Finite Group Databasebeta

← C127×#10515 #44574188

Subgroups — C127×#10515 #44574188

Order 32512.

366 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 94 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×#3246C211yesmaximal, characteristic
2C127×#1832C211yesmaximal, characteristic, complemented
3C127×#1552C211yesmaximal, characteristic, complemented
4C127×#1551C211yesmaximal, characteristic
5C127×#1506C211yesmaximal, characteristic, complemented
6C127×#1505C211yesmaximal, characteristic
7C127×#1290C211yesmaximal, characteristic, complemented
8#533×C127C2211yescharacteristic
9#532×C127C2211yescharacteristic
10#476×C127—21no—
11#475×C127—21no—
12#465×C127—22no—
13#461×C127—21no—
14#460×C127—21no—
15#432×C127C2211yescharacteristic
16#432×C127C2211yescharacteristic
17C2×C4⋊C8×C127C2211yescharacteristic
18C2×C4⋊C8×C127C2211yescharacteristic
19#376×C127C2211yescharacteristic
20#336×C127—22no—
21#333×C127—21no—
22#332×C127—21no—
23#332×C127—21no—
24#331×C127—21no—
25#328×C127—22no—
26#327×C127—22no—
27C2×#41×C127—21no—
28C2×SD16×C127—22no—
29C22×C8×C127—21no—
30C22×C8×C127D412yes—
31C4⋊Q8×C127D411yescharacteristic
32C4⋊Q8×C127D411yescharacteristic
33C4⋊Q8×C127D411yescharacteristic
34#127×C127D411yescharacteristic
35#125×C127—21no—
36#124×C127—21no—
37C22⋊Q8×C127—21no—
38C22⋊Q8×C127—21no—
39#121×C127—21no—
40#121×C127—21no—
41C4×Q8×C127—21no—
42C4×Q8×C127—21no—
43C4×D4×C127—21no—
44C4×D4×C127—21no—
45#117×C127—21no—
46C2×C42×C127—21no—
47C2×C42×C127C2311yescharacteristic
48#106×C127—22no—
49C4⋊C8×C127—21no—
50C4⋊C8×C127—21no—
51C4⋊C8×C127—21no—
52C4⋊C8×C127—21no—
53C4⋊C8×C127—21no—
54C4⋊C8×C127—21no—
55#103×C127—22no—
56#102×C127—22no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59#97×C127—21no—
60#97×C127—21no—
61#95×C127—21no—
62#95×C127D412yes—
63#41×C127—82no—
64C2×Q8×C127—21no—
65C2×Q8×C127—21no—
66C2×Q8×C127—21no—
67C2×D4×C127—21no—
68C2×D4×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—41no—
71C22×C4×C127—22no—
72C22×C4×C127—41no—
73C22×C4×C127C2×D411yescharacteristic
74C22×C4×C127C2×D411yescharacteristic
75C22×C4×C127#4111yescharacteristic
76SD16×C127—82no—
77C2×C8×C127—22no—
78C2×C8×C127—42no—
79C2×C8×C127—42no—
80C2×C8×C127—41no—
81C2×C8×C127—41no—
82C2×C8×C127—41no—
83C4⋊C4×C127—21no—
84C4⋊C4×C127—21no—
85C4⋊C4×C127—21no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—21no—
88C4⋊C4×C127—21no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C22⋊C4×C127—41no—
92C22⋊C4×C127—41no—
93C22⋊C4×C127—41no—
94C42×C127—41no—
95C42×C127—41no—
96C42×C127—41no—
97C42×C127C2×D411yescharacteristic
98C42×C127C2×D411yescharacteristic
99C42×C127#4111yescharacteristic
100C42×C127#4111yescharacteristic
101C23×C127—41no—
102C23×C127#12111yescharacteristic
103Q8×C127—41no—
104Q8×C127—41no—
105Q8×C127—81no—
106Q8×C127—81no—
107D4×C127—41no—
108D4×C127—41no—
109D4×C127—81no—
110D4×C127—81no—
111D4×C127—82no—
112C2×C4×C127—22no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—41no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—41no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—42no—
122C2×C4×C127—82no—
123C2×C4×C127#12011yescharacteristic
124C2×C4×C127#12111yescharacteristic
125C2×C4×C127#12111yescharacteristic
126C2×C4×C127#12311yescharacteristic
127C2×C4×C127#12311yescharacteristic
128C2×C4×C127#12411yescharacteristic
129C8×C127—42no—
130C8×C127—81no—
131C8×C127—81no—
132C8×C127—81no—
133C22×C127—21no—
134C22×C127—21no—
135C22×C127—81no—
136C22×C127—81no—
137C22×C127—81no—
138C22×C127#39411yescharacteristic
139C22×C127#47112yes—
140C4×C127—21no—
141C4×C127—21no—
142C4×C127—21no—
143C4×C127—21no—
144C4×C127—21no—
145C4×C127—82no—
146C4×C127—81no—
147C4×C127—81no—
