Finite Group Databasebeta

← C127×#8394 #44572067

Subgroups — C127×#8394 #44572067

Order 32512.

738 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556×C127C211yesmaximal, characteristic
2C127×#1801C211yesmaximal, characteristic, complemented
3C127×#1548C212yesmaximal, complemented
4C127×#1486C212yesmaximal
5C127×#1461C211yesmaximal, characteristic, complemented
6#556×C127—24no—
7#556×C127—24no—
8C2×C4⋊Q8×C127C2211yescharacteristic
9C2×#126×C127—22no—
10C2×#124×C127C412yes—
11C2×C22⋊Q8×C127—21no—
12C2×C4×Q8×C127C412yes—
13C2×C4×Q8×C127C2211yescharacteristic
14C2×C4×D4×C127C2211yescharacteristic
15C2×#117×C127—21no—
16#431×C127C2212yes—
17#423×C127C2212yes—
18C2×#101×C127—21no—
19#343×C127—21no—
20#333×C127—24no—
21D4⋊C8×C127—24no—
22#324×C127—22no—
23C22×Q8×C127C2×C412yes—
24C22×Q8×C127D411yescharacteristic
25C22×D4×C127C2×C411yescharacteristic
26C23×C4×C127—21no—
27C2×M16×C127—22no—
28C4⋊Q8×C127—22no—
29C4⋊Q8×C127D414yes—
30#126×C127—216no—
31#124×C127—24no—
32#124×C127—24no—
33C22⋊Q8×C127—28no—
34C4×Q8×C127—24no—
35C4×Q8×C127—24no—
36C4×Q8×C127—22no—
37C4×Q8×C127—22no—
38C4×D4×C127—22no—
39C4×D4×C127D414yes—
40#117×C127—28no—
41C2×C4⋊C4×C127—22no—
42C2×C4⋊C4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127C2×C411yescharacteristic
45C2×C4⋊C4×C127D411yescharacteristic
46C2×C4⋊C4×C127D412yes—
47C2×C22⋊C4×C127—22no—
48C2×C22⋊C4×C127—22no—
49C2×C22⋊C4×C127—21no—
50C2×C42×C127—22no—
51C2×C42×C127C2×C412yes—
52C2×C42×C127C2311yescharacteristic
53C4⋊C8×C127—22no—
54C4⋊C8×C127—22no—
55C4⋊C8×C127—22no—
56#101×C127—42no—
57C22⋊C8×C127—22no—
58C4×C8×C127—22no—
59C24×C127—21no—
60C2×Q8×C127—22no—
61C2×Q8×C127—22no—
62C2×Q8×C127—22no—
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127—44no—
66C2×Q8×C127C22⋊C414yes—
67C2×D4×C127—21no—
68C2×D4×C127—42no—
69C2×D4×C127C22⋊C412yes—
70C22×C4×C127—21no—
71C22×C4×C127—22no—
72C22×C4×C127—24no—
73C22×C4×C127—42no—
74C22×C4×C127—41no—
75C22×C4×C127—41no—
76C22×C4×C127C22⋊C412yes—
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—24no—
88C4⋊C4×C127—24no—
89C4⋊C4×C127—22no—
90C4⋊C4×C127—24no—
91C4⋊C4×C127C22⋊C412yes—
92C22⋊C4×C127—28no—
93C22⋊C4×C127—28no—
94C22⋊C4×C127—24no—
95C42×C127—28no—
96C42×C127—22no—
97C42×C127—22no—
98C42×C127C2×D412yes—
99C42×C127C2×D412yes—
100C23×C127—22no—
101C23×C127—42no—
102C23×C127—42no—
103C23×C127—41no—
104C23×C127—41no—
105C23×C127C2×C22⋊C411yescharacteristic
106Q8×C127—44no—
107Q8×C127—44no—
108Q8×C127—42no—
109Q8×C127—42no—
110D4×C127—24no—
111D4×C127—42no—
112C2×C4×C127—22no—
113C2×C4×C127—22no—
114C2×C4×C127—22no—
115C2×C4×C127—24no—
116C2×C4×C127—22no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—41no—
120C2×C4×C127—44no—
121C2×C4×C127—41no—
122C2×C4×C127—41no—
123C2×C4×C127—44no—
124C2×C4×C127—44no—
125C2×C4×C127—42no—
126C2×C4×C127C2×C22⋊C411yescharacteristic
127C2×C4×C127C2×C22⋊C411yescharacteristic
128C2×C4×C127#12012yes—
129C2×C4×C127#12012yes—
130C2×C4×C127#13512yes—
131C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C127#13711yescharacteristic
133C8×C127—82no—
134C8×C127—82no—
135C22×C127—21no—
136C22×C127—21no—
137C22×C127—42no—
138C22×C127—42no—
139C22×C127—42no—
140C22×C127—82no—
141C22×C127—82no—
142C22×C127#37911yescharacteristic
143C22×C127#41611yescharacteristic
144C22×C127#41711yescharacteristic
145C4×C127—22no—
146C4×C127—21no—
147C4×C127—42no—
148C4×C127—44no—
149C4×C127—42no—
150C4×C127—42no—
151C4×C127#41312yes—
152C4×C127#44912yes—
153C4×C127#45212yes—
154#8394C12711yesmaximal, characteristic, direct product factor, Sylow
