Finite Group Databasebeta

← C127×#9976 #44573649

Subgroups — C127×#9976 #44573649

Order 32512.

716 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 102 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×#3007C212yesmaximal
3C127×#1780C211yesmaximal, characteristic
4C127×#1472C212yesmaximal, complemented
5C127×#1293C211yesmaximal, characteristic, complemented
6C22×#41×C127C2211yescharacteristic
7C2×#137×C127—22no—
8C2×C22⋊Q8×C127C2212yes—
9#484×C127—22no—
10#467×C127—22no—
11#452×C127—24no—
12#451×C127—22no—
13#451×C127—22no—
14#450×C127—22no—
15#417×C127—22no—
16#407×C127C2212yes—
17C2×C22⋊C8×C127—22no—
18C2×C22⋊C8×C127C2211yescharacteristic
19C2×C22⋊C8×C127C2211yescharacteristic
20#324×C127—24no—
21C2×#41×C127—22no—
22C2×#41×C127—22no—
23C2×#41×C127—44no—
24C22×Q8×C127D412yes—
25C22×Q8×C127D411yescharacteristic
26C22×D4×C127—21no—
27C22×D4×C127D411yescharacteristic
28C23×C4×C127—21no—
29C23×C4×C127C2311yescharacteristic
30#137×C127—48no—
31C2×Q16×C127—24no—
32C2×Q16×C127—22no—
33C2×SD16×C127—24no—
34C2×M16×C127—22no—
35C22×C8×C127—21no—
36C22×C8×C127D412yes—
37C22⋊Q8×C127—22no—
38C22⋊Q8×C127—22no—
39C22⋊Q8×C127—22no—
40C22⋊Q8×C127—22no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127D412yes—
43C2×C22⋊C4×C127—22no—
44#107×C127—22no—
45#103×C127—24no—
46#103×C127—24no—
47#102×C127—24no—
48C22⋊C8×C127—22no—
49C22⋊C8×C127—22no—
50C22⋊C8×C127—22no—
51C22⋊C8×C127—44no—
52C22⋊C8×C127D414yes—
53C24×C127—21no—
54C24×C127#4111yescharacteristic
55#41×C127—28no—
56#41×C127—44no—
57#41×C127—44no—
58#41×C127—44no—
59C2×Q8×C127—22no—
60C2×Q8×C127—22no—
61C2×Q8×C127—22no—
62C2×Q8×C127—42no—
63C2×Q8×C127—44no—
64C2×D4×C127—22no—
65C2×D4×C127—42no—
66C2×D4×C127—42no—
67C2×D4×C127—42no—
68C2×D4×C127—42no—
69C22×C4×C127—24no—
70C22×C4×C127—21no—
71C22×C4×C127—22no—
72C22×C4×C127—22no—
73C22×C4×C127—42no—
74C22×C4×C127—42no—
75C22×C4×C127—41no—
76C22×C4×C127—41no—
77C22×C4×C127C2×D412yes—
78C22×C4×C127C2×D411yescharacteristic
79C22×C4×C127C2×D411yescharacteristic
80C22×C4×C127C2×D412yes—
81Q16×C127—48no—
82Q16×C127—82no—
83SD16×C127—48no—
84M16×C127—44no—
85C2×C8×C127—22no—
86C2×C8×C127—24no—
87C2×C8×C127—42no—
88C2×C8×C127—42no—
89C2×C8×C127—41no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127—22no—
92C4⋊C4×C127—42no—
93C4⋊C4×C127—42no—
94C22⋊C4×C127—42no—
95C22⋊C4×C127—42no—
96C23×C127—22no—
97C23×C127—22no—
98C23×C127—41no—
99C23×C127—41no—
100C23×C127—41no—
101C23×C127—42no—
102C23×C127—41no—
103C23×C127—42no—
104C23×C127#12112yes—
105C23×C127#12111yescharacteristic
106Q8×C127—24no—
107Q8×C127—42no—
108Q8×C127—44no—
109Q8×C127—44no—
110D4×C127—24no—
111D4×C127—42no—
112D4×C127—48no—
113C2×C4×C127—22no—
114C2×C4×C127—24no—
115C2×C4×C127—22no—
116C2×C4×C127—41no—
117C2×C4×C127—42no—
118C2×C4×C127—44no—
119C2×C4×C127—41no—
120C2×C4×C127—41no—
121C2×C4×C127—44no—
122C2×C4×C127—44no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127—82no—
126C2×C4×C127#12012yes—
127C2×C4×C127#12011yescharacteristic
128C2×C4×C127#12711yescharacteristic
129C8×C127—44no—
130C8×C127—42no—
131C8×C127—81no—
132C22×C127—21no—
133C22×C127—21no—
134C22×C127—22no—
135C22×C127—22no—
136C22×C127—22no—
137C22×C127—44no—
138C22×C127—42no—
139C22×C127—82no—
140C22×C127—82no—
141C22×C127—81no—
142C22×C127—81no—
143C22×C127#39211yescharacteristic
144C22×C127#47312yes—
145C4×C127—22no—
146C4×C127—21no—
147C4×C127—42no—
