Finite Group Databasebeta

← C127×#9444 #44573117

Subgroups — C127×#9444 #44573117

Order 32512.

474 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2921C211yesmaximal, characteristic
2C127×#2499C211yesmaximal, characteristic
3C127×#1820C211yesmaximal, characteristic
4C127×#1511C212yesmaximal
5C127×#1284C212yesmaximal
6C2×C22⋊Q8×C127C2211yescharacteristic
7C2×#117×C127C2211yescharacteristic
8#433×C127—24no—
9#423×C127—21no—
10#407×C127C2212yes—
11C4×M16×C127—21no—
12#397×C127—21no—
13#397×C127—21no—
14#391×C127C411yescharacteristic, complemented
15#391×C127C411yescharacteristic, complemented
16#387×C127—21no—
17#387×C127—21no—
18#384×C127C411yescharacteristic, complemented
19#382×C127C411yescharacteristic, complemented
20#381×C127C2212yes—
21C4×C22⋊C4×C127C2211yescharacteristic
22#342×C127—21no—
23#342×C127—21no—
24#328×C127—24no—
25C22×Q8×C127C2×C411yescharacteristic
26C23×C4×C127C2311yescharacteristic
27C2×M16×C127—22no—
28C22⋊Q8×C127—22no—
29C22⋊Q8×C127—22no—
30#117×C127—22no—
31#117×C127—22no—
32C2×C4⋊C4×C127—21no—
33C2×C4⋊C4×C127—21no—
34C2×C4⋊C4×C127—21no—
35C2×C4⋊C4×C127C2×C411yescharacteristic
36C2×C4⋊C4×C127C2×C411yescharacteristic
37C2×C4⋊C4×C127C2×C411yescharacteristic
38C2×C22⋊C4×C127—21no—
39C2×C22⋊C4×C127—21no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127D411yescharacteristic
42C2×C22⋊C4×C127D411yescharacteristic
43C2×C42×C127C2×C411yescharacteristic
44C2×C42×C127C2×C411yescharacteristic
45C2×C42×C127D411yescharacteristic
46C2×C42×C127D411yescharacteristic
47C4⋊C8×C127—24no—
48C22⋊C8×C127—22no—
49C22⋊C8×C127D414yes—
50#97×C127—22no—
51C4×C8×C127—22no—
52#95×C127—21no—
53#95×C127—21no—
54#95×C127—21no—
55#95×C127—21no—
56#95×C127—21no—
57#95×C127—21no—
58#95×C127—21no—
59#95×C127—22no—
60#95×C127—22no—
61C24×C127C2×D411yescharacteristic
62C2×Q8×C127—22no—
63C2×Q8×C127—41no—
64C2×Q8×C127—41no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—21no—
68C22×C4×C127—21no—
69C22×C4×C127—41no—
70C22×C4×C127—41no—
71C22×C4×C127—42no—
72C22×C4×C127C22⋊C411yescharacteristic
73C22×C4×C127C22⋊C411yescharacteristic
74C22×C4×C127C22⋊C411yescharacteristic
75C22×C4×C127C22⋊C411yescharacteristic
76C22×C4×C127C22×C411yescharacteristic
77C22×C4×C127C2×D411yescharacteristic
78C22×C4×C127C2×D412yes—
79C22×C4×C127#4112yes—
80M16×C127—42no—
81M16×C127—42no—
82C2×C8×C127—24no—
83C4⋊C4×C127—22no—
84C4⋊C4×C127—22no—
85C4⋊C4×C127—22no—
86C4⋊C4×C127—42no—
87C4⋊C4×C127—42no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C22⋊C4×C127—22no—
91C22⋊C4×C127—22no—
92C22⋊C4×C127—42no—
93C22⋊C4×C127—42no—
94C22⋊C4×C127—42no—
95C22⋊C4×C127—42no—
96C42×C127—21no—
97C42×C127—21no—
98C42×C127—21no—
99C42×C127—21no—
100C42×C127—41no—
101C42×C127—41no—
102C23×C127—21no—
103C23×C127—21no—
104C23×C127—41no—
105C23×C127—41no—
106C23×C127C2×C22⋊C411yescharacteristic
107C23×C127#12112yes—
108Q8×C127—82no—
109C2×C4×C127—21no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—22no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—42no—
119C2×C4×C127—42no—
120C2×C4×C127—41no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—41no—
128C2×C4×C127—41no—
129C2×C4×C127—41no—
130C2×C4×C127—41no—
131C2×C4×C127—41no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—82no—
136C2×C4×C127—82no—
137C2×C4×C127—82no—
138C2×C4×C127C4×D412yes—
139C2×C4×C127#12011yescharacteristic
140C2×C4×C127#12311yescharacteristic
141C8×C127—44no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—22no—
145C22×C127—22no—
146C22×C127—22no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127#38611yescharacteristic
