Finite Group Databasebeta

← C127×#8353 #44572026

Subgroups — C127×#8353 #44572026

Order 32512.

1054 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#3496C211yesmaximal, characteristic
2C127×#1954C211yesmaximal, characteristic, complemented
3C127×#1800C211yesmaximal, characteristic, complemented
4C127×#1547C212yesmaximal, complemented
5C127×#1492C212yesmaximal, complemented
6C2×H(5,𝔽2)×C127—21no—
7#547×C127—22no—
8#546×C127—24no—
9D42×C127—22no—
10#537×C127—21no—
11#534×C127C414yes—
12#532×C127C2212yes—
13C2×#121×C127—22no—
14C2×C4×D4×C127C2211yescharacteristic
15#423×C127C2212yescomplemented
16C2×C4⋊C8×C127C2211yescharacteristic, complemented
17C2×#100×C127—21no—
18C2×C22⋊C8×C127—22no—
19C2×C4×C8×C127C2211yescharacteristic, complemented
20#343×C127—21no—
21#332×C127—22no—
22#331×C127—22no—
23D4⋊C8×C127—24no—
24H(5,𝔽2)×C127—44no—
25H(5,𝔽2)×C127—42no—
26H(5,𝔽2)×C127—42no—
27C2×#41×C127—24no—
28C2×#41×C127—22no—
29C22×D4×C127—24no—
30C22×D4×C127—24no—
31C22×D4×C127—22no—
32C22×D4×C127D411yescharacteristic
33C23×C4×C127D412yes—
34C2×M16×C127—22no—
35C22×C8×C127—21no—
36C22×C8×C127—21no—
37C4⋊Q8×C127D412yes—
38#127×C127D412yes—
39#124×C127—22no—
40#123×C127—28no—
41C22⋊Q8×C127—24no—
42#121×C127—24no—
43#121×C127—24no—
44#121×C127—24no—
45#121×C127—24no—
46#121×C127—22no—
47#121×C127C2×C414yes—
48#120×C127—28no—
49C4×D4×C127—22no—
50C4×D4×C127—22no—
51C4×D4×C127—22no—
52C4×D4×C127D414yes—
53C2×C4⋊C4×C127—22no—
54C2×C4⋊C4×C127D411yescharacteristic
55C2×C22⋊C4×C127—24no—
56C2×C22⋊C4×C127C2×C412yes—
57C2×C42×C127C2311yescharacteristic
58C4⋊C8×C127—22no—
59C4⋊C8×C127—22no—
60C4⋊C8×C127—22no—
61#100×C127—42no—
62C22⋊C8×C127—44no—
63C4×C8×C127—22no—
64C24×C127—24no—
65C24×C127C22⋊C412yes—
66#41×C127—48no—
67#41×C127—48no—
68#41×C127—42no—
69#41×C127—42no—
70#41×C127—42no—
71#41×C127—42no—
72C2×Q8×C127—22no—
73C2×D4×C127—28no—
74C2×D4×C127—24no—
75C2×D4×C127—24no—
76C2×D4×C127—22no—
77C2×D4×C127—22no—
78C2×D4×C127—22no—
79C2×D4×C127—22no—
80C2×D4×C127—48no—
81C2×D4×C127—48no—
82C2×D4×C127—48no—
83C2×D4×C127—48no—
84C2×D4×C127—82no—
85C2×D4×C127—82no—
86C2×D4×C127C22⋊C414yes—
87C2×D4×C127D812yes—
88C2×D4×C127SD1612yes—
89C22×C4×C127—24no—
90C22×C4×C127—22no—
91C22×C4×C127—24no—
92C22×C4×C127—22no—
93C22×C4×C127—42no—
94C22×C4×C127—42no—
95C22×C4×C127C22⋊C412yes—
96C22×C4×C127C22×C411yescharacteristic
97C22×C4×C127C2×D411yescharacteristic
98C22×C4×C127C2×D411yescharacteristic
99M16×C127—42no—
100M16×C127—42no—
101C2×C8×C127—22no—
102C2×C8×C127—41no—
103C2×C8×C127—41no—
104C2×C8×C127—42no—
105C2×C8×C127—41no—
106C2×C8×C127—42no—
107C4⋊C4×C127—24no—
108C4⋊C4×C127—22no—
109C4⋊C4×C127—22no—
110C4⋊C4×C127C22⋊C414yes—
111C22⋊C4×C127—28no—
112C22⋊C4×C127—24no—
113C22⋊C4×C127—24no—
114C22⋊C4×C127—24no—
115C42×C127C2×D412yes—
116C42×C127C2×D412yes—
117C23×C127—22no—
118C23×C127—22no—
119C23×C127—24no—
