Finite Group Databasebeta

← C127×#12921 #44576594

Subgroups — C127×#12921 #44576594

Order 32512.

442 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#427×C127C211yesmaximal, characteristic
2C2×#367×C127C211yesmaximal, characteristic, complemented
3C2×#366×C127C211yesmaximal, characteristic, complemented
4C2×C22⋊C16×C127C211yesmaximal, characteristic
5C127×#1808C211yesmaximal, characteristic
6C127×#1380C211yesmaximal, characteristic, complemented
7C127×#1380C211yesmaximal, characteristic, complemented
8C23×C8×C127C2211yescharacteristic
9C2×#117×C127—21no—
10C22×C16×C127C2211yescharacteristic
11C22×C16×C127C2211yescharacteristic
12#427×C127—22no—
13#427×C127—22no—
14C2×#107×C127C2211yescharacteristic
15C2×#107×C127C2211yescharacteristic
16C2×#107×C127C2211yescharacteristic
17C2×#107×C127C2211yescharacteristic
18C2×#106×C127—21no—
19#380×C127—21no—
20#367×C127—22no—
21#366×C127—22no—
22C22⋊C16×C127C414yescomplemented
23#340×C127—21no—
24C23×C4×C127D411yescharacteristic
25C22×C8×C127—21no—
26C22×C8×C127—21no—
27C22×C8×C127—21no—
28C22×C8×C127—21no—
29C22×C8×C127C2×C412yes—
30C22×C8×C127D411yescharacteristic
31C22×C8×C127Q812yes—
32C22×C8×C127C2311yescharacteristic
33#117×C127—24no—
34#117×C127—42no—
35C2×C4⋊C4×C127—21no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C4⋊C4×C127—21no—
39C2×C22⋊C4×C127—41no—
40C2×C22⋊C4×C127—41no—
41C2×C42×C127—41no—
42C2×C16×C127—21no—
43C2×C16×C127—21no—
44C2×C16×C127—21no—
45C2×C16×C127—21no—
46C2×C16×C127C2×C412yes—
47C2×C16×C127C2×C412yes—
48#107×C127—21no—
49#107×C127—21no—
50#107×C127—21no—
51#107×C127—21no—
52#107×C127—21no—
53#107×C127—21no—
54#107×C127D412yes—
55#107×C127D412yes—
56#106×C127—24no—
57#95×C127—41no—
58C24×C127D811yescharacteristic
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—21no—
62C22×C4×C127—21no—
63C22×C4×C127—41no—
64C22×C4×C127—41no—
65C22×C4×C127—41no—
66C22×C4×C127—41no—
67C22×C4×C127C4⋊C412yes—
68C22×C4×C127D811yescharacteristic
69C22×C4×C127Q1612yes—
70C22×C4×C127C2×D411yescharacteristic
71C2×C8×C127—21no—
72C2×C8×C127—21no—
73C2×C8×C127—22no—
74C2×C8×C127—22no—
75C2×C8×C127—22no—
76C2×C8×C127—22no—
77C2×C8×C127C4⋊C412yes—
78C2×C8×C127C22×C411yescharacteristic
79C2×C8×C127C2×D411yescharacteristic
80C2×C8×C127C2×D411yescharacteristic
81C2×C8×C127C2×Q811yescharacteristic
82C2×C8×C127#4111yescharacteristic
83C2×C8×C127#4111yescharacteristic
84C4⋊C4×C127—22no—
85C4⋊C4×C127—22no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127—41no—
89C4⋊C4×C127—41no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—41no—
92C22⋊C4×C127—44no—
93C22⋊C4×C127—82no—
94C42×C127—44no—
95C16×C127—22no—
96C16×C127—22no—
97C23×C127—21no—
98C23×C127—21no—
99C23×C127—21no—
100C23×C127—21no—
101C23×C127—21no—
102C23×C127—21no—
103C23×C127#10712yes—
104C23×C127C2×D811yescharacteristic
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—42no—
112C2×C4×C127—42no—
113C2×C4×C127—81no—
114C2×C4×C127—81no—
115C2×C4×C127—81no—
116C2×C4×C127—81no—
117C2×C4×C127—81no—
118C2×C4×C127—81no—
119C2×C4×C127—81no—
120C2×C4×C127—81no—
121C2×C4×C127—81no—
122C2×C4×C127—81no—
123C2×C4×C127#10712yes—
124C2×C4×C127C2×C4⋊C411yescharacteristic
125C2×C4×C127#12011yescharacteristic
126C2×C4×C127#12311yescharacteristic
127C2×C4×C127C2×Q1611yescharacteristic
128C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C127#13711yescharacteristic
130C8×C127—22no—
131C8×C127C4×D411yescharacteristic
132C8×C127C4×D411yescharacteristic
133C8×C127C22⋊Q811yescharacteristic
134C8×C127C22⋊Q811yescharacteristic
135C22×C127—22no—
136C22×C127—22no—
137C22×C127—22no—
138C22×C127—41no—
139C22×C127—41no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127C2×#10711yescharacteristic
143C22×C127#44711yescharacteristic
144C22×C127#48011yescharacteristic
145C22×C127#50811yescharacteristic
146C22×C127#50811yescharacteristic
147C22×C127#50911yescharacteristic
148C22×C127#51011yescharacteristic
149C4×C127—22no—
150C4×C127—82no—
151C4×C127—82no—
152C4×C127—161no—
153C4×C127—161no—
154C4×C127#38511yescharacteristic
155C4×C127#42811yescharacteristic
156C4×C127#45111yescharacteristic
157C4×C127#48411yescharacteristic
158#12921C12711yesmaximal, characteristic, direct product factor, Sylow
159C2×C127—42no—
