Finite Group Databasebeta

← C127×#12169 #44575842

Subgroups — C127×#12169 #44575842

Order 32512.

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

normal subgroups only

This group has 342 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×#3262C211yesmaximal, characteristic
2C127×#2188C211yesmaximal, characteristic, complemented
3C127×#2185C211yesmaximal, characteristic
4C127×#2047C211yesmaximal, characteristic, complemented
5C127×#1350C211yesmaximal, characteristic, complemented
6C127×#1349C211yesmaximal, characteristic
7C127×#1323C211yesmaximal, characteristic, complemented
8C22×Q16×C127C2211yescharacteristic
9#542×C127—21no—
10C2×Q32×C127—21no—
11C2×SD32×C127—21no—
12#477×C127—21no—
13#474×C127—21no—
14#468×C127—21no—
15#467×C127C2211yescharacteristic
16#466×C127C2211yescharacteristic
17#464×C127—21no—
18#463×C127—21no—
19#424×C127C2211yescharacteristic
20#419×C127—21no—
21C2×#101×C127—21no—
22#358×C127—21no—
23#358×C127—21no—
24C22⋊C16×C127C2211yescharacteristic
25C22⋊C16×C127C2211yescharacteristic
26#341×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
34C2×M16×C127—21no—
35C2×M16×C127D411yescharacteristic
36C22×C8×C127C2311yescharacteristic
37C4⋊Q8×C127—21no—
38#126×C127—41no—
39#124×C127—21no—
40#124×C127—41no—
41#123×C127—41no—
42C22⋊Q8×C127—21no—
43C22⋊Q8×C127—41no—
44#121×C127—21no—
45C4×Q8×C127—21no—
46C4×D4×C127—21no—
47#117×C127D411yescharacteristic
48Q32×C127—42no—
49SD32×C127—42no—
50C2×C16×C127—21no—
51C2×C16×C127—21no—
52#107×C127—21no—
53#107×C127D411yescharacteristic
54#107×C127D411yescharacteristic
55C4⋊C8×C127—21no—
56C4⋊C8×C127—21no—
57#103×C127—21no—
58#103×C127—21no—
59#103×C127—41no—
60#103×C127—41no—
61#102×C127—21no—
62#102×C127—21no—
63#101×C127—42no—
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×Q8×C127—41no—
72C2×D4×C127—21no—
73C2×D4×C127—41no—
74C22×C4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127C2×D411yescharacteristic
77Q16×C127—22no—
78Q16×C127—22no—
79Q16×C127—41no—
80Q16×C127—41no—
81Q16×C127—41no—
82SD16×C127—81no—
83D8×C127—41no—
84M16×C127—41no—
85M16×C127—41no—
86M16×C127—41no—
87C2×C8×C127—21no—
88C2×C8×C127—41no—
89C2×C8×C127—41no—
90C2×C8×C127C2×D411yescharacteristic
91C2×C8×C127C2×D411yescharacteristic
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127D811yescharacteristic
98C4⋊C4×C127D811yescharacteristic
99C22⋊C4×C127—21no—
100C22⋊C4×C127—41no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—41no—
103C22⋊C4×C127—41no—
104C22⋊C4×C127—41no—
105C42×C127—21no—
106C42×C127—41no—
107C16×C127—41no—
108C16×C127—41no—
109C23×C127—41no—
110C23×C127AGL(1,ℤ/8)11yescharacteristic
111Q8×C127—42no—
112Q8×C127—41no—
113Q8×C127—41no—
114Q8×C127—42no—
115Q8×C127—81no—
116Q8×C127—81no—
117Q8×C127—161no—
118D4×C127—41no—
119D4×C127—41no—
120D4×C127—161no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—21no—
124C2×C4×C127—21no—
125C2×C4×C127—42no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—81no—
130C2×C4×C127—81no—
131C2×C4×C127—161no—
132C2×C4×C127#12011yescharacteristic
133C2×C4×C127C2×D811yescharacteristic
134C8×C127—21no—
135C8×C127—21no—
136C8×C127—41no—
137C8×C127—81no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—81no—
142C22×C127—81no—
143C22×C127—81no—
144C22×C127#44711yescharacteristic
145C4×C127—21no—
146C4×C127—41no—
147C4×C127—41no—
148C4×C127—82no—
149C4×C127—81no—
150C4×C127—81no—
151C4×C127—81no—
152C4×C127#45411yescharacteristic
153C4×C127B(4,𝔽2)11yescharacteristic
154#12169C12711yesmaximal, characteristic, direct product factor, Sylow
