Finite Group Databasebeta

← C127×#17675 #44581348

Subgroups — C127×#17675 #44581348

Order 32512.

1234 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×#532×C127C211yesmaximal, characteristic
2C2×#432×C127C212yesmaximal, complemented
3C127×#1533C214yesmaximal, complemented
4C127×#1530C218yesmaximal, complemented
5C22×#41×C127C412yes—
6#532×C127—22no—
7#532×C127—22no—
8#532×C127C414yes—
9#532×C127C2214yes—
10C2×C4⋊Q8×C127C411yescharacteristic
11C2×#127×C127C411yescharacteristic
12C2×#124×C127C2212yes—
13C2×#121×C127—22no—
14C2×C4×D4×C127—22no—
15C22×C42×C127C2211yescharacteristic
16#432×C127—22no—
17#432×C127—22no—
18#432×C127C2214yes—
19#432×C127C2214yescomplemented
20C2×C4⋊C8×C127—22no—
21C2×C22⋊C8×C127—22no—
22C2×#97×C127C2214yescomplemented
23#329×C127C22116yescomplemented
24C2×#41×C127—24no—
25C2×#41×C127—48no—
26C2×#41×C127C2×C418yes—
27C22×Q8×C127C2×C412yes—
28C22×D4×C127—22no—
29C22×D4×C127—22no—
30C22×D4×C127C2×C412yes—
31C23×C4×C127—22no—
32C23×C4×C127C2×C411yescharacteristic
33C23×C4×C127D412yes—
34C22×C8×C127—24no—
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127—21no—
37C4⋊Q8×C127C2×C414yes—
38#127×C127—21no—
39#127×C127—21no—
40#127×C127C2×C414yes—
41#124×C127—24no—
42#124×C127C2×C414yes—
43#124×C127C2314yes—
44#121×C127—24no—
45#121×C127—24no—
46#121×C127—24no—
47#121×C127—24no—
48C4×D4×C127—24no—
49C4×D4×C127—24no—
50C4×D4×C127—24no—
51C4×D4×C127—24no—
52C2×C4⋊C4×C127—22no—
53C2×C22⋊C4×C127—24no—
54C2×C42×C127C2×C411yescharacteristic
55C2×C42×C127C2×C412yes—
56C2×C42×C127D412yes—
57C2×C42×C127D412yes—
58C2×C42×C127D412yes—
59C2×C42×C127C2311yescharacteristic
60C2×C42×C127C2312yes—
61C4⋊C8×C127—24no—
62C4⋊C8×C127—24no—
63C22⋊C8×C127—24no—
64C22⋊C8×C127—24no—
65#97×C127—24no—
66#97×C127C2318yes—
67C24×C127—22no—
68C24×C127C22⋊C411yescharacteristic
69#41×C127—416no—
70#41×C127—416no—
71C2×Q8×C127—22no—
72C2×Q8×C127—44no—
73C2×Q8×C127C22×C414yes—
74C2×D4×C127—28no—
75C2×D4×C127—24no—
76C2×D4×C127—24no—
77C2×D4×C127—22no—
78C2×D4×C127—44no—
79C2×D4×C127—48no—
80C2×D4×C127—48no—
81C2×D4×C127C22×C414yes—
82C22×C4×C127—24no—
83C22×C4×C127—22no—
84C22×C4×C127—28no—
85C22×C4×C127—44no—
86C22×C4×C127—48no—
87C22×C4×C127C22⋊C412yes—
88C22×C4×C127C22⋊C412yes—
89C22×C4×C127C22⋊C411yescharacteristic
90C22×C4×C127C22⋊C412yes—
91C22×C4×C127C22⋊C412yes—
92C22×C4×C127C22×C411yescharacteristic
93C22×C4×C127C22×C412yes—
94C22×C4×C127C2×D412yes—
95C22×C4×C127C2×D412yes—
96C22×C4×C127C2×D412yes—
97C2×C8×C127—28no—
98C2×C8×C127—48no—
99C4⋊C4×C127—24no—
100C4⋊C4×C127—24no—
101C22⋊C4×C127—28no—
102C22⋊C4×C127—28no—
103C42×C127C22⋊C412yes—
104C42×C127C22⋊C412yes—
105C42×C127C22⋊C411yescharacteristic
106C42×C127C22⋊C411yescharacteristic
107C42×C127C22×C411yescharacteristic
108C42×C127C22×C412yes—
109C42×C127C2×D412yes—
