Finite Group Databasebeta

← C127×#13048 #44576721

Subgroups — C127×#13048 #44576721

Order 32512.

752 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 118 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
1C2×#455×C127C211yesmaximal, characteristic
2C127×#2372C211yesmaximal, characteristic, complemented
3C2×#354×C127C211yesmaximal, characteristic, complemented
4C127×#1790C212yesmaximal
5C127×#1390C212yesmaximal, complemented
6C2×H(5,𝔽2)×C127C2211yescharacteristic
7C2×#137×C127—21no—
8C2×#124×C127C2211yescharacteristic
9#455×C127—22no—
10#455×C127C414yescomplemented
11C2×#104×C127—21no—
12C2×#100×C127C2211yescharacteristic
13C2×#99×C127—22no—
14C2×#99×C127C2212yes—
15#391×C127—21no—
16#388×C127—22no—
17#386×C127—22no—
18#386×C127C2212yes—
19C4×C22⋊C4×C127—22no—
20#376×C127—21no—
21#354×C127—22no—
22H(5,𝔽2)×C127—22no—
23H(5,𝔽2)×C127—21no—
24H(5,𝔽2)×C127—42no—
25H(5,𝔽2)×C127C2×C412yes—
26C2×#41×C127—21no—
27C2×#41×C127—41no—
28C22×Q8×C127D411yescharacteristic
29C22×D4×C127—22no—
30C22×D4×C127—41no—
31C22×D4×C127C2311yescharacteristic
32C23×C4×C127D412yes—
33#137×C127—24no—
34#137×C127—42no—
35C2×Q16×C127—41no—
36C2×SD16×C127—41no—
37C2×M16×C127—21no—
38#124×C127—22no—
39#124×C127C2×C412yes—
40#124×C127D412yes—
41C2×C4⋊C4×C127—21no—
42C2×C22⋊C4×C127—22no—
43C2×C22⋊C4×C127—22no—
44C2×C22⋊C4×C127—22no—
45C2×C22⋊C4×C127—42no—
46C2×C22⋊C4×C127D412yes—
47C2×C42×C127—22no—
48C2×C42×C127—21no—
49C2×C42×C127D411yescharacteristic
50#104×C127—24no—
51#100×C127—21no—
52#100×C127C2×C412yes—
53#99×C127—22no—
54#99×C127—22no—
55#99×C127—44no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—22no—
59#95×C127—22no—
60#95×C127—42no—
61C24×C127—22no—
62C24×C127C2×D412yes—
63#41×C127—22no—
64#41×C127—22no—
65#41×C127—42no—
66#41×C127—41no—
67#41×C127—44no—
68#41×C127—42no—
69#41×C127—41no—
70C2×Q8×C127—21no—
71C2×Q8×C127—21no—
72C2×Q8×C127—41no—
73C2×Q8×C127—41no—
74C2×Q8×C127C22⋊C412yescomplemented
75C2×D4×C127—21no—
76C2×D4×C127—24no—
77C2×D4×C127—24no—
78C2×D4×C127—41no—
79C2×D4×C127—41no—
80C2×D4×C127—42no—
81C2×D4×C127—42no—
82C2×D4×C127—42no—
83C2×D4×C127—42no—
84C2×D4×C127—44no—
85C2×D4×C127—41no—
86C2×D4×C127—42no—
87C2×D4×C127C22×C411yescharacteristic
88C2×D4×C127C2×D411yescharacteristic
89C2×D4×C127#4112yes—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—21no—
93C22×C4×C127—21no—
94C22×C4×C127—22no—
95C22×C4×C127—41no—
96C22×C4×C127—44no—
97C22×C4×C127—42no—
98C22×C4×C127—42no—
99C22×C4×C127C2×D411yescharacteristic
100Q16×C127—44no—
101SD16×C127—44no—
102M16×C127—22no—
103M16×C127—41no—
104C2×C8×C127—41no—
105C4⋊C4×C127—81no—
106C22⋊C4×C127—24no—
107C22⋊C4×C127—24no—
108C22⋊C4×C127—44no—
109C22⋊C4×C127—42no—
110C22⋊C4×C127—42no—
111C22⋊C4×C127—42no—
112C22⋊C4×C127—84no—
113C42×C127—21no—
114C42×C127—44no—
115C42×C127—81no—
116C42×C127C22⋊C412yescomplemented
117C23×C127—22no—
118C23×C127—22no—
119C23×C127—24no—
120C23×C127—42no—
121C23×C127—42no—
122C23×C127—42no—
123C23×C127—41no—
124C23×C127—82no—
125C23×C127—82no—
126C23×C127C4×D412yes—
127C23×C127#12011yescharacteristic
128C23×C127#12112yes—
129Q8×C127—22no—
130Q8×C127—42no—
131Q8×C127—41no—
132Q8×C127—81no—
133D4×C127—22no—
134D4×C127—41no—
135D4×C127—41no—
136D4×C127—42no—
137D4×C127—41no—
138D4×C127—44no—
139D4×C127—44no—
140D4×C127—42no—
141D4×C127—42no—
142C2×C4×C127—21no—
143C2×C4×C127—21no—
144C2×C4×C127—24no—
145C2×C4×C127—24no—
146C2×C4×C127—22no—
147C2×C4×C127—22no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
152C2×C4×C127—42no—
153C2×C4×C127—41no—
154C2×C4×C127—42no—
155C2×C4×C127—41no—
156C2×C4×C127—41no—
157C2×C4×C127—44no—
158C2×C4×C127—42no—
159C2×C4×C127—42no—
160C2×C4×C127—42no—
161C2×C4×C127—82no—
162C2×C4×C127—82no—
163C2×C4×C127—82no—
164C2×C4×C127—82no—
165C2×C4×C127C2×C22⋊C411yescharacteristic
166C2×C4×C127#12311yescharacteristic
167C8×C127—42no—
168C22×C127—22no—
169C22×C127—22no—
170C22×C127—22no—
171C22×C127—42no—
172C22×C127—42no—
173C22×C127—42no—
174C22×C127—42no—
175C22×C127—42no—
176C22×C127—82no—
177C22×C127—84no—
178C22×C127—82no—
179C22×C127—162no—
180C22×C127#38611yescharacteristic
181C22×C127B(4,𝔽2)11yescharacteristic
