Finite Group Databasebeta

← C127×#12828 #44576501

Subgroups — C127×#12828 #44576501

Order 32512.

390 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#2952C211yesmaximal, characteristic
2C127×#2153C211yesmaximal, characteristic, complemented
3C127×#2145C211yesmaximal, characteristic
4C127×#2143C211yesmaximal, characteristic
5C127×#1833C211yesmaximal, characteristic, complemented
6C127×#1386C211yesmaximal, characteristic, complemented
7C127×#1383C211yesmaximal, characteristic, complemented
8C2×#135×C127C2211yescharacteristic
9#520×C127—21no—
10C2×M32×C127C2211yescharacteristic
11C2×M32×C127C2211yescharacteristic
12#442×C127C412yescomplemented
13#441×C127C412yescomplemented
14C4×SD16×C127—22no—
15C2×#108×C127C2211yescharacteristic
16#427×C127C2211yescharacteristic
17C2×#106×C127C2211yescharacteristic
18#417×C127—21no—
19#405×C127C2211yescharacteristic
20#364×C127—22no—
21#360×C127—22no—
22#359×C127—22no—
23#344×C127—21no—
24H(3,ℤ/4)×C127—21no—
25C2×#41×C127—21no—
26#135×C127—21no—
27#135×C127—21no—
28#135×C127D412yes—
29#135×C127Q812yes—
30C2×Q16×C127C2×C411yescharacteristic
31C2×SD16×C127—21no—
32C2×D8×C127C2×C411yescharacteristic
33C2×M16×C127—21no—
34C2×M16×C127D411yescharacteristic
35C22×C8×C127C2311yescharacteristic
36C4×Q8×C127—22no—
37C4×D4×C127—22no—
38C4×D4×C127—42no—
39#117×C127—21no—
40#117×C127—41no—
41#117×C127D411yescharacteristic
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—41no—
44M32×C127—21no—
45M32×C127—21no—
46M32×C127—21no—
47M32×C127—21no—
48C2×C16×C127—21no—
49C2×C16×C127—21no—
50#108×C127—21no—
51#108×C127—21no—
52#107×C127—21no—
53#106×C127—21no—
54#106×C127—21no—
55#106×C127C2×C412yes—
56#103×C127—22no—
57#102×C127—22no—
58#97×C127C2×C412yes—
59C4×C8×C127D412yes—
60#41×C127—22no—
61#41×C127—22no—
62#41×C127—41no—
63#41×C127—41no—
64C2×Q8×C127—21no—
65C2×D4×C127—21no—
66C2×D4×C127—41no—
67C22×C4×C127—42no—
68C22×C4×C127—41no—
69C22×C4×C127—41no—
70C22×C4×C127C2×D411yescharacteristic
71Q16×C127—21no—
72Q16×C127C4⋊C412yes—
73SD16×C127—22no—
74SD16×C127—22no—
75D8×C127—21no—
76D8×C127C4⋊C412yes—
77M16×C127—21no—
78M16×C127—21no—
79M16×C127—41no—
80M16×C127—41no—
81C2×C8×C127—41no—
82C2×C8×C127C22⋊C412yes—
83C2×C8×C127C22×C411yescharacteristic
84C2×C8×C127C2×D411yescharacteristic
85C2×C8×C127C2×D411yescharacteristic
86C2×C8×C127C2×Q811yescharacteristic
87C2×C8×C127#4111yescharacteristic
88C2×C8×C127#4111yescharacteristic
89C4⋊C4×C127—21no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127—42no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C22⋊C4×C127—21no—
95C22⋊C4×C127—42no—
96C22⋊C4×C127—41no—
97C42×C127—42no—
98C42×C127—41no—
99C42×C127C22⋊C412yes—
100C16×C127—41no—
101C16×C127—41no—
102C23×C127—41no—
103C23×C127AGL(1,ℤ/8)11yescharacteristic
104Q8×C127—22no—
105Q8×C127—41no—
106D4×C127—22no—
107D4×C127—42no—
108D4×C127—42no—
109D4×C127—41no—
110C2×C4×C127—22no—
111C2×C4×C127—41no—
112C2×C4×C127—42no—
113C2×C4×C127—42no—
114C2×C4×C127—42no—
115C2×C4×C127—81no—
116C2×C4×C127—162no—
117C2×C4×C127—161no—
118C2×C4×C127C2×C22⋊C411yescharacteristic
119C2×C4×C127#12111yescharacteristic
120C2×C4×C127C22⋊Q811yescharacteristic
121C2×C4×C127#13511yescharacteristic
122C2×C4×C127#13511yescharacteristic
123C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
124C8×C127—22no—
125C8×C127—81no—
126C8×C127C2×C4⋊C411yescharacteristic
127C8×C127#11711yescharacteristic
128C8×C127#12111yescharacteristic
129C8×C127C22⋊Q811yescharacteristic
130C22×C127—41no—
131C22×C127—42no—
132C22×C127—81no—
133C22×C127—81no—
134C22×C127#41811yescharacteristic
135C22×C127#46311yescharacteristic
136C22×C127#47811yescharacteristic
137C4×C127—42no—
138C4×C127—42no—
139C4×C127—82no—
140C4×C127—161no—
141C4×C127#38011yescharacteristic
142C4×C127#41611yescharacteristic
143C4×C127#46311yescharacteristic
144C4×C127#47811yescharacteristic
145#12828C12711yesmaximal, characteristic, direct product factor, Sylow
146C2×C127—21no—
147C2×C127—21no—
148C2×C127—21no—
149C2×C127—82no—
