Finite Group Databasebeta

← C2×C4×#100×C127 #44584067

Subgroups — C2×C4×#100×C127 #44584067

Order 32512.

1970 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×D4×C127C211yesmaximal, characteristic
2C22×#100×C127C212yesmaximal
3C2×C4×M16×C127C212yesmaximal, complemented
4C4×#100×C127C218yesmaximal, direct product factor
5C2×#343×C127C212yesmaximal, complemented
6C23×D4×C127C2211yescharacteristic
7C24×C4×C127C412yes—
8C22×M16×C127C2214yes—
9C2×C4×D4×C127—22no—
10C2×C4×D4×C127—24no—
11C2×C4×D4×C127—22no—
12C2×C4×D4×C127C414yes—
13C2×C4×D4×C127C2214yes—
14C22×C4⋊C4×C127C2211yescharacteristic
15C22×C22⋊C4×C127C412yes—
16C22×C42×C127C2211yescharacteristic
17C2×#100×C127C4116yesdirect product factor
18C2×#100×C127C2218yes—
19C4×M16×C127—24no—
20C4×M16×C127C2218yescomplemented
21C2×#97×C127—22no—
22C2×C4×C8×C127—22no—
23#343×C127C2218yescomplemented
24C25×C127C2×C412yes—
25C22×D4×C127—22no—
26C22×D4×C127—22no—
27C22×D4×C127—24no—
28C22×D4×C127C2×C412yes—
29C22×D4×C127C2×C418yes—
30C22×D4×C127C2312yes—
31C23×C4×C127—24no—
32C23×C4×C127—22no—
33C23×C4×C127—22no—
34C23×C4×C127—48no—
35C23×C4×C127C2×C411yescharacteristic
36C23×C4×C127C2×C412yes—
37C23×C4×C127C2×C418yes—
38C23×C4×C127C2×C411yescharacteristic
39C23×C4×C127C2311yescharacteristic
40C2×M16×C127—24no—
41C2×M16×C127—28no—
42C2×M16×C127C2×C4116yescomplemented
43C2×M16×C127C2318yes—
44C22×C8×C127—24no—
45C4×D4×C127—28no—
46C4×D4×C127—28no—
47C4×D4×C127—28no—
48C4×D4×C127—28no—
49C2×C4⋊C4×C127—22no—
50C2×C4⋊C4×C127—22no—
51C2×C4⋊C4×C127C2×C412yes—
52C2×C4⋊C4×C127C2312yes—
53C2×C22⋊C4×C127—24no—
54C2×C22⋊C4×C127—24no—
55C2×C22⋊C4×C127C2×C418yes—
56C2×C42×C127C2×C412yes—
57C2×C42×C127D412yes—
58C2×C42×C127D414yes—
59C2×C42×C127D412yes—
60C2×C42×C127C2312yes—
61#100×C127C2×C4132yesdirect product factor
62#97×C127—28no—
63C4×C8×C127—28no—
64C24×C127—22no—
65C24×C127—22no—
66C24×C127—24no—
67C24×C127—44no—
68C24×C127—44no—
69C24×C127C4218yes—
70C24×C127C22×C411yescharacteristic
71C24×C127C22×C414yes—
72C2×D4×C127—24no—
73C2×D4×C127—24no—
74C2×D4×C127—216no—
75C2×D4×C127—216no—
76C2×D4×C127—24no—
77C2×D4×C127—24no—
78C2×D4×C127C4218yes—
79C2×D4×C127C22×C418yes—
80C22×C4×C127—24no—
81C22×C4×C127—22no—
82C22×C4×C127—22no—
83C22×C4×C127—28no—
84C22×C4×C127—28no—
85C22×C4×C127—42no—
86C22×C4×C127—48no—
87C22×C4×C127—416no—
88C22×C4×C127—416no—
89C22×C4×C127—48no—
90C22×C4×C127—44no—
91C22×C4×C127C22⋊C412yes—
92C22×C4×C127C22⋊C411yescharacteristic
93C22×C4×C127C22⋊C412yes—
94C22×C4×C127C22⋊C412yes—
95C22×C4×C127C22⋊C411yescharacteristic
96C22×C4×C127C22×C412yes—
97C22×C4×C127C22×C414yes—
98C22×C4×C127C22×C412yes—
99C22×C4×C127C22×C411yescharacteristic
100C22×C4×C127C22×C414yes—
