Finite Group Databasebeta

← C127×#12778 #44576451

Subgroups — C127×#12778 #44576451

Order 32512.

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

normal subgroups only

This group has 860 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×#3549C211yesmaximal, characteristic
2C127×#2128C212yesmaximal, complemented
3C127×#2126C211yesmaximal, characteristic, complemented
4C127×#1910C212yesmaximal, complemented
5C127×#1884C211yesmaximal, characteristic, complemented
6#568×C127C2211yescharacteristic, complemented
7H(3,𝔽4)×C127—21no—
8H(3,𝔽4)×C127C412yescomplemented
9#555×C127—22no—
10#555×C127C412yescomplemented
11D42×C127—22no—
12D42×C127—21no—
13#535×C127—21no—
14#535×C127—24no—
15#535×C127C2212yes—
16#532×C127—21no—
17#532×C127C2211yescharacteristic
18#421×C127—21no—
19#418×C127—22no—
20#410×C127C2212yescomplemented
21#410×C127C2211yescharacteristic, complemented
22#354×C127—21no—
23#353×C127—21no—
24#353×C127—24no—
25#339×C127—21no—
26H(3,ℤ/4)×C127—22no—
27C2×#41×C127—22no—
28C2×#41×C127D411yescharacteristic
29C2×#41×C127C2311yescharacteristic
30C22×D4×C127—24no—
31C22×D4×C127—21no—
32C22×D4×C127D412yes—
33#131×C127—42no—
34C2×M16×C127—22no—
35C2×M16×C127—21no—
36C4⋊Q8×C127—21no—
37C4⋊Q8×C127C2×C411yescharacteristic
38#127×C127—21no—
39#127×C127—22no—
40#127×C127D411yescharacteristic
41#127×C127D414yes—
42#124×C127—24no—
43#124×C127—22no—
44#124×C127—22no—
45#124×C127—24no—
46#124×C127C2×C411yescharacteristic
47#124×C127C2×C414yes—
48#124×C127D411yescharacteristic
49#121×C127—22no—
50#121×C127—22no—
51#121×C127—24no—
52#121×C127—24no—
53#121×C127—22no—
54#121×C127—22no—
55#121×C127—22no—
56#120×C127—22no—
57#120×C127—24no—
58#120×C127—22no—
59#120×C127—24no—
60#120×C127—22no—
61#120×C127—22no—
62C4×D4×C127—22no—
63C4×D4×C127—22no—
64C4×D4×C127—22no—
65#117×C127—21no—
66#117×C127—24no—
67#117×C127—22no—
68#117×C127—21no—
69#117×C127D412yes—
70C2×C42×C127D411yescharacteristic
71#104×C127—42no—
72#102×C127—44no—
73#99×C127—22no—
74#99×C127—22no—
75#99×C127—22no—
76C24×C127—22no—
77C24×C127—22no—
78#41×C127—41no—
79#41×C127—41no—
80#41×C127—41no—
81#41×C127—41no—
82#41×C127—84no—
83C2×Q8×C127—22no—
84C2×Q8×C127C22⋊C411yescharacteristic
85C2×Q8×C127C22×C411yescharacteristic
86C2×D4×C127—22no—
87C2×D4×C127—22no—
88C2×D4×C127—22no—
89C2×D4×C127—22no—
90C2×D4×C127—24no—
91C2×D4×C127—24no—
92C2×D4×C127—21no—
93C2×D4×C127—24no—
94C2×D4×C127—22no—
95C2×D4×C127—42no—
96C2×D4×C127—42no—
97C2×D4×C127—88no—
98C2×D4×C127—81no—
99C2×D4×C127—81no—
100C2×D4×C127C22⋊C411yescharacteristic
101C2×D4×C127C22⋊C414yes—
102C2×D4×C127C2×D411yescharacteristic
103C2×D4×C127C2×D412yes—
104C22×C4×C127—24no—
105C22×C4×C127—21no—
106C22×C4×C127—21no—
107C22×C4×C127C2×D411yescharacteristic
108C22×C4×C127C2×D412yes—
109M16×C127—22no—
110M16×C127—42no—
111M16×C127—41no—
112C2×C8×C127—24no—
113C2×C8×C127—41no—
114C4⋊C4×C127—22no—
115C4⋊C4×C127—22no—
116C4⋊C4×C127—22no—
117C4⋊C4×C127—42no—
118C4⋊C4×C127—41no—
119C22⋊C4×C127—24no—
120C22⋊C4×C127—22no—
121C22⋊C4×C127—24no—
122C22⋊C4×C127—22no—
123C22⋊C4×C127—22no—
124C22⋊C4×C127—24no—
125C22⋊C4×C127—22no—
