C127C2×D411yescharacteristic
85C4⋊C4×C127—21no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—41no—
93C22⋊C4×C127—41no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—82no—
99C42×C127—41no—
100C16×C127—41no—
101C16×C127—41no—
102C23×C127—21no—
103C23×C127—21no—
104C23×C127—21no—
105C23×C127—41no—
106C23×C127—41no—
107C23×C127—41no—
108C23×C127C2×D811yescharacteristic
109Q8×C127—41no—
110Q8×C127—41no—
111D4×C127—41no—
112D4×C127—41no—
113D4×C127—82no—
114D4×C127—82no—
115D4×C127—81no—
116D4×C127—81no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—41no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—81no—
127C2×C4×C127—82no—
128C2×C4×C127—81no—
129C2×C4×C127—81no—
130C2×C4×C127—81no—
131C2×C4×C127—81no—
132C2×C4×C127—81no—
133C2×C4×C127—81no—
134C2×C4×C127#12011yescharacteristic
135C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
136C8×C127—21no—
137C8×C127—21no—
138C8×C127—41no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—42no—
142C22×C127—41no—
143C22×C127—41no—
144C22×C127—41no—
145C22×C127—81no—
146C22×C127—81no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127#44711yescharacteristic
150C22×C127#50811yescharacteristic
151C22×C127#50911yescharacteristic
152C4×C127—21no—
153C4×C127—41no—
154C4×C127—81no—
155C4×C127—81no—
156C4×C127—81no—
157C4×C127—161no—
158C4×C127—161no—
159C4×C127#45511yescharacteristic
160C4×C127#45811yescharacteristic
161#12280C12711yesmaximal, characteristic, direct product factor, Sylow
162C2×C127—21no—
163C2×C127—21no—
164C2×C127—81no—
165C2×C127—161no—
166C2×C127#160711yescharacteristic
167C2×C127#218611yescharacteristic
168C2×C127#219111yescharacteristic
169#3191C2×C12711yescharacteristic
170#2235C2×C12711yescharacteristic, complemented
171#2232C2×C12711yescharacteristic, complemented
172#2075C2×C12711yescharacteristic, complemented
173#1348C2×C12711yescharacteristic, complemented
174#1347C2×C12711yescharacteristic, complemented
175#1314C2×C12711yescharacteristic, complemented
176C127#1228011yescharacteristic, direct product factor, Sylow
177#525—21no—
178#485—21no—
179#482—21no—
180#467C22×C12711yescharacteristic
181#466C22×C12711yescharacteristic
182#465—21no—
183#452—21no—
184#449—21no—
185C2×#107C22×C12711yescharacteristic
186C2×C22⋊C8C22×C12711yescharacteristic
187#380—21no—
188#367—21no—
189#366—21no—
190#358—21no—
191#357—21no—
192C22⋊C16C22×C12711yescharacteristic
193C22⋊C16C22×C12711yescharacteristic
194#340—21no—
195#340C22×C12711yescharacteristic
196C2×#41—21no—
197C23×C4D4×C12711yescharacteristic
198C2×Q16D4×C12711yescharacteristic
199C2×SD16—21no—
200C2×D8D4×C12711yescharacteristic
201C22×C8D4×C12711yescharacteristic
202C22×C8C23×C12711yescharacteristic
203#123—41no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206#121—21no—
207#121—21no—
208#120—41no—
209C4×D4—41no—
210C4×D4—41no—
211#117—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C22⋊C4—41no—
215C2×C16—21no—
216C2×C16—21no—
217#107—21no—
218#107—21no—
219#107D4×C12711yescharacteristic
220#107D4×C12711yescharacteristic
221#106—21no—
222#103—21no—
223#103—21no—
224#102—21no—
225#102—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228#95—41no—
229C24D8×C12711yescharacteristic
230#41—81no—
231#41—81no—
232C2×Q8—21no—
233C2×D4—21no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4D8×C12711yescharacteristic
245C22×C4C2×D4×C12711yescharacteristic
246Q16—41no—
247SD16—81no—
248D8—41no—
249C2×C8—21no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8C2×D4×C12711yescharacteristic
253C2×C8C2×D4×C12711yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—82no—
268C42—41no—
269C16—41no—
270C16—41no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×D8×C12711yescharacteristic
278Q8—41no—
279Q8—41no—
280D4—41no—
281D4—41no—
282D4—82no—
283D4—82no—
284D4—81no—
285D4—81no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4#120×C12711yescharacteristic
304C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
305C8—21no—
306C8—21no—
307C8—41no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22#447×C12711yescharacteristic
319C22#508×C12711yescharacteristic
320C22#509×C12711yescharacteristic
321C4—21no—
322C4—41no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—161no—
327C4—161no—
328C4#455×C12711yescharacteristic
329C4#458×C12711yescharacteristic
330C2—21no—
331C2—21no—
332C2—81no—
333C2—161no—
334C2C127×#160711yescharacteristic
335C2C127×#218611yescharacteristic
336C2C127×#219111yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3191C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C22⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2235C2semidirect
3. C127×#2232C2semidirect
4. C127×#2075C2semidirect
5. C127×#1348C2semidirect
6. C127×#1347C2semidirect
7. C127×#1314C2semidirect
161. #12280C127direct (N × H)
170. #2235C2×C127semidirect
171. #2232C2×C127semidirect
172. #2075C2×C127semidirect
173. #1348C2×C127semidirect
174. #1347C2×C127semidirect
175. #1314C2×C127semidirect
176. C127#12280direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31912
C127×#22352
C127×#22322
C127×#20752
C127×#13482
C127×#13472
C127×#13142
#12280127