Finite Group Databasebeta

← C127×#9782 #44573455

Subgroups — C127×#9782 #44573455

Order 32512.

522 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467×C127C211yesmaximal, characteristic
2C127×#1996C212yesmaximal
3C127×#1808C211yesmaximal, characteristic
4C8×C22⋊C4×C127C211yesmaximal, characteristic
5C127×#1520C212yesmaximal
6C22×Q16×C127C411yescharacteristic, complemented
7C23×C8×C127C2211yescharacteristic
8C2×C22⋊Q8×C127C2212yes—
9#467×C127—21no—
10#467×C127—21no—
11#467×C127—21no—
12#467×C127—21no—
13C2×#107×C127—21no—
14C2×#107×C127C411yescharacteristic, complemented
15C2×#103×C127C412yescomplemented
16C2×C22⋊C8×C127C2211yescharacteristic
17C2×C4×C8×C127—21no—
18#380×C127C2212yes—
19C4×C22⋊C4×C127C2211yescharacteristic
20#342×C127—24no—
21#340×C127—21no—
22#336×C127—21no—
23#328×C127—24no—
24C22×Q8×C127C2×C412yes—
25C23×C4×C127C2311yescharacteristic
26C2×Q16×C127—21no—
27C2×Q16×C127—21no—
28C2×Q16×C127—42no—
29C22×C8×C127—21no—
30C22×C8×C127—42no—
31C22×C8×C127C2×C411yescharacteristic
32C22×C8×C127C2×C411yescharacteristic
33C22×C8×C127D411yescharacteristic
34C22×C8×C127D411yescharacteristic
35C22×C8×C127D411yescharacteristic
36C22×C8×C127D411yescharacteristic
37C22⋊Q8×C127—22no—
38C22⋊Q8×C127—22no—
39C22⋊Q8×C127—22no—
40C22⋊Q8×C127—22no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—22no—
43C2×C4⋊C4×C127C2×C412yes—
44C2×C22⋊C4×C127—22no—
45C2×C22⋊C4×C127—22no—
46C2×C22⋊C4×C127D412yes—
47C2×C42×C127—21no—
48#107×C127—21no—
49#107×C127—21no—
50#107×C127—42no—
51#103×C127—22no—
52#103×C127—22no—
53C22⋊C8×C127—21no—
54C22⋊C8×C127D412yes—
55C4×C8×C127—42no—
56#95×C127—21no—
57#95×C127—22no—
58C24×C127C2×D411yescharacteristic
59C2×Q8×C127—22no—
60C2×Q8×C127—22no—
61C2×Q8×C127—42no—
62C2×Q8×C127—42no—
63C22×C4×C127—22no—
64C22×C4×C127—22no—
65C22×C4×C127—22no—
66C22×C4×C127—41no—
67C22×C4×C127—41no—
68C22×C4×C127—42no—
69C22×C4×C127C22×C411yescharacteristic
70C22×C4×C127C2×D411yescharacteristic
71C22×C4×C127C2×D411yescharacteristic
72C22×C4×C127C2×D411yescharacteristic
73C22×C4×C127#4111yescharacteristic
74C22×C4×C127#4111yescharacteristic
75Q16×C127—42no—
76Q16×C127—42no—
77C2×C8×C127—22no—
78C2×C8×C127—22no—
79C2×C8×C127—22no—
80C2×C8×C127—22no—
81C2×C8×C127—41no—
82C2×C8×C127—41no—
83C2×C8×C127—41no—
84C2×C8×C127—41no—
85C2×C8×C127—41no—
86C2×C8×C127C22⋊C411yescharacteristic
87C2×C8×C127C22⋊C411yescharacteristic
88C2×C8×C127C22⋊C411yescharacteristic
89C2×C8×C127C22⋊C411yescharacteristic
90C4⋊C4×C127—22no—
91C4⋊C4×C127—22no—
92C4⋊C4×C127—42no—
93C4⋊C4×C127—42no—
94C4⋊C4×C127—42no—
95C4⋊C4×C127—42no—
96C22⋊C4×C127—22no—
97C22⋊C4×C127—42no—
98C22⋊C4×C127—42no—
99C22⋊C4×C127—44no—
100C22⋊C4×C127Q1614yescomplemented
101C42×C127—41no—
102C42×C127—41no—
103C23×C127—41no—
104C23×C127—41no—
105C23×C127C4×D411yescharacteristic
106C23×C127#12111yescharacteristic
107C23×C127#12711yescharacteristic
108C23×C127C2×Q1612yes—
109C23×C127#13512yes—
110Q8×C127—42no—
111Q8×C127—42no—
112Q8×C127—82no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—21no—
118C2×C4×C127—21no—
119C2×C4×C127—24no—
120C2×C4×C127—41no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—42no—
124C2×C4×C127—42no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127—82no—
128C2×C4×C127—82no—
129C2×C4×C127—82no—
130C2×C4×C127—82no—
131C2×C4×C127—82no—
132C2×C4×C127C2×C22⋊C411yescharacteristic
133C2×C4×C127C4×D411yescharacteristic
134C2×C4×C127#12111yescharacteristic
135C2×C4×C127#12411yescharacteristic
136C8×C127—21no—
137C8×C127—21no—
138C8×C127—21no—
139C8×C127—42no—
140C8×C127#9912yes—
141C22×C127—21no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—21no—
147C22×C127—42no—
148C22×C127—42no—
149C22×C127#39011yescharacteristic
150C22×C127C4×Q1612yes—
151C22×C127#46712yes—
152C22×C127#49411yescharacteristic
153C22×C127#49511yescharacteristic
154C4×C127—21no—
155C4×C127—21no—
156C4×C127—21no—
157C4×C127—44no—
158C4×C127—82no—
