Finite Group Databasebeta

← C127×#12722 #44576395

Subgroups — C127×#12722 #44576395

Order 32512.

790 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 790 conjugacy classes of subgroups — too many to draw. Only its 138 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×#3508C211yesmaximal, characteristic
2C127×#2134C211yesmaximal, characteristic, complemented
3C127×#2128C211yesmaximal, characteristic, complemented
4C127×#2122C212yesmaximal, complemented
5C127×#1827C212yesmaximal, complemented
6#559×C127—22no—
7#550×C127—22no—
8#546×C127—24no—
9#540×C127—22no—
10#535×C127C2211yescharacteristic
11#535×C127C2211yescharacteristic
12#533×C127C411yescharacteristic, complemented
13#532×C127—22no—
14C2×#127×C127C411yescharacteristic, complemented
15#518×C127C411yescharacteristic, complemented
16#518×C127C411yescharacteristic, complemented
17C4×#41×C127C2211yescharacteristic
18#413×C127C2212yes—
19#410×C127C2212yes—
20#355×C127—22no—
21#353×C127—22no—
22AΣL(1,𝔽16)×C127—24no—
23#339×C127—22no—
24H(3,ℤ/4)×C127—22no—
25C2×#41×C127—24no—
26C2×#41×C127C2311yescharacteristic
27C22×D4×C127—22no—
28C22×D4×C127C2×C411yescharacteristic
29C22×D4×C127C2×C411yescharacteristic
30C2×M16×C127—22no—
31C4⋊Q8×C127—21no—
32#127×C127—22no—
33#127×C127—21no—
34#127×C127—21no—
35#127×C127—21no—
36#127×C127D412yes—
37#125×C127—21no—
38#124×C127—24no—
39#124×C127—21no—
40#124×C127D412yes—
41#123×C127—24no—
42#123×C127—24no—
43C22⋊Q8×C127—22no—
44#121×C127—24no—
45#121×C127—24no—
46#121×C127—24no—
47#121×C127—22no—
48#121×C127—22no—
49#120×C127—22no—
50#120×C127D414yes—
51C4×Q8×C127—21no—
52C4×Q8×C127—22no—
53C4×D4×C127—24no—
54C4×D4×C127—22no—
55C4×D4×C127—22no—
56C4×D4×C127—21no—
57C4×D4×C127—22no—
58C4×D4×C127—22no—
59C4×D4×C127—22no—
60#117×C127—22no—
61#117×C127C2×C411yescharacteristic
62#117×C127C2×C411yescharacteristic
63#117×C127C2×C411yescharacteristic
64#117×C127D412yes—
65C2×C22⋊C4×C127—22no—
66C2×C22⋊C4×C127—22no—
67C2×C42×C127C2×C411yescharacteristic
68C2×C42×C127D412yes—
69#101×C127—22no—
70#100×C127—22no—
71#99×C127—22no—
72#99×C127—22no—
73C24×C127—21no—
74C24×C127C22⋊C412yes—
75#41×C127—42no—
76#41×C127—88no—
77C2×Q8×C127—22no—
78C2×Q8×C127C2×D411yescharacteristic
79C2×D4×C127—24no—
80C2×D4×C127—24no—
81C2×D4×C127—24no—
82C2×D4×C127—24no—
83C2×D4×C127—21no—
84C2×D4×C127—21no—
85C2×D4×C127—21no—
86C2×D4×C127—42no—
87C2×D4×C127—42no—
88C2×D4×C127—84no—
89C2×D4×C127C22⋊C412yes—
90C2×D4×C127C2×D411yescharacteristic
91C2×D4×C127C2×D412yes—
92C22×C4×C127—21no—
93C22×C4×C127—22no—
94C22×C4×C127—24no—
95C22×C4×C127—24no—
96C22×C4×C127C22×C411yescharacteristic
97C22×C4×C127C2×D412yes—
98M16×C127—42no—
99M16×C127—42no—
100C2×C8×C127—42no—
101C4⋊C4×C127—22no—
102C4⋊C4×C127—22no—
103C4⋊C4×C127—22no—
104C4⋊C4×C127—21no—
105C4⋊C4×C127—21no—
106C4⋊C4×C127—21no—
107C4⋊C4×C127—22no—
108C4⋊C4×C127—42no—
