Finite Group Databasebeta

← C127×#12116 #44575789

Subgroups — C127×#12116 #44575789

Order 32512.

426 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3197C211yesmaximal, characteristic, complemented
2C127×#2191C211yesmaximal, characteristic, complemented
3C127×#2190C211yesmaximal, characteristic, complemented
4C127×#2010C211yesmaximal, characteristic, complemented
5C127×#1340C211yesmaximal, characteristic, complemented
6C127×#1339C211yesmaximal, characteristic, complemented
7C127×#1320C211yesmaximal, characteristic, complemented
8#575×C127C2211yescharacteristic
9#525×C127—21no—
10#525×C127—21no—
11#510×C127—21no—
12#509×C127—21no—
13#485×C127—21no—
14#482×C127—21no—
15#470×C127C2211yescharacteristic
16#469×C127C2211yescharacteristic
17#468×C127—21no—
18#452×C127—21no—
19#449×C127—21no—
20#421×C127—21no—
21#407×C127C2211yescharacteristic
22#360×C127—21no—
23#360×C127—21no—
24#349×C127C2211yescharacteristic
25#349×C127C2211yescharacteristic
26H(3,ℤ/4)×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C2×#41×C127—21no—
29C23×C4×C127D411yescharacteristic
30#137×C127—41no—
31AGL(1,ℤ/8)×C127—41no—
32#135×C127—21no—
33#135×C127—21no—
34C2×Q16×C127D411yescharacteristic, complemented
35C2×SD16×C127—21no—
36C2×D8×C127D411yescharacteristic, complemented
37C2×M16×C127D411yescharacteristic
38C2×M16×C127C2311yescharacteristic
39#123×C127—41no—
40#123×C127—41no—
41C22⋊Q8×C127—21no—
42C22⋊Q8×C127—21no—
43C22⋊Q8×C127—41no—
44#121×C127—21no—
45#121×C127—21no—
46#121×C127—41no—
47#120×C127—41no—
48#120×C127—41no—
49C4×D4×C127—42no—
50C4×D4×C127—41no—
51C4×D4×C127—41no—
52#117×C127—21no—
53#117×C127—21no—
54Q32×C127—41no—
55SD32×C127—41no—
56SD32×C127—41no—
57D16×C127—41no—
58M32×C127—21no—
59M32×C127—21no—
60#107×C127—21no—
61#106×C127D411yescharacteristic, complemented
62#106×C127D411yescharacteristic, complemented
63#104×C127—42no—
64#103×C127—21no—
65#103×C127—21no—
66#102×C127—21no—
67#102×C127—21no—
68C22⋊C8×C127—21no—
69C22⋊C8×C127—21no—
70C24×C127D811yescharacteristic
71#41×C127—41no—
72#41×C127—41no—
73#41×C127—41no—
74#41×C127—41no—
75#41×C127—81no—
76#41×C127—81no—
77C2×Q8×C127—21no—
78C2×Q8×C127—21no—
79C2×D4×C127—21no—
80C2×D4×C127—21no—
81C2×D4×C127—41no—
82C2×D4×C127—41no—
83C2×D4×C127—42no—
84C2×D4×C127—41no—
85C2×D4×C127—41no—
86C22×C4×C127—21no—
87C22×C4×C127—21no—
88C22×C4×C127—41no—
89C22×C4×C127—41no—
90C22×C4×C127—41no—
91C22×C4×C127—41no—
92C22×C4×C127D811yescharacteristic
93C22×C4×C127C2×D411yescharacteristic
94Q16×C127—21no—
95Q16×C127—21no—
96Q16×C127—41no—
97SD16×C127—41no—
98SD16×C127—41no—
99SD16×C127—81no—
100D8×C127—21no—
101D8×C127—21no—
102D8×C127—41no—
103M16×C127—41no—
104M16×C127—41no—
105C2×C8×C127—21no—
106C2×C8×C127C2×D411yescharacteristic
107C2×C8×C127C2×D411yescharacteristic
108C4⋊C4×C127—21no—
109C4⋊C4×C127—21no—
110C4⋊C4×C127—41no—
111C4⋊C4×C127—41no—
112C4⋊C4×C127—42no—
113C4⋊C4×C127—41no—
114C22⋊C4×C127—41no—
115C22⋊C4×C127—41no—
116C22⋊C4×C127—41no—
117C22⋊C4×C127—42no—
118C22⋊C4×C127—42no—
119C22⋊C4×C127—41no—
120C22⋊C4×C127—41no—
121C22⋊C4×C127—41no—
122C42×C127—22no—
123C42×C127—41no—
124C16×C127—41no—
125C16×C127—41no—
126C23×C127—21no—
127C23×C127—41no—
128C23×C127—41no—
129C23×C127—41no—
130C23×C127—81no—
131C23×C127C2×D811yescharacteristic
132Q8×C127—41no—
133Q8×C127—41no—
134Q8×C127—41no—
135Q8×C127—41no—
136D4×C127—41no—
137D4×C127—41no—
138D4×C127—41no—
139D4×C127—41no—
140D4×C127—81no—
141D4×C127—82no—
142D4×C127—82no—
143D4×C127—84no—
144D4×C127—82no—
145D4×C127—81no—
146C2×C4×C127—21no—
147C2×C4×C127—21no—
148C2×C4×C127—21no—
149C2×C4×C127—41no—
150C2×C4×C127—41no—
151C2×C4×C127—41no—
152C2×C4×C127—42no—
153C2×C4×C127—41no—
154C2×C4×C127—41no—
155C2×C4×C127—41no—
156C2×C4×C127—41no—
157C2×C4×C127—81no—
158C2×C4×C127—81no—
159C2×C4×C127—82no—
160C2×C4×C127—81no—
161C2×C4×C127#12011yescharacteristic
162C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
163C8×C127—21no—
164C8×C127—21no—
165C8×C127—41no—
166C22×C127—21no—
167C22×C127—41no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—42no—
171C22×C127—41no—
172C22×C127—81no—
173C22×C127—81no—
174C22×C127—81no—
175C22×C127—82no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127#44711yescharacteristic
179C4×C127—21no—
180C4×C127—41no—
181C4×C127—81no—
182C4×C127—81no—
183C4×C127—82no—
184C4×C127—81no—