182C22×C127B(4,𝔽2)11yescharacteristic
183C4×C127—21no—
184C4×C127—21no—
185C4×C127—42no—
186C4×C127—42no—
187C4×C127—42no—
188C4×C127—44no—
189C4×C127—82no—
190C4×C127—162no—
191#13048C12711yesmaximal, characteristic, direct product factor, Sylow
192C2×C127—21no—
193C2×C127—21no—
194C2×C127—42no—
195C2×C127—42no—
196C2×C127—82no—
197C2×C127#188911yescharacteristic
198C2×C127#219712yes—
199C2×#455C2×C12711yescharacteristic
200#2372C2×C12711yescharacteristic, complemented
201C2×#354C2×C12711yescharacteristic, complemented
202#1790C2×C12712yes—
203#1390C2×C12712yescomplemented
204C127#1304811yescharacteristic, direct product factor, Sylow
205C2×H(5,𝔽2)C22×C12711yescharacteristic
206C2×#137—21no—
207C2×#124C22×C12711yescharacteristic
208#455—22no—
209#455C4×C12714yescomplemented
210C2×#104—21no—
211C2×#100C22×C12711yescharacteristic
212C2×#99—22no—
213C2×#99C22×C12712yes—
214#391—21no—
215#388—22no—
216#386—22no—
217#386C22×C12712yes—
218C4×C22⋊C4—22no—
219#376—21no—
220#354—22no—
221H(5,𝔽2)—22no—
222H(5,𝔽2)—21no—
223H(5,𝔽2)—42no—
224H(5,𝔽2)C2×C4×C12712yes—
225C2×#41—21no—
226C2×#41—41no—
227C22×Q8D4×C12711yescharacteristic
228C22×D4—22no—
229C22×D4—41no—
230C22×D4C23×C12711yescharacteristic
231C23×C4D4×C12712yes—
232#137—24no—
233#137—42no—
234C2×Q16—41no—
235C2×SD16—41no—
236C2×M16—21no—
237#124—22no—
238#124C2×C4×C12712yes—
239#124D4×C12712yes—
240C2×C4⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4D4×C12712yes—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42D4×C12711yescharacteristic
249#104—24no—
250#100—21no—
251#100C2×C4×C12712yes—
252#99—22no—
253#99—22no—
254#99—44no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—42no—
260C24—22no—
261C24C2×D4×C12712yes—
262#41—22no—
263#41—22no—
264#41—42no—
265#41—41no—
266#41—44no—
267#41—42no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8C22⋊C4×C12712yescomplemented
274C2×D4—21no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4C22×C4×C12711yescharacteristic
287C2×D4C2×D4×C12711yescharacteristic
288C2×D4#41×C12712yes—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—41no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4C2×D4×C12711yescharacteristic
299Q16—44no—
300SD16—44no—
301M16—22no—
302M16—41no—
303C2×C8—41no—
304C4⋊C4—81no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—44no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—84no—
312C42—21no—
313C42—44no—
314C42—81no—
315C42C22⋊C4×C12712yescomplemented
316C23—22no—
317C23—22no—
318C23—24no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—41no—
323C23—82no—
324C23—82no—
325C23C4×D4×C12712yes—
326C23#120×C12711yescharacteristic
327C23#121×C12712yes—
328Q8—22no—
329Q8—42no—
330Q8—41no—
331Q8—81no—
332D4—22no—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—41no—
337D4—44no—
338D4—44no—
339D4—42no—
340D4—42no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4C2×C22⋊C4×C12711yescharacteristic
365C2×C4#123×C12711yescharacteristic
366C8—42no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—82no—
376C22—84no—
377C22—82no—
378C22—162no—
379C22#386×C12711yescharacteristic
380C22B(4,𝔽2)×C12711yescharacteristic
381C22B(4,𝔽2)×C12711yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4—44no—
388C4—82no—
389C4—162no—
390C2—21no—
391C2—21no—
392C2—42no—
393C2—42no—
394C2—82no—
395C2C127×#188911yescharacteristic
396C2C127×#219712yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutatorC2×D4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC2×H(5,𝔽2)×C127C22
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C22×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2372C2semidirect
3. C2×#354×C127C2semidirect
5. C127×#1390C2semidirect
10. #455×C127C4semidirect
74. C2×Q8×C127C22⋊C4semidirect
116. C42×C127C22⋊C4semidirect
191. #13048C127direct (N × H)
200. #2372C2×C127semidirect
201. C2×#354C2×C127semidirect
203. #1390C2×C127semidirect
204. C127#13048direct (N × H)
209. #455C4×C127semidirect
273. C2×Q8C22⋊C4×C127semidirect
315. C42C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#455×C1272
C127×#23722
C2×#354×C1272
C127×#17902
C127×#13902
#13048127