110C42×C127C2×D412yes—
111C42×C127C2×D412yes—
112C42×C127C2411yescharacteristic
113C23×C127—24no—
114C23×C127—42no—
115C23×C127—48no—
116C23×C127—44no—
117C23×C127#10014yes—
118C23×C127C2×C22⋊C411yescharacteristic
119C23×C127C2×C22⋊C412yes—
120Q8×C127—44no—
121Q8×C127—44no—
122D4×C127—44no—
123D4×C127—48no—
124D4×C127—44no—
125D4×C127—416no—
126D4×C127—48no—
127C2×C4×C127—22no—
128C2×C4×C127—24no—
129C2×C4×C127—22no—
130C2×C4×C127—24no—
131C2×C4×C127—24no—
132C2×C4×C127—24no—
133C2×C4×C127—24no—
134C2×C4×C127—24no—
135C2×C4×C127—42no—
136C2×C4×C127—42no—
137C2×C4×C127—42no—
138C2×C4×C127—416no—
139C2×C4×C127—44no—
140C2×C4×C127C2×C22⋊C412yes—
141C2×C4×C127C2×C22⋊C412yes—
142C2×C4×C127C2×C22⋊C412yes—
143C2×C4×C127C2×C22⋊C411yescharacteristic
144C2×C4×C127C2×C22⋊C412yes—
145C2×C4×C127C23×C411yescharacteristic
146C2×C4×C127C22×D412yes—
147C8×C127—48no—
148C22×C127—22no—
149C22×C127—22no—
150C22×C127—22no—
151C22×C127—44no—
152C22×C127—44no—
153C22×C127—48no—
154C22×C127—88no—
155C22×C127C2×#10012yes—
156C22×C127C2×#10014yes—
157C22×C127C22×C22⋊C411yescharacteristic
158C4×C127—22no—
159C4×C127—21no—
160C4×C127—21no—
161C4×C127—22no—
162C4×C127—22no—
163C4×C127—22no—
164C4×C127—44no—
165C4×C127#42114yes—
166#17675C12711yesmaximal, characteristic, direct product factor, Sylow
167C2×C127—22no—
168C2×C127—24no—
169C2×C127—84no—
170C2×C127C22×#10011yescharacteristic
171C2×C127C2×#42112yes—
172C2×#532C2×C12711yescharacteristic
173C2×#432C2×C12712yescomplemented
174#1533C2×C12714yescomplemented
175#1530C2×C12718yescomplemented
176C127#1767511yescharacteristic, direct product factor, Sylow
177C22×#41C4×C12712yes—
178#532—22no—
179#532—22no—
180#532C4×C12714yes—
181#532C22×C12714yes—
182C2×C4⋊Q8C4×C12711yescharacteristic
183C2×#127C4×C12711yescharacteristic
184C2×#124C22×C12712yes—
185C2×#121—22no—
186C2×C4×D4—22no—
187C22×C42C22×C12711yescharacteristic
188#432—22no—
189#432—22no—
190#432C22×C12714yes—
191#432C22×C12714yescomplemented
192C2×C4⋊C8—22no—
193C2×C22⋊C8—22no—
194C2×#97C22×C12714yescomplemented
195#329C22×C127116yescomplemented
196C2×#41—24no—
197C2×#41—48no—
198C2×#41C2×C4×C12718yes—
199C22×Q8C2×C4×C12712yes—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4C2×C4×C12712yes—
203C23×C4—22no—
204C23×C4C2×C4×C12711yescharacteristic
205C23×C4D4×C12712yes—
206C22×C8—24no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C2×C4×C12714yes—
210#127—21no—
211#127—21no—
212#127C2×C4×C12714yes—
213#124—24no—
214#124C2×C4×C12714yes—
215#124C23×C12714yes—
216#121—24no—
217#121—24no—
218#121—24no—
219#121—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—24no—
224C2×C4⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C42C2×C4×C12711yescharacteristic
227C2×C42C2×C4×C12712yes—
228C2×C42D4×C12712yes—
229C2×C42D4×C12712yes—
230C2×C42D4×C12712yes—
