Finite Group Databasebeta

← C127×#10533 #44574206

Subgroups — C127×#10533 #44574206

Order 32512.

364 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3254C211yesmaximal, characteristic
2C127×#1837C211yesmaximal, characteristic, complemented
3C127×#1561C211yesmaximal, characteristic, complemented
4C127×#1560C211yesmaximal, characteristic
5C127×#1493C211yesmaximal, characteristic, complemented
6C127×#1492C211yesmaximal, characteristic
7C127×#1309C211yesmaximal, characteristic, complemented
8#533×C127C2211yescharacteristic
9#532×C127C2211yescharacteristic
10#477×C127—21no—
11#474×C127—21no—
12#468×C127—22no—
13#461×C127—21no—
14#460×C127—21no—
15#423×C127C2211yescharacteristic
16#423×C127C2211yescharacteristic
17#423×C127C2211yescharacteristic
18#423×C127C2211yescharacteristic
19C2×C4×C8×C127C2211yescharacteristic
20#341×C127—22no—
21#333×C127—21no—
22#332×C127—21no—
23#332×C127—21no—
24#331×C127—21no—
25Q8⋊C8×C127—22no—
26D4⋊C8×C127—22no—
27C2×#41×C127—21no—
28C2×SD16×C127—22no—
29C2×M16×C127—21no—
30C2×M16×C127—21no—
31C2×M16×C127—21no—
32C2×M16×C127D412yes—
33C22×C8×C127—21no—
34C4⋊Q8×C127D411yescharacteristic
35C4⋊Q8×C127D411yescharacteristic
36C4⋊Q8×C127D411yescharacteristic
37#127×C127D411yescharacteristic
38#125×C127—21no—
39#124×C127—21no—
40C22⋊Q8×C127—21no—
41C22⋊Q8×C127—21no—
42#121×C127—21no—
43#121×C127—21no—
44C4×Q8×C127—21no—
45C4×Q8×C127—21no—
46C4×D4×C127—21no—
47C4×D4×C127—21no—
48#117×C127—21no—
49C2×C42×C127C2311yescharacteristic
50#107×C127—22no—
51C4⋊C8×C127—21no—
52C4⋊C8×C127—21no—
53C4⋊C8×C127—21no—
54C4⋊C8×C127—21no—
55C4⋊C8×C127—21no—
56C4⋊C8×C127—21no—
57C4⋊C8×C127—21no—
58C4⋊C8×C127—21no—
59#103×C127—22no—
60#102×C127—22no—
61C4×C8×C127—21no—
62C4×C8×C127D412yes—
63#41×C127—81no—
64#41×C127—81no—
65C2×Q8×C127—21no—
66C2×Q8×C127—21no—
67C2×Q8×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—41no—
72C22×C4×C127—41no—
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127#4111yescharacteristic
75C22×C4×C127#4111yescharacteristic
76SD16×C127—82no—
77M16×C127—42no—
78M16×C127—42no—
79M16×C127—81no—
80M16×C127—81no—
81C2×C8×C127—22no—
82C2×C8×C127—22no—
83C2×C8×C127—41no—
84C2×C8×C127—41no—
85C2×C8×C127—41no—
86C2×C8×C127—41no—
87C2×C8×C127—41no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—21no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—21no—
93C4⋊C4×C127—21no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—41no—
99C42×C127—41no—
100C42×C127C2×D411yescharacteristic
101C42×C127C2×D411yescharacteristic
102C42×C127C2×D411yescharacteristic
103C42×C127#4111yescharacteristic
104C23×C127—41no—
105C23×C127#12411yescharacteristic
106Q8×C127—41no—
107Q8×C127—41no—
108Q8×C127—41no—
109Q8×C127—41no—
110Q8×C127—81no—
111D4×C127—41no—
112D4×C127—41no—
113D4×C127—41no—
114D4×C127—41no—
115D4×C127—81no—
116D4×C127—82no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—81no—
127C2×C4×C127—81no—
128C2×C4×C127#12011yescharacteristic
129C2×C4×C127#12111yescharacteristic
130C2×C4×C127#12111yescharacteristic
131C2×C4×C127#12111yescharacteristic
132C2×C4×C127#12311yescharacteristic
133C2×C4×C127#12311yescharacteristic
134C8×C127—42no—
135C8×C127—42no—
136C8×C127—81no—
137C8×C127—81no—
138C8×C127—81no—
139C22×C127—21no—
140C22×C127—21no—
141C22×C127—21no—
142C22×C127—81no—
143C22×C127—81no—
144C22×C127—81no—
145C22×C127#39411yescharacteristic
146C4×C127—21no—
147C4×C127—21no—
148C4×C127—21no—
149C4×C127—81no—
150C4×C127—81no—
151C4×C127—81no—
152C4×C127#45411yescharacteristic
153C4×C127#45511yescharacteristic
154C4×C127#47111yescharacteristic
155C4×C127#47111yescharacteristic
156C4×C127#47211yescharacteristic
157C4×C127#47311yescharacteristic