126C22⋊C4×C127—24no—
127C22⋊C4×C127—42no—
128C22⋊C4×C127—41no—
129C42×C127—22no—
130C42×C127—21no—
131C42×C127—41no—
132C42×C127—42no—
133C42×C127C22⋊C411yescharacteristic
134C42×C127C22⋊C411yescharacteristic
135C42×C127C22⋊C414yes—
136C23×C127—24no—
137C23×C127—21no—
138C23×C127—82no—
139C23×C127—84no—
140C23×C127—84no—
141C23×C127—82no—
142C23×C127#12012yes—
143C23×C127#12011yescharacteristic
144Q8×C127—41no—
145Q8×C127—41no—
146Q8×C127—82no—
147D4×C127—42no—
148D4×C127—41no—
149D4×C127—44no—
150D4×C127—41no—
151D4×C127—44no—
152D4×C127—44no—
153D4×C127—42no—
154D4×C127—42no—
155D4×C127—81no—
156D4×C127—84no—
157D4×C127—84no—
158D4×C127—82no—
159D4×C127—84no—
160D4×C127—82no—
161C2×C4×C127—21no—
162C2×C4×C127—21no—
163C2×C4×C127—22no—
164C2×C4×C127—22no—
165C2×C4×C127—21no—
166C2×C4×C127—21no—
167C2×C4×C127—21no—
168C2×C4×C127—24no—
169C2×C4×C127—48no—
170C2×C4×C127—42no—
171C2×C4×C127—41no—
172C2×C4×C127—41no—
173C2×C4×C127—81no—
174C2×C4×C127—82no—
175C2×C4×C127C2×C22⋊C411yescharacteristic
176C2×C4×C127C2×C22⋊C412yes—
177C2×C4×C127#12011yescharacteristic
178C8×C127—42no—
179C22×C127—21no—
180C22×C127—22no—
181C22×C127—41no—
182C22×C127—42no—
183C22×C127—44no—
184C22×C127—44no—
185C22×C127—42no—
186C22×C127—41no—
187C22×C127—44no—
188C22×C127—164no—
189C22×C127—164no—
190C22×C127#37911yescharacteristic
191C4×C127—22no—
192C4×C127—22no—
193C4×C127—21no—
194C4×C127—44no—
195C4×C127—41no—
196C4×C127—41no—
197C4×C127—161no—
198C4×C127—162no—
199C4×C127B(4,𝔽2)11yescharacteristic
200C4×C127#45811yescharacteristic
201#12778C12711yesmaximal, characteristic, direct product factor, Sylow
202C2×C127—21no—
203C2×C127—41no—
204C2×C127—42no—
205C2×C127—84no—
206C2×C127—81no—
207C2×C127#179011yescharacteristic
208#3549C2×C12711yescharacteristic
209#2128C2×C12712yescomplemented
210#2126C2×C12711yescharacteristic, complemented
211#1910C2×C12712yescomplemented
212#1884C2×C12711yescharacteristic, complemented
213C127#1277811yescharacteristic, direct product factor, Sylow
214#568C22×C12711yescharacteristic, complemented
215H(3,𝔽4)—21no—
216H(3,𝔽4)C4×C12712yescomplemented
217#555—22no—
218#555C4×C12712yescomplemented
219D42—22no—
220D42—21no—
221#535—21no—
222#535—24no—
223#535C22×C12712yes—
224#532—21no—
225#532C22×C12711yescharacteristic
226#421—21no—
227#418—22no—
228#410C22×C12712yescomplemented
229#410C22×C12711yescharacteristic, complemented
230#354—21no—
231#353—21no—
232#353—24no—
233#339—21no—
234H(3,ℤ/4)—22no—
235C2×#41—22no—
236C2×#41D4×C12711yescharacteristic
237C2×#41C23×C12711yescharacteristic
238C22×D4—24no—
239C22×D4—21no—
240C22×D4D4×C12712yes—
241#131—42no—
242C2×M16—22no—
243C2×M16—21no—
244C4⋊Q8—21no—
245C4⋊Q8C2×C4×C12711yescharacteristic
246#127—21no—
247#127—22no—
248#127D4×C12711yescharacteristic
249#127D4×C12714yes—
250#124—24no—
251#124—22no—
252#124—22no—
253#124—24no—
254#124C2×C4×C12711yescharacteristic
255#124C2×C4×C12714yes—
256#124D4×C12711yescharacteristic
257#121—22no—
258#121—22no—
259#121—24no—
260#121—24no—
261#121—22no—
262#121—22no—
263#121—22no—
264#120—22no—
265#120—24no—
266#120—22no—
267#120—24no—
268#120—22no—