159C4×C127—82no—
160C4×C127—162no—
161C4×C127C2×#9911yescharacteristic
162C4×C127#41011yescharacteristic
163#9782C12711yesmaximal, characteristic, direct product factor, Sylow
164C2×C127—22no—
165C2×C127—21no—
166C2×C127—22no—
167C2×C127—21no—
168C2×C127#196411yescharacteristic
169C2×C127#196911yescharacteristic
170C2×C127#197611yescharacteristic
171C2×#467C2×C12711yescharacteristic
172#1996C2×C12712yes—
173#1808C2×C12711yescharacteristic
174C8×C22⋊C4C2×C12711yescharacteristic
175#1520C2×C12712yes—
176C127#978211yescharacteristic, direct product factor, Sylow
177C22×Q16C4×C12711yescharacteristic, complemented
178C23×C8C22×C12711yescharacteristic
179C2×C22⋊Q8C22×C12712yes—
180#467—21no—
181#467—21no—
182#467—21no—
183#467—21no—
184C2×#107—21no—
185C2×#107C4×C12711yescharacteristic, complemented
186C2×#103C4×C12712yescomplemented
187C2×C22⋊C8C22×C12711yescharacteristic
188C2×C4×C8—21no—
189#380C22×C12712yes—
190C4×C22⋊C4C22×C12711yescharacteristic
191#342—24no—
192#340—21no—
193#336—21no—
194#328—24no—
195C22×Q8C2×C4×C12712yes—
196C23×C4C23×C12711yescharacteristic
197C2×Q16—21no—
198C2×Q16—21no—
199C2×Q16—42no—
200C22×C8—21no—
201C22×C8—42no—
202C22×C8C2×C4×C12711yescharacteristic
203C22×C8C2×C4×C12711yescharacteristic
204C22×C8D4×C12711yescharacteristic
205C22×C8D4×C12711yescharacteristic
206C22×C8D4×C12711yescharacteristic
207C22×C8D4×C12711yescharacteristic
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4C2×C4×C12712yes—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4D4×C12712yes—
218C2×C42—21no—
219#107—21no—
220#107—21no—
221#107—42no—
222#103—22no—
223#103—22no—
224C22⋊C8—21no—
225C22⋊C8D4×C12712yes—
226C4×C8—42no—
227#95—21no—
228#95—22no—
229C24C2×D4×C12711yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—42no—
233C2×Q8—42no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4C22×C4×C12711yescharacteristic
241C22×C4C2×D4×C12711yescharacteristic
242C22×C4C2×D4×C12711yescharacteristic
243C22×C4C2×D4×C12711yescharacteristic
244C22×C4#41×C12711yescharacteristic
245C22×C4#41×C12711yescharacteristic
246Q16—42no—
247Q16—42no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C22⋊C4×C12711yescharacteristic
258C2×C8C22⋊C4×C12711yescharacteristic
259C2×C8C22⋊C4×C12711yescharacteristic
260C2×C8C22⋊C4×C12711yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—44no—
271C22⋊C4Q16×C12714yescomplemented
272C42—41no—
273C42—41no—
274C23—41no—
275C23—41no—
276C23C4×D4×C12711yescharacteristic
277C23#121×C12711yescharacteristic
278C23#127×C12711yescharacteristic
279C23C2×Q16×C12712yes—
280C23#135×C12712yes—
281Q8—42no—
282Q8—42no—
283Q8—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4C2×C22⋊C4×C12711yescharacteristic
304C2×C4C4×D4×C12711yescharacteristic
305C2×C4#121×C12711yescharacteristic
306C2×C4#124×C12711yescharacteristic
307C8—21no—
308C8—21no—
309C8—21no—
310C8—42no—
311C8#99×C12712yes—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22#390×C12711yescharacteristic
321C22C4×Q16×C12712yes—
322C22#467×C12712yes—
323C22#494×C12711yescharacteristic
324C22#495×C12711yescharacteristic
325C4—21no—
326C4—21no—
327C4—21no—
328C4—44no—
329C4—82no—
330C4—82no—
331C4—162no—
332C4C2×#99×C12711yescharacteristic
333C4#410×C12711yescharacteristic
334C2—22no—
335C2—21no—
336C2—22no—
337C2—21no—
338C2C127×#196411yescharacteristic
339C2C127×#196911yescharacteristic
340C2C127×#197611yescharacteristic
subgroupG/H
center, socleC22×C127#390
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
6. C22×Q16×C127C4semidirect
14. C2×#107×C127C4semidirect
15. C2×#103×C127C4semidirect
100. C22⋊C4×C127Q16semidirect
163. #9782C127direct (N × H)
176. C127#9782direct (N × H)
177. C22×Q16C4×C127semidirect
185. C2×#107C4×C127semidirect
186. C2×#103C4×C127semidirect
271. C22⋊C4Q16×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#467×C1272
C127×#19962
C127×#18082
C8×C22⋊C4×C1272
C127×#15202
#9782127