Finite Group Databasebeta

← C127×#10458 #44574131

Subgroups — C127×#10458 #44574131

Order 32512.

428 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3037C211yesmaximal, characteristic
2C127×#1801C211yesmaximal, characteristic, complemented
3C127×#1567C211yesmaximal, characteristic, complemented
4C127×#1547C211yesmaximal, characteristic
5C127×#1492C211yesmaximal, characteristic
6C2×Q8⋊C8×C127C211yesmaximal, characteristic, complemented
7C127×#1292C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9#532×C127C2211yescharacteristic
10C2×C4×Q8×C127C2211yescharacteristic
11#464×C127—22no—
12#462×C127—21no—
13#460×C127—21no—
14#423×C127C2211yescharacteristic
15#423×C127C2211yescharacteristic
16#423×C127C2211yescharacteristic
17C2×C4⋊C8×C127C2211yescharacteristic
18#417×C127—21no—
19C2×#101×C127—21no—
20C2×C4×C8×C127C2211yescharacteristic
21#343×C127—21no—
22#335×C127—22no—
23#332×C127—21no—
24#331×C127—21no—
25Q8⋊C8×C127—21no—
26Q8⋊C8×C127—21no—
27D4⋊C8×C127—21no—
28D4⋊C8×C127—21no—
29C2×#41×C127—21no—
30C22×Q8×C127D411yescharacteristic
31#137×C127—42no—
32#137×C127—42no—
33C2×Q16×C127—22no—
34C2×SD16×C127—22no—
35C2×M16×C127—21no—
36C2×M16×C127—21no—
37C2×M16×C127—21no—
38C22×C8×C127—21no—
39C22×C8×C127—21no—
40C4⋊Q8×C127D411yescharacteristic
41#127×C127D411yescharacteristic
42#124×C127—21no—
43#121×C127—21no—
44#121×C127—21no—
45C4×Q8×C127—21no—
46C4×Q8×C127—21no—
47C4×Q8×C127—21no—
48C4×Q8×C127—21no—
49C4×D4×C127—21no—
50C4×D4×C127—21no—
51C2×C4⋊C4×C127—21no—
52C2×C4⋊C4×C127D411yescharacteristic
53C2×C42×C127—21no—
54C2×C42×C127C2311yescharacteristic
55C4⋊C8×C127—21no—
56C4⋊C8×C127—21no—
57C4⋊C8×C127—21no—
58C4⋊C8×C127—21no—
59C4⋊C8×C127—21no—
60C4⋊C8×C127—21no—
61C4⋊C8×C127—21no—
62C4⋊C8×C127D412yes—
63#103×C127—22no—
64#102×C127—22no—
65#101×C127—42no—
66C4×C8×C127—21no—
67C4×C8×C127D411yescharacteristic
68C4×C8×C127D411yescharacteristic
69#41×C127—41no—
70#41×C127—41no—
71#41×C127—41no—
72#41×C127—41no—
73C2×Q8×C127—21no—
74C2×Q8×C127—21no—
75C2×Q8×C127—21no—
76C2×Q8×C127—21no—
77C2×Q8×C127—21no—
78C2×Q8×C127—21no—
79C2×Q8×C127D812yes—
80C2×D4×C127—21no—
81C2×D4×C127—21no—
82C2×D4×C127—21no—
83C2×D4×C127—41no—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—41no—
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127#4111yescharacteristic
89C22×C4×C127#4111yescharacteristic
90Q16×C127—44no—
91SD16×C127—44no—
92M16×C127—41no—
93M16×C127—41no—
94M16×C127—41no—
95M16×C127—41no—
96M16×C127—41no—
97M16×C127—41no—
98C2×C8×C127—21no—
99C2×C8×C127—22no—
100C2×C8×C127—21no—
101C2×C8×C127—42no—
102C2×C8×C127—41no—
103C2×C8×C127—41no—
104C2×C8×C127—41no—
105C2×C8×C127—81no—
106C4⋊C4×C127—21no—
107C4⋊C4×C127—21no—
108C4⋊C4×C127—21no—
109C4⋊C4×C127—21no—
110C4⋊C4×C127—41no—
111C4⋊C4×C127—41no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C42×C127—41no—
115C42×C127—41no—
116C42×C127C2×D411yescharacteristic
117C42×C127C2×D411yescharacteristic
118C42×C127C2×D411yescharacteristic
119C42×C127#4111yescharacteristic
120C23×C127—41no—
121C23×C127#12311yescharacteristic
122Q8×C127—22no—
123Q8×C127—22no—
124Q8×C127—41no—
125Q8×C127—41no—
126Q8×C127—41no—
127Q8×C127—41no—
128D4×C127—41no—
129D4×C127—41no—
130D4×C127—41no—
131D4×C127—41no—
132D4×C127—81no—
133D4×C127—81no—
134D4×C127—81no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—22no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—81no—
144C2×C4×C127—81no—
145C2×C4×C127—81no—
146C2×C4×C127#12011yescharacteristic
147C2×C4×C127#12111yescharacteristic
148C2×C4×C127#12111yescharacteristic
149C2×C4×C127#12111yescharacteristic
150C2×C4×C127#12311yescharacteristic
151C2×C4×C127#12411yescharacteristic
152C2×C4×C127C2×D811yescharacteristic
153C2×C4×C127#13511yescharacteristic
154C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C127#13711yescharacteristic
156C8×C127—22no—
157C8×C127—41no—
158C8×C127—42no—
159C8×C127—81no—
160C8×C127—81no—
161C8×C127—81no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—81no—
165C22×C127—81no—
166C22×C127—81no—
167C22×C127#39411yescharacteristic
168C22×C127#48011yescharacteristic
169C22×C127#48211yescharacteristic
170C4×C127—21no—
171C4×C127—21no—
172C4×C127—21no—
173C4×C127—42no—
174C4×C127—81no—
175C4×C127—81no—
176C4×C127#44711yescharacteristic
177C4×C127#45211yescharacteristic
178C4×C127#46611yescharacteristic
179C4×C127#46811yescharacteristic
180C4×C127#47111yescharacteristic
