Finite Group Databasebeta

← C127×#12163 #44575836

Subgroups — C127×#12163 #44575836

Order 32512.

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

normal subgroups only

This group has 360 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×#3250C211yesmaximal, characteristic
2C127×#2188C211yesmaximal, characteristic, complemented
3C127×#2185C211yesmaximal, characteristic
4C127×#2056C211yesmaximal, characteristic, complemented
5C127×#1348C211yesmaximal, characteristic, complemented
6C127×#1347C211yesmaximal, characteristic
7C127×#1322C211yesmaximal, characteristic, complemented
8C22×Q16×C127C2211yescharacteristic
9#529×C127—21no—
10C2×Q32×C127—21no—
11C2×SD32×C127—21no—
12#479×C127—21no—
13#478×C127—21no—
14#467×C127C2211yescharacteristic
15#466×C127C2211yescharacteristic
16#465×C127—21no—
17#464×C127—21no—
18#463×C127—21no—
19C2×C4⋊C8×C127C2211yescharacteristic
20C2×#103×C127—21no—
21#391×C127—21no—
22#358×C127—21no—
23#358×C127—21no—
24C22⋊C16×C127C2211yescharacteristic
25C22⋊C16×C127C2211yescharacteristic
26#340×C127C2211yescharacteristic
27C22×Q8×C127—21no—
28C2×Q16×C127—21no—
29C2×Q16×C127—21no—
30C2×Q16×C127—42no—
31C2×Q16×C127D411yescharacteristic
32C2×SD16×C127—21no—
33C2×D8×C127D411yescharacteristic
34C22×C8×C127D411yescharacteristic
35C22×C8×C127C2311yescharacteristic
36#126×C127—41no—
37#125×C127—21no—
38#124×C127—21no—
39#123×C127—41no—
40C22⋊Q8×C127—21no—
41#121×C127—21no—
42C4×Q8×C127—21no—
43C4×D4×C127—21no—
44C4×D4×C127—41no—
45#117×C127—41no—
46C2×C4⋊C4×C127—21no—
47C2×C42×C127D411yescharacteristic
48Q32×C127—42no—
49SD32×C127—42no—
50C2×C16×C127—21no—
51C2×C16×C127—21no—
52#107×C127D411yescharacteristic
53#107×C127D411yescharacteristic
54#106×C127—21no—
55C4⋊C8×C127—21no—
56C4⋊C8×C127—21no—
57#103×C127—21no—
58#103×C127—21no—
59#103×C127—42no—
60#102×C127—21no—
61#102×C127—21no—
62#95×C127—41no—
63#95×C127—41no—
64C2×Q8×C127—22no—
65C2×Q8×C127—21no—
66C2×Q8×C127—41no—
67C2×Q8×C127—41no—
68C2×Q8×C127—41no—
69C2×Q8×C127—41no—
70C2×Q8×C127—41no—
71C2×D4×C127—21no—
72C2×D4×C127—41no—
73C22×C4×C127—41no—
74C22×C4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127D811yescharacteristic
77C22×C4×C127D811yescharacteristic
78C22×C4×C127C2×D411yescharacteristic
79Q16×C127—22no—
80Q16×C127—22no—
81Q16×C127—41no—
82Q16×C127—41no—
83Q16×C127—41no—
84SD16×C127—81no—
85D8×C127—41no—
86C2×C8×C127—21no—
87C2×C8×C127—41no—
88C2×C8×C127—41no—
89C2×C8×C127C2×D411yescharacteristic
90C2×C8×C127C2×D411yescharacteristic
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—41no—
98C4⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C42×C127—21no—
105C42×C127—21no—
106C42×C127—41no—
107C16×C127—41no—
108C16×C127—41no—
109C23×C127—41no—
110C23×C127C2×D811yescharacteristic
111Q8×C127—42no—
112Q8×C127—41no—
113Q8×C127—41no—
114Q8×C127—42no—
115Q8×C127—81no—
116Q8×C127—81no—
117D4×C127—41no—
118D4×C127—41no—
119D4×C127—81no—
120D4×C127—81no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—21no—
126C2×C4×C127—21no—
127C2×C4×C127—41no—
128C2×C4×C127—42no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—82no—
133C2×C4×C127—81no—
134C2×C4×C127—81no—
135C2×C4×C127—81no—
136C2×C4×C127—81no—
137C2×C4×C127—81no—
138C2×C4×C127#12011yescharacteristic
139C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
140C8×C127—21no—
141C8×C127—21no—
142C8×C127—41no—
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—81no—
146C22×C127—81no—
147C22×C127—81no—
148C22×C127#44711yescharacteristic
149C22×C127#50811yescharacteristic
150C22×C127#51011yescharacteristic
151C4×C127—21no—
152C4×C127—41no—
153C4×C127—41no—
154C4×C127—82no—
155C4×C127—81no—
156C4×C127—81no—
157C4×C127—81no—
158C4×C127—161no—
159C4×C127#45411yescharacteristic
