Finite Group Databasebeta

← C127×#12339 #44576012

Subgroups — C127×#12339 #44576012

Order 32512.

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

normal subgroups only

This group has 308 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×#3402C211yesmaximal, characteristic
2C127×#2251C211yesmaximal, characteristic
3C127×#2209C211yesmaximal, characteristic, complemented
4C127×#1690C211yesmaximal, characteristic, complemented
5C127×#1360C211yesmaximal, characteristic
6C127×#1338C211yesmaximal, characteristic, complemented
7C127×#1334C211yesmaximal, characteristic, complemented
8#556×C127—21no—
9C2×Q32×C127—21no—
10C2×Q32×C127—21no—
11C2×SD32×C127—21no—
12#497×C127—21no—
13#495×C127C2211yescharacteristic
14#494×C127C2211yescharacteristic
15#475×C127—21no—
16#464×C127—21no—
17#463×C127—21no—
18#462×C127—21no—
19#446×C127C2211yescharacteristic
20#440×C127—21no—
21C4×Q16×C127C2211yescharacteristic
22C4⋊C16×C127C2211yescharacteristic
23#358×C127—21no—
24#346×C127C2211yescharacteristic
25#334×C127C2211yescharacteristic
26#332×C127—21no—
27C2×Q16×C127—21no—
28C2×Q16×C127—21no—
29C2×Q16×C127—21no—
30C2×Q16×C127D411yescharacteristic
31C2×Q16×C127D411yescharacteristic
32C2×SD16×C127—21no—
33C2×SD16×C127—21no—
34C2×D8×C127D411yescharacteristic
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127—21no—
37#126×C127—41no—
38#126×C127—41no—
39#124×C127—21no—
40#124×C127—21no—
41C22⋊Q8×C127—41no—
42C4×Q8×C127—21no—
43C4×Q8×C127—21no—
44C4×Q8×C127D411yescharacteristic
45C4×D4×C127—21no—
46#117×C127—41no—
47Q32×C127—42no—
48Q32×C127—42no—
49SD32×C127—42no—
50C2×C16×C127—21no—
51C2×C16×C127—21no—
52#107×C127D411yescharacteristic
53#106×C127—41no—
54C4⋊C8×C127—21no—
55C4⋊C8×C127—21no—
56C4⋊C8×C127D411yescharacteristic
57#103×C127—21no—
58#103×C127—21no—
59#103×C127—41no—
60#102×C127—21no—
61#97×C127—21no—
62C4×C8×C127C2311yescharacteristic
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—41no—
66C2×Q8×C127D811yescharacteristic
67C2×D4×C127—21no—
68C22×C4×C127—41no—
69Q16×C127—21no—
70Q16×C127—21no—
71Q16×C127—41no—
72Q16×C127—41no—
73Q16×C127—41no—
74Q16×C127—41no—
75Q16×C127—41no—
76Q16×C127—81no—
77SD16×C127—41no—
78SD16×C127—41no—
79SD16×C127—81no—
80D8×C127—41no—
81C2×C8×C127—21no—
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—
93C4⋊C4×C127D811yescharacteristic
94C22⋊C4×C127—41no—
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—41no—
99C42×C127—21no—
100C42×C127—41no—
101C42×C127—41no—
102C42×C127C2×D411yescharacteristic
103C16×C127—22no—
104C16×C127—41no—
105C23×C127—41no—
106Q8×C127—41no—
107Q8×C127—41no—
108Q8×C127—41no—
109Q8×C127—41no—
110Q8×C127—41no—
111Q8×C127—81no—
112Q8×C127—81no—
113D4×C127—41no—
114D4×C127—41no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—82no—
121C2×C4×C127—81no—
122C2×C4×C127#12011yescharacteristic
123C2×C4×C127C2×D811yescharacteristic
124C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
125C8×C127—21no—
126C8×C127—41no—
127C8×C127—81no—
128C8×C127#13511yescharacteristic
129C8×C127AGL(1,ℤ/8)11yescharacteristic
130C22×C127—81no—
131C22×C127—81no—
132C22×C127—81no—
133C22×C127#44711yescharacteristic
134C4×C127—21no—
135C4×C127—21no—
136C4×C127—41no—
137C4×C127—41no—
138C4×C127—81no—
139C4×C127—81no—
140C4×C127—81no—
141C4×C127—161no—
142C4×C127#44911yescharacteristic
143C4×C127#45311yescharacteristic
144#12339C12711yesmaximal, characteristic, direct product factor, Sylow
145C2×C127—161no—
146C2×C127#162111yescharacteristic
147C2×C127#219111yescharacteristic
