Finite Group Databasebeta

← C127×#15026 #44578699

Subgroups — C127×#15026 #44578699

Order 32512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#385×C127C211yesmaximal, characteristic
2C22×#95×C127C211yesmaximal, characteristic
3C127×#1443C214yesmaximal, complemented
4C127×#1442C214yesmaximal, complemented
5C127×C22⋊#95C211yesmaximal, characteristic
6C127×C22⋊#95C212yesmaximal
7C127×C22⋊#95C212yesmaximal
8C24×C4×C127C2212yes—
9C24×C4×C127C2211yescharacteristic
10C22×C4⋊C4×C127C2212yes—
11C22×C22⋊C4×C127C2212yes—
12C22×C22⋊C4×C127C2211yescharacteristic
13C22×C22⋊C4×C127C2211yescharacteristic
14#385×C127—24no—
15C2×#95×C127—24no—
16C2×#95×C127C2214yes—
17C2×#95×C127C2214yes—
18C2×#95×C127C2214yes—
19C2×#95×C127C2212yes—
20C2×#95×C127C2214yes—
21C2×#95×C127C2214yes—
22C2×#95×C127C2214yes—
23C25×C127C2311yescharacteristic
24C23×C4×C127—24no—
25C23×C4×C127—24no—
26C23×C4×C127—28no—
27C23×C4×C127—24no—
28C23×C4×C127—24no—
29C23×C4×C127—44no—
30C23×C4×C127—44no—
31C23×C4×C127D414yes—
32C23×C4×C127C2311yescharacteristic
33C23×C4×C127C2311yescharacteristic
34C23×C4×C127C2314yes—
35C23×C4×C127C2312yes—
36C23×C4×C127C2312yes—
37C23×C4×C127C2314yes—
38C2×C4⋊C4×C127—24no—
39C2×C4⋊C4×C127—44no—
40C2×C22⋊C4×C127—22no—
41C2×C22⋊C4×C127—22no—
42C2×C22⋊C4×C127—22no—
43C2×C22⋊C4×C127—24no—
44C2×C22⋊C4×C127—24no—
45C2×C22⋊C4×C127—24no—
46C2×C22⋊C4×C127—24no—
47C2×C22⋊C4×C127Q814yes—
48#95×C127—28no—
49#95×C127—24no—
50#95×C127—28no—
51#95×C127—28no—
52#95×C127—44no—
53#95×C127—44no—
54#95×C127—44no—
55C24×C127—24no—
56C24×C127—24no—
57C24×C127C2×D412yes—
58C24×C127C2×Q812yes—
59C24×C127#4112yes—
60C24×C127#4114yes—
61C24×C127#4114yes—
62C24×C127C2411yescharacteristic
63C22×C4×C127—216no—
64C22×C4×C127—28no—
65C22×C4×C127—24no—
66C22×C4×C127—22no—
67C22×C4×C127—24no—
68C22×C4×C127—22no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—24no—
72C22×C4×C127—24no—
73C22×C4×C127—24no—
74C22×C4×C127—24no—
75C22×C4×C127—24no—
76C22×C4×C127—24no—
77C22×C4×C127—24no—
78C22×C4×C127—21no—
79C22×C4×C127—21no—
80C22×C4×C127—24no—
81C22×C4×C127—24no—
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127—22no—
85C22×C4×C127—24no—
86C22×C4×C127—24no—
87C22×C4×C127—24no—
88C22×C4×C127—24no—
89C22×C4×C127—24no—
90C22×C4×C127—24no—
91C22×C4×C127—48no—
92C22×C4×C127—416no—
93C22×C4×C127—416no—
94C22×C4×C127—48no—
95C22×C4×C127—416no—
96C22×C4×C127—48no—
97C22×C4×C127C2×D414yes—
98C22×C4×C127C2×Q814yes—
99C22×C4×C127#4118yes—
100C4⋊C4×C127—48no—
101C22⋊C4×C127—28no—
102C22⋊C4×C127—44no—
103C22⋊C4×C127—48no—
104C23×C127—24no—
105C23×C127—28no—
106C23×C127—24no—
107C23×C127—28no—
108C23×C127—28no—
109C23×C127—28no—
110C23×C127—28no—
111C23×C127—28no—
112C23×C127C22⋊Q814yes—
113C23×C127#12314yes—
114C23×C127#12414yes—
115C23×C127#12518yes—
116C23×C127#12618yes—
117C23×C127C22×D411yescharacteristic
118C23×C127C22×Q811yescharacteristic
119C23×C127C2×#4111yescharacteristic
120C23×C127C2×#4112yes—
121C23×C127C2×#4114yes—
122C23×C127C2×#4112yes—
123C23×C127H(5,𝔽2)14yes—
124C2×C4×C127—24no—
125C2×C4×C127—24no—
126C2×C4×C127—28no—
127C2×C4×C127—24no—
128C2×C4×C127—24no—
129C2×C4×C127—28no—
130C2×C4×C127—28no—
131C2×C4×C127—28no—
132C2×C4×C127—432no—
133C2×C4×C127—416no—
134C2×C4×C127—42no—
135C2×C4×C127—44no—
136C2×C4×C127—42no—
137C2×C4×C127—44no—
138C2×C4×C127—48no—
139C2×C4×C127—44no—
140C2×C4×C127—44no—
141C2×C4×C127—44no—
142C2×C4×C127—41no—
143C2×C4×C127—41no—
144C2×C4×C127—42no—
145C2×C4×C127—44no—
146C2×C4×C127—44no—
147C2×C4×C127—44no—
148C2×C4×C127—42no—
149C2×C4×C127—44no—
150C2×C4×C127—44no—
151C2×C4×C127—44no—
152C2×C4×C127—44no—
153C2×C4×C127—44no—
154C2×C4×C127—44no—
155C22×C127—28no—
156C22×C127—28no—
157C22×C127—24no—
158C22×C127—24no—
159C22×C127—216no—
160C22×C127—216no—
161C22×C127#40018yes—
162C22×C127C2×C22⋊Q811yescharacteristic
163C22×C127C2×#12311yescharacteristic
164C22×C127#52512yes—
165C22×C127C2×#12411yescharacteristic