269#120—22no—
270C4×D4—22no—
271C4×D4—22no—
272C4×D4—22no—
273#117—21no—
274#117—24no—
275#117—22no—
276#117—21no—
277#117D4×C12712yes—
278C2×C42D4×C12711yescharacteristic
279#104—42no—
280#102—44no—
281#99—22no—
282#99—22no—
283#99—22no—
284C24—22no—
285C24—22no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—84no—
291C2×Q8—22no—
292C2×Q8C22⋊C4×C12711yescharacteristic
293C2×Q8C22×C4×C12711yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—24no—
299C2×D4—24no—
300C2×D4—21no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—88no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4C22⋊C4×C12711yescharacteristic
309C2×D4C22⋊C4×C12714yes—
310C2×D4C2×D4×C12711yescharacteristic
311C2×D4C2×D4×C12712yes—
312C22×C4—24no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4C2×D4×C12711yescharacteristic
316C22×C4C2×D4×C12712yes—
317M16—22no—
318M16—42no—
319M16—41no—
320C2×C8—24no—
321C2×C8—41no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—24no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—24no—
333C22⋊C4—22no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—41no—
337C42—22no—
338C42—21no—
339C42—41no—
340C42—42no—
341C42C22⋊C4×C12711yescharacteristic
342C42C22⋊C4×C12711yescharacteristic
343C42C22⋊C4×C12714yes—
344C23—24no—
345C23—21no—
346C23—82no—
347C23—84no—
348C23—84no—
349C23—82no—
350C23#120×C12712yes—
351C23#120×C12711yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—82no—
355D4—42no—
356D4—41no—
357D4—44no—
358D4—41no—
359D4—44no—
360D4—44no—
361D4—42no—
362D4—42no—
363D4—81no—
364D4—84no—
365D4—84no—
366D4—82no—
367D4—84no—
368D4—82no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—24no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4C2×C22⋊C4×C12711yescharacteristic
384C2×C4C2×C22⋊C4×C12712yes—
385C2×C4#120×C12711yescharacteristic
386C8—42no—
387C22—21no—
388C22—22no—
389C22—41no—
390C22—42no—
391C22—44no—
392C22—44no—
393C22—42no—
394C22—41no—
395C22—44no—
396C22—164no—
397C22—164no—
398C22#379×C12711yescharacteristic
399C4—22no—
400C4—22no—
401C4—21no—
402C4—44no—
403C4—41no—
404C4—41no—
405C4—162no—
406C4—161no—
407C4B(4,𝔽2)×C12711yescharacteristic
408C4#458×C12711yescharacteristic
409C2—21no—
410C2—42no—
411C2—41no—
412C2—84no—
413C2—81no—
414C2C127×#179011yescharacteristic
subgroupG/H
center, socleC2×C127#1790
commutatorC2×Q8C22×C4×C127
Frattini, cosocleC2×#41C23×C127
prime radicalC127×#3549C2
LCSG ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2128C2semidirect
3. C127×#2126C2semidirect
4. C127×#1910C2semidirect
5. C127×#1884C2semidirect
6. #568×C127C22semidirect
8. H(3,𝔽4)×C127C4semidirect
10. #555×C127C4semidirect
20. #410×C127C22semidirect
21. #410×C127C22semidirect
201. #12778C127direct (N × H)
209. #2128C2×C127semidirect
210. #2126C2×C127semidirect
211. #1910C2×C127semidirect
212. #1884C2×C127semidirect
213. C127#12778direct (N × H)
214. #568C22×C127semidirect
216. H(3,𝔽4)C4×C127semidirect
218. #555C4×C127semidirect
228. #410C22×C127semidirect
229. #410C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#35492
C127×#21282
C127×#21262
C127×#19102
C127×#18842
#12778127