Finite Group Databasebeta

← C127×#12104 #44575777

Subgroups — C127×#12104 #44575777

Order 32512.

480 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3191C211yesmaximal, characteristic, complemented
2C127×#2187C211yesmaximal, characteristic, complemented
3C127×#2184C211yesmaximal, characteristic, complemented
4C127×#1999C211yesmaximal, characteristic, complemented
5C127×#1348C211yesmaximal, characteristic, complemented
6C127×#1347C211yesmaximal, characteristic, complemented
7C127×#1314C211yesmaximal, characteristic, complemented
8C22×D8×C127C2211yescharacteristic
9#525×C127—21no—
10C2×#121×C127—21no—
11C2×SD32×C127—21no—
12C2×D16×C127—21no—
13#485×C127—21no—
14#482×C127—21no—
15#467×C127C2211yescharacteristic
16#466×C127C2211yescharacteristic
17#465×C127—21no—
18#452×C127—21no—
19#449×C127—21no—
20C2×#102×C127—21no—
21C2×C22⋊C8×C127C2211yescharacteristic
22#357×C127—21no—
23#357×C127—21no—
24C22⋊C16×C127C2211yescharacteristic
25C22⋊C16×C127C2211yescharacteristic
26#340×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C22×D4×C127—21no—
29C22×D4×C127—41no—
30C23×C4×C127D411yescharacteristic
31C2×Q16×C127D411yescharacteristic, complemented
32C2×SD16×C127—21no—
33C2×D8×C127—21no—
34C2×D8×C127—21no—
35C2×D8×C127—42no—
36C2×D8×C127D411yescharacteristic, complemented
37C22×C8×C127D411yescharacteristic
38C22×C8×C127C2311yescharacteristic
39#123×C127—41no—
40C22⋊Q8×C127—21no—
41C22⋊Q8×C127—21no—
42#121×C127—21no—
43#121×C127—21no—
44#121×C127—42no—
45#121×C127—42no—
46#120×C127—41no—
47C4×D4×C127—41no—
48C4×D4×C127—41no—
49#117×C127—21no—
50C2×C4⋊C4×C127—21no—
51C2×C22⋊C4×C127—41no—
52SD32×C127—42no—
53D16×C127—42no—
54C2×C16×C127—21no—
55C2×C16×C127—21no—
56#107×C127D411yescharacteristic, complemented
57#107×C127D411yescharacteristic, complemented
58#106×C127—21no—
59#103×C127—21no—
60#103×C127—21no—
61#102×C127—21no—
62#102×C127—21no—
63#102×C127—42no—
64C22⋊C8×C127—21no—
65C22⋊C8×C127—21no—
66C24×C127—41no—
67C24×C127D811yescharacteristic
68#41×C127—81no—
69#41×C127—81no—
70C2×Q8×C127—21no—
71C2×D4×C127—22no—
72C2×D4×C127—21no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127—41no—
77C2×D4×C127—41no—
78C2×D4×C127—44no—
79C2×D4×C127—41no—
80C2×D4×C127—41no—
81C2×D4×C127—41no—
82C2×D4×C127—82no—
83C2×D4×C127—82no—
84C22×C4×C127—21no—
85C22×C4×C127—21no—
86C22×C4×C127—41no—
87C22×C4×C127—41no—
88C22×C4×C127—41no—
89C22×C4×C127—41no—
90C22×C4×C127D811yescharacteristic
91C22×C4×C127C2×D411yescharacteristic
92Q16×C127—41no—
93SD16×C127—81no—
94D8×C127—22no—
95D8×C127—22no—
96D8×C127—41no—
97D8×C127—41no—
98D8×C127—41no—
99C2×C8×C127—21no—
100C2×C8×C127—41no—
101C2×C8×C127—41no—
102C2×C8×C127C2×D411yescharacteristic
103C2×C8×C127C2×D411yescharacteristic
104C4⋊C4×C127—22no—
105C4⋊C4×C127—21no—
106C4⋊C4×C127—21no—
107C4⋊C4×C127—41no—
108C4⋊C4×C127—41no—
109C4⋊C4×C127—41no—
110C22⋊C4×C127—41no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—44no—
114C22⋊C4×C127—41no—
115C22⋊C4×C127—41no—
116C42×C127—41no—
117C16×C127—41no—
118C16×C127—41no—
119C23×C127—21no—
120C23×C127—21no—
121C23×C127—21no—
122C23×C127—41no—
123C23×C127—41no—
124C23×C127—42no—
125C23×C127—41no—
126C23×C127—81no—
127C23×C127—81no—
128C23×C127—81no—
129C23×C127—81no—
130C23×C127—81no—
131C23×C127—81no—
132C23×C127C2×D811yescharacteristic
133Q8×C127—41no—
134Q8×C127—41no—
135D4×C127—42no—
136D4×C127—41no—
137D4×C127—42no—
138D4×C127—41no—
139D4×C127—81no—
140D4×C127—82no—
141D4×C127—88no—
142D4×C127—82no—
143D4×C127—81no—
144D4×C127—81no—
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—81no—
157C2×C4×C127—82no—
158C2×C4×C127—81no—
159C2×C4×C127—81no—
160C2×C4×C127#12011yescharacteristic
161C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
162C8×C127—21no—
163C8×C127—21no—
164C8×C127—41no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—42no—
168C22×C127—41no—
169C22×C127—41no—
170C22×C127—41no—
171C22×C127—81no—
172C22×C127—82no—
173C22×C127—82no—
174C22×C127—81no—
175C22×C127—82no—
176C22×C127—81no—
177C22×C127—81no—
178C22×C127—161no—
179C22×C127—161no—
180C22×C127—161no—
181C22×C127—161no—
182C22×C127#44711yescharacteristic
183C22×C127#50811yescharacteristic
184C22×C127#50911yescharacteristic
185C4×C127—21no—
186C4×C127—41no—
187C4×C127—82no—
188C4×C127—81no—
189C4×C127—81no—
190C4×C127—81no—
191C4×C127#45311yescharacteristic
192C4×C127B(4,𝔽2)11yescharacteristic
193#12104C12711yesmaximal, characteristic, direct product factor, Sylow
194C2×C127—21no—
