Finite Group Databasebeta

← C127×#8389 #44572062

Subgroups — C127×#8389 #44572062

Order 32512.

738 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556×C127C211yesmaximal, characteristic
2C127×#1800C211yesmaximal, characteristic
3C127×#1546C212yesmaximal, complemented
4C127×#1487C212yesmaximal, complemented
5C127×#1460C211yesmaximal, characteristic
6#556×C127—24no—
7#556×C127—24no—
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#126×C127—22no—
10C2×#124×C127—21no—
11C2×C22⋊Q8×C127C412yes—
12C2×C4×Q8×C127—21no—
13C2×C4×Q8×C127C2211yescharacteristic
14C2×C4×D4×C127C2211yescharacteristic
15C2×#117×C127C412yes—
16#431×C127C2212yes—
17#423×C127C2212yes—
18C2×#100×C127—21no—
19#343×C127—21no—
20#332×C127—24no—
21Q8⋊C8×C127—24no—
22#323×C127—22no—
23C22×Q8×C127—21no—
24C22×Q8×C127C2×C411yescharacteristic
25C22×D4×C127D411yescharacteristic
26C23×C4×C127C2×C412yes—
27C2×M16×C127—22no—
28C4⋊Q8×C127—22no—
29C4⋊Q8×C127D414yes—
30#126×C127—216no—
31#124×C127—28no—
32C22⋊Q8×C127—24no—
33C22⋊Q8×C127—24no—
34C4×Q8×C127—28no—
35C4×Q8×C127—22no—
36C4×Q8×C127D414yes—
37C4×D4×C127—22no—
38C4×D4×C127—22no—
39#117×C127—24no—
40#117×C127—24no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127C2×C411yescharacteristic
45C2×C4⋊C4×C127C2×C412yes—
46C2×C4⋊C4×C127D411yescharacteristic
47C2×C22⋊C4×C127—22no—
48C2×C22⋊C4×C127—22no—
49C2×C22⋊C4×C127D412yes—
50C2×C42×C127—22no—
51C2×C42×C127—21no—
52C2×C42×C127C2311yescharacteristic
53C4⋊C8×C127—22no—
54C4⋊C8×C127—22no—
55C4⋊C8×C127—22no—
56#100×C127—42no—
57C22⋊C8×C127—22no—
58C4×C8×C127—22no—
59C24×C127C22⋊C412yes—
60C2×Q8×C127—21no—
61C2×Q8×C127—24no—
62C2×Q8×C127—42no—
63C2×Q8×C127—42no—
64C2×Q8×C127—41no—
65C2×Q8×C127—41no—
66C2×Q8×C127C22⋊C412yes—
67C2×D4×C127—22no—
68C2×D4×C127—22no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C22×C4×C127—21no—
72C22×C4×C127—21no—
73C22×C4×C127—22no—
74C22×C4×C127—22no—
75C22×C4×C127—22no—
76C22×C4×C127—44no—
77C22×C4×C127C22⋊C412yes—
78C22×C4×C127C22×C411yescharacteristic
79C22×C4×C127C2×D411yescharacteristic
80C22×C4×C127C2×D411yescharacteristic
81M16×C127—42no—
82M16×C127—42no—
83C2×C8×C127—42no—
84C2×C8×C127—42no—
85C4⋊C4×C127—28no—
86C4⋊C4×C127—21no—
87C4⋊C4×C127—22no—
88C4⋊C4×C127—24no—
89C4⋊C4×C127—24no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127C22⋊C412yes—
92C4⋊C4×C127C22⋊C414yes—
93C22⋊C4×C127—28no—
94C22⋊C4×C127—28no—
95C22⋊C4×C127—24no—
96C42×C127—28no—
97C42×C127—24no—
98C42×C127C2×D412yes—
99C42×C127C2×D412yes—
100C23×C127—22no—
101C23×C127—22no—
102C23×C127—22no—
103C23×C127—42no—
104C23×C127—42no—
105C23×C127C2×C22⋊C411yescharacteristic
106Q8×C127—24no—
107Q8×C127—42no—
108Q8×C127—44no—
109Q8×C127—44no—
110D4×C127—42no—
111D4×C127—42no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—22no—
116C2×C4×C127—24no—
117C2×C4×C127—41no—
118C2×C4×C127—44no—
119C2×C4×C127—42no—
120C2×C4×C127—42no—
121C2×C4×C127—42no—
122C2×C4×C127—42no—
123C2×C4×C127—44no—
124C2×C4×C127—42no—
125C2×C4×C127C2×C22⋊C411yescharacteristic
126C2×C4×C127C2×C22⋊C411yescharacteristic
127C2×C4×C127#12012yes—
128C2×C4×C127#12012yes—
129C2×C4×C127#13512yes—
130C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
131C2×C4×C127#13711yescharacteristic
132C8×C127—82no—
133C8×C127—82no—
134C22×C127—21no—
135C22×C127—21no—
136C22×C127—42no—
137C22×C127—42no—
138C22×C127—42no—
139C22×C127—82no—
140C22×C127—82no—
141C22×C127#37911yescharacteristic
142C22×C127#41611yescharacteristic
143C22×C127#41711yescharacteristic
144C4×C127—22no—
145C4×C127—21no—
146C4×C127—42no—
147C4×C127—42no—
148C4×C127—42no—
149C4×C127—44no—
150C4×C127#41312yes—
151C4×C127#44912yes—
152C4×C127#45212yes—