155C2×C127—41no—
156C2×C127—161no—
157C2×C127#159811yescharacteristic
158C2×C127#219411yescharacteristic
159C2×C127#219511yescharacteristic
160#3262C2×C12711yescharacteristic
161#2188C2×C12711yescharacteristic, complemented
162#2185C2×C12711yescharacteristic
163#2047C2×C12711yescharacteristic, complemented
164#1350C2×C12711yescharacteristic, complemented
165#1349C2×C12711yescharacteristic
166#1323C2×C12711yescharacteristic, complemented
167C127#1216911yescharacteristic, direct product factor, Sylow
168C22×Q16C22×C12711yescharacteristic
169#542—21no—
170C2×Q32—21no—
171C2×SD32—21no—
172#477—21no—
173#474—21no—
174#468—21no—
175#467C22×C12711yescharacteristic
176#466C22×C12711yescharacteristic
177#464—21no—
178#463—21no—
179#424C22×C12711yescharacteristic
180#419—21no—
181C2×#101—21no—
182#358—21no—
183#358—21no—
184C22⋊C16C22×C12711yescharacteristic
185C22⋊C16C22×C12711yescharacteristic
186#341C22×C12711yescharacteristic
187C22×Q8—21no—
188C2×Q16—21no—
189C2×Q16—21no—
190C2×Q16—42no—
191C2×Q16D4×C12711yescharacteristic
192C2×SD16—21no—
193C2×D8D4×C12711yescharacteristic
194C2×M16—21no—
195C2×M16D4×C12711yescharacteristic
196C22×C8C23×C12711yescharacteristic
197C4⋊Q8—21no—
198#126—41no—
199#124—21no—
200#124—41no—
201#123—41no—
202C22⋊Q8—21no—
203C22⋊Q8—41no—
204#121—21no—
205C4×Q8—21no—
206C4×D4—21no—
207#117D4×C12711yescharacteristic
208Q32—42no—
209SD32—42no—
210C2×C16—21no—
211C2×C16—21no—
212#107—21no—
213#107D4×C12711yescharacteristic
214#107D4×C12711yescharacteristic
215C4⋊C8—21no—
216C4⋊C8—21no—
217#103—21no—
218#103—21no—
219#103—41no—
220#103—41no—
221#102—21no—
222#102—21no—
223#101—42no—
224C2×Q8—22no—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—41no—
232C2×D4—21no—
233C2×D4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4C2×D4×C12711yescharacteristic
237Q16—22no—
238Q16—22no—
239Q16—41no—
240Q16—41no—
241Q16—41no—
242SD16—81no—
243D8—41no—
244M16—41no—
245M16—41no—
246M16—41no—
247C2×C8—21no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8C2×D4×C12711yescharacteristic
251C2×C8C2×D4×C12711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4D8×C12711yescharacteristic
258C4⋊C4D8×C12711yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C42—21no—
266C42—41no—
267C16—41no—
268C16—41no—
269C23—41no—
270C23AGL(1,ℤ/8)×C12711yescharacteristic
271Q8—42no—
272Q8—41no—
273Q8—41no—
274Q8—42no—
275Q8—81no—
276Q8—81no—
277Q8—161no—
278D4—41no—
279D4—41no—
280D4—161no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—161no—
292C2×C4#120×C12711yescharacteristic
293C2×C4C2×D8×C12711yescharacteristic
294C8—21no—
295C8—21no—
296C8—41no—
297C8—81no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#447×C12711yescharacteristic
305C4—21no—
306C4—41no—
307C4—41no—
308C4—82no—
309C4—81no—
310C4—81no—
311C4—81no—
312C4#454×C12711yescharacteristic
313C4B(4,𝔽2)×C12711yescharacteristic
314C2—41no—
315C2—161no—
316C2C127×#159811yescharacteristic
317C2C127×#219411yescharacteristic
318C2C127×#219511yescharacteristic
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 ◁ #424×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2188C2semidirect
4. C127×#2047C2semidirect
5. C127×#1350C2semidirect
7. C127×#1323C2semidirect
154. #12169C127direct (N × H)
161. #2188C2×C127semidirect
163. #2047C2×C127semidirect
164. #1350C2×C127semidirect
166. #1323C2×C127semidirect
167. C127#12169direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32622
C127×#21882
C127×#21852
C127×#20472
C127×#13502
C127×#13492
C127×#13232
#12169127