120C23×C127—42no—
121C23×C127—42no—
122C23×C127—48no—
123C23×C127—84no—
124C23×C127—84no—
125C23×C127#10214yes—
126C23×C127C2×C22⋊C411yescharacteristic
127Q8×C127—42no—
128Q8×C127—42no—
129D4×C127—24no—
130D4×C127—24no—
131D4×C127—44no—
132D4×C127—416no—
133D4×C127—44no—
134D4×C127—44no—
135D4×C127—42no—
136D4×C127—42no—
137D4×C127—42no—
138D4×C127—42no—
139D4×C127—84no—
140C2×C4×C127—21no—
141C2×C4×C127—21no—
142C2×C4×C127—22no—
143C2×C4×C127—24no—
144C2×C4×C127—41no—
145C2×C4×C127—48no—
146C2×C4×C127—44no—
147C2×C4×C127—44no—
148C2×C4×C127—42no—
149C2×C4×C127—42no—
150C2×C4×C127—42no—
151C2×C4×C127—82no—
152C2×C4×C127C2×C22⋊C412yes—
153C2×C4×C127#12011yescharacteristic
154C2×C4×C127#12012yes—
155C2×C4×C127#12011yescharacteristic
156C2×C4×C127C2×D811yescharacteristic
157C2×C4×C127C2×SD1611yescharacteristic
158C2×C4×C127AGL(1,ℤ/8)12yes—
159C8×C127—42no—
160C8×C127—82no—
161C8×C127—81no—
162C22×C127—21no—
163C22×C127—21no—
164C22×C127—24no—
165C22×C127—44no—
166C22×C127—42no—
167C22×C127—42no—
168C22×C127—44no—
169C22×C127—44no—
170C22×C127—44no—
171C22×C127—82no—
172C22×C127—168no—
173C22×C127#37911yescharacteristic
174C22×C127C2×#10211yescharacteristic
175C22×C127#41811yescharacteristic
176C4×C127—21no—
177C4×C127—22no—
178C4×C127—42no—
179C4×C127—44no—
180C4×C127#44712yes—
181C4×C127#45012yes—
182C4×C127#45311yescharacteristic
183C4×C127#45511yescharacteristic
184#8353C12711yesmaximal, characteristic, direct product factor, Sylow
185C2×C127—21no—
186C2×C127—21no—
187C2×C127—44no—
188C2×C127—84no—
189C2×C127#178611yescharacteristic
190C2×C127#179111yescharacteristic
191C2×C127#179211yescharacteristic
192#3496C2×C12711yescharacteristic
193#1954C2×C12711yescharacteristic, complemented
194#1800C2×C12711yescharacteristic, complemented
195#1547C2×C12712yescomplemented
196#1492C2×C12712yescomplemented
197C127#835311yescharacteristic, direct product factor, Sylow
198C2×H(5,𝔽2)—21no—
199#547—22no—
200#546—24no—
201D42—22no—
202#537—21no—
203#534C4×C12714yes—
204#532C22×C12712yes—
205C2×#121—22no—
206C2×C4×D4C22×C12711yescharacteristic
207#423C22×C12712yescomplemented
208C2×C4⋊C8C22×C12711yescharacteristic, complemented
209C2×#100—21no—
210C2×C22⋊C8—22no—
211C2×C4×C8C22×C12711yescharacteristic, complemented
212#343—21no—
213#332—22no—
214#331—22no—
215D4⋊C8—24no—
216H(5,𝔽2)—44no—
217H(5,𝔽2)—42no—
218H(5,𝔽2)—42no—
219C2×#41—24no—
220C2×#41—22no—
221C22×D4—24no—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4D4×C12711yescharacteristic
225C23×C4D4×C12712yes—
226C2×M16—22no—
227C22×C8—21no—
228C22×C8—21no—
229C4⋊Q8D4×C12712yes—
230#127D4×C12712yes—
231#124—22no—
232#123—28no—
233C22⋊Q8—24no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121—24no—
238#121—22no—
239#121C2×C4×C12714yes—
240#120—28no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4D4×C12714yes—
245C2×C4⋊C4—22no—
246C2×C4⋊C4D4×C12711yescharacteristic