101C22×C4×C127C2×D412yes—
102C22×C4×C127C2×D412yes—
103C22×C4×C127#4114yes—
104C22×C4×C127C2411yescharacteristic
105M16×C127—216no—
106M16×C127—28no—
107M16×C127C22×C4116yescomplemented
108C2×C8×C127—216no—
109C2×C8×C127—28no—
110C4⋊C4×C127—24no—
111C4⋊C4×C127—24no—
112C22⋊C4×C127—216no—
113C42×C127C22⋊C414yes—
114C42×C127C22⋊C414yes—
115C42×C127C2×D414yes—
116C42×C127C2×D414yes—
117C23×C127—22no—
118C23×C127—22no—
119C23×C127—24no—
120C23×C127—24no—
121C23×C127—28no—
122C23×C127—28no—
123C23×C127—44no—
124C23×C127—416no—
125C23×C127—44no—
126C23×C127—44no—
127C23×C127—44no—
128C23×C127—416no—
129C23×C127—44no—
130C23×C127—44no—
131C23×C127C2×C4212yes—
132C23×C127C2×C4218yes—
133C23×C127C2×C22⋊C411yescharacteristic
134C23×C127C2×C22⋊C411yescharacteristic
135C23×C127#11712yes—
136C23×C127C23×C411yescharacteristic
137D4×C127—216no—
138D4×C127—216no—
139C2×C4×C127—22no—
140C2×C4×C127—24no—
141C2×C4×C127—24no—
142C2×C4×C127—24no—
143C2×C4×C127—24no—
144C2×C4×C127—42no—
145C2×C4×C127—42no—
146C2×C4×C127—416no—
147C2×C4×C127—44no—
148C2×C4×C127—44no—
149C2×C4×C127—416no—
150C2×C4×C127—44no—
151C2×C4×C127#10014yesdirect product factor
152C2×C4×C127C2×C4212yes—
153C2×C4×C127C2×C22⋊C414yes—
154C2×C4×C127C2×C22⋊C412yes—
155C2×C4×C127C2×C22⋊C412yes—
156C2×C4×C127C2×C22⋊C412yes—
157C2×C4×C127#11712yes—
158C2×C4×C127C4×D418yes—
159C2×C4×C127C4×D418yes—
160C2×C4×C127C23×C412yes—
161C2×C4×C127C22×D412yes—
162C2×C4×C127C2×#4112yes—
163C8×C127—216no—
164C22×C127—21no—
165C22×C127—21no—
166C22×C127—21no—
167C22×C127—21no—
168C22×C127—24no—
169C22×C127—24no—
170C22×C127—28no—
171C22×C127—48no—
172C22×C127—48no—
173C22×C127—416no—
174C22×C127—44no—
175C22×C127—44no—
176C22×C127—416no—
177C22×C127C4×C22⋊C412yes—
178C22×C127C4×C22⋊C412yes—
179C22×C127C2×#10012yes—
180C22×C127#41312yes—
181C22×C127C22×C4211yescharacteristic
182C22×C127C22×C22⋊C411yescharacteristic
183C22×C127C2×#11711yescharacteristic
184C4×C127—22no—
185C4×C127—24no—
186C4×C127—44no—
187C4×C127C4×C22⋊C414yes—
188C4×C127C2×#10014yesdirect product factor
189C4×C127C2×C4×D414yes—
190C2×C4×#100C12711yesmaximal, characteristic, direct product factor, Sylow
191C2×C127—21no—
192C2×C127—22no—
193C2×C127—21no—
194C2×C127—48no—
195C2×C127C4×#10014yesdirect product factor
196C2×C127C2×C4×C22⋊C411yescharacteristic
197C2×C127C22×#10011yescharacteristic
198C2×C127C2×#41311yescharacteristic
199C22×C4×D4C2×C12711yescharacteristic
200C22×#100C2×C12712yes—
201C2×C4×M16C2×C12712yescomplemented
202C4×#100C2×C12718yesdirect product factor
203C2×#343C2×C12712yescomplemented
204C127C2×C4×#10011yescharacteristic, direct product factor, Sylow
205C23×D4C22×C12711yescharacteristic
