Finite Group Databasebeta

← C127×#13343 #44577016

Subgroups — C127×#13343 #44577016

Order 32512.

1084 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 1084 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
1C2×B(4,𝔽2)×C127C212yesmaximal, complemented
2C127×#2679C211yesmaximal, characteristic, complemented
3C2×AΣL(1,𝔽16)×C127C212yesmaximal, complemented
4C127×#1792C211yesmaximal, characteristic, complemented
5C127×#1393C211yesmaximal, characteristic, complemented
6C2×H(5,𝔽2)×C127C2211yescharacteristic
7C2×#120×C127—22no—
8C2×#120×C127—22no—
9C2×#120×C127C2212yes—
10B(4,𝔽2)×C127—24no—
11B(4,𝔽2)×C127—24no—
12C2×#104×C127—21no—
13#417×C127—21no—
14C2×#100×C127C2211yescharacteristic
15C2×#99×C127—22no—
16C2×#99×C127C2212yes—
17#394×C127—21no—
18#394×C127—22no—
19#390×C127—21no—
20#390×C127C2211yescharacteristic
21AΣL(1,𝔽16)×C127—24no—
22C25×C127D412yescomplemented
23H(5,𝔽2)×C127—22no—
24H(5,𝔽2)×C127—21no—
25H(5,𝔽2)×C127—42no—
26H(5,𝔽2)×C127D412yescomplemented
27C2×#41×C127—21no—
28C2×#41×C127—41no—
29C22×D4×C127—22no—
30C22×D4×C127—21no—
31C22×D4×C127—21no—
32C22×D4×C127—22no—
33C22×D4×C127—41no—
34C22×D4×C127—42no—
35C22×D4×C127C2311yescharacteristic
36C2×M16×C127—21no—
37#120×C127—28no—
38#120×C127—22no—
39#120×C127—22no—
40#120×C127—22no—
41#120×C127—22no—
42#120×C127—48no—
43#120×C127—44no—
44C2×C4⋊C4×C127—21no—
45C2×C4⋊C4×C127D411yescharacteristic, complemented
46C2×C22⋊C4×C127—22no—
47C2×C22⋊C4×C127—22no—
48C2×C22⋊C4×C127—22no—
49C2×C22⋊C4×C127—22no—
50C2×C22⋊C4×C127—22no—
51C2×C22⋊C4×C127—42no—
52C2×C22⋊C4×C127D412yescomplemented
53C2×C42×C127D411yescharacteristic, complemented
54#104×C127—42no—
55#103×C127—41no—
56#102×C127—41no—
57#100×C127—22no—
58#99×C127—28no—
59#99×C127—22no—
60#99×C127D414yescomplemented
61#95×C127—21no—
62C24×C127—22no—
63C24×C127—22no—
64C24×C127—22no—
65C24×C127—22no—
66C24×C127—42no—
67C24×C127—42no—
68C24×C127—44no—
69C24×C127—82no—
70C24×C127C2×D412yes—
71#41×C127—22no—
72#41×C127—22no—
73#41×C127—42no—
74#41×C127—41no—
75#41×C127—44no—
76#41×C127—42no—
77#41×C127—41no—
78C2×Q8×C127—21no—
79C2×D4×C127—21no—
80C2×D4×C127—24no—
81C2×D4×C127—24no—
82C2×D4×C127—24no—
83C2×D4×C127—41no—
84C2×D4×C127—41no—
85C2×D4×C127—42no—
86C2×D4×C127—42no—
87C2×D4×C127—42no—
88C2×D4×C127—42no—
89C2×D4×C127—44no—
90C2×D4×C127—41no—
91C2×D4×C127—42no—
92C2×D4×C127—48no—
93C2×D4×C127—44no—
94C2×D4×C127—44no—
95C2×D4×C127—42no—
96C2×D4×C127—42no—
97C2×D4×C127—81no—
98C2×D4×C127—81no—
99C2×D4×C127—82no—
100C2×D4×C127—81no—
101C2×D4×C127—84no—
102C2×D4×C127—84no—
103C2×D4×C127C2×D411yescharacteristic
104C2×D4×C127C2×D412yes—
105C2×D4×C127#4111yescharacteristic
106C22×C4×C127—22no—
107C22×C4×C127—21no—
108C22×C4×C127—21no—
109C22×C4×C127—22no—
110C22×C4×C127—41no—
111C22×C4×C127—42no—
112C22×C4×C127C2×D411yescharacteristic
113M16×C127—41no—
114M16×C127—41no—
115C2×C8×C127—41no—
116C4⋊C4×C127—41no—
117C4⋊C4×C127—81no—
118C22⋊C4×C127—22no—
119C22⋊C4×C127—24no—
120C22⋊C4×C127—22no—
121C22⋊C4×C127—44no—
122C22⋊C4×C127—44no—
123C22⋊C4×C127—48no—
124C22⋊C4×C127—42no—
125C22⋊C4×C127—44no—
126C22⋊C4×C127—42no—
127C22⋊C4×C127—42no—
128C42×C127—41no—
129C23×C127—22no—
130C23×C127—22no—
131C23×C127—24no—
132C23×C127—24no—
133C23×C127—42no—
134C23×C127—42no—
135C23×C127—42no—
136C23×C127—44no—
137C23×C127—42no—
138C23×C127—42no—
139C23×C127—42no—