181C4×C127#47311yescharacteristic
182#10458C12711yesmaximal, characteristic, direct product factor, Sylow
183C2×C127—21no—
184C2×C127—21no—
185C2×C127—161no—
186C2×C127#203311yescharacteristic
187C2×C127#205011yescharacteristic
188C2×C127#205211yescharacteristic
189#3037C2×C12711yescharacteristic
190#1801C2×C12711yescharacteristic, complemented
191#1567C2×C12711yescharacteristic, complemented
192#1547C2×C12711yescharacteristic
193#1492C2×C12711yescharacteristic
194C2×Q8⋊C8C2×C12711yescharacteristic, complemented
195#1292C2×C12711yescharacteristic, complemented
196C127#1045811yescharacteristic, direct product factor, Sylow
197C2×#137—21no—
198#532C22×C12711yescharacteristic
199C2×C4×Q8C22×C12711yescharacteristic
200#464—22no—
201#462—21no—
202#460—21no—
203#423C22×C12711yescharacteristic
204#423C22×C12711yescharacteristic
205#423C22×C12711yescharacteristic
206C2×C4⋊C8C22×C12711yescharacteristic
207#417—21no—
208C2×#101—21no—
209C2×C4×C8C22×C12711yescharacteristic
210#343—21no—
211#335—22no—
212#332—21no—
213#331—21no—
214Q8⋊C8—21no—
215Q8⋊C8—21no—
216D4⋊C8—21no—
217D4⋊C8—21no—
218C2×#41—21no—
219C22×Q8D4×C12711yescharacteristic
220#137—42no—
221#137—42no—
222C2×Q16—22no—
223C2×SD16—22no—
224C2×M16—21no—
225C2×M16—21no—
226C2×M16—21no—
227C22×C8—21no—
228C22×C8—21no—
229C4⋊Q8D4×C12711yescharacteristic
230#127D4×C12711yescharacteristic
231#124—21no—
232#121—21no—
233#121—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×D4—21no—
239C4×D4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4D4×C12711yescharacteristic
242C2×C42—21no—
243C2×C42C23×C12711yescharacteristic
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8D4×C12712yes—
252#103—22no—
253#102—22no—
254#101—42no—
255C4×C8—21no—
256C4×C8D4×C12711yescharacteristic
257C4×C8D4×C12711yescharacteristic
258#41—41no—
259#41—41no—
260#41—41no—
261#41—41no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8D8×C12712yes—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—41no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4C2×D4×C12711yescharacteristic
277C22×C4#41×C12711yescharacteristic
278C22×C4#41×C12711yescharacteristic
279Q16—44no—
280SD16—44no—
281M16—41no—
282M16—41no—
283M16—41no—
284M16—41no—
285M16—41no—
286M16—41no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—21no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—81no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—41no—
303C42—41no—
304C42—41no—
305C42C2×D4×C12711yescharacteristic
306C42C2×D4×C12711yescharacteristic
307C42C2×D4×C12711yescharacteristic
308C42#41×C12711yescharacteristic
309C23—41no—
310C23#123×C12711yescharacteristic
311Q8—22no—
312Q8—22no—
313Q8—41no—
314Q8—41no—
315Q8—41no—
316Q8—41no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—41no—
321D4—81no—
322D4—81no—
323D4—81no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4#120×C12711yescharacteristic
336C2×C4#121×C12711yescharacteristic
337C2×C4#121×C12711yescharacteristic
338C2×C4#121×C12711yescharacteristic
339C2×C4#123×C12711yescharacteristic
340C2×C4#124×C12711yescharacteristic
341C2×C4C2×D8×C12711yescharacteristic
342C2×C4#135×C12711yescharacteristic
343C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
344C2×C4#137×C12711yescharacteristic
345C8—22no—
346C8—41no—
347C8—42no—
348C8—81no—
349C8—81no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#394×C12711yescharacteristic
357C22#480×C12711yescharacteristic
358C22#482×C12711yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—42no—
363C4—81no—
364C4—81no—
365C4#447×C12711yescharacteristic
366C4#452×C12711yescharacteristic
367C4#466×C12711yescharacteristic
368C4#468×C12711yescharacteristic
369C4#471×C12711yescharacteristic
370C4#473×C12711yescharacteristic
371C2—21no—
372C2—21no—
373C2—161no—
374C2C127×#203311yescharacteristic
375C2C127×#205011yescharacteristic
376C2C127×#205211yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC2×C42C23×C127
prime radical#532×C127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1801C2semidirect
3. C127×#1567C2semidirect
6. C2×Q8⋊C8×C127C2semidirect
7. C127×#1292C2semidirect
182. #10458C127direct (N × H)
190. #1801C2×C127semidirect
191. #1567C2×C127semidirect
194. C2×Q8⋊C8C2×C127semidirect
195. #1292C2×C127semidirect
196. C127#10458direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#30372
C127×#18012
C127×#15672
C127×#15472
C127×#14922
C2×Q8⋊C8×C1272
C127×#12922
#10458127