Finite Group Databasebeta

← C127×#12324 #44575997

Subgroups — C127×#12324 #44575997

Order 32512.

406 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3273C211yesmaximal, characteristic, complemented
2C127×#2250C211yesmaximal, characteristic, complemented
3C127×#2206C211yesmaximal, characteristic, complemented
4C127×#1687C211yesmaximal, characteristic, complemented
5C127×#1360C211yesmaximal, characteristic, complemented
6C127×#1333C211yesmaximal, characteristic, complemented
7C127×#1330C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C127—21no—
9#546×C127—21no—
10C2×SD32×C127—21no—
11C2×D16×C127—21no—
12C2×D16×C127—21no—
13#495×C127C2211yescharacteristic
14#493×C127C2211yescharacteristic, complemented
15#468×C127—21no—
16#463×C127—21no—
17#459×C127—21no—
18#452×C127—21no—
19#447×C127—21no—
20C4×D8×C127C2211yescharacteristic
21#436×C127C2211yescharacteristic
22C4⋊C16×C127C2211yescharacteristic, complemented
23#357×C127—21no—
24#346×C127C2211yescharacteristic, complemented
25#334×C127C2211yescharacteristic, complemented
26#331×C127—21no—
27C2×#41×C127—21no—
28C22×D4×C127—21no—
29AGL(1,ℤ/8)×C127—42no—
30AGL(1,ℤ/8)×C127—42no—
31C2×Q16×C127D411yescharacteristic
32C2×SD16×C127—21no—
33C2×SD16×C127—21no—
34C2×D8×C127—21no—
35C2×D8×C127—21no—
36C2×D8×C127—21no—
37C2×D8×C127—41no—
38C2×D8×C127D411yescharacteristic
39C2×D8×C127D411yescharacteristic
40C2×M16×C127—21no—
41#127×C127—21no—
42#124×C127—21no—
43#123×C127—41no—
44C22⋊Q8×C127—21no—
45#121×C127—21no—
46#121×C127—41no—
47#120×C127—41no—
48C4×D4×C127—21no—
49C4×D4×C127D411yescharacteristic
50C2×C22⋊C4×C127—41no—
51SD32×C127—42no—
52D16×C127—42no—
53D16×C127—42no—
54C2×C16×C127—21no—
55C2×C16×C127—21no—
56#107×C127D411yescharacteristic
57C4⋊C8×C127—21no—
58C4⋊C8×C127D411yescharacteristic
59#103×C127—21no—
60#102×C127—21no—
61#102×C127—21no—
62#102×C127—41no—
63C22⋊C8×C127—21no—
64C4×C8×C127C2311yescharacteristic
65C24×C127—41no—
66#41×C127—42no—
67#41×C127—81no—
68C2×Q8×C127—21no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—41no—
72C2×D4×C127—42no—
73C2×D4×C127—41no—
74C2×D4×C127—41no—
75C2×D4×C127—41no—
76C2×D4×C127—41no—
77C2×D4×C127—41no—
78C2×D4×C127—81no—
79C2×D4×C127D811yescharacteristic
80C22×C4×C127—21no—
81C22×C4×C127—41no—
82C22×C4×C127—41no—
83Q16×C127—41no—
84SD16×C127—42no—
85SD16×C127—41no—
86SD16×C127—41no—
87D8×C127—21no—
88D8×C127—41no—
89D8×C127—41no—
90D8×C127—41no—
91D8×C127—42no—
92D8×C127—41no—
93D8×C127—41no—
94D8×C127—82no—
95D8×C127D812yescomplemented
96M16×C127—41no—
97M16×C127—41no—
98C2×C8×C127—21no—
99C2×C8×C127—41no—
100C2×C8×C127C2×D411yescharacteristic
101C2×C8×C127C2×D411yescharacteristic
102C4⋊C4×C127—21no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127D811yescharacteristic
105C22⋊C4×C127—21no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C22⋊C4×C127—41no—
109C22⋊C4×C127—41no—
110C22⋊C4×C127—41no—
111C42×C127C2×D411yescharacteristic
112C16×C127—22no—
113C16×C127—41no—
114C23×C127—21no—
115C23×C127—41no—
116C23×C127—41no—
117C23×C127—82no—
118C23×C127—82no—
119C23×C127—81no—
120C23×C127—161no—
121Q8×C127—41no—
122Q8×C127—41no—
123D4×C127—22no—
124D4×C127—41no—
125D4×C127—41no—
126D4×C127—41no—
127D4×C127—41no—
128D4×C127—81no—
129D4×C127—81no—
130D4×C127—82no—
131D4×C127—82no—
132D4×C127—81no—
133D4×C127—81no—
134D4×C127—161no—
135D4×C127—161no—
136C2×C4×C127—21no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—41no—
140C2×C4×C127—41no—
141C2×C4×C127—82no—
142C2×C4×C127—161no—
143C2×C4×C127#12011yescharacteristic
144C2×C4×C127C2×D811yescharacteristic
145C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
146C8×C127—21no—
147C8×C127—41no—
148C8×C127—81no—
149C8×C127C2×D811yescharacteristic
150C8×C127AGL(1,ℤ/8)11yescharacteristic
151C22×C127—41no—
152C22×C127—42no—
153C22×C127—41no—
154C22×C127—41no—
155C22×C127—81no—
156C22×C127—81no—
157C22×C127—81no—
158C22×C127—81no—
159C22×C127—81no—
160C22×C127—164no—
161C22×C127—161no—
162C22×C127—161no—
163C22×C127—161no—
164C22×C127#44711yescharacteristic
165C4×C127—21no—
166C4×C127—21no—
167C4×C127—41no—
168C4×C127—81no—
169C4×C127—81no—
170C4×C127#44711yescharacteristic
171C4×C127#45311yescharacteristic
172#12324C12711yesmaximal, characteristic, direct product factor, Sylow
173C2×C127—81no—
174C2×C127—82no—
175C2×C127—161no—
176C2×C127—321no—
177C2×C127#161911yescharacteristic