247C2×C22⋊C4—24no—
248C2×C22⋊C4C2×C4×C12712yes—
249C2×C42C23×C12711yescharacteristic
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253#100—42no—
254C22⋊C8—44no—
255C4×C8—22no—
256C24—24no—
257C24C22⋊C4×C12712yes—
258#41—48no—
259#41—48no—
260#41—42no—
261#41—42no—
262#41—42no—
263#41—42no—
264C2×Q8—22no—
265C2×D4—28no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—48no—
275C2×D4—48no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4C22⋊C4×C12714yes—
279C2×D4D8×C12712yes—
280C2×D4SD16×C12712yes—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C22⋊C4×C12712yes—
288C22×C4C22×C4×C12711yescharacteristic
289C22×C4C2×D4×C12711yescharacteristic
290C22×C4C2×D4×C12711yescharacteristic
291M16—42no—
292M16—42no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4C22⋊C4×C12714yes—
303C22⋊C4—28no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C42C2×D4×C12712yes—
308C42C2×D4×C12712yes—
309C23—22no—
310C23—22no—
311C23—24no—
312C23—42no—
313C23—42no—
314C23—48no—
315C23—84no—
316C23—84no—
317C23#102×C12714yes—
318C23C2×C22⋊C4×C12711yescharacteristic
319Q8—42no—
320Q8—42no—
321D4—24no—
322D4—24no—
323D4—44no—
324D4—416no—
325D4—44no—
326D4—44no—
327D4—42no—
328D4—42no—
329D4—42no—
330D4—42no—
331D4—84no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—41no—
337C2×C4—48no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—82no—
344C2×C4C2×C22⋊C4×C12712yes—
345C2×C4#120×C12711yescharacteristic
346C2×C4#120×C12712yes—
347C2×C4#120×C12711yescharacteristic
348C2×C4C2×D8×C12711yescharacteristic
349C2×C4C2×SD16×C12711yescharacteristic
350C2×C4AGL(1,ℤ/8)×C12712yes—
351C8—42no—
352C8—82no—
353C8—81no—
354C22—21no—
355C22—21no—
356C22—24no—
357C22—44no—
358C22—42no—
359C22—42no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—168no—
365C22#379×C12711yescharacteristic
366C22C2×#102×C12711yescharacteristic
367C22#418×C12711yescharacteristic
368C4—22no—
369C4—21no—
370C4—44no—
371C4—42no—
372C4#447×C12712yes—
373C4#450×C12712yes—
374C4#453×C12711yescharacteristic
375C4#455×C12711yescharacteristic
376C2—21no—
377C2—21no—
378C2—44no—
379C2—84no—
380C2C127×#178611yescharacteristic
381C2C127×#179111yescharacteristic
382C2C127×#179211yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC127×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1954C2semidirect
3. C127×#1800C2semidirect
4. C127×#1547C2semidirect
5. C127×#1492C2semidirect
15. #423×C127C22semidirect
16. C2×C4⋊C8×C127C22semidirect
19. C2×C4×C8×C127C22semidirect
184. #8353C127direct (N × H)
193. #1954C2×C127semidirect
194. #1800C2×C127semidirect
195. #1547C2×C127semidirect
196. #1492C2×C127semidirect
197. C127#8353direct (N × H)
207. #423C22×C127semidirect
208. C2×C4⋊C8C22×C127semidirect
211. C2×C4×C8C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34962
C127×#19542
C127×#18002
C127×#15472
C127×#14922
#8353127