148C2×C127#219511yescharacteristic
149#3402C2×C12711yescharacteristic
150#2251C2×C12711yescharacteristic
151#2209C2×C12711yescharacteristic, complemented
152#1690C2×C12711yescharacteristic, complemented
153#1360C2×C12711yescharacteristic
154#1338C2×C12711yescharacteristic, complemented
155#1334C2×C12711yescharacteristic, complemented
156C127#1233911yescharacteristic, direct product factor, Sylow
157#556—21no—
158C2×Q32—21no—
159C2×Q32—21no—
160C2×SD32—21no—
161#497—21no—
162#495C22×C12711yescharacteristic
163#494C22×C12711yescharacteristic
164#475—21no—
165#464—21no—
166#463—21no—
167#462—21no—
168#446C22×C12711yescharacteristic
169#440—21no—
170C4×Q16C22×C12711yescharacteristic
171C4⋊C16C22×C12711yescharacteristic
172#358—21no—
173#346C22×C12711yescharacteristic
174#334C22×C12711yescharacteristic
175#332—21no—
176C2×Q16—21no—
177C2×Q16—21no—
178C2×Q16—21no—
179C2×Q16D4×C12711yescharacteristic
180C2×Q16D4×C12711yescharacteristic
181C2×SD16—21no—
182C2×SD16—21no—
183C2×D8D4×C12711yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186#126—41no—
187#126—41no—
188#124—21no—
189#124—21no—
190C22⋊Q8—41no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8D4×C12711yescharacteristic
194C4×D4—21no—
195#117—41no—
196Q32—42no—
197Q32—42no—
198SD32—42no—
199C2×C16—21no—
200C2×C16—21no—
201#107D4×C12711yescharacteristic
202#106—41no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8D4×C12711yescharacteristic
206#103—21no—
207#103—21no—
208#103—41no—
209#102—21no—
210#97—21no—
211C4×C8C23×C12711yescharacteristic
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—41no—
215C2×Q8D8×C12711yescharacteristic
216C2×D4—21no—
217C22×C4—41no—
218Q16—21no—
219Q16—21no—
220Q16—41no—
221Q16—41no—
222Q16—41no—
223Q16—41no—
224Q16—41no—
225Q16—81no—
226SD16—41no—
227SD16—41no—
228SD16—81no—
229D8—41no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8C2×D4×C12711yescharacteristic
233C2×C8C2×D4×C12711yescharacteristic
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4D8×C12711yescharacteristic
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C42—21no—
249C42—41no—
250C42—41no—
251C42C2×D4×C12711yescharacteristic
252C16—22no—
253C16—41no—
254C23—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—81no—
261Q8—81no—
262D4—41no—
263D4—41no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4#120×C12711yescharacteristic
272C2×C4C2×D8×C12711yescharacteristic
273C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
274C8—21no—
275C8—41no—
276C8—81no—
277C8#135×C12711yescharacteristic
278C8AGL(1,ℤ/8)×C12711yescharacteristic
279C22—81no—
280C22—81no—
281C22—81no—
282C22#447×C12711yescharacteristic
283C4—21no—
284C4—21no—
285C4—41no—
286C4—41no—
287C4—81no—
288C4—81no—
289C4—81no—
290C4—161no—
291C4#449×C12711yescharacteristic
292C4#453×C12711yescharacteristic
293C2—161no—
294C2C127×#162111yescharacteristic
295C2C127×#219111yescharacteristic
296C2C127×#219511yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC2×D8×C127D4
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ #446×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#2209C2semidirect
4. C127×#1690C2semidirect
6. C127×#1338C2semidirect
7. C127×#1334C2semidirect
144. #12339C127direct (N × H)
151. #2209C2×C127semidirect
152. #1690C2×C127semidirect
154. #1338C2×C127semidirect
155. #1334C2×C127semidirect
156. C127#12339direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34022
C127×#22512
C127×#22092
C127×#16902
C127×#13602
C127×#13382
C127×#13342
#12339127