Finite Group Databasebeta

← C2×C127×#1354 #44575393

Subgroups — C2×C127×#1354 #44575393

Order 32512.

652 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#395×C127C211yesmaximal, characteristic
2C2×#343×C127C211yesmaximal, characteristic
3C2×#324×C127C211yesmaximal, characteristic
4C127×#1354C214yesmaximal, direct product factor
5C23×Q8×C127C411yescharacteristic
6C22×M16×C127—21no—
7C22×C4⋊C4×C127—21no—
8C22×C4⋊C4×C127C2211yescharacteristic
9C2×C22⋊C8×C127—21no—
10#395×C127—21no—
11#395×C127—21no—
12#395×C127C412yes—
13#395×C127C2212yes—
14C2×#95×C127—21no—
15C2×#95×C127C411yescharacteristic
16#343×C127—21no—
17#343×C127C2212yes—
18#324×C127—21no—
19#324×C127C2212yes—
20C22×Q8×C127—21no—
21C22×Q8×C127—44no—
22C22×Q8×C127—41no—
23C22×Q8×C127—41no—
24C22×Q8×C127C2×C412yes—
25C23×C4×C127—21no—
26C23×C4×C127—21no—
27C23×C4×C127C2×C411yescharacteristic
28C23×C4×C127D411yescharacteristic
29C23×C4×C127D411yescharacteristic
30C2×M16×C127—24no—
31C2×M16×C127—22no—
32C2×M16×C127—41no—
33C2×M16×C127—42no—
34C22×C8×C127—41no—
35C22×C8×C127—41no—
36C2×C4⋊C4×C127—22no—
37C2×C4⋊C4×C127—22no—
38C2×C4⋊C4×C127—41no—
39C2×C4⋊C4×C127—41no—
40C2×C4⋊C4×C127—44no—
41C2×C4⋊C4×C127C2×C411yescharacteristic
42C2×C4⋊C4×C127D411yescharacteristic
43C2×C4⋊C4×C127D411yescharacteristic
44C2×C4⋊C4×C127C2311yescharacteristic
45C22⋊C8×C127—22no—
46C22⋊C8×C127—41no—
47#95×C127—22no—
48#95×C127—21no—
49#95×C127—22no—
50#95×C127C2×C412yes—
51C24×C127C22⋊C411yescharacteristic
52C2×Q8×C127—44no—
53C2×Q8×C127—44no—
54C2×Q8×C127—42no—
55C2×Q8×C127—42no—
56C2×Q8×C127—48no—
57C2×Q8×C127—44no—
58C2×Q8×C127—44no—
59C22×C4×C127—21no—
60C22×C4×C127—21no—
61C22×C4×C127—22no—
62C22×C4×C127—24no—
63C22×C4×C127—24no—
64C22×C4×C127—21no—
65C22×C4×C127—21no—
66C22×C4×C127—21no—
67C22×C4×C127—22no—
68C22×C4×C127—22no—
69C22×C4×C127—21no—
70C22×C4×C127—42no—
71C22×C4×C127—42no—
72C22×C4×C127—41no—
73C22×C4×C127—41no—
74C22×C4×C127—42no—
75C22×C4×C127—42no—
76C22×C4×C127—42no—
77C22×C4×C127C22⋊C411yescharacteristic
78C22×C4×C127C22⋊C411yescharacteristic
79C22×C4×C127C22⋊C411yescharacteristic
80C22×C4×C127SD1612yes—
81C22×C4×C127Q1612yes—
82C22×C4×C127C22×C411yescharacteristic
83C22×C4×C127C2×D411yescharacteristic
84C22×C4×C127C2×D411yescharacteristic
85M16×C127—24no—
86M16×C127—44no—
87M16×C127—42no—
88C2×C8×C127—44no—
89C2×C8×C127—42no—
90C2×C8×C127—42no—
91C2×C8×C127—82no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127—41no—
95C4⋊C4×C127—41no—
96C4⋊C4×C127—44no—
97C4⋊C4×C127—44no—
98C23×C127—22no—
99C23×C127—22no—
100C23×C127#9911yescharacteristic
101C23×C127#9911yescharacteristic
102C23×C127#10312yes—
103C23×C127#10412yes—
104C23×C127C2×C22⋊C411yescharacteristic
105Q8×C127—44no—
106Q8×C127—44no—
107Q8×C127—48no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—24no—
117C2×C4×C127—24no—
118C2×C4×C127—21no—
119C2×C4×C127—21no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—42no—
123C2×C4×C127—48no—
124C2×C4×C127—42no—
125C2×C4×C127—44no—
126C2×C4×C127—42no—
127C2×C4×C127—42no—
128C2×C4×C127—44no—
129C2×C4×C127—42no—
130C2×C4×C127—42no—
131C2×C4×C127—42no—
132C2×C4×C127—41no—
133C2×C4×C127—41no—
134C2×C4×C127—41no—
135C2×C4×C127—41no—
136C2×C4×C127—81no—
137C2×C4×C127#10312yes—
138C2×C4×C127C2×SD1611yescharacteristic
139C2×C4×C127C2×Q1611yescharacteristic
140C8×C127—44no—
141C8×C127—82no—
142C22×C127—21no—
143C22×C127—21no—
144C22×C127—24no—
145C22×C127—24no—
146C22×C127—21no—
147C22×C127—21no—
148C22×C127#32812yes—
149C22×C127#32812yes—
150C22×C127AΣL(1,𝔽16)12yes—
151C22×C127#35612yes—
152C22×C127C2×#9911yescharacteristic
153C22×C127C2×#10311yescharacteristic
154C22×C127C2×#10411yescharacteristic
155C4×C127—21no—
156C4×C127—22no—
157C4×C127—21no—
158C4×C127—44no—