153#8389C12711yesmaximal, characteristic, direct product factor, Sylow
154C2×C127—21no—
155C2×C127—21no—
156C2×C127—82no—
157C2×C127#178511yescharacteristic
158C2×C127#178811yescharacteristic
159C2×C127#179611yescharacteristic
160C2×#556C2×C12711yescharacteristic
161#1800C2×C12711yescharacteristic
162#1546C2×C12712yescomplemented
163#1487C2×C12712yescomplemented
164#1460C2×C12711yescharacteristic
165C127#838911yescharacteristic, direct product factor, Sylow
166#556—24no—
167#556—24no—
168C2×C4⋊Q8C22×C12711yescharacteristic
169C2×#126—22no—
170C2×#124—21no—
171C2×C22⋊Q8C4×C12712yes—
172C2×C4×Q8—21no—
173C2×C4×Q8C22×C12711yescharacteristic
174C2×C4×D4C22×C12711yescharacteristic
175C2×#117C4×C12712yes—
176#431C22×C12712yes—
177#423C22×C12712yes—
178C2×#100—21no—
179#343—21no—
180#332—24no—
181Q8⋊C8—24no—
182#323—22no—
183C22×Q8—21no—
184C22×Q8C2×C4×C12711yescharacteristic
185C22×D4D4×C12711yescharacteristic
186C23×C4C2×C4×C12712yes—
187C2×M16—22no—
188C4⋊Q8—22no—
189C4⋊Q8D4×C12714yes—
190#126—216no—
191#124—28no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C4×Q8—28no—
195C4×Q8—22no—
196C4×Q8D4×C12714yes—
197C4×D4—22no—
198C4×D4—22no—
199#117—24no—
200#117—24no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C2×C4×C12711yescharacteristic
205C2×C4⋊C4C2×C4×C12712yes—
206C2×C4⋊C4D4×C12711yescharacteristic
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4D4×C12712yes—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42C23×C12711yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216#100—42no—
217C22⋊C8—22no—
218C4×C8—22no—
219C24C22⋊C4×C12712yes—
220C2×Q8—21no—
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8C22⋊C4×C12712yes—
227C2×D4—22no—
228C2×D4—22no—
229C2×D4—21no—
230C2×D4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—44no—
237C22×C4C22⋊C4×C12712yes—
238C22×C4C22×C4×C12711yescharacteristic
239C22×C4C2×D4×C12711yescharacteristic
240C22×C4C2×D4×C12711yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—42no—
244C2×C8—42no—
245C4⋊C4—28no—
246C4⋊C4—21no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4C22⋊C4×C12712yes—
252C4⋊C4C22⋊C4×C12714yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—24no—
258C42C2×D4×C12712yes—
259C42C2×D4×C12712yes—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—42no—
264C23—42no—
265C23C2×C22⋊C4×C12711yescharacteristic
266Q8—24no—
267Q8—42no—
268Q8—44no—
269Q8—44no—
270D4—42no—
271D4—42no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—41no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4C2×C22⋊C4×C12711yescharacteristic
286C2×C4C2×C22⋊C4×C12711yescharacteristic
287C2×C4#120×C12712yes—
288C2×C4#120×C12712yes—
289C2×C4#135×C12712yes—
290C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
291C2×C4#137×C12711yescharacteristic
292C8—82no—
293C8—82no—
294C22—21no—
295C22—21no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—82no—
300C22—82no—
301C22#379×C12711yescharacteristic
302C22#416×C12711yescharacteristic
303C22#417×C12711yescharacteristic
304C4—21no—
305C4—22no—
306C4—42no—
307C4—42no—
308C4—44no—
309C4—42no—
310C4#413×C12712yes—
311C4#449×C12712yes—
312C4#452×C12712yes—
313C2—21no—
314C2—21no—
315C2—82no—
316C2C127×#178511yescharacteristic
317C2C127×#178811yescharacteristic
318C2C127×#179611yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC22×D4×C127D4
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
3. C127×#1546C2semidirect
4. C127×#1487C2semidirect
153. #8389C127direct (N × H)
162. #1546C2×C127semidirect
163. #1487C2×C127semidirect
165. C127#8389direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#556×C1272
C127×#18002
C127×#15462
C127×#14872
C127×#14602
#8389127