195C2×C127—21no—
196C2×C127—81no—
197C2×C127—161no—
198C2×C127—162no—
199C2×C127#159511yescharacteristic
200C2×C127#218611yescharacteristic
201C2×C127#219011yescharacteristic
202#3191C2×C12711yescharacteristic, complemented
203#2187C2×C12711yescharacteristic, complemented
204#2184C2×C12711yescharacteristic, complemented
205#1999C2×C12711yescharacteristic, complemented
206#1348C2×C12711yescharacteristic, complemented
207#1347C2×C12711yescharacteristic, complemented
208#1314C2×C12711yescharacteristic, complemented
209C127#1210411yescharacteristic, direct product factor, Sylow
210C22×D8C22×C12711yescharacteristic
211#525—21no—
212C2×#121—21no—
213C2×SD32—21no—
214C2×D16—21no—
215#485—21no—
216#482—21no—
217#467C22×C12711yescharacteristic
218#466C22×C12711yescharacteristic
219#465—21no—
220#452—21no—
221#449—21no—
222C2×#102—21no—
223C2×C22⋊C8C22×C12711yescharacteristic
224#357—21no—
225#357—21no—
226C22⋊C16C22×C12711yescharacteristic
227C22⋊C16C22×C12711yescharacteristic
228#340C22×C12711yescharacteristic
229C2×#41—21no—
230C22×D4—21no—
231C22×D4—41no—
232C23×C4D4×C12711yescharacteristic
233C2×Q16D4×C12711yescharacteristic, complemented
234C2×SD16—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8—42no—
238C2×D8D4×C12711yescharacteristic, complemented
239C22×C8D4×C12711yescharacteristic
240C22×C8C23×C12711yescharacteristic
241#123—41no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244#121—21no—
245#121—21no—
246#121—42no—
247#121—42no—
248#120—41no—
249C4×D4—41no—
250C4×D4—41no—
251#117—21no—
252C2×C4⋊C4—21no—
253C2×C22⋊C4—41no—
254SD32—42no—
255D16—42no—
256C2×C16—21no—
257C2×C16—21no—
258#107D4×C12711yescharacteristic, complemented
259#107D4×C12711yescharacteristic, complemented
260#106—21no—
261#103—21no—
262#103—21no—
263#102—21no—
264#102—21no—
265#102—42no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C24—41no—
269C24D8×C12711yescharacteristic
270#41—81no—
271#41—81no—
272C2×Q8—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—44no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—82no—
285C2×D4—82no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4D8×C12711yescharacteristic
293C22×C4C2×D4×C12711yescharacteristic
294Q16—41no—
295SD16—81no—
296D8—22no—
297D8—22no—
298D8—41no—
299D8—41no—
300D8—41no—
301C2×C8—21no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8C2×D4×C12711yescharacteristic
305C2×C8C2×D4×C12711yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—44no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—41no—
319C16—41no—
320C16—41no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23C2×D8×C12711yescharacteristic
335Q8—41no—
336Q8—41no—
337D4—42no—
338D4—41no—
339D4—42no—
340D4—41no—
341D4—81no—
342D4—88no—
343D4—82no—
344D4—82no—
345D4—81no—
346D4—81no—
347D4—81no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4#120×C12711yescharacteristic
363C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
364C8—21no—
365C8—21no—
366C8—41no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—81no—
374C22—82no—
375C22—82no—
376C22—81no—
377C22—82no—
378C22—81no—
379C22—81no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22#447×C12711yescharacteristic
385C22#508×C12711yescharacteristic
386C22#509×C12711yescharacteristic
387C4—21no—
388C4—41no—
389C4—82no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4#453×C12711yescharacteristic
394C4B(4,𝔽2)×C12711yescharacteristic
395C2—21no—
396C2—21no—
397C2—81no—
398C2—162no—
399C2—161no—
400C2C127×#159511yescharacteristic
401C2C127×#218611yescharacteristic
402C2C127×#219011yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C22⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3191C2semidirect
2. C127×#2187C2semidirect
3. C127×#2184C2semidirect
4. C127×#1999C2semidirect
5. C127×#1348C2semidirect
6. C127×#1347C2semidirect
7. C127×#1314C2semidirect
31. C2×Q16×C127D4semidirect
36. C2×D8×C127D4semidirect
56. #107×C127D4semidirect
57. #107×C127D4semidirect
193. #12104C127direct (N × H)
202. #3191C2×C127semidirect
203. #2187C2×C127semidirect
204. #2184C2×C127semidirect
205. #1999C2×C127semidirect
206. #1348C2×C127semidirect
207. #1347C2×C127semidirect
208. #1314C2×C127semidirect
209. C127#12104direct (N × H)
233. C2×Q16D4×C127semidirect
238. C2×D8D4×C127semidirect
258. #107D4×C127semidirect
259. #107D4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31912
C127×#21872
C127×#21842
C127×#19992
C127×#13482
C127×#13472
C127×#13142
#12104127