160C4×C127B(4,𝔽2)11yescharacteristic
161#12163C12711yesmaximal, characteristic, direct product factor, Sylow
162C2×C127—21no—
163C2×C127—21no—
164C2×C127—161no—
165C2×C127#159811yescharacteristic
166C2×C127#218911yescharacteristic
167C2×C127#219111yescharacteristic
168#3250C2×C12711yescharacteristic
169#2188C2×C12711yescharacteristic, complemented
170#2185C2×C12711yescharacteristic
171#2056C2×C12711yescharacteristic, complemented
172#1348C2×C12711yescharacteristic, complemented
173#1347C2×C12711yescharacteristic
174#1322C2×C12711yescharacteristic, complemented
175C127#1216311yescharacteristic, direct product factor, Sylow
176C22×Q16C22×C12711yescharacteristic
177#529—21no—
178C2×Q32—21no—
179C2×SD32—21no—
180#479—21no—
181#478—21no—
182#467C22×C12711yescharacteristic
183#466C22×C12711yescharacteristic
184#465—21no—
185#464—21no—
186#463—21no—
187C2×C4⋊C8C22×C12711yescharacteristic
188C2×#103—21no—
189#391—21no—
190#358—21no—
191#358—21no—
192C22⋊C16C22×C12711yescharacteristic
193C22⋊C16C22×C12711yescharacteristic
194#340C22×C12711yescharacteristic
195C22×Q8—21no—
196C2×Q16—21no—
197C2×Q16—21no—
198C2×Q16—42no—
199C2×Q16D4×C12711yescharacteristic
200C2×SD16—21no—
201C2×D8D4×C12711yescharacteristic
202C22×C8D4×C12711yescharacteristic
203C22×C8C23×C12711yescharacteristic
204#126—41no—
205#125—21no—
206#124—21no—
207#123—41no—
208C22⋊Q8—21no—
209#121—21no—
210C4×Q8—21no—
211C4×D4—21no—
212C4×D4—41no—
213#117—41no—
214C2×C4⋊C4—21no—
215C2×C42D4×C12711yescharacteristic
216Q32—42no—
217SD32—42no—
218C2×C16—21no—
219C2×C16—21no—
220#107D4×C12711yescharacteristic
221#107D4×C12711yescharacteristic
222#106—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225#103—21no—
226#103—21no—
227#103—42no—
228#102—21no—
229#102—21no—
230#95—41no—
231#95—41no—
232C2×Q8—22no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×D4—21no—
240C2×D4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4D8×C12711yescharacteristic
245C22×C4D8×C12711yescharacteristic
246C22×C4C2×D4×C12711yescharacteristic
247Q16—22no—
248Q16—22no—
249Q16—41no—
250Q16—41no—
251Q16—41no—
252SD16—81no—
253D8—41no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C2×D4×C12711yescharacteristic
258C2×C8C2×D4×C12711yescharacteristic
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C42—21no—
273C42—21no—
274C42—41no—
275C16—41no—
276C16—41no—
277C23—41no—
278C23C2×D8×C12711yescharacteristic
279Q8—42no—
280Q8—41no—
281Q8—41no—
282Q8—42no—
283Q8—81no—
284Q8—81no—
285D4—41no—
286D4—41no—
287D4—81no—
288D4—81no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—82no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4#120×C12711yescharacteristic
307C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
308C8—21no—
309C8—21no—
310C8—41no—
311C22—21no—
312C22—21no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22#447×C12711yescharacteristic
317C22#508×C12711yescharacteristic
318C22#510×C12711yescharacteristic
319C4—21no—
320C4—41no—
321C4—41no—
322C4—82no—
323C4—81no—
324C4—81no—
325C4—81no—
326C4—161no—
327C4#454×C12711yescharacteristic
328C4B(4,𝔽2)×C12711yescharacteristic
329C2—21no—
330C2—21no—
331C2—161no—
332C2C127×#159811yescharacteristic
333C2C127×#218911yescharacteristic
334C2C127×#219111yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radical#466×C127C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2188C2semidirect
4. C127×#2056C2semidirect
5. C127×#1348C2semidirect
7. C127×#1322C2semidirect
161. #12163C127direct (N × H)
169. #2188C2×C127semidirect
171. #2056C2×C127semidirect
172. #1348C2×C127semidirect
174. #1322C2×C127semidirect
175. C127#12163direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32502
C127×#21882
C127×#21852
C127×#20562
C127×#13482
C127×#13472
C127×#13222
#12163127