155C2×C127—21no—
156C2×C127—21no—
157C2×C127—82no—
158C2×C127#178511yescharacteristic
159C2×C127#178811yescharacteristic
160C2×C127#179611yescharacteristic
161C2×#556C2×C12711yescharacteristic
162#1801C2×C12711yescharacteristic, complemented
163#1548C2×C12712yescomplemented
164#1486C2×C12712yes—
165#1461C2×C12711yescharacteristic, complemented
166C127#839411yescharacteristic, direct product factor, Sylow
167#556—24no—
168#556—24no—
169C2×C4⋊Q8C22×C12711yescharacteristic
170C2×#126—22no—
171C2×#124C4×C12712yes—
172C2×C22⋊Q8—21no—
173C2×C4×Q8C4×C12712yes—
174C2×C4×Q8C22×C12711yescharacteristic
175C2×C4×D4C22×C12711yescharacteristic
176C2×#117—21no—
177#431C22×C12712yes—
178#423C22×C12712yes—
179C2×#101—21no—
180#343—21no—
181#333—24no—
182D4⋊C8—24no—
183#324—22no—
184C22×Q8C2×C4×C12712yes—
185C22×Q8D4×C12711yescharacteristic
186C22×D4C2×C4×C12711yescharacteristic
187C23×C4—21no—
188C2×M16—22no—
189C4⋊Q8—22no—
190C4⋊Q8D4×C12714yes—
191#126—216no—
192#124—24no—
193#124—24no—
194C22⋊Q8—28no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×D4—22no—
200C4×D4D4×C12714yes—
201#117—28no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×C4×C12711yescharacteristic
206C2×C4⋊C4D4×C12711yescharacteristic
207C2×C4⋊C4D4×C12712yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C42—22no—
212C2×C42C2×C4×C12712yes—
213C2×C42C23×C12711yescharacteristic
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217#101—42no—
218C22⋊C8—22no—
219C4×C8—22no—
220C24—21no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—44no—
227C2×Q8C22⋊C4×C12714yes—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4C22⋊C4×C12712yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4C22⋊C4×C12712yes—
238C22×C4C22⋊C4×C12712yes—
239C22×C4C22×C4×C12711yescharacteristic
240C22×C4C2×D4×C12711yescharacteristic
241C22×C4C2×D4×C12711yescharacteristic
242M16—42no—
243M16—42no—
244C2×C8—42no—
245C2×C8—42no—
246C4⋊C4—28no—
247C4⋊C4—21no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4C22⋊C4×C12712yes—
253C22⋊C4—28no—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C42—28no—
257C42—22no—
258C42—22no—
259C42C2×D4×C12712yes—
260C42C2×D4×C12712yes—
261C23—22no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊C4×C12711yescharacteristic
267Q8—44no—
268Q8—44no—
269Q8—42no—
270Q8—42no—
271D4—24no—
272D4—42no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—42no—
287C2×C4C2×C22⋊C4×C12711yescharacteristic
288C2×C4C2×C22⋊C4×C12711yescharacteristic
289C2×C4#120×C12712yes—
290C2×C4#120×C12712yes—
291C2×C4#135×C12712yes—
292C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
293C2×C4#137×C12711yescharacteristic
294C8—82no—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—82no—
302C22—82no—
303C22#379×C12711yescharacteristic
304C22#416×C12711yescharacteristic
305C22#417×C12711yescharacteristic
306C4—21no—
307C4—22no—
308C4—44no—
309C4—42no—
310C4—42no—
311C4—42no—
312C4#413×C12712yes—
313C4#449×C12712yes—
314C4#452×C12712yes—
315C2—21no—
316C2—21no—
317C2—82no—
318C2C127×#178511yescharacteristic
319C2C127×#178811yescharacteristic
320C2C127×#179611yescharacteristic
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC22×D4×C127C2×C4
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
2. C127×#1801C2semidirect
3. C127×#1548C2semidirect
5. C127×#1461C2semidirect
154. #8394C127direct (N × H)
162. #1801C2×C127semidirect
163. #1548C2×C127semidirect
165. #1461C2×C127semidirect
166. C127#8394direct (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×#18012
C127×#15482
C127×#14862
C127×#14612
#8394127