Finite Group Databasebeta

← C127×#12856 #44576529

Subgroups — C127×#12856 #44576529

Order 32512.

914 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C2×#390×C127C211yesmaximal, characteristic
2C127×#1781C212yesmaximal
3C127×#1393C214yesmaximal, complemented
4C23×D4×C127C2211yescharacteristic
5C22×C4⋊C4×C127C411yescharacteristic, complemented
6C22×C22⋊C4×C127—22no—
7C22×C42×C127C411yescharacteristic, complemented
8C2×#99×C127—22no—
9C2×#99×C127C2214yes—
10#390×C127—22no—
11#390×C127C412yescomplemented
12#390×C127C2212yes—
13#379×C127—22no—
14C25×C127D412yes—
15C22×D4×C127—24no—
16C22×D4×C127—42no—
17C22×D4×C127—41no—
18C22×D4×C127—41no—
19C22×D4×C127C2×C411yescharacteristic
20C22×D4×C127D412yes—
21C22×D4×C127C2311yescharacteristic
22C23×C4×C127—21no—
23C23×C4×C127—21no—
24C23×C4×C127C2×C411yescharacteristic
25C2×C4⋊C4×C127—21no—
26C2×C4⋊C4×C127—41no—
27C2×C4⋊C4×C127—41no—
28C2×C4⋊C4×C127C2×C412yes—
29C2×C22⋊C4×C127—24no—
30C2×C22⋊C4×C127—24no—
31C2×C22⋊C4×C127—24no—
32C2×C22⋊C4×C127—48no—
33C2×C22⋊C4×C127—42no—
34C2×C22⋊C4×C127—42no—
35C2×C42×C127—21no—
36C2×C42×C127—42no—
37C2×C42×C127C2×C412yes—
38#99×C127—28no—
39#99×C127—44no—
40C24×C127—24no—
41C24×C127—22no—
42C24×C127—22no—
43C24×C127—42no—
44C24×C127—44no—
45C24×C127—44no—
46C24×C127C22⋊C412yes—
47C24×C127C22⋊C411yescharacteristic
48C24×C127C2×D412yes—
49C2×D4×C127—24no—
50C2×D4×C127—24no—
51C2×D4×C127—22no—
52C2×D4×C127—41no—
53C2×D4×C127—41no—
54C2×D4×C127—42no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—48no—
58C2×D4×C127—48no—
59C2×D4×C127#4114yes—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127—24no—
65C22×C4×C127—22no—
66C22×C4×C127—22no—
67C22×C4×C127—42no—
68C22×C4×C127—42no—
69C22×C4×C127—44no—
70C22×C4×C127—41no—
71C22×C4×C127—41no—
72C22×C4×C127—44no—
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127C22×C411yescharacteristic
75C4⋊C4×C127—42no—
76C4⋊C4×C127—42no—
77C22⋊C4×C127—48no—
78C22⋊C4×C127—48no—
79C22⋊C4×C127—44no—
80C22⋊C4×C127—44no—
81C22⋊C4×C127—84no—
82C22⋊C4×C127—84no—
83C42×C127—42no—
84C42×C127—42no—
85C23×C127—22no—
86C23×C127—22no—
87C23×C127—21no—
88C23×C127—21no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—22no—
94C23×C127—44no—
95C23×C127—44no—
96C23×C127—42no—
97C23×C127—42no—
98C23×C127—42no—
99C23×C127—42no—
100C23×C127—416no—
101C23×C127—44no—
102C23×C127—44no—
103C23×C127—82no—
104C23×C127—82no—
105C23×C127—88no—
106C23×C127#9911yescharacteristic
107C23×C127#9911yescharacteristic
108C23×C127C2×C22⋊C411yescharacteristic
109C23×C127#12314yes—
110D4×C127—44no—
111D4×C127—48no—
112D4×C127—44no—
113C2×C4×C127—24no—
114C2×C4×C127—24no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—24no—
120C2×C4×C127—48no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—44no—
124C2×C4×C127—42no—
125C2×C4×C127—82no—
126C2×C4×C127—82no—
127C2×C4×C127—84no—
128C2×C4×C127—84no—
129C2×C4×C127—84no—
130C2×C4×C127#9912yes—
131C2×C4×C127#11712yes—
132C22×C127—24no—
133C22×C127—24no—
134C22×C127—21no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127—21no—
138C22×C127—21no—
139C22×C127—21no—
140C22×C127—24no—
141C22×C127—44no—
142C22×C127—44no—
143C22×C127—42no—
144C22×C127—42no—
145C22×C127—42no—
146C22×C127—42no—
147C22×C127—42no—
148C22×C127—82no—
149C22×C127—88no—
150C22×C127—88no—
151C22×C127—82no—
152C22×C127#35312yes—
153C22×C127#35412yes—
154C22×C127#38111yescharacteristic
155C22×C127C2×#9911yescharacteristic
156C22×C127C2×#9911yescharacteristic
157C4×C127—22no—
158C4×C127—22no—
159C4×C127—44no—
160C4×C127—82no—
161C4×C127—164no—