206C24×C4C4×C12712yes—
207C22×M16C22×C12714yes—
208C2×C4×D4—22no—
209C2×C4×D4—22no—
210C2×C4×D4—24no—
211C2×C4×D4C4×C12714yes—
212C2×C4×D4C22×C12714yes—
213C22×C4⋊C4C22×C12711yescharacteristic
214C22×C22⋊C4C4×C12712yes—
215C22×C42C22×C12711yescharacteristic
216C2×#100C4×C127116yesdirect product factor
217C2×#100C22×C12718yes—
218C4×M16—24no—
219C4×M16C22×C12718yescomplemented
220C2×#97—22no—
221C2×C4×C8—22no—
222#343C22×C12718yescomplemented
223C25C2×C4×C12712yes—
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—24no—
227C22×D4C2×C4×C12712yes—
228C22×D4C2×C4×C12718yes—
229C22×D4C23×C12712yes—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—48no—
234C23×C4C2×C4×C12711yescharacteristic
235C23×C4C2×C4×C12712yes—
236C23×C4C2×C4×C12718yes—
237C23×C4C2×C4×C12711yescharacteristic
238C23×C4C23×C12711yescharacteristic
239C2×M16—24no—
240C2×M16—28no—
241C2×M16C2×C4×C127116yescomplemented
242C2×M16C23×C12718yes—
243C22×C8—24no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—28no—
247C4×D4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4C2×C4×C12712yes—
251C2×C4⋊C4C23×C12712yes—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4C2×C4×C12718yes—
255C2×C42C2×C4×C12712yes—
256C2×C42D4×C12712yes—
257C2×C42D4×C12712yes—
258C2×C42D4×C12714yes—
259C2×C42C23×C12712yes—
260#100C2×C4×C127132yesdirect product factor
261#97—28no—
262C4×C8—28no—
263C24—22no—
264C24—22no—
265C24—24no—
266C24—44no—
267C24—44no—
268C24C42×C12718yes—
269C24C22×C4×C12711yescharacteristic
270C24C22×C4×C12714yes—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—216no—
274C2×D4—216no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4C42×C12718yes—
278C2×D4C22×C4×C12718yes—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—28no—
283C22×C4—28no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—416no—
287C22×C4—416no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4C22⋊C4×C12712yes—
291C22×C4C22⋊C4×C12711yescharacteristic
292C22×C4C22⋊C4×C12712yes—
293C22×C4C22⋊C4×C12712yes—
294C22×C4C22⋊C4×C12711yescharacteristic
295C22×C4C22×C4×C12712yes—
296C22×C4C22×C4×C12714yes—
297C22×C4C22×C4×C12712yes—
298C22×C4C22×C4×C12711yescharacteristic
299C22×C4C22×C4×C12714yes—
300C22×C4C2×D4×C12712yes—
301C22×C4C2×D4×C12712yes—
302C22×C4#41×C12714yes—
303C22×C4C24×C12711yescharacteristic
304M16—216no—
305M16—28no—
306M16C22×C4×C127116yescomplemented
307C2×C8—216no—
308C2×C8—28no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C22⋊C4—216no—
312C42C22⋊C4×C12714yes—
313C42C22⋊C4×C12714yes—
314C42C2×D4×C12714yes—
315C42C2×D4×C12714yes—
316C23—22no—
317C23—22no—
318C23—24no—
319C23—24no—
320C23—28no—
321C23—28no—
322C23—44no—
