Finite Group Databasebeta

← C127×#12103 #44575776

Subgroups — C127×#12103 #44575776

Order 32512.

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

normal subgroups only

This group has 476 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×#3189C211yesmaximal, characteristic, complemented
2C127×#2187C212yesmaximal, complemented
3C127×#1999C211yesmaximal, characteristic
4C127×#1348C212yesmaximal, complemented
5C127×#1314C211yesmaximal, characteristic, complemented
6C22×D8×C127C2211yescharacteristic
7C2×C22⋊Q8×C127—21no—
8C2×#121×C127—21no—
9C2×SD32×C127—22no—
10#484×C127—22no—
11#467×C127—21no—
12#467×C127C2212yes—
13#451×C127—22no—
14C2×#102×C127—21no—
15C2×C22⋊C8×C127C2211yescharacteristic
16#357×C127—22no—
17C22⋊C16×C127C2212yes—
18#340×C127C2211yescharacteristic
19C22×Q8×C127—21no—
20C22×D4×C127—21no—
21C22×D4×C127—41no—
22C23×C4×C127D411yescharacteristic
23C2×Q16×C127—21no—
24C2×Q16×C127D412yescomplemented
25C2×D8×C127—22no—
26C2×D8×C127—42no—
27C22×C8×C127D411yescharacteristic
28C22×C8×C127C2311yescharacteristic
29C22⋊Q8×C127—22no—
30C22⋊Q8×C127—22no—
31C22⋊Q8×C127—41no—
32C22⋊Q8×C127—41no—
33#121×C127—42no—
34#121×C127—42no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—41no—
38C2×C22⋊C4×C127—41no—
39C2×C22⋊C4×C127—41no—
40SD32×C127—44no—
41C2×C16×C127—22no—
42#107×C127—21no—
43#107×C127D412yescomplemented
44#103×C127—22no—
45#103×C127—22no—
46#102×C127—42no—
47C22⋊C8×C127—22no—
48C24×C127—41no—
49C24×C127D811yescharacteristic
50C2×Q8×C127—22no—
51C2×Q8×C127—41no—
52C2×Q8×C127—81no—
53C2×Q8×C127—81no—
54C2×D4×C127—22no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—41no—
58C2×D4×C127—44no—
59C2×D4×C127—41no—
60C2×D4×C127—41no—
61C2×D4×C127—82no—
62C2×D4×C127—82no—
63C22×C4×C127—21no—
64C22×C4×C127—21no—
65C22×C4×C127—41no—
66C22×C4×C127—41no—
67C22×C4×C127—41no—
68C22×C4×C127—41no—
69C22×C4×C127—41no—
70C22×C4×C127D811yescharacteristic
71C22×C4×C127C2×D411yescharacteristic
72Q16×C127—42no—
73Q16×C127—81no—
74D8×C127—24no—
75D8×C127—42no—
76C2×C8×C127—21no—
77C2×C8×C127—41no—
78C2×C8×C127—41no—
79C2×C8×C127C2×D412yes—
80C4⋊C4×C127—22no—
81C4⋊C4×C127—22no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—42no—
85C4⋊C4×C127—42no—
86C22⋊C4×C127—42no—
87C22⋊C4×C127—44no—
88C22⋊C4×C127—42no—
89C16×C127—42no—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—41no—
93C23×C127—41no—
94C23×C127—42no—
95C23×C127—81no—
96C23×C127—81no—
97C23×C127—81no—
98C23×C127—81no—
99C23×C127—81no—
100C23×C127—81no—
101C23×C127SD3212yescomplemented
102C23×C127C2×D811yescharacteristic
103Q8×C127—42no—
104Q8×C127—42no—
105Q8×C127—82no—
106D4×C127—42no—
107D4×C127—42no—
108D4×C127—81no—
109D4×C127—88no—
110D4×C127—81no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127—81no—
121C2×C4×C127—81no—
122C2×C4×C127—82no—
123C2×C4×C127—82no—
124C2×C4×C127—81no—
125C2×C4×C127#12011yescharacteristic
126C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
127C8×C127—22no—
128C8×C127—41no—
129C22×C127—21no—
130C22×C127—21no—
131C22×C127—22no—
132C22×C127—42no—
133C22×C127—41no—
134C22×C127—41no—
135C22×C127—81no—
136C22×C127—82no—
137C22×C127—82no—
138C22×C127—82no—
139C22×C127—161no—
140C22×C127—161no—
141C22×C127—161no—
142C22×C127—161no—
143C22×C127#44711yescharacteristic
144C22×C127C2×SD3211yescharacteristic
145C22×C127#51011yescharacteristic
146C4×C127—21no—
147C4×C127—41no—
148C4×C127—82no—
149C4×C127—82no—
150C4×C127—82no—
151C4×C127#45311yescharacteristic
152C4×C127B(4,𝔽2)11yescharacteristic
153#12103C12711yesmaximal, characteristic, direct product factor, Sylow