148C4×C127—81no—
149C4×C127#45411yescharacteristic
150C4×C127#45511yescharacteristic
151#10515C12711yesmaximal, characteristic, direct product factor, Sylow
152C2×C127—22no—
153C2×C127—161no—
154C2×C127#204011yescharacteristic
155C2×C127#204611yescharacteristic
156C2×C127#205311yescharacteristic
157#3246C2×C12711yescharacteristic
158#1832C2×C12711yescharacteristic, complemented
159#1552C2×C12711yescharacteristic, complemented
160#1551C2×C12711yescharacteristic
161#1506C2×C12711yescharacteristic, complemented
162#1505C2×C12711yescharacteristic
163#1290C2×C12711yescharacteristic, complemented
164C127#1051511yescharacteristic, direct product factor, Sylow
165#533C22×C12711yescharacteristic
166#532C22×C12711yescharacteristic
167#476—21no—
168#475—21no—
169#465—22no—
170#461—21no—
171#460—21no—
172#432C22×C12711yescharacteristic
173#432C22×C12711yescharacteristic
174C2×C4⋊C8C22×C12711yescharacteristic
175C2×C4⋊C8C22×C12711yescharacteristic
176#376C22×C12711yescharacteristic
177#336—22no—
178#333—21no—
179#332—21no—
180#332—21no—
181#331—21no—
182#328—22no—
183#327—22no—
184C2×#41—21no—
185C2×SD16—22no—
186C22×C8—21no—
187C22×C8D4×C12712yes—
188C4⋊Q8D4×C12711yescharacteristic
189C4⋊Q8D4×C12711yescharacteristic
190C4⋊Q8D4×C12711yescharacteristic
191#127D4×C12711yescharacteristic
192#125—21no—
193#124—21no—
194C22⋊Q8—21no—
195C22⋊Q8—21no—
196#121—21no—
197#121—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×D4—21no—
201C4×D4—21no—
202#117—21no—
203C2×C42—21no—
204C2×C42C23×C12711yescharacteristic
205#106—22no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212#103—22no—
213#102—22no—
214C22⋊C8—21no—
215C22⋊C8—21no—
216#97—21no—
217#97—21no—
218#95—21no—
219#95D4×C12712yes—
220#41—82no—
221C2×Q8—21no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×D4—21no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—41no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4C2×D4×C12711yescharacteristic
231C22×C4C2×D4×C12711yescharacteristic
232C22×C4#41×C12711yescharacteristic
233SD16—82no—
234C2×C8—22no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C42—41no—
252C42—41no—
253C42—41no—
254C42C2×D4×C12711yescharacteristic
255C42C2×D4×C12711yescharacteristic
256C42#41×C12711yescharacteristic
257C42#41×C12711yescharacteristic
258C23—41no—
259C23#121×C12711yescharacteristic
260Q8—41no—
261Q8—41no—
262Q8—81no—
263Q8—81no—
264D4—41no—
265D4—41no—
266D4—81no—
267D4—81no—
268D4—82no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—82no—
280C2×C4#120×C12711yescharacteristic
281C2×C4#121×C12711yescharacteristic
282C2×C4#121×C12711yescharacteristic
283C2×C4#123×C12711yescharacteristic
284C2×C4#123×C12711yescharacteristic
285C2×C4#124×C12711yescharacteristic
286C8—42no—
287C8—81no—
288C8—81no—
289C8—81no—
290C22—21no—
291C22—21no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22#394×C12711yescharacteristic
296C22#471×C12712yes—
297C4—21no—
298C4—21no—
299C4—21no—
300C4—21no—
301C4—21no—
302C4—82no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4#454×C12711yescharacteristic
307C4#455×C12711yescharacteristic
308C2—22no—
309C2—161no—
310C2C127×#204011yescharacteristic
311C2C127×#204611yescharacteristic
312C2C127×#205311yescharacteristic
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×#1832C2semidirect
3. C127×#1552C2semidirect
5. C127×#1506C2semidirect
7. C127×#1290C2semidirect
151. #10515C127direct (N × H)
158. #1832C2×C127semidirect
159. #1552C2×C127semidirect
161. #1506C2×C127semidirect
163. #1290C2×C127semidirect
164. C127#10515direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32462
C127×#18322
C127×#15522
C127×#15512
C127×#15062
C127×#15052
C127×#12902
#10515127