Finite Group Databasebeta

← C127×#10086 #44573759

Subgroups — C127×#10086 #44573759

Order 32512.

542 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 114 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×#501×C127C211yesmaximal, characteristic, complemented
2C2×#489×C127C211yesmaximal, characteristic
3C127×#2820C211yesmaximal, characteristic
4C127×#1960C211yesmaximal, characteristic, complemented
5C2×#330×C127C211yesmaximal, characteristic, complemented
6C2×#329×C127C211yesmaximal, characteristic
7C127×#1295C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#125×C127C2211yescharacteristic
10C2×#124×C127C2211yescharacteristic
11#501×C127—22no—
12#501×C127—22no—
13#489×C127—22no—
14#489×C127—22no—
15C2×#107×C127—21no—
16C2×#106×C127—21no—
17C2×#103×C127—21no—
18C2×#103×C127—22no—
19C2×#102×C127—21no—
20C2×#97×C127C2211yescharacteristic
21C2×#97×C127C2211yescharacteristic
22C2×#97×C127C2211yescharacteristic
23#399×C127—21no—
24#397×C127—21no—
25#396×C127—22no—
26#395×C127—21no—
27#394×C127—21no—
28#382×C127C2211yescharacteristic
29#330×C127—22no—
30#329×C127—22no—
31C22×Q8×C127—21no—
32C22×Q8×C127D411yescharacteristic
33C22×D4×C127D411yescharacteristic
34C22×C8×C127—21no—
35C22×C8×C127—21no—
36C22×C8×C127—21no—
37C4⋊Q8×C127—21no—
38C4⋊Q8×C127—21no—
39C4⋊Q8×C127—21no—
40C4⋊Q8×C127D412yes—
41#125×C127—21no—
42#125×C127—21no—
43#125×C127—21no—
44#125×C127—21no—
45#124×C127—21no—
46#124×C127—21no—
47#124×C127—21no—
48#124×C127—21no—
49C2×C4⋊C4×C127—21no—
50C2×C4⋊C4×C127—21no—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127—21no—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127D411yescharacteristic
55C2×C4⋊C4×C127D411yescharacteristic
56C2×C4⋊C4×C127D412yes—
57C2×C22⋊C4×C127—21no—
58C2×C22⋊C4×C127—21no—
59C2×C22⋊C4×C127—22no—
60C2×C22⋊C4×C127—22no—
61C2×C42×C127C2311yescharacteristic
62#107×C127—24no—
63#106×C127—24no—
64#103×C127—24no—
65#103×C127—44no—
66#102×C127—24no—
67#97×C127—21no—
68#97×C127—21no—
69#97×C127—21no—
70#97×C127—21no—
71#97×C127D412yes—
72#97×C127D412yes—
73#95×C127—22no—
74#95×C127—21no—
75#95×C127—21no—
76C24×C127—21no—
77C2×Q8×C127—21no—
78C2×Q8×C127—22no—
79C2×Q8×C127—21no—
80C2×Q8×C127—42no—
81C2×Q8×C127—42no—
82C2×Q8×C127—42no—
83C2×Q8×C127—41no—
84C2×D4×C127—21no—
85C2×D4×C127—21no—
86C2×D4×C127—42no—
87C22×C4×C127—22no—
88C22×C4×C127—21no—
89C22×C4×C127—21no—
90C22×C4×C127—21no—
91C22×C4×C127—41no—
92C22×C4×C127C2×D411yescharacteristic
93C22×C4×C127C2×D411yescharacteristic
94C22×C4×C127C2×D411yescharacteristic
95C2×C8×C127—24no—
96C2×C8×C127—22no—
97C2×C8×C127—22no—
98C2×C8×C127—22no—
99C2×C8×C127—41no—
100C2×C8×C127—42no—
101C2×C8×C127—41no—
102C4⋊C4×C127—22no—
103C4⋊C4×C127—21no—
104C4⋊C4×C127—22no—
105C4⋊C4×C127—21no—
106C4⋊C4×C127—21no—
107C4⋊C4×C127—21no—
108C4⋊C4×C127—42no—
109C4⋊C4×C127—41no—
110C4⋊C4×C127—41no—
111C4⋊C4×C127—41no—
112C4⋊C4×C127—41no—
113C4⋊C4×C127—41no—
114C4⋊C4×C127—41no—
115C4⋊C4×C127—82no—
116C22⋊C4×C127—41no—
117C22⋊C4×C127—41no—
118C22⋊C4×C127—41no—
119C22⋊C4×C127—41no—
120C22⋊C4×C127—82no—
121C22⋊C4×C127—82no—
122C42×C127C2×D411yescharacteristic
123C42×C127C2×D411yescharacteristic
124C42×C127C2×D411yescharacteristic
125C42×C127#4111yescharacteristic
126C23×C127—41no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—42no—
130C23×C127—42no—
131C23×C127#12011yescharacteristic
132Q8×C127—42no—
133Q8×C127—44no—
134Q8×C127—81no—
135Q8×C127—82no—
136D4×C127—42no—
137D4×C127—81no—
138C2×C4×C127—21no—
139C2×C4×C127—21no—
140C2×C4×C127—22no—
141C2×C4×C127—22no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—41no—
145C2×C4×C127—42no—
146C2×C4×C127—41no—
147C2×C4×C127—42no—
148C2×C4×C127—41no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—42no—
152C2×C4×C127—42no—
153C2×C4×C127—81no—
154C2×C4×C127—81no—
155C2×C4×C127#12011yescharacteristic
156C2×C4×C127#12011yescharacteristic
157C2×C4×C127#12111yescharacteristic
158C2×C4×C127#12111yescharacteristic
159C2×C4×C127#12111yescharacteristic
160C2×C4×C127#12711yescharacteristic
161C8×C127—24no—
162C8×C127—42no—
163C8×C127—42no—
164C22×C127—82no—
165C22×C127—82no—
166C22×C127—81no—
167C22×C127—81no—
168C22×C127—81no—
169C22×C127#39211yescharacteristic
170C22×C127#45311yescharacteristic