150C22×C127#41011yescharacteristic
151C22×C127#41011yescharacteristic
152C4×C127—21no—
153C4×C127—22no—
154C4×C127—21no—
155C4×C127—42no—
156C4×C127—41no—
157C4×C127—41no—
158C4×C127—41no—
159C4×C127—41no—
160C4×C127—81no—
161C4×C127—81no—
162C4×C127—81no—
163C4×C127—162no—
164#9444C12711yesmaximal, characteristic, direct product factor, Sylow
165C2×C127—21no—
166C2×C127—21no—
167C2×C127—42no—
168C2×C127#188411yescharacteristic
169C2×C127#189911yescharacteristic
170C2×C127#190911yescharacteristic
171#2921C2×C12711yescharacteristic
172#2499C2×C12711yescharacteristic
173#1820C2×C12711yescharacteristic
174#1511C2×C12712yes—
175#1284C2×C12712yes—
176C127#944411yescharacteristic, direct product factor, Sylow
177C2×C22⋊Q8C22×C12711yescharacteristic
178C2×#117C22×C12711yescharacteristic
179#433—24no—
180#423—21no—
181#407C22×C12712yes—
182C4×M16—21no—
183#397—21no—
184#397—21no—
185#391C4×C12711yescharacteristic, complemented
186#391C4×C12711yescharacteristic, complemented
187#387—21no—
188#387—21no—
189#384C4×C12711yescharacteristic, complemented
190#382C4×C12711yescharacteristic, complemented
191#381C22×C12712yes—
192C4×C22⋊C4C22×C12711yescharacteristic
193#342—21no—
194#342—21no—
195#328—24no—
196C22×Q8C2×C4×C12711yescharacteristic
197C23×C4C23×C12711yescharacteristic
198C2×M16—22no—
199C22⋊Q8—22no—
200C22⋊Q8—22no—
201#117—22no—
202#117—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C2×C4×C12711yescharacteristic
207C2×C4⋊C4C2×C4×C12711yescharacteristic
208C2×C4⋊C4C2×C4×C12711yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4D4×C12711yescharacteristic
213C2×C22⋊C4D4×C12711yescharacteristic
214C2×C42C2×C4×C12711yescharacteristic
215C2×C42C2×C4×C12711yescharacteristic
216C2×C42D4×C12711yescharacteristic
217C2×C42D4×C12711yescharacteristic
218C4⋊C8—24no—
219C22⋊C8—22no—
220C22⋊C8D4×C12714yes—
221#97—22no—
222C4×C8—22no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—22no—
232C24C2×D4×C12711yescharacteristic
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4C22⋊C4×C12711yescharacteristic
244C22×C4C22⋊C4×C12711yescharacteristic
245C22×C4C22⋊C4×C12711yescharacteristic
246C22×C4C22⋊C4×C12711yescharacteristic
247C22×C4C22×C4×C12711yescharacteristic
248C22×C4C2×D4×C12711yescharacteristic
249C22×C4C2×D4×C12712yes—
250C22×C4#41×C12712yes—
251M16—42no—
252M16—42no—
253C2×C8—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—41no—
272C42—41no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊C4×C12711yescharacteristic
278C23#121×C12712yes—
279Q8—82no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4C4×D4×C12712yes—
310C2×C4#120×C12711yescharacteristic
311C2×C4#123×C12711yescharacteristic
312C8—44no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22#386×C12711yescharacteristic
321C22#410×C12711yescharacteristic
322C22#410×C12711yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—162no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C127×#188411yescharacteristic
339C2C127×#189911yescharacteristic
340C2C127×#190911yescharacteristic
subgroupG/H
center, socleC22×C127#386
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC24×C127C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
14. #391×C127C4semidirect
15. #391×C127C4semidirect
18. #384×C127C4semidirect
19. #382×C127C4semidirect
164. #9444C127direct (N × H)
176. C127#9444direct (N × H)
185. #391C4×C127semidirect
186. #391C4×C127semidirect
189. #384C4×C127semidirect
190. #382C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#29212
C127×#24992
C127×#18202
C127×#15112
C127×#12842
#9444127