178C2×C127#219011yescharacteristic
179C2×C127#219211yescharacteristic
180#3273C2×C12711yescharacteristic, complemented
181#2250C2×C12711yescharacteristic, complemented
182#2206C2×C12711yescharacteristic, complemented
183#1687C2×C12711yescharacteristic, complemented
184#1360C2×C12711yescharacteristic, complemented
185#1333C2×C12711yescharacteristic, complemented
186#1330C2×C12711yescharacteristic, complemented
187C127#1232411yescharacteristic, direct product factor, Sylow
188C2×AGL(1,ℤ/8)—21no—
189#546—21no—
190C2×SD32—21no—
191C2×D16—21no—
192C2×D16—21no—
193#495C22×C12711yescharacteristic
194#493C22×C12711yescharacteristic, complemented
195#468—21no—
196#463—21no—
197#459—21no—
198#452—21no—
199#447—21no—
200C4×D8C22×C12711yescharacteristic
201#436C22×C12711yescharacteristic
202C4⋊C16C22×C12711yescharacteristic, complemented
203#357—21no—
204#346C22×C12711yescharacteristic, complemented
205#334C22×C12711yescharacteristic, complemented
206#331—21no—
207C2×#41—21no—
208C22×D4—21no—
209AGL(1,ℤ/8)—42no—
210AGL(1,ℤ/8)—42no—
211C2×Q16D4×C12711yescharacteristic
212C2×SD16—21no—
213C2×SD16—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—41no—
218C2×D8D4×C12711yescharacteristic
219C2×D8D4×C12711yescharacteristic
220C2×M16—21no—
221#127—21no—
222#124—21no—
223#123—41no—
224C22⋊Q8—21no—
225#121—21no—
226#121—41no—
227#120—41no—
228C4×D4—21no—
229C4×D4D4×C12711yescharacteristic
230C2×C22⋊C4—41no—
231SD32—42no—
232D16—42no—
233D16—42no—
234C2×C16—21no—
235C2×C16—21no—
236#107D4×C12711yescharacteristic
237C4⋊C8—21no—
238C4⋊C8D4×C12711yescharacteristic
239#103—21no—
240#102—21no—
241#102—21no—
242#102—41no—
243C22⋊C8—21no—
244C4×C8C23×C12711yescharacteristic
245C24—41no—
246#41—42no—
247#41—81no—
248C2×Q8—21no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—81no—
259C2×D4D8×C12711yescharacteristic
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—41no—
263Q16—41no—
264SD16—42no—
265SD16—41no—
266SD16—41no—
267D8—21no—
268D8—41no—
269D8—41no—
270D8—41no—
271D8—42no—
272D8—41no—
273D8—41no—
274D8—82no—
275D8D8×C12712yescomplemented
276M16—41no—
277M16—41no—
278C2×C8—21no—
279C2×C8—41no—
280C2×C8C2×D4×C12711yescharacteristic
281C2×C8C2×D4×C12711yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—41no—
284C4⋊C4D8×C12711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C42C2×D4×C12711yescharacteristic
292C16—22no—
293C16—41no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—82no—
298C23—82no—
299C23—81no—
300C23—161no—
301Q8—41no—
302Q8—41no—
303D4—22no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—81no—
309D4—81no—
310D4—82no—
311D4—82no—
312D4—81no—
313D4—81no—
314D4—161no—
315D4—161no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—161no—
323C2×C4#120×C12711yescharacteristic
324C2×C4C2×D8×C12711yescharacteristic
325C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
326C8—21no—
327C8—41no—
328C8—81no—
329C8C2×D8×C12711yescharacteristic
330C8AGL(1,ℤ/8)×C12711yescharacteristic
331C22—41no—
332C22—42no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—164no—
341C22—161no—
342C22—161no—
343C22—161no—
344C22#447×C12711yescharacteristic
345C4—21no—
346C4—21no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4#447×C12711yescharacteristic
351C4#453×C12711yescharacteristic
352C2—81no—
353C2—82no—
354C2—161no—
355C2—321no—
356C2C127×#161911yescharacteristic
357C2C127×#219011yescharacteristic
358C2C127×#219211yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ #436×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3273C2semidirect
2. C127×#2250C2semidirect
3. C127×#2206C2semidirect
4. C127×#1687C2semidirect
5. C127×#1360C2semidirect
6. C127×#1333C2semidirect
7. C127×#1330C2semidirect
14. #493×C127C22semidirect
22. C4⋊C16×C127C22semidirect
24. #346×C127C22semidirect
25. #334×C127C22semidirect
95. D8×C127D8semidirect
172. #12324C127direct (N × H)
180. #3273C2×C127semidirect
181. #2250C2×C127semidirect
182. #2206C2×C127semidirect
183. #1687C2×C127semidirect
184. #1360C2×C127semidirect
185. #1333C2×C127semidirect
186. #1330C2×C127semidirect
187. C127#12324direct (N × H)
194. #493C22×C127semidirect
202. C4⋊C16C22×C127semidirect
204. #346C22×C127semidirect
205. #334C22×C127semidirect
275. D8D8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32732
C127×#22502
C127×#22062
C127×#16872
C127×#13602
C127×#13332
C127×#13302
#12324127