140C23×C127—44no—
141C23×C127—44no—
142C23×C127—42no—
143C23×C127—42no—
144C23×C127—41no—
145C23×C127—42no—
146C23×C127—82no—
147C23×C127—84no—
148C23×C127—82no—
149C23×C127—84no—
150C23×C127—88no—
151C23×C127—82no—
152C23×C127—82no—
153C23×C127—82no—
154C23×C127#12012yes—
155C23×C127#12011yescharacteristic
156C23×C127#12112yes—
157Q8×C127—22no—
158Q8×C127—41no—
159D4×C127—22no—
160D4×C127—42no—
161D4×C127—41no—
162D4×C127—41no—
163D4×C127—41no—
164D4×C127—44no—
165D4×C127—44no—
166D4×C127—42no—
167D4×C127—42no—
168D4×C127—84no—
169D4×C127—84no—
170D4×C127—84no—
171D4×C127—84no—
172D4×C127—816no—
173C2×C4×C127—21no—
174C2×C4×C127—21no—
175C2×C4×C127—22no—
176C2×C4×C127—24no—
177C2×C4×C127—42no—
178C2×C4×C127—42no—
179C2×C4×C127—41no—
180C2×C4×C127—42no—
181C2×C4×C127—41no—
182C2×C4×C127—41no—
183C2×C4×C127—44no—
184C2×C4×C127—42no—
185C2×C4×C127—42no—
186C2×C4×C127—42no—
187C2×C4×C127—42no—
188C2×C4×C127—41no—
189C2×C4×C127—82no—
190C2×C4×C127—82no—
191C2×C4×C127#12111yescharacteristic
192C2×C4×C127#12711yescharacteristic
193C8×C127—81no—
194C22×C127—21no—
195C22×C127—21no—
196C22×C127—22no—
197C22×C127—22no—
198C22×C127—42no—
199C22×C127—42no—
200C22×C127—44no—
201C22×C127—44no—
202C22×C127—42no—
203C22×C127—42no—
204C22×C127—48no—
205C22×C127—44no—
206C22×C127—42no—
207C22×C127—82no—
208C22×C127—82no—
209C22×C127—82no—
210C22×C127—84no—
211C22×C127—84no—
212C22×C127—84no—
213C22×C127—84no—
214C22×C127—81no—
215C22×C127—82no—
216C22×C127—81no—
217C22×C127—82no—
218C22×C127—162no—
219C22×C127#39211yescharacteristic
220C22×C127B(4,𝔽2)11yescharacteristic
221C22×C127B(4,𝔽2)11yescharacteristic
222C4×C127—21no—
223C4×C127—21no—
224C4×C127—42no—
225C4×C127—42no—
226C4×C127—82no—
227C4×C127—84no—
228C4×C127—81no—
229#13343C12711yesmaximal, characteristic, direct product factor, Sylow
230C2×C127—21no—
231C2×C127—21no—
232C2×C127—42no—
233C2×C127—42no—
234C2×C127—84no—
235C2×C127—82no—
236C2×C127#202111yescharacteristic
237C2×C127#219611yescharacteristic
238C2×C127#219911yescharacteristic
239C2×B(4,𝔽2)C2×C12712yescomplemented
240#2679C2×C12711yescharacteristic, complemented
241C2×AΣL(1,𝔽16)C2×C12712yescomplemented
242#1792C2×C12711yescharacteristic, complemented
243#1393C2×C12711yescharacteristic, complemented
244C127#1334311yescharacteristic, direct product factor, Sylow
245C2×H(5,𝔽2)C22×C12711yescharacteristic
246C2×#120—22no—
247C2×#120—22no—
248C2×#120C22×C12712yes—
249B(4,𝔽2)—24no—
250B(4,𝔽2)—24no—
251C2×#104—21no—
252#417—21no—
253C2×#100C22×C12711yescharacteristic
254C2×#99—22no—
255C2×#99C22×C12712yes—
256#394—21no—
257#394—22no—
258#390—21no—
259#390C22×C12711yescharacteristic
260AΣL(1,𝔽16)—24no—
261C25D4×C12712yescomplemented
262H(5,𝔽2)—22no—
263H(5,𝔽2)—21no—
264H(5,𝔽2)—42no—
265H(5,𝔽2)D4×C12712yescomplemented
266C2×#41—21no—
267C2×#41—41no—
268C22×D4—22no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—22no—
272C22×D4—41no—
273C22×D4—42no—
274C22×D4C23×C12711yescharacteristic
275C2×M16—21no—
276#120—28no—
277#120—22no—
278#120—22no—
279#120—22no—
280#120—22no—
281#120—48no—
282#120—44no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4D4×C12711yescharacteristic, complemented
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4D4×C12712yescomplemented
292C2×C42D4×C12711yescharacteristic, complemented
293#104—42no—
294#103—41no—
295#102—41no—