166C22×C127C2×#12512yes—
167C22×C127C2×#12612yes—
168C22×C127#52914yes—
169C22×C127#53412yes—
170C22×C127#53814yes—
171C22×C127#55114yes—
172C22×C127#55114yes—
173C22×C127#55314yes—
174C22×C127#55414yes—
175C4×C127—48no—
176C4×C127—44no—
177C4×C127—44no—
178C4×C127—81no—
179C4×C127—81no—
180C4×C127—84no—
181#15026C12711yesmaximal, characteristic, direct product factor, Sylow
182C2×C127—28no—
183C2×C127C2×#40011yescharacteristic
184C2×C127#250514yes—
185C2×C127#251014yes—
186C2×C127#256112yes—
187C2×C127#257912yes—
188C2×C127#262211yescharacteristic
189C2×C127#285311yescharacteristic
190C2×#385C2×C12711yescharacteristic
191C22×#95C2×C12711yescharacteristic
192#1443C2×C12714yescomplemented
193#1442C2×C12714yescomplemented
194C22⋊#95C2×C12711yescharacteristic
195C22⋊#95C2×C12712yes—
196C22⋊#95C2×C12712yes—
197C127#1502611yescharacteristic, direct product factor, Sylow
198C24×C4C22×C12712yes—
199C24×C4C22×C12711yescharacteristic
200C22×C4⋊C4C22×C12712yes—
201C22×C22⋊C4C22×C12712yes—
202C22×C22⋊C4C22×C12711yescharacteristic
203C22×C22⋊C4C22×C12711yescharacteristic
204#385—24no—
205C2×#95—24no—
206C2×#95C22×C12714yes—
207C2×#95C22×C12714yes—
208C2×#95C22×C12714yes—
209C2×#95C22×C12712yes—
210C2×#95C22×C12714yes—
211C2×#95C22×C12714yes—
212C2×#95C22×C12714yes—
213C25C23×C12711yescharacteristic
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4D4×C12714yes—
222C23×C4C23×C12711yescharacteristic
223C23×C4C23×C12711yescharacteristic
224C23×C4C23×C12714yes—
225C23×C4C23×C12712yes—
226C23×C4C23×C12712yes—
227C23×C4C23×C12714yes—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—44no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4Q8×C12714yes—
238#95—28no—
239#95—24no—
240#95—28no—
241#95—28no—
242#95—44no—
243#95—44no—
244#95—44no—
245C24—24no—
246C24—24no—
247C24C2×D4×C12712yes—
248C24C2×Q8×C12712yes—
249C24#41×C12712yes—
250C24#41×C12714yes—
251C24#41×C12714yes—
252C24C24×C12711yescharacteristic
253C22×C4—216no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—48no—
282C22×C4—416no—
283C22×C4—416no—
284C22×C4—48no—
285C22×C4—416no—
286C22×C4—48no—
287C22×C4C2×D4×C12714yes—
288C22×C4C2×Q8×C12714yes—
289C22×C4#41×C12718yes—
290C4⋊C4—48no—
291C22⋊C4—28no—
292C22⋊C4—44no—
293C22⋊C4—48no—
294C23—24no—
295C23—28no—
296C23—24no—
297C23—28no—
298C23—28no—
299C23—28no—
300C23—28no—
301C23—28no—
302C23C22⋊Q8×C12714yes—
303C23#123×C12714yes—
304C23#124×C12714yes—
305C23#125×C12718yes—
306C23#126×C12718yes—
307C23C22×D4×C12711yescharacteristic
308C23C22×Q8×C12711yescharacteristic
309C23C2×#41×C12711yescharacteristic
310C23C2×#41×C12712yes—
311C23C2×#41×C12714yes—
312C23C2×#41×C12712yes—
313C23H(5,𝔽2)×C12714yes—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—416no—
323C2×C4—432no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—48no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C22—28no—
346C22—24no—
347C22—24no—
348C22—216no—
349C22—216no—
350C22—28no—
351C22#400×C12718yes—
352C22C2×C22⋊Q8×C12711yescharacteristic
353C22C2×#123×C12711yescharacteristic
354C22#525×C12712yes—
355C22C2×#124×C12711yescharacteristic
356C22C2×#125×C12712yes—
357C22C2×#126×C12712yes—
358C22#529×C12714yes—
359C22#534×C12712yes—
360C22#538×C12714yes—
361C22#551×C12714yes—
362C22#551×C12714yes—
363C22#553×C12714yes—
364C22#554×C12714yes—
365C4—44no—
366C4—44no—
367C4—48no—
368C4—81no—
369C4—81no—
370C4—84no—
371C2—28no—
372C2C2×#400×C12711yescharacteristic
373C2C127×#250514yes—
374C2C127×#251014yes—
375C2C127×#256112yes—
376C2C127×#257912yes—
377C2C127×#262211yescharacteristic
378C2C127×#285311yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
3. C127×#1443C2semidirect
4. C127×#1442C2semidirect
181. #15026C127direct (N × H)
192. #1443C2×C127semidirect
193. #1442C2×C127semidirect
197. C127#15026direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#385×C1272
C22×#95×C1272
C127×#14432
C127×#14422
C127×C22⋊#952
#15026127