231C2×C42C23×C12711yescharacteristic
232C2×C42C23×C12712yes—
233C4⋊C8—24no—
234C4⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—24no—
237#97—24no—
238#97C23×C12718yes—
239C24—22no—
240C24C22⋊C4×C12711yescharacteristic
241#41—416no—
242#41—416no—
243C2×Q8—22no—
244C2×Q8—44no—
245C2×Q8C22×C4×C12714yes—
246C2×D4—28no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—48no—
252C2×D4—48no—
253C2×D4C22×C4×C12714yes—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—44no—
258C22×C4—48no—
259C22×C4C22⋊C4×C12712yes—
260C22×C4C22⋊C4×C12712yes—
261C22×C4C22⋊C4×C12711yescharacteristic
262C22×C4C22⋊C4×C12712yes—
263C22×C4C22⋊C4×C12712yes—
264C22×C4C22×C4×C12711yescharacteristic
265C22×C4C22×C4×C12712yes—
266C22×C4C2×D4×C12712yes—
267C22×C4C2×D4×C12712yes—
268C22×C4C2×D4×C12712yes—
269C2×C8—28no—
270C2×C8—48no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C22⋊C4—28no—
274C22⋊C4—28no—
275C42C22⋊C4×C12712yes—
276C42C22⋊C4×C12712yes—
277C42C22⋊C4×C12711yescharacteristic
278C42C22⋊C4×C12711yescharacteristic
279C42C22×C4×C12711yescharacteristic
280C42C22×C4×C12712yes—
281C42C2×D4×C12712yes—
282C42C2×D4×C12712yes—
283C42C2×D4×C12712yes—
284C42C24×C12711yescharacteristic
285C23—24no—
286C23—42no—
287C23—48no—
288C23—44no—
289C23#100×C12714yes—
290C23C2×C22⋊C4×C12711yescharacteristic
291C23C2×C22⋊C4×C12712yes—
292Q8—44no—
293Q8—44no—
294D4—44no—
295D4—44no—
296D4—48no—
297D4—416no—
298D4—48no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—416no—
311C2×C4—44no—
312C2×C4C2×C22⋊C4×C12712yes—
313C2×C4C2×C22⋊C4×C12712yes—
314C2×C4C2×C22⋊C4×C12712yes—
315C2×C4C2×C22⋊C4×C12711yescharacteristic
316C2×C4C2×C22⋊C4×C12712yes—
317C2×C4C23×C4×C12711yescharacteristic
318C2×C4C22×D4×C12712yes—
319C8—48no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—44no—
324C22—44no—
325C22—48no—
326C22—88no—
327C22C2×#100×C12712yes—
328C22C2×#100×C12714yes—
329C22C22×C22⋊C4×C12711yescharacteristic
330C4—22no—
331C4—21no—
332C4—21no—
333C4—22no—
334C4—22no—
335C4—22no—
336C4—44no—
337C4#421×C12714yes—
338C2—22no—
339C2—24no—
340C2—84no—
341C2C22×#100×C12711yescharacteristic
342C2C2×#421×C12712yes—
subgroupG/H
center, socleC22×C127C22×C22⋊C4
commutatorC2×C4C23×C4×C127
Frattini, cosocleC42C24×C127
prime radicalC2×#532×C127C2
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#432×C127C2semidirect
3. C127×#1533C2semidirect
4. C127×#1530C2semidirect
19. #432×C127C22semidirect
22. C2×#97×C127C22semidirect
23. #329×C127C22semidirect
166. #17675C127direct (N × H)
173. C2×#432C2×C127semidirect
174. #1533C2×C127semidirect
175. #1530C2×C127semidirect
176. C127#17675direct (N × H)
191. #432C22×C127semidirect
194. C2×#97C22×C127semidirect
195. #329C22×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#532×C1272
C2×#432×C1272
C127×#15332
C127×#15302
#17675127