159C4×C127—41no—
160C4×C127—44no—
161C4×C127—41no—
162C4×C127—41no—
163C4×C127—41no—
164C2×#1354C12711yesmaximal, characteristic, direct product factor, Sylow
165C2×C127—21no—
166C2×C127—22no—
167C2×C127—21no—
168C2×C127#135414yesdirect product factor
169C2×C127C2×#32811yescharacteristic
170C2×C127C2×AΣL(1,𝔽16)11yescharacteristic
171C2×C127C2×#35611yescharacteristic
172C2×#395C2×C12711yescharacteristic
173C2×#343C2×C12711yescharacteristic
174C2×#324C2×C12711yescharacteristic
175#1354C2×C12714yesdirect product factor
176C127C2×#135411yescharacteristic, direct product factor, Sylow
177C23×Q8C4×C12711yescharacteristic
178C22×M16—21no—
179C22×C4⋊C4—21no—
180C22×C4⋊C4C22×C12711yescharacteristic
181C2×C22⋊C8—21no—
182#395—21no—
183#395—21no—
184#395C4×C12712yes—
185#395C22×C12712yes—
186C2×#95—21no—
187C2×#95C4×C12711yescharacteristic
188#343—21no—
189#343C22×C12712yes—
190#324—21no—
191#324C22×C12712yes—
192C22×Q8—21no—
193C22×Q8—44no—
194C22×Q8—41no—
195C22×Q8—41no—
196C22×Q8C2×C4×C12712yes—
197C23×C4—21no—
198C23×C4—21no—
199C23×C4C2×C4×C12711yescharacteristic
200C23×C4D4×C12711yescharacteristic
201C23×C4D4×C12711yescharacteristic
202C2×M16—24no—
203C2×M16—22no—
204C2×M16—41no—
205C2×M16—42no—
206C22×C8—41no—
207C22×C8—41no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4C2×C4×C12711yescharacteristic
214C2×C4⋊C4D4×C12711yescharacteristic
215C2×C4⋊C4D4×C12711yescharacteristic
216C2×C4⋊C4C23×C12711yescharacteristic
217C22⋊C8—22no—
218C22⋊C8—41no—
219#95—22no—
220#95—21no—
221#95—22no—
222#95C2×C4×C12712yes—
223C24C22⋊C4×C12711yescharacteristic
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—48no—
229C2×Q8—44no—
230C2×Q8—44no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4C22⋊C4×C12711yescharacteristic
250C22×C4C22⋊C4×C12711yescharacteristic
251C22×C4C22⋊C4×C12711yescharacteristic
252C22×C4SD16×C12712yes—
253C22×C4Q16×C12712yes—
254C22×C4C22×C4×C12711yescharacteristic
255C22×C4C2×D4×C12711yescharacteristic
256C22×C4C2×D4×C12711yescharacteristic
257M16—24no—
258M16—44no—
259M16—42no—
260C2×C8—44no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—82no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C23—22no—
271C23—22no—
272C23#99×C12711yescharacteristic
273C23#99×C12711yescharacteristic
274C23#103×C12712yes—
275C23#104×C12712yes—
276C23C2×C22⋊C4×C12711yescharacteristic
277Q8—44no—
278Q8—44no—
279Q8—48no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—42no—
298C2×C4—48no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4#103×C12712yes—
310C2×C4C2×SD16×C12711yescharacteristic
311C2×C4C2×Q16×C12711yescharacteristic
312C8—44no—
313C8—82no—
314C22—21no—
315C22—21no—
316C22—24no—
317C22—24no—
318C22—21no—
319C22—21no—
320C22#328×C12712yes—
321C22#328×C12712yes—
322C22AΣL(1,𝔽16)×C12712yes—
323C22#356×C12712yes—
324C22C2×#99×C12711yescharacteristic
325C22C2×#103×C12711yescharacteristic
326C22C2×#104×C12711yescharacteristic
327C4—21no—
328C4—22no—
329C4—21no—
330C4—44no—
331C4—41no—
332C4—44no—
333C4—41no—
334C4—41no—
335C4—41no—
336C2—21no—
337C2—22no—
338C2—21no—
339C2C127×#135414yesdirect product factor
340C2C2×#328×C12711yescharacteristic
341C2C2×AΣL(1,𝔽16)×C12711yescharacteristic
342C2C2×#356×C12711yescharacteristic
subgroupG/H
center, socleC23×C127#99
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC24×C127C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C24×C127 ◁ C22×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C127×#1354C2direct (N × H)
164. C2×#1354C127direct (N × H)
168. C2×C127#1354direct (N × H)
175. #1354C2×C127direct (N × H)
176. C127C2×#1354direct (N × H)
339. C2C127×#1354direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#395×C1272
C2×#343×C1272
C2×#324×C1272
C127×#13542
C2×#1354127