109C22⋊C4×C127—24no—
110C22⋊C4×C127—24no—
111C22⋊C4×C127—21no—
112C22⋊C4×C127—21no—
113C22⋊C4×C127—22no—
114C22⋊C4×C127—24no—
115C22⋊C4×C127—24no—
116C22⋊C4×C127—42no—
117C22⋊C4×C127C22⋊C412yes—
118C42×C127—21no—
119C42×C127—21no—
120C42×C127—22no—
121C42×C127—42no—
122C42×C127C22⋊C412yes—
123C42×C127C22⋊C411yescharacteristic
124C42×C127C22⋊C411yescharacteristic
125C42×C127C22⋊C412yes—
126C23×C127—22no—
127C23×C127—22no—
128C23×C127—42no—
129C23×C127—42no—
130C23×C127—82no—
131C23×C127—82no—
132C23×C127C2×C22⋊C411yescharacteristic
133C23×C127#12012yes—
134Q8×C127—41no—
135Q8×C127—44no—
136D4×C127—42no—
137D4×C127—41no—
138D4×C127—42no—
139D4×C127—42no—
140D4×C127—48no—
141D4×C127—42no—
142D4×C127—42no—
143D4×C127—84no—
144D4×C127—84no—
145D4×C127—84no—
146C2×C4×C127—21no—
147C2×C4×C127—22no—
148C2×C4×C127—22no—
149C2×C4×C127—22no—
150C2×C4×C127—22no—
151C2×C4×C127—21no—
152C2×C4×C127—22no—
153C2×C4×C127—22no—
154C2×C4×C127—42no—
155C2×C4×C127—44no—
156C2×C4×C127—84no—
157C2×C4×C127—84no—
158C2×C4×C127—82no—
159C2×C4×C127C2×C22⋊C411yescharacteristic
160C2×C4×C127C2×C22⋊C411yescharacteristic
161C2×C4×C127#12012yes—
162C8×C127—82no—
163C22×C127—22no—
164C22×C127—21no—
165C22×C127—42no—
166C22×C127—41no—
167C22×C127—44no—
168C22×C127—44no—
169C22×C127—42no—
170C22×C127—42no—
171C22×C127—162no—
172C22×C127—162no—
173C22×C127#37911yescharacteristic
174C4×C127—22no—
175C4×C127—22no—
176C4×C127—21no—
177C4×C127—42no—
178C4×C127—41no—
179C4×C127—42no—
180C4×C127—42no—
181C4×C127—162no—
182C4×C127#41012yes—
183#12722C12711yesmaximal, characteristic, direct product factor, Sylow
184C2×C127—21no—
185C2×C127—42no—
186C2×C127—41no—
187C2×C127—82no—
188C2×C127—82no—
189C2×C127#178211yescharacteristic
190#3508C2×C12711yescharacteristic
191#2134C2×C12711yescharacteristic, complemented
192#2128C2×C12711yescharacteristic, complemented
193#2122C2×C12712yescomplemented
194#1827C2×C12712yescomplemented
195C127#1272211yescharacteristic, direct product factor, Sylow
196#559—22no—
197#550—22no—
198#546—24no—
199#540—22no—
200#535C22×C12711yescharacteristic
201#535C22×C12711yescharacteristic
202#533C4×C12711yescharacteristic, complemented
203#532—22no—
204C2×#127C4×C12711yescharacteristic, complemented
205#518C4×C12711yescharacteristic, complemented
206#518C4×C12711yescharacteristic, complemented
207C4×#41C22×C12711yescharacteristic
208#413C22×C12712yes—
209#410C22×C12712yes—
210#355—22no—
211#353—22no—
212AΣL(1,𝔽16)—24no—
213#339—22no—
214H(3,ℤ/4)—22no—
215C2×#41—24no—
216C2×#41C23×C12711yescharacteristic
217C22×D4—22no—
218C22×D4C2×C4×C12711yescharacteristic
219C22×D4C2×C4×C12711yescharacteristic
220C2×M16—22no—
221C4⋊Q8—21no—
222#127—22no—
223#127—21no—
224#127—21no—
225#127—21no—
226#127D4×C12712yes—
227#125—21no—
228#124—24no—
229#124—21no—
230#124D4×C12712yes—
231#123—24no—
232#123—24no—