150C2×C127#180411yescharacteristic
151#2952C2×C12711yescharacteristic
152#2153C2×C12711yescharacteristic, complemented
153#2145C2×C12711yescharacteristic
154#2143C2×C12711yescharacteristic
155#1833C2×C12711yescharacteristic, complemented
156#1386C2×C12711yescharacteristic, complemented
157#1383C2×C12711yescharacteristic, complemented
158C127#1282811yescharacteristic, direct product factor, Sylow
159C2×#135C22×C12711yescharacteristic
160#520—21no—
161C2×M32C22×C12711yescharacteristic
162C2×M32C22×C12711yescharacteristic
163#442C4×C12712yescomplemented
164#441C4×C12712yescomplemented
165C4×SD16—22no—
166C2×#108C22×C12711yescharacteristic
167#427C22×C12711yescharacteristic
168C2×#106C22×C12711yescharacteristic
169#417—21no—
170#405C22×C12711yescharacteristic
171#364—22no—
172#360—22no—
173#359—22no—
174#344—21no—
175H(3,ℤ/4)—21no—
176C2×#41—21no—
177#135—21no—
178#135—21no—
179#135D4×C12712yes—
180#135Q8×C12712yes—
181C2×Q16C2×C4×C12711yescharacteristic
182C2×SD16—21no—
183C2×D8C2×C4×C12711yescharacteristic
184C2×M16—21no—
185C2×M16D4×C12711yescharacteristic
186C22×C8C23×C12711yescharacteristic
187C4×Q8—22no—
188C4×D4—22no—
189C4×D4—42no—
190#117—21no—
191#117—41no—
192#117D4×C12711yescharacteristic
193C2×C4⋊C4—21no—
194C2×C4⋊C4—41no—
195M32—21no—
196M32—21no—
197M32—21no—
198M32—21no—
199C2×C16—21no—
200C2×C16—21no—
201#108—21no—
202#108—21no—
203#107—21no—
204#106—21no—
205#106—21no—
206#106C2×C4×C12712yes—
207#103—22no—
208#102—22no—
209#97C2×C4×C12712yes—
210C4×C8D4×C12712yes—
211#41—22no—
212#41—22no—
213#41—41no—
214#41—41no—
215C2×Q8—21no—
216C2×D4—21no—
217C2×D4—41no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4C2×D4×C12711yescharacteristic
222Q16—21no—
223Q16C4⋊C4×C12712yes—
224SD16—22no—
225SD16—22no—
226D8—21no—
227D8C4⋊C4×C12712yes—
228M16—21no—
229M16—21no—
230M16—41no—
231M16—41no—
232C2×C8—41no—
233C2×C8C22⋊C4×C12712yes—
234C2×C8C22×C4×C12711yescharacteristic
235C2×C8C2×D4×C12711yescharacteristic
236C2×C8C2×D4×C12711yescharacteristic
237C2×C8C2×Q8×C12711yescharacteristic
238C2×C8#41×C12711yescharacteristic
239C2×C8#41×C12711yescharacteristic
240C4⋊C4—21no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C22⋊C4—21no—
246C22⋊C4—42no—
247C22⋊C4—41no—
248C42—42no—
249C42—41no—
250C42C22⋊C4×C12712yes—
251C16—41no—
252C16—41no—
253C23—41no—
254C23AGL(1,ℤ/8)×C12711yescharacteristic
255Q8—22no—
256Q8—41no—
257D4—22no—
258D4—42no—
259D4—42no—
260D4—41no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4—162no—
268C2×C4—161no—
269C2×C4C2×C22⋊C4×C12711yescharacteristic
270C2×C4#121×C12711yescharacteristic
271C2×C4C22⋊Q8×C12711yescharacteristic
272C2×C4#135×C12711yescharacteristic
273C2×C4#135×C12711yescharacteristic
274C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
275C8—22no—
276C8—81no—
277C8C2×C4⋊C4×C12711yescharacteristic
278C8#117×C12711yescharacteristic
279C8#121×C12711yescharacteristic
280C8C22⋊Q8×C12711yescharacteristic
281C22—41no—
282C22—42no—
283C22—81no—
284C22—81no—
285C22#418×C12711yescharacteristic
286C22#463×C12711yescharacteristic
287C22#478×C12711yescharacteristic
288C4—42no—
289C4—42no—
290C4—82no—
291C4—161no—
292C4#380×C12711yescharacteristic
293C4#416×C12711yescharacteristic
294C4#463×C12711yescharacteristic
295C4#478×C12711yescharacteristic
296C2—21no—
297C2—21no—
298C2—21no—
299C2—82no—
300C2C127×#180411yescharacteristic
subgroupG/H
center, socleC2×C127#1804
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC2×#135×C127C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2153C2semidirect
5. C127×#1833C2semidirect
6. C127×#1386C2semidirect
7. C127×#1383C2semidirect
12. #442×C127C4semidirect
13. #441×C127C4semidirect
145. #12828C127direct (N × H)
152. #2153C2×C127semidirect
155. #1833C2×C127semidirect
156. #1386C2×C127semidirect
157. #1383C2×C127semidirect
158. C127#12828direct (N × H)
163. #442C4×C127semidirect
164. #441C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#29522
C127×#21532
C127×#21452
C127×#21432
C127×#18332
C127×#13862
C127×#13832
#12828127