296#100—22no—
297#99—28no—
298#99—22no—
299#99D4×C12714yescomplemented
300#95—21no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—42no—
306C24—42no—
307C24—44no—
308C24—82no—
309C24C2×D4×C12712yes—
310#41—22no—
311#41—22no—
312#41—42no—
313#41—41no—
314#41—44no—
315#41—42no—
316#41—41no—
317C2×Q8—21no—
318C2×D4—21no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—44no—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—48no—
332C2×D4—44no—
333C2×D4—44no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4—82no—
339C2×D4—81no—
340C2×D4—84no—
341C2×D4—84no—
342C2×D4C2×D4×C12711yescharacteristic
343C2×D4C2×D4×C12712yes—
344C2×D4#41×C12711yescharacteristic
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4C2×D4×C12711yescharacteristic
352M16—41no—
353M16—41no—
354C2×C8—41no—
355C4⋊C4—41no—
356C4⋊C4—81no—
357C22⋊C4—22no—
358C22⋊C4—24no—
359C22⋊C4—22no—
360C22⋊C4—44no—
361C22⋊C4—44no—
362C22⋊C4—48no—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C42—41no—
368C23—22no—
369C23—22no—
370C23—24no—
371C23—24no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—44no—
381C23—42no—
382C23—41no—
383C23—42no—
384C23—42no—
385C23—82no—
386C23—84no—
387C23—82no—
388C23—84no—
389C23—88no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23#120×C12711yescharacteristic
394C23#120×C12712yes—
395C23#121×C12712yes—
396Q8—22no—
397Q8—41no—
398D4—22no—
399D4—42no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—44no—
404D4—44no—
405D4—42no—
406D4—42no—
407D4—84no—
408D4—84no—
409D4—84no—
410D4—84no—
411D4—816no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4#121×C12711yescharacteristic
431C2×C4#127×C12711yescharacteristic
432C8—81no—
433C22—21no—
434C22—21no—
435C22—22no—
436C22—22no—
437C22—42no—
438C22—42no—
439C22—44no—
440C22—44no—
441C22—42no—
442C22—42no—
443C22—48no—
444C22—44no—
445C22—42no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—84no—
450C22—84no—
451C22—84no—
452C22—84no—
453C22—81no—
454C22—82no—
455C22—81no—
456C22—82no—
457C22—162no—
458C22#392×C12711yescharacteristic
459C22B(4,𝔽2)×C12711yescharacteristic
460C22B(4,𝔽2)×C12711yescharacteristic
461C4—21no—
462C4—21no—
463C4—42no—
464C4—42no—
465C4—82no—
466C4—84no—
467C4—81no—
468C2—21no—
469C2—21no—
470C2—42no—
471C2—42no—
472C2—84no—
473C2—82no—
474C2C127×#202111yescharacteristic
475C2C127×#219611yescharacteristic
476C2C127×#219911yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC22×D4C23×C127
LCSG ▷ C22×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
1. C2×B(4,𝔽2)×C127C2semidirect
2. C127×#2679C2semidirect
3. C2×AΣL(1,𝔽16)×C127C2semidirect
4. C127×#1792C2semidirect
5. C127×#1393C2semidirect
22. C25×C127D4semidirect
26. H(5,𝔽2)×C127D4semidirect
45. C2×C4⋊C4×C127D4semidirect
52. C2×C22⋊C4×C127D4semidirect
53. C2×C42×C127D4semidirect
60. #99×C127D4semidirect
229. #13343C127direct (N × H)
239. C2×B(4,𝔽2)C2×C127semidirect
240. #2679C2×C127semidirect
241. C2×AΣL(1,𝔽16)C2×C127semidirect
242. #1792C2×C127semidirect
243. #1393C2×C127semidirect
244. C127#13343direct (N × H)
261. C25D4×C127semidirect
265. H(5,𝔽2)D4×C127semidirect
284. C2×C4⋊C4D4×C127semidirect
291. C2×C22⋊C4D4×C127semidirect
292. C2×C42D4×C127semidirect
299. #99D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×B(4,𝔽2)×C1272
C127×#26792
C2×AΣL(1,𝔽16)×C1272
C127×#17922
C127×#13932
#13343127