Finite Group Databasebeta

← C127×#8334 #44572007

Subgroups — C127×#8334 #44572007

Order 32512.

1170 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 182 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#124×C127C211yesmaximal, characteristic
2C127×#1749C212yesmaximal, complemented
3C2×#329×C127C214yesmaximal, complemented
4C23×Q8×C127C411yescharacteristic
5C23×D4×C127C411yescharacteristic
6C2×#124×C127—28no—
7C2×#124×C127—22no—
8C2×#124×C127C412yes—
9C2×#124×C127C2212yes—
10C22×C22⋊C4×C127—22no—
11C22×C42×C127C2211yescharacteristic
12C2×C22⋊C8×C127—22no—
13C2×#97×C127C2214yescomplemented
14#336×C127—22no—
15#329×C127—28no—
16C25×C127—21no—
17C22×Q8×C127—24no—
18C22×Q8×C127—21no—
19C22×Q8×C127—42no—
20C22×Q8×C127—42no—
21C22×Q8×C127C2×C412yes—
22C22×D4×C127—24no—
23C22×D4×C127—21no—
24C22×D4×C127—42no—
25C22×D4×C127—42no—
26C22×D4×C127C2×C412yes—
27C23×C4×C127—21no—
28C23×C4×C127C2×C411yescharacteristic
29C23×C4×C127D412yes—
30C22×C8×C127—24no—
31#124×C127—216no—
32#124×C127—24no—
33#124×C127—24no—
34#124×C127—24no—
35#124×C127D418yes—
36C2×C22⋊C4×C127—216no—
37C2×C22⋊C4×C127—24no—
38C2×C22⋊C4×C127—24no—
39C2×C42×C127—24no—
40C2×C42×C127C2×C411yescharacteristic
41C2×C42×C127D412yes—
42C2×C42×C127C2311yescharacteristic
43C22⋊C8×C127—28no—
44#97×C127—28no—
45C24×C127—24no—
46C24×C127—22no—
47C24×C127—42no—
48C24×C127—44no—
49C24×C127—42no—
50C24×C127—44no—
51C24×C127C22⋊C411yescharacteristic
52C2×Q8×C127—24no—
53C2×Q8×C127—22no—
54C2×Q8×C127—416no—
55C2×Q8×C127—42no—
56C2×Q8×C127—42no—
57C2×Q8×C127—42no—
58C2×Q8×C127—42no—
59C2×Q8×C127C22⋊C414yes—
60C2×D4×C127—24no—
61C2×D4×C127—22no—
62C2×D4×C127—42no—
63C2×D4×C127—42no—
64C2×D4×C127—416no—
65C2×D4×C127—42no—
66C2×D4×C127—42no—
67C2×D4×C127C22⋊C414yes—
68C22×C4×C127—22no—
69C22×C4×C127—22no—
70C22×C4×C127—22no—
71C22×C4×C127—24no—
72C22×C4×C127—24no—
73C22×C4×C127—22no—
74C22×C4×C127—24no—
75C22×C4×C127—42no—
76C22×C4×C127—42no—
77C22×C4×C127—42no—
78C22×C4×C127C22⋊C412yes—
79C22×C4×C127C22⋊C411yescharacteristic
80C22×C4×C127C22×C411yescharacteristic
81C22×C4×C127C2×D412yes—
82C2×C8×C127—28no—
83C2×C8×C127—44no—
84C2×C8×C127—44no—
85C22⋊C4×C127—216no—
86C22⋊C4×C127—216no—
87C42×C127—22no—
88C42×C127—24no—
89C42×C127C2×D414yes—
90C23×C127—21no—
91C23×C127—21no—
92C23×C127—22no—
93C23×C127—24no—
94C23×C127—48no—
95C23×C127—416no—
96C23×C127—42no—
97C23×C127—42no—
98C23×C127—44no—
99C23×C127—44no—
100C23×C127—88no—
101C23×C127—84no—
102C23×C127—84no—
103C23×C127#10012yes—
104C23×C127#10414yes—
105C23×C127C2×C22⋊C411yescharacteristic
106Q8×C127—44no—
107Q8×C127—48no—
108Q8×C127—44no—
109D4×C127—44no—
110D4×C127—48no—
111D4×C127—44no—
112C2×C4×C127—28no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—21no—
116C2×C4×C127—21no—
117C2×C4×C127—22no—
118C2×C4×C127—22no—
119C2×C4×C127—22no—
120C2×C4×C127—24no—
121C2×C4×C127—22no—
122C2×C4×C127—22no—
123C2×C4×C127—48no—
124C2×C4×C127—48no—
125C2×C4×C127—42no—
126C2×C4×C127—42no—
127C2×C4×C127#10012yes—
128C2×C4×C127C2×C22⋊C412yes—
129C2×C4×C127#12014yes—
130C8×C127—48no—
131C22×C127—24no—
132C22×C127—22no—
133C22×C127—22no—
134C22×C127—24no—
135C22×C127—44no—
136C22×C127—48no—
137C22×C127—816no—
138C22×C127—82no—
139C22×C127—84no—
140C22×C127—82no—
141C22×C127#32914yescomplemented
142C22×C127#37911yescharacteristic
143C22×C127C2×#10011yescharacteristic
144C22×C127C2×#10011yescharacteristic
145C22×C127C2×#10412yes—
146C22×C127#42112yes—
147C4×C127—22no—
148C4×C127—22no—
149C4×C127—24no—
150C4×C127—44no—
151C4×C127—42no—
152#8334C12711yesmaximal, characteristic, direct product factor, Sylow
153C2×C127—24no—
154C2×C127—84no—
