Finite Group Databasebeta

← C127×#10337 #44574010

Subgroups — C127×#10337 #44574010

Order 32512.

1274 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2405C212yesmaximal, complemented
2C127×#2403C211yesmaximal, characteristic
3C127×#1819C211yesmaximal, characteristic, complemented
4C127×#1781C212yesmaximal, complemented
5C127×#1304C211yesmaximal, characteristic, complemented
6C23×D4×C127C2212yes—
7C2×#123×C127—22no—
8C2×#120×C127—22no—
9C22×C22⋊C4×C127C2211yescharacteristic
10C2×#99×C127—24no—
11#394×C127—24no—
12#392×C127—22no—
13#386×C127—22no—
14#386×C127—22no—
15#385×C127—22no—
16#379×C127—22no—
17#379×C127C2211yescharacteristic
18#379×C127C2211yescharacteristic
19#379×C127C2212yes—
20#342×C127—22no—
21C25×C127D412yes—
22C25×C127C2311yescharacteristic
23C22×D4×C127—22no—
24C22×D4×C127—22no—
25C22×D4×C127—22no—
26C22×D4×C127—22no—
27C22×D4×C127—22no—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127—48no—
31C23×C4×C127D412yes—
32C23×C4×C127D412yescomplemented
33#123×C127—48no—
34#120×C127—48no—
35C2×C4⋊C4×C127—24no—
36C2×C22⋊C4×C127—24no—
37C2×C22⋊C4×C127—24no—
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—22no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127—22no—
43C2×C22⋊C4×C127—22no—
44C2×C22⋊C4×C127—21no—
45C2×C22⋊C4×C127—21no—
46C2×C22⋊C4×C127—22no—
47C2×C22⋊C4×C127—24no—
48C2×C22⋊C4×C127D412yes—
49C2×C22⋊C4×C127D414yescomplemented
50#99×C127—48no—
51#95×C127—24no—
52#95×C127—22no—
53C24×C127—22no—
54C24×C127—22no—
55C24×C127—22no—
56C24×C127—44no—
57C24×C127—44no—
58C24×C127—42no—
59C24×C127—44no—
60C24×C127—44no—
61C24×C127C2×D412yes—
62C24×C127C2×D411yescharacteristic
63C24×C127C2×D411yescharacteristic
64C24×C127C2×D412yes—
65C24×C127#4111yescharacteristic
66C2×D4×C127—28no—
67C2×D4×C127—28no—
68C2×D4×C127—28no—
69C2×D4×C127—44no—
70C2×D4×C127—44no—
71C2×D4×C127—44no—
72C2×D4×C127—44no—
73C2×D4×C127—44no—
74C2×D4×C127—44no—
75C2×D4×C127—44no—
76C2×D4×C127—44no—
77C2×D4×C127—44no—
78C2×D4×C127—42no—
79C2×D4×C127—42no—
80C2×D4×C127—42no—
81C2×D4×C127—42no—
82C2×D4×C127—48no—
83C2×D4×C127—82no—
84C22×C4×C127—22no—
85C22×C4×C127—22no—
86C22×C4×C127—22no—
87C22×C4×C127—22no—
88C22×C4×C127—22no—
89C22×C4×C127—24no—
90C22×C4×C127—22no—
91C22×C4×C127—22no—
92C22×C4×C127—44no—
93C22×C4×C127—44no—
94C22×C4×C127—41no—
95C4⋊C4×C127—48no—
96C22⋊C4×C127—48no—
97C22⋊C4×C127—44no—
98C22⋊C4×C127—44no—
99C22⋊C4×C127—44no—
100C22⋊C4×C127—48no—
101C22⋊C4×C127—41no—
102C22⋊C4×C127—42no—
103C22⋊C4×C127—41no—
104C22⋊C4×C127—44no—
105C22⋊C4×C127—42no—
106C22⋊C4×C127—44no—
107C22⋊C4×C127—42no—
108C22⋊C4×C127—84no—
109C23×C127—24no—
110C23×C127—24no—
111C23×C127—24no—
112C23×C127—24no—
113C23×C127—22no—
114C23×C127—24no—
115C23×C127—24no—
116C23×C127—24no—
117C23×C127—42no—
118C23×C127—41no—
119C23×C127—42no—
120C23×C127—42no—
121C23×C127—42no—
122C23×C127—42no—
123C23×C127—41no—
124C23×C127—44no—
125C23×C127—42no—
126C23×C127—42no—
127C23×C127—42no—
128C23×C127—44no—
129C23×C127—42no—
130C23×C127—44no—
131C23×C127—44no—
132C23×C127—82no—
133C23×C127—82no—
134C23×C127—88no—
135C23×C127—816no—
136C23×C127#12012yes—
137C23×C127#12011yescharacteristic
138C23×C127#12112yes—
139C23×C127#12111yescharacteristic
140C23×C127#12711yescharacteristic
141D4×C127—48no—
142D4×C127—48no—
143D4×C127—48no—
144D4×C127—48no—
145D4×C127—88no—
146C2×C4×C127—24no—
147C2×C4×C127—24no—
148C2×C4×C127—24no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
152C2×C4×C127—42no—
153C2×C4×C127—44no—
154C2×C4×C127—44no—
155C2×C4×C127—44no—
156C2×C4×C127—44no—
157C2×C4×C127—42no—
158C2×C4×C127—44no—
159C2×C4×C127—44no—
160C2×C4×C127—42no—
161C2×C4×C127—82no—
162C2×C4×C127—82no—
163C2×C4×C127—81no—
164C22×C127—22no—
165C22×C127—22no—
166C22×C127—22no—
167C22×C127—22no—
168C22×C127—22no—
169C22×C127—22no—
170C22×C127—22no—
171C22×C127—21no—
172C22×C127—22no—
173C22×C127—21no—
174C22×C127—44no—
175C22×C127—48no—
176C22×C127—48no—
177C22×C127—48no—
178C22×C127—44no—
179C22×C127—82no—