160C2×C127#185211yescharacteristic
161C2×C127#215811yescharacteristic
162C2×C127#215911yescharacteristic
163C2×C127#218611yescharacteristic
164C2×C127#218911yescharacteristic
165C2×C127#223411yescharacteristic
166C2×C127#223711yescharacteristic
167C2×#427C2×C12711yescharacteristic
168C2×#367C2×C12711yescharacteristic, complemented
169C2×#366C2×C12711yescharacteristic, complemented
170C2×C22⋊C16C2×C12711yescharacteristic
171#1808C2×C12711yescharacteristic
172#1380C2×C12711yescharacteristic, complemented
173#1380C2×C12711yescharacteristic, complemented
174C127#1292111yescharacteristic, direct product factor, Sylow
175C23×C8C22×C12711yescharacteristic
176C2×#117—21no—
177C22×C16C22×C12711yescharacteristic
178C22×C16C22×C12711yescharacteristic
179#427—22no—
180#427—22no—
181C2×#107C22×C12711yescharacteristic
182C2×#107C22×C12711yescharacteristic
183C2×#107C22×C12711yescharacteristic
184C2×#107C22×C12711yescharacteristic
185C2×#106—21no—
186#380—21no—
187#367—22no—
188#366—22no—
189C22⋊C16C4×C12714yescomplemented
190#340—21no—
191C23×C4D4×C12711yescharacteristic
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×C4×C12712yes—
197C22×C8D4×C12711yescharacteristic
198C22×C8Q8×C12712yes—
199C22×C8C23×C12711yescharacteristic
200#117—24no—
201#117—42no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C42—41no—
209C2×C16—21no—
210C2×C16—21no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16C2×C4×C12712yes—
214C2×C16C2×C4×C12712yes—
215#107—21no—
216#107—21no—
217#107—21no—
218#107—21no—
219#107—21no—
220#107—21no—
221#107D4×C12712yes—
222#107D4×C12712yes—
223#106—24no—
224#95—41no—
225C24D8×C12711yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4C4⋊C4×C12712yes—
235C22×C4D8×C12711yescharacteristic
236C22×C4Q16×C12712yes—
237C22×C4C2×D4×C12711yescharacteristic
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8C4⋊C4×C12712yes—
245C2×C8C22×C4×C12711yescharacteristic
246C2×C8C2×D4×C12711yescharacteristic
247C2×C8C2×D4×C12711yescharacteristic
248C2×C8C2×Q8×C12711yescharacteristic
249C2×C8#41×C12711yescharacteristic
250C2×C8#41×C12711yescharacteristic
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C22⋊C4—44no—
260C22⋊C4—82no—
261C42—44no—
262C16—22no—
263C16—22no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23#107×C12712yes—
271C23C2×D8×C12711yescharacteristic
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4#107×C12712yes—
291C2×C4C2×C4⋊C4×C12711yescharacteristic
292C2×C4#120×C12711yescharacteristic
293C2×C4#123×C12711yescharacteristic
294C2×C4C2×Q16×C12711yescharacteristic
295C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
296C2×C4#137×C12711yescharacteristic
297C8—22no—
298C8C4×D4×C12711yescharacteristic
299C8C4×D4×C12711yescharacteristic
300C8C22⋊Q8×C12711yescharacteristic
301C8C22⋊Q8×C12711yescharacteristic
302C22—22no—
303C22—22no—
304C22—22no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22C2×#107×C12711yescharacteristic
310C22#447×C12711yescharacteristic
311C22#480×C12711yescharacteristic
312C22#508×C12711yescharacteristic
313C22#508×C12711yescharacteristic
314C22#509×C12711yescharacteristic
315C22#510×C12711yescharacteristic
316C4—22no—
317C4—82no—
318C4—82no—
319C4—161no—
320C4—161no—
321C4#385×C12711yescharacteristic
322C4#428×C12711yescharacteristic
323C4#451×C12711yescharacteristic
324C4#484×C12711yescharacteristic
325C2—42no—
326C2C127×#185211yescharacteristic
327C2C127×#215811yescharacteristic
328C2C127×#215911yescharacteristic
329C2C127×#218611yescharacteristic
330C2C127×#218911yescharacteristic
331C2C127×#223411yescharacteristic
332C2C127×#223711yescharacteristic
subgroupG/H
center, socleC23×C127C2×D8
commutatorC2×C8C22×C4×C127
Frattini, cosocleC22×C8C23×C127
prime radicalC24×C127D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#367×C127C2semidirect
3. C2×#366×C127C2semidirect
6. C127×#1380C2semidirect
7. C127×#1380C2semidirect
22. C22⋊C16×C127C4semidirect
158. #12921C127direct (N × H)
168. C2×#367C2×C127semidirect
169. C2×#366C2×C127semidirect
172. #1380C2×C127semidirect
173. #1380C2×C127semidirect
174. C127#12921direct (N × H)
189. C22⋊C16C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#427×C1272
C2×#367×C1272
C2×#366×C1272
C2×C22⋊C16×C1272
C127×#18082
C127×#13802
#12921127