155C2×C127C2×#32911yescharacteristic
156C2×C127#152712yes—
157C2×C127#153311yescharacteristic
158C2×C127#178311yescharacteristic
159C2×C127#178912yes—
160C22×#124C2×C12711yescharacteristic
161#1749C2×C12712yescomplemented
162C2×#329C2×C12714yescomplemented
163C127#833411yescharacteristic, direct product factor, Sylow
164C23×Q8C4×C12711yescharacteristic
165C23×D4C4×C12711yescharacteristic
166C2×#124—28no—
167C2×#124—22no—
168C2×#124C4×C12712yes—
169C2×#124C22×C12712yes—
170C22×C22⋊C4—22no—
171C22×C42C22×C12711yescharacteristic
172C2×C22⋊C8—22no—
173C2×#97C22×C12714yescomplemented
174#336—22no—
175#329—28no—
176C25—21no—
177C22×Q8—24no—
178C22×Q8—21no—
179C22×Q8—42no—
180C22×Q8—42no—
181C22×Q8C2×C4×C12712yes—
182C22×D4—24no—
183C22×D4—21no—
184C22×D4—42no—
185C22×D4—42no—
186C22×D4C2×C4×C12712yes—
187C23×C4—21no—
188C23×C4C2×C4×C12711yescharacteristic
189C23×C4D4×C12712yes—
190C22×C8—24no—
191#124—216no—
192#124—24no—
193#124—24no—
194#124—24no—
195#124D4×C12718yes—
196C2×C22⋊C4—216no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C42—24no—
200C2×C42C2×C4×C12711yescharacteristic
201C2×C42D4×C12712yes—
202C2×C42C23×C12711yescharacteristic
203C22⋊C8—28no—
204#97—28no—
205C24—24no—
206C24—22no—
207C24—42no—
208C24—44no—
209C24—42no—
210C24—44no—
211C24C22⋊C4×C12711yescharacteristic
212C2×Q8—24no—
213C2×Q8—22no—
214C2×Q8—416no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8C22⋊C4×C12714yes—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—416no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4C22⋊C4×C12714yes—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4C22⋊C4×C12712yes—
239C22×C4C22⋊C4×C12711yescharacteristic
240C22×C4C22×C4×C12711yescharacteristic
241C22×C4C2×D4×C12712yes—
242C2×C8—28no—
243C2×C8—44no—
244C2×C8—44no—
245C22⋊C4—216no—
246C22⋊C4—216no—
247C42—22no—
248C42—24no—
249C42C2×D4×C12714yes—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—24no—
254C23—48no—
255C23—416no—
256C23—42no—
257C23—42no—
258C23—44no—
259C23—44no—
260C23—88no—
261C23—84no—
262C23—84no—
263C23#100×C12712yes—
264C23#104×C12714yes—
265C23C2×C22⋊C4×C12711yescharacteristic
266Q8—44no—
267Q8—48no—
268Q8—44no—
269D4—44no—
270D4—48no—
271D4—44no—
272C2×C4—28no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—48no—
284C2×C4—48no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4#100×C12712yes—
288C2×C4C2×C22⋊C4×C12712yes—
289C2×C4#120×C12714yes—
290C8—48no—
291C22—24no—
292C22—22no—
293C22—22no—
294C22—24no—
295C22—44no—
296C22—48no—
297C22—816no—
298C22—82no—
299C22—84no—
300C22—82no—
301C22#329×C12714yescomplemented
302C22#379×C12711yescharacteristic
303C22C2×#100×C12711yescharacteristic
304C22C2×#100×C12711yescharacteristic
305C22C2×#104×C12712yes—
306C22#421×C12712yes—
307C4—22no—
308C4—22no—
309C4—24no—
310C4—44no—
311C4—42no—
312C2—24no—
313C2—84no—
314C2C2×#329×C12711yescharacteristic
315C2C127×#152712yes—
316C2C127×#153311yescharacteristic
317C2C127×#178311yescharacteristic
318C2C127×#178912yes—
subgroupG/H
center, socleC23×C127C2×C22⋊C4
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×C42C23×C127
prime radicalC23×D4×C127C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C22×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1749C2semidirect
3. C2×#329×C127C2semidirect
13. C2×#97×C127C22semidirect
141. C22×C127#329semidirect
152. #8334C127direct (N × H)
161. #1749C2×C127semidirect
162. C2×#329C2×C127semidirect
163. C127#8334direct (N × H)
173. C2×#97C22×C127semidirect
301. C22#329×C127semidirect

Maximal subgroups

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

subgroupindex
C22×#124×C1272
C127×#17492
C2×#329×C1272
#8334127