154C2×C127—21no—
155C2×C127—21no—
156C2×C127—42no—
157C2×C127—162no—
158C2×C127#159511yescharacteristic
159C2×C127#218511yescharacteristic
160C2×C127#219011yescharacteristic
161#3189C2×C12711yescharacteristic, complemented
162#2187C2×C12712yescomplemented
163#1999C2×C12711yescharacteristic
164#1348C2×C12712yescomplemented
165#1314C2×C12711yescharacteristic, complemented
166C127#1210311yescharacteristic, direct product factor, Sylow
167C22×D8C22×C12711yescharacteristic
168C2×C22⋊Q8—21no—
169C2×#121—21no—
170C2×SD32—22no—
171#484—22no—
172#467—21no—
173#467C22×C12712yes—
174#451—22no—
175C2×#102—21no—
176C2×C22⋊C8C22×C12711yescharacteristic
177#357—22no—
178C22⋊C16C22×C12712yes—
179#340C22×C12711yescharacteristic
180C22×Q8—21no—
181C22×D4—21no—
182C22×D4—41no—
183C23×C4D4×C12711yescharacteristic
184C2×Q16—21no—
185C2×Q16D4×C12712yescomplemented
186C2×D8—22no—
187C2×D8—42no—
188C22×C8D4×C12711yescharacteristic
189C22×C8C23×C12711yescharacteristic
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194#121—42no—
195#121—42no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201SD32—44no—
202C2×C16—22no—
203#107—21no—
204#107D4×C12712yescomplemented
205#103—22no—
206#103—22no—
207#102—42no—
208C22⋊C8—22no—
209C24—41no—
210C24D8×C12711yescharacteristic
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—81no—
214C2×Q8—81no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—44no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—82no—
223C2×D4—82no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4D8×C12711yescharacteristic
232C22×C4C2×D4×C12711yescharacteristic
233Q16—42no—
234Q16—81no—
235D8—24no—
236D8—42no—
237C2×C8—21no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8C2×D4×C12712yes—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—44no—
249C22⋊C4—42no—
250C16—42no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—42no—
256C23—81no—
257C23—81no—
258C23—81no—
259C23—81no—
260C23—81no—
261C23—81no—
262C23SD32×C12712yescomplemented
263C23C2×D8×C12711yescharacteristic
264Q8—42no—
265Q8—42no—
266Q8—82no—
267D4—42no—
268D4—42no—
269D4—81no—
270D4—88no—
271D4—81no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—81no—
286C2×C4#120×C12711yescharacteristic
287C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
288C8—22no—
289C8—41no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—81no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—161no—
301C22—161no—
302C22—161no—
303C22—161no—
304C22#447×C12711yescharacteristic
305C22C2×SD32×C12711yescharacteristic
306C22#510×C12711yescharacteristic
307C4—21no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4—82no—
312C4#453×C12711yescharacteristic
313C4B(4,𝔽2)×C12711yescharacteristic
314C2—21no—
315C2—21no—
316C2—42no—
317C2—162no—
318C2C127×#159511yescharacteristic
319C2C127×#218511yescharacteristic
320C2C127×#219011yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#1999C2
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×#3189C2semidirect
2. C127×#2187C2semidirect
4. C127×#1348C2semidirect
5. C127×#1314C2semidirect
24. C2×Q16×C127D4semidirect
43. #107×C127D4semidirect
101. C23×C127SD32semidirect
153. #12103C127direct (N × H)
161. #3189C2×C127semidirect
162. #2187C2×C127semidirect
164. #1348C2×C127semidirect
165. #1314C2×C127semidirect
166. C127#12103direct (N × H)
185. C2×Q16D4×C127semidirect
204. #107D4×C127semidirect
262. C23SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31892
C127×#21872
C127×#19992
C127×#13482
C127×#13142
#12103127