185C4×C127—81no—
186C4×C127#45311yescharacteristic
187C4×C127B(4,𝔽2)11yescharacteristic
188#12116C12711yesmaximal, characteristic, direct product factor, Sylow
189C2×C127—21no—
190C2×C127—41no—
191C2×C127—81no—
192C2×C127—161no—
193C2×C127—161no—
194C2×C127#159511yescharacteristic
195#3197C2×C12711yescharacteristic, complemented
196#2191C2×C12711yescharacteristic, complemented
197#2190C2×C12711yescharacteristic, complemented
198#2010C2×C12711yescharacteristic, complemented
199#1340C2×C12711yescharacteristic, complemented
200#1339C2×C12711yescharacteristic, complemented
201#1320C2×C12711yescharacteristic, complemented
202C127#1211611yescharacteristic, direct product factor, Sylow
203#575C22×C12711yescharacteristic
204#525—21no—
205#525—21no—
206#510—21no—
207#509—21no—
208#485—21no—
209#482—21no—
210#470C22×C12711yescharacteristic
211#469C22×C12711yescharacteristic
212#468—21no—
213#452—21no—
214#449—21no—
215#421—21no—
216#407C22×C12711yescharacteristic
217#360—21no—
218#360—21no—
219#349C22×C12711yescharacteristic
220#349C22×C12711yescharacteristic
221H(3,ℤ/4)C22×C12711yescharacteristic
222C2×#41—21no—
223C2×#41—21no—
224C23×C4D4×C12711yescharacteristic
225#137—41no—
226AGL(1,ℤ/8)—41no—
227#135—21no—
228#135—21no—
229C2×Q16D4×C12711yescharacteristic, complemented
230C2×SD16—21no—
231C2×D8D4×C12711yescharacteristic, complemented
232C2×M16D4×C12711yescharacteristic
233C2×M16C23×C12711yescharacteristic
234#123—41no—
235#123—41no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—41no—
239#121—21no—
240#121—21no—
241#121—41no—
242#120—41no—
243#120—41no—
244C4×D4—42no—
245C4×D4—41no—
246C4×D4—41no—
247#117—21no—
248#117—21no—
249Q32—41no—
250SD32—41no—
251SD32—41no—
252D16—41no—
253M32—21no—
254M32—21no—
255#107—21no—
256#106D4×C12711yescharacteristic, complemented
257#106D4×C12711yescharacteristic, complemented
258#104—42no—
259#103—21no—
260#103—21no—
261#102—21no—
262#102—21no—
263C22⋊C8—21no—
264C22⋊C8—21no—
265C24D8×C12711yescharacteristic
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—81no—
271#41—81no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4D8×C12711yescharacteristic
288C22×C4C2×D4×C12711yescharacteristic
289Q16—21no—
290Q16—21no—
291Q16—41no—
292SD16—41no—
293SD16—41no—
294SD16—81no—
295D8—21no—
296D8—21no—
297D8—41no—
298M16—41no—
299M16—41no—
300C2×C8—21no—
301C2×C8C2×D4×C12711yescharacteristic
302C2×C8C2×D4×C12711yescharacteristic
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—41no—
319C16—41no—
320C16—41no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—81no—
326C23C2×D8×C12711yescharacteristic
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—41no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—81no—
336D4—82no—
337D4—82no—
338D4—82no—
339D4—84no—
340D4—81no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4#120×C12711yescharacteristic
357C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
358C8—21no—
359C8—21no—
360C8—41no—
361C22—21no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—42no—
366C22—41no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—82no—
371C22—81no—
372C22—81no—
373C22#447×C12711yescharacteristic
374C4—21no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—82no—
379C4—81no—
380C4—81no—
381C4#453×C12711yescharacteristic
382C4B(4,𝔽2)×C12711yescharacteristic
383C2—21no—
384C2—41no—
385C2—81no—
386C2—161no—
387C2—161no—
388C2C127×#159511yescharacteristic
subgroupG/H
center, socleC2×C127#1595
commutator, Frattini, cosocleC2×M16C23×C127
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3197C2semidirect
2. C127×#2191C2semidirect
3. C127×#2190C2semidirect
4. C127×#2010C2semidirect
5. C127×#1340C2semidirect
6. C127×#1339C2semidirect
7. C127×#1320C2semidirect
34. C2×Q16×C127D4semidirect
36. C2×D8×C127D4semidirect
61. #106×C127D4semidirect
62. #106×C127D4semidirect
188. #12116C127direct (N × H)
195. #3197C2×C127semidirect
196. #2191C2×C127semidirect
197. #2190C2×C127semidirect
198. #2010C2×C127semidirect
199. #1340C2×C127semidirect
200. #1339C2×C127semidirect
201. #1320C2×C127semidirect
202. C127#12116direct (N × H)
229. C2×Q16D4×C127semidirect
231. C2×D8D4×C127semidirect
256. #106D4×C127semidirect
257. #106D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31972
C127×#21912
C127×#21902
C127×#20102
C127×#13402
C127×#13392
C127×#13202
#12116127