323C23—416no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—416no—
328C23—44no—
329C23—44no—
330C23C2×C42×C12712yes—
331C23C2×C42×C12718yes—
332C23C2×C22⋊C4×C12711yescharacteristic
333C23C2×C22⋊C4×C12711yescharacteristic
334C23#117×C12712yes—
335C23C23×C4×C12711yescharacteristic
336D4—216no—
337D4—216no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—416no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—416no—
349C2×C4—44no—
350C2×C4#100×C12714yesdirect product factor
351C2×C4C2×C42×C12712yes—
352C2×C4C2×C22⋊C4×C12714yes—
353C2×C4C2×C22⋊C4×C12712yes—
354C2×C4C2×C22⋊C4×C12712yes—
355C2×C4C2×C22⋊C4×C12712yes—
356C2×C4#117×C12712yes—
357C2×C4C4×D4×C12718yes—
358C2×C4C4×D4×C12718yes—
359C2×C4C23×C4×C12712yes—
360C2×C4C22×D4×C12712yes—
361C2×C4C2×#41×C12712yes—
362C8—216no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—24no—
368C22—24no—
369C22—28no—
370C22—48no—
371C22—48no—
372C22—416no—
373C22—44no—
374C22—44no—
375C22—416no—
376C22C4×C22⋊C4×C12712yes—
377C22C4×C22⋊C4×C12712yes—
378C22C2×#100×C12712yes—
379C22#413×C12712yes—
380C22C22×C42×C12711yescharacteristic
381C22C22×C22⋊C4×C12711yescharacteristic
382C22C2×#117×C12711yescharacteristic
383C4—22no—
384C4—24no—
385C4—44no—
386C4C4×C22⋊C4×C12714yes—
387C4C2×#100×C12714yesdirect product factor
388C4C2×C4×D4×C12714yes—
389C2—21no—
390C2—21no—
391C2—22no—
392C2—48no—
393C2C4×#100×C12714yesdirect product factor
394C2C2×C4×C22⋊C4×C12711yescharacteristic
395C2C22×#100×C12711yescharacteristic
396C2C2×#413×C12711yescharacteristic
subgroupG/H
centerC22×C4×C127C22⋊C4
commutatorC22C22×C42×C127
Frattini, cosocleC22×C4C24×C127
socleC23×C127C2×C22⋊C4
prime radicalC23×D4×C127C22
LCSG ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C2×C4×M16×C127C2semidirect
4. C4×#100×C127C2direct (N × H)
5. C2×#343×C127C2semidirect
17. C2×#100×C127C4direct (N × H)
20. C4×M16×C127C22semidirect
23. #343×C127C22semidirect
42. C2×M16×C127C2×C4semidirect
61. #100×C127C2×C4direct (N × H)
107. M16×C127C22×C4semidirect
151. C2×C4×C127#100direct (N × H)
188. C4×C127C2×#100direct (N × H)
190. C2×C4×#100C127direct (N × H)
195. C2×C127C4×#100direct (N × H)
201. C2×C4×M16C2×C127semidirect
202. C4×#100C2×C127direct (N × H)
203. C2×#343C2×C127semidirect
204. C127C2×C4×#100direct (N × H)
216. C2×#100C4×C127direct (N × H)
219. C4×M16C22×C127semidirect
222. #343C22×C127semidirect
241. C2×M16C2×C4×C127semidirect
260. #100C2×C4×C127direct (N × H)
306. M16C22×C4×C127semidirect
350. C2×C4#100×C127direct (N × H)
387. C4C2×#100×C127direct (N × H)
393. C2C4×#100×C127direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×D4×C1272
C22×#100×C1272
C2×C4×M16×C1272
C4×#100×C1272
C2×#343×C1272
C2×C4×#100127