180C22×C127—84no—
181C22×C127—84no—
182C22×C127—84no—
183C22×C127—82no—
184C22×C127—84no—
185C22×C127—84no—
186C22×C127—82no—
187C22×C127—82no—
188C22×C127#39211yescharacteristic
189C22×C127B(4,𝔽2)14yes—
190C22×C127#45812yes—
191C4×C127—44no—
192C4×C127—84no—
193C4×C127—82no—
194C4×C127—82no—
195C4×C127—161no—
196#10337C12711yesmaximal, characteristic, direct product factor, Sylow
197C2×C127—22no—
198C2×C127—24no—
199C2×C127—41no—
200C2×C127—41no—
201C2×C127—42no—
202C2×C127—84no—
203C2×C127#202112yes—
204C2×C127#202211yescharacteristic
205#2405C2×C12712yescomplemented
206#2403C2×C12711yescharacteristic
207#1819C2×C12711yescharacteristic, complemented
208#1781C2×C12712yescomplemented
209#1304C2×C12711yescharacteristic, complemented
210C127#1033711yescharacteristic, direct product factor, Sylow
211C23×D4C22×C12712yes—
212C2×#123—22no—
213C2×#120—22no—
214C22×C22⋊C4C22×C12711yescharacteristic
215C2×#99—24no—
216#394—24no—
217#392—22no—
218#386—22no—
219#386—22no—
220#385—22no—
221#379—22no—
222#379C22×C12711yescharacteristic
223#379C22×C12711yescharacteristic
224#379C22×C12712yes—
225#342—22no—
226C25D4×C12712yes—
227C25C23×C12711yescharacteristic
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—22no—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—48no—
236C23×C4D4×C12712yes—
237C23×C4D4×C12712yescomplemented
238#123—48no—
239#120—48no—
240C2×C4⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4D4×C12712yes—
254C2×C22⋊C4D4×C12714yescomplemented
255#99—48no—
256#95—24no—
257#95—22no—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—44no—
262C24—44no—
263C24—42no—
264C24—44no—
265C24—44no—
266C24C2×D4×C12712yes—
267C24C2×D4×C12711yescharacteristic
268C24C2×D4×C12711yescharacteristic
269C24C2×D4×C12712yes—
270C24#41×C12711yescharacteristic
271C2×D4—28no—
272C2×D4—28no—
273C2×D4—28no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—48no—
288C2×D4—82no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—41no—
300C4⋊C4—48no—
301C22⋊C4—48no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—48no—
306C22⋊C4—41no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—42no—
313C22⋊C4—84no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—22no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—42no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23—44no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—44no—
334C23—42no—
335C23—44no—
336C23—44no—
337C23—82no—
338C23—82no—
339C23—88no—
340C23—816no—
341C23#120×C12712yes—
342C23#120×C12711yescharacteristic
343C23#121×C12712yes—
344C23#121×C12711yescharacteristic
345C23#127×C12711yescharacteristic
346D4—48no—
347D4—48no—
348D4—48no—
349D4—48no—
350D4—88no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—81no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—21no—
377C22—22no—
378C22—21no—
379C22—48no—
380C22—44no—
381C22—48no—
382C22—48no—
383C22—44no—
384C22—82no—
385C22—84no—
386C22—84no—
387C22—84no—
388C22—82no—
389C22—84no—
390C22—84no—
391C22—82no—
392C22—82no—
393C22#392×C12711yescharacteristic
394C22B(4,𝔽2)×C12714yes—
395C22#458×C12712yes—
396C4—44no—
397C4—82no—
398C4—84no—
399C4—82no—
400C4—161no—
401C2—22no—
402C2—24no—
403C2—41no—
404C2—42no—
405C2—41no—
406C2—84no—
407C2C127×#202112yes—
408C2C127×#202211yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC25C23×C127
prime radicalC127×#2403C2
LCSG ▷ C25 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C25×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2405C2semidirect
3. C127×#1819C2semidirect
4. C127×#1781C2semidirect
5. C127×#1304C2semidirect
32. C23×C4×C127D4semidirect
49. C2×C22⋊C4×C127D4semidirect
196. #10337C127direct (N × H)
205. #2405C2×C127semidirect
207. #1819C2×C127semidirect
208. #1781C2×C127semidirect
209. #1304C2×C127semidirect
210. C127#10337direct (N × H)
237. C23×C4D4×C127semidirect
254. C2×C22⋊C4D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#24052
C127×#24032
C127×#18192
C127×#17812
C127×#13042
#10337127