158#10533C12711yesmaximal, characteristic, direct product factor, Sylow
159C2×C127—41no—
160C2×C127—161no—
161C2×C127#204011yescharacteristic
162C2×C127#205111yescharacteristic
163C2×C127#205211yescharacteristic
164#3254C2×C12711yescharacteristic
165#1837C2×C12711yescharacteristic, complemented
166#1561C2×C12711yescharacteristic, complemented
167#1560C2×C12711yescharacteristic
168#1493C2×C12711yescharacteristic, complemented
169#1492C2×C12711yescharacteristic
170#1309C2×C12711yescharacteristic, complemented
171C127#1053311yescharacteristic, direct product factor, Sylow
172#533C22×C12711yescharacteristic
173#532C22×C12711yescharacteristic
174#477—21no—
175#474—21no—
176#468—22no—
177#461—21no—
178#460—21no—
179#423C22×C12711yescharacteristic
180#423C22×C12711yescharacteristic
181#423C22×C12711yescharacteristic
182#423C22×C12711yescharacteristic
183C2×C4×C8C22×C12711yescharacteristic
184#341—22no—
185#333—21no—
186#332—21no—
187#332—21no—
188#331—21no—
189Q8⋊C8—22no—
190D4⋊C8—22no—
191C2×#41—21no—
192C2×SD16—22no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16D4×C12712yes—
197C22×C8—21no—
198C4⋊Q8D4×C12711yescharacteristic
199C4⋊Q8D4×C12711yescharacteristic
200C4⋊Q8D4×C12711yescharacteristic
201#127D4×C12711yescharacteristic
202#125—21no—
203#124—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206#121—21no—
207#121—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×D4—21no—
211C4×D4—21no—
212#117—21no—
213C2×C42C23×C12711yescharacteristic
214#107—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223#103—22no—
224#102—22no—
225C4×C8—21no—
226C4×C8D4×C12712yes—
227#41—81no—
228#41—81no—
229C2×Q8—21no—
230C2×Q8—21no—
231C2×Q8—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C22×C4—41no—
237C22×C4C2×D4×C12711yescharacteristic
238C22×C4#41×C12711yescharacteristic
239C22×C4#41×C12711yescharacteristic
240SD16—82no—
241M16—42no—
242M16—42no—
243M16—81no—
244M16—81no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—41no—
264C42C2×D4×C12711yescharacteristic
265C42C2×D4×C12711yescharacteristic
266C42C2×D4×C12711yescharacteristic
267C42#41×C12711yescharacteristic
268C23—41no—
269C23#124×C12711yescharacteristic
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274Q8—81no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—81no—
280D4—82no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4#120×C12711yescharacteristic
293C2×C4#121×C12711yescharacteristic
294C2×C4#121×C12711yescharacteristic
295C2×C4#121×C12711yescharacteristic
296C2×C4#123×C12711yescharacteristic
297C2×C4#123×C12711yescharacteristic
298C8—42no—
299C8—42no—
300C8—81no—
301C8—81no—
302C8—81no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—81no—
307C22—81no—
308C22—81no—
309C22#394×C12711yescharacteristic
310C4—21no—
311C4—21no—
312C4—21no—
313C4—81no—
314C4—81no—
315C4—81no—
316C4#454×C12711yescharacteristic
317C4#455×C12711yescharacteristic
318C4#471×C12711yescharacteristic
319C4#471×C12711yescharacteristic
320C4#472×C12711yescharacteristic
321C4#473×C12711yescharacteristic
322C2—41no—
323C2—161no—
324C2C127×#204011yescharacteristic
325C2C127×#205111yescharacteristic
326C2C127×#205211yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC2×C42C23×C127
prime radical#532×C127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1837C2semidirect
3. C127×#1561C2semidirect
5. C127×#1493C2semidirect
7. C127×#1309C2semidirect
158. #10533C127direct (N × H)
165. #1837C2×C127semidirect
166. #1561C2×C127semidirect
168. #1493C2×C127semidirect
170. #1309C2×C127semidirect
171. C127#10533direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32542
C127×#18372
C127×#15612
C127×#15602
C127×#14932
C127×#14922
C127×#13092
#10533127