162#12856C12711yesmaximal, characteristic, direct product factor, Sylow
163C2×C127—21no—
164C2×C127—21no—
165C2×C127—22no—
166C2×C127—22no—
167C2×C127—44no—
168C2×C127—82no—
169C2×C127#181911yescharacteristic
170C2×C127C2×#35311yescharacteristic
171C2×C127C2×#35411yescharacteristic
172C2×#390C2×C12711yescharacteristic
173#1781C2×C12712yes—
174#1393C2×C12714yescomplemented
175C127#1285611yescharacteristic, direct product factor, Sylow
176C23×D4C22×C12711yescharacteristic
177C22×C4⋊C4C4×C12711yescharacteristic, complemented
178C22×C22⋊C4—22no—
179C22×C42C4×C12711yescharacteristic, complemented
180C2×#99—22no—
181C2×#99C22×C12714yes—
182#390—22no—
183#390C4×C12712yescomplemented
184#390C22×C12712yes—
185#379—22no—
186C25D4×C12712yes—
187C22×D4—24no—
188C22×D4—42no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4C2×C4×C12711yescharacteristic
192C22×D4D4×C12712yes—
193C22×D4C23×C12711yescharacteristic
194C23×C4—21no—
195C23×C4—21no—
196C23×C4C2×C4×C12711yescharacteristic
197C2×C4⋊C4—21no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4C2×C4×C12712yes—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—48no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42C2×C4×C12712yes—
210#99—28no—
211#99—44no—
212C24—24no—
213C24—22no—
214C24—22no—
215C24—42no—
216C24—44no—
217C24—44no—
218C24C22⋊C4×C12711yescharacteristic
219C24C22⋊C4×C12712yes—
220C24C2×D4×C12712yes—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—42no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—48no—
230C2×D4—48no—
231C2×D4#41×C12714yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—44no—
245C22×C4C22⋊C4×C12711yescharacteristic
246C22×C4C22×C4×C12711yescharacteristic
247C4⋊C4—42no—
248C4⋊C4—42no—
249C22⋊C4—48no—
250C22⋊C4—48no—
251C22⋊C4—44no—
252C22⋊C4—44no—
253C22⋊C4—84no—
254C22⋊C4—84no—
255C42—42no—
256C42—42no—
257C23—22no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—44no—
267C23—44no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—416no—
273C23—44no—
274C23—44no—
275C23—82no—
276C23—82no—
277C23—88no—
278C23#99×C12711yescharacteristic
279C23#99×C12711yescharacteristic
280C23C2×C22⋊C4×C12711yescharacteristic
281C23#123×C12714yes—
282D4—44no—
283D4—48no—
284D4—44no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—48no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4#99×C12712yes—
303C2×C4#117×C12712yes—
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—24no—
313C22—44no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—42no—
319C22—42no—
320C22—82no—
321C22—88no—
322C22—88no—
323C22—82no—
324C22#353×C12712yes—
325C22#354×C12712yes—
326C22#381×C12711yescharacteristic
327C22C2×#99×C12711yescharacteristic
328C22C2×#99×C12711yescharacteristic
329C4—22no—
330C4—22no—
331C4—44no—
332C4—82no—
333C4—164no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—22no—
338C2—44no—
339C2—82no—
340C2C127×#181911yescharacteristic
341C2C2×#353×C12711yescharacteristic
342C2C2×#354×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC22×D4C23×C127
prime radicalC23×D4×C127C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C24×C127 ◁ C23×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1393C2semidirect
5. C22×C4⋊C4×C127C4semidirect
7. C22×C42×C127C4semidirect
11. #390×C127C4semidirect
162. #12856C127direct (N × H)
174. #1393C2×C127semidirect
175. C127#12856direct (N × H)
177. C22×C4⋊C4C4×C127semidirect
179. C22×C42C4×C127semidirect
183. #390C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#390×C1272
C127×#17812
C127×#13932
#12856127