233C22⋊Q8—22no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121—22no—
238#121—22no—
239#120—22no—
240#120D4×C12714yes—
241C4×Q8—21no—
242C4×Q8—22no—
243C4×D4—24no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250#117—22no—
251#117C2×C4×C12711yescharacteristic
252#117C2×C4×C12711yescharacteristic
253#117C2×C4×C12711yescharacteristic
254#117D4×C12712yes—
255C2×C22⋊C4—22no—
256C2×C22⋊C4—22no—
257C2×C42C2×C4×C12711yescharacteristic
258C2×C42D4×C12712yes—
259#101—22no—
260#100—22no—
261#99—22no—
262#99—22no—
263C24—21no—
264C24C22⋊C4×C12712yes—
265#41—42no—
266#41—88no—
267C2×Q8—22no—
268C2×Q8C2×D4×C12711yescharacteristic
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—84no—
279C2×D4C22⋊C4×C12712yes—
280C2×D4C2×D4×C12711yescharacteristic
281C2×D4C2×D4×C12712yes—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4C22×C4×C12711yescharacteristic
287C22×C4C2×D4×C12712yes—
288M16—42no—
289M16—42no—
290C2×C8—42no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—22no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—42no—
307C22⋊C4C22⋊C4×C12712yes—
308C42—21no—
309C42—21no—
310C42—22no—
311C42—42no—
312C42C22⋊C4×C12711yescharacteristic
313C42C22⋊C4×C12711yescharacteristic
314C42C22⋊C4×C12712yes—
315C42C22⋊C4×C12712yes—
316C23—22no—
317C23—22no—
318C23—42no—
319C23—42no—
320C23—82no—
321C23—82no—
322C23C2×C22⋊C4×C12711yescharacteristic
323C23#120×C12712yes—
324Q8—41no—
325Q8—44no—
326D4—42no—
327D4—41no—
328D4—42no—
329D4—42no—
330D4—48no—
331D4—42no—
332D4—42no—
333D4—84no—
334D4—84no—
335D4—84no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4C2×C22⋊C4×C12711yescharacteristic
350C2×C4C2×C22⋊C4×C12711yescharacteristic
351C2×C4#120×C12712yes—
352C8—82no—
353C22—22no—
354C22—21no—
355C22—42no—
356C22—41no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22—162no—
362C22—162no—
363C22#379×C12711yescharacteristic
364C4—22no—
365C4—22no—
366C4—21no—
367C4—42no—
368C4—41no—
369C4—42no—
370C4—42no—
371C4—162no—
372C4#410×C12712yes—
373C2—21no—
374C2—42no—
375C2—41no—
376C2—82no—
377C2—82no—
378C2C127×#178211yescharacteristic
subgroupG/H
center, socleC2×C127#1782
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×#41C23×C127
prime radicalC127×#3508C2
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2134C2semidirect
3. C127×#2128C2semidirect
4. C127×#2122C2semidirect
5. C127×#1827C2semidirect
12. #533×C127C4semidirect
14. C2×#127×C127C4semidirect
15. #518×C127C4semidirect
16. #518×C127C4semidirect
183. #12722C127direct (N × H)
191. #2134C2×C127semidirect
192. #2128C2×C127semidirect
193. #2122C2×C127semidirect
194. #1827C2×C127semidirect
195. C127#12722direct (N × H)
202. #533C4×C127semidirect
204. C2×#127C4×C127semidirect
205. #518C4×C127semidirect
206. #518C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35082
C127×#21342
C127×#21282
C127×#21222
C127×#18272
#12722127