148C4×C127—82no—
149C4×C127—82no—
150C4×C127B(4,𝔽2)12yes—
151#9976C12711yesmaximal, characteristic, direct product factor, Sylow
152C2×C127—22no—
153C2×C127—42no—
154C2×C127—82no—
155C2×C127#201711yescharacteristic
156C2×C127#202011yescharacteristic
157C2×C127#202111yescharacteristic
158#3189C2×C12711yescharacteristic, complemented
159#3007C2×C12712yes—
160#1780C2×C12711yescharacteristic
161#1472C2×C12712yescomplemented
162#1293C2×C12711yescharacteristic, complemented
163C127#997611yescharacteristic, direct product factor, Sylow
164C22×#41C22×C12711yescharacteristic
165C2×#137—22no—
166C2×C22⋊Q8C22×C12712yes—
167#484—22no—
168#467—22no—
169#452—24no—
170#451—22no—
171#451—22no—
172#450—22no—
173#417—22no—
174#407C22×C12712yes—
175C2×C22⋊C8—22no—
176C2×C22⋊C8C22×C12711yescharacteristic
177C2×C22⋊C8C22×C12711yescharacteristic
178#324—24no—
179C2×#41—22no—
180C2×#41—22no—
181C2×#41—44no—
182C22×Q8D4×C12712yes—
183C22×Q8D4×C12711yescharacteristic
184C22×D4—21no—
185C22×D4D4×C12711yescharacteristic
186C23×C4—21no—
187C23×C4C23×C12711yescharacteristic
188#137—48no—
189C2×Q16—24no—
190C2×Q16—22no—
191C2×SD16—24no—
192C2×M16—22no—
193C22×C8—21no—
194C22×C8D4×C12712yes—
195C22⋊Q8—22no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4D4×C12712yes—
201C2×C22⋊C4—22no—
202#107—22no—
203#103—24no—
204#103—24no—
205#102—24no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—44no—
210C22⋊C8D4×C12714yes—
211C24—21no—
212C24#41×C12711yescharacteristic
213#41—28no—
214#41—44no—
215#41—44no—
216#41—44no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—22no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C22×C4—24no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4C2×D4×C12712yes—
236C22×C4C2×D4×C12711yescharacteristic
237C22×C4C2×D4×C12711yescharacteristic
238C22×C4C2×D4×C12712yes—
239Q16—48no—
240Q16—82no—
241SD16—48no—
242M16—44no—
243C2×C8—22no—
244C2×C8—24no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—41no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C23—22no—
255C23—22no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—42no—
260C23—41no—
261C23—42no—
262C23#121×C12712yes—
263C23#121×C12711yescharacteristic
264Q8—24no—
265Q8—42no—
266Q8—44no—
267Q8—44no—
268D4—24no—
269D4—42no—
270D4—48no—
271C2×C4—22no—
272C2×C4—24no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4#120×C12712yes—
285C2×C4#120×C12711yescharacteristic
286C2×C4#127×C12711yescharacteristic
287C8—44no—
288C8—42no—
289C8—81no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—44no—
296C22—42no—
297C22—82no—
298C22—82no—
299C22—81no—
300C22—81no—
301C22#392×C12711yescharacteristic
302C22#473×C12712yes—
303C4—22no—
304C4—21no—
305C4—42no—
306C4—82no—
307C4—82no—
308C4B(4,𝔽2)×C12712yes—
309C2—22no—
310C2—42no—
311C2—82no—
312C2C127×#201711yescharacteristic
313C2C127×#202011yescharacteristic
314C2C127×#202111yescharacteristic
subgroupG/H
center, socleC22×C127#392
commutator, Frattini, cosocleC23×C4C23×C127
prime radicalC22×#41×C127C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3189C2semidirect
4. C127×#1472C2semidirect
5. C127×#1293C2semidirect
151. #9976C127direct (N × H)
158. #3189C2×C127semidirect
161. #1472C2×C127semidirect
162. #1293C2×C127semidirect
163. C127#9976direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31892
C127×#30072
C127×#17802
C127×#14722
C127×#12932
#9976127