171C22×C127#45411yescharacteristic
172C22×C127#45512yes—
173C22×C127#45612yes—
174C4×C127—21no—
175C4×C127—21no—
176C4×C127—21no—
177C4×C127—21no—
178C4×C127—21no—
179C4×C127—21no—
180C4×C127—82no—
181C4×C127—81no—
182C4×C127—81no—
183C4×C127—81no—
184C4×C127—82no—
185#10086C12711yesmaximal, characteristic, direct product factor, Sylow
186C2×C127—161no—
187C2×C127#166112yes—
188C2×C127#166412yes—
189C2×C127#200411yescharacteristic
190C2×C127#200511yescharacteristic
191C2×C127#200611yescharacteristic
192C2×#501C2×C12711yescharacteristic, complemented
193C2×#489C2×C12711yescharacteristic
194#2820C2×C12711yescharacteristic
195#1960C2×C12711yescharacteristic, complemented
196C2×#330C2×C12711yescharacteristic, complemented
197C2×#329C2×C12711yescharacteristic
198#1295C2×C12711yescharacteristic, complemented
199C127#1008611yescharacteristic, direct product factor, Sylow
200C2×C4⋊Q8C22×C12711yescharacteristic
201C2×#125C22×C12711yescharacteristic
202C2×#124C22×C12711yescharacteristic
203#501—22no—
204#501—22no—
205#489—22no—
206#489—22no—
207C2×#107—21no—
208C2×#106—21no—
209C2×#103—21no—
210C2×#103—22no—
211C2×#102—21no—
212C2×#97C22×C12711yescharacteristic
213C2×#97C22×C12711yescharacteristic
214C2×#97C22×C12711yescharacteristic
215#399—21no—
216#397—21no—
217#396—22no—
218#395—21no—
219#394—21no—
220#382C22×C12711yescharacteristic
221#330—22no—
222#329—22no—
223C22×Q8—21no—
224C22×Q8D4×C12711yescharacteristic
225C22×D4D4×C12711yescharacteristic
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C4⋊Q8—21no—
230C4⋊Q8—21no—
231C4⋊Q8—21no—
232C4⋊Q8D4×C12712yes—
233#125—21no—
234#125—21no—
235#125—21no—
236#125—21no—
237#124—21no—
238#124—21no—
239#124—21no—
240#124—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4D4×C12711yescharacteristic
247C2×C4⋊C4D4×C12711yescharacteristic
248C2×C4⋊C4D4×C12712yes—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42C23×C12711yescharacteristic
254#107—24no—
255#106—24no—
256#103—24no—
257#103—44no—
258#102—24no—
259#97—21no—
260#97—21no—
261#97—21no—
262#97—21no—
263#97D4×C12712yes—
264#97D4×C12712yes—
265#95—22no—
266#95—21no—
267#95—21no—
268C24—21no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—21no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—41no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—42no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4C2×D4×C12711yescharacteristic
285C22×C4C2×D4×C12711yescharacteristic
286C22×C4C2×D4×C12711yescharacteristic
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—82no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C42C2×D4×C12711yescharacteristic
315C42C2×D4×C12711yescharacteristic
316C42C2×D4×C12711yescharacteristic
317C42#41×C12711yescharacteristic
318C23—41no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23#120×C12711yescharacteristic
324Q8—42no—
325Q8—44no—
326Q8—81no—
327Q8—82no—
328D4—42no—
329D4—81no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4#120×C12711yescharacteristic
348C2×C4#120×C12711yescharacteristic
349C2×C4#121×C12711yescharacteristic
350C2×C4#121×C12711yescharacteristic
351C2×C4#121×C12711yescharacteristic
352C2×C4#127×C12711yescharacteristic
353C8—24no—
354C8—42no—
355C8—42no—
356C22—82no—
357C22—82no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#392×C12711yescharacteristic
362C22#453×C12711yescharacteristic
363C22#454×C12711yescharacteristic
364C22#455×C12712yes—
365C22#456×C12712yes—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—82no—
377C2—161no—
378C2C127×#166112yes—
379C2C127×#166412yes—
380C2C127×#200411yescharacteristic
381C2C127×#200511yescharacteristic
382C2C127×#200611yescharacteristic
subgroupG/H
center, socleC23×C127#120
commutator, Frattini, cosocleC2×C42C23×C127
prime radicalC22×D4×C127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#501×C127C2semidirect
4. C127×#1960C2semidirect
5. C2×#330×C127C2semidirect
7. C127×#1295C2semidirect
185. #10086C127direct (N × H)
192. C2×#501C2×C127semidirect
195. #1960C2×C127semidirect
196. C2×#330C2×C127semidirect
198. #1295C2×C127semidirect
199. C127#10086direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#501×C1272
C2×#489×C1272
C127×#28202
C127×#19602
C2×#330×C1272
C2×#329×C1272
C127×#12952
#10086127