Finite Group Databasebeta

← C127×#8462 #44572135

Subgroups — C127×#8462 #44572135

Order 32512.

1014 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1014 conjugacy classes of subgroups — too many to draw. Only its 150 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
1C127×#3485C211yesmaximal, characteristic
2C127×#1782C211yesmaximal, characteristic, complemented
3C127×#1780C211yesmaximal, characteristic, complemented
4C127×#1516C214yesmaximal, complemented
5C2×H(5,𝔽2)×C127C411yescharacteristic, complemented
6C22×#41×C127C2211yescharacteristic
7#547×C127—24no—
8#546×C127—24no—
9#536×C127C412yescomplemented
10#534×C127—22no—
11#525×C127C2212yes—
12C2×#123×C127—22no—
13#518×C127C411yescharacteristic, complemented
14C2×#99×C127—22no—
15C2×C22⋊C8×C127—22no—
16C2×C22⋊C8×C127C2212yescomplemented
17#381×C127C2212yescomplemented
18#328×C127—24no—
19#327×C127—24no—
20H(5,𝔽2)×C127—44no—
21C2×#41×C127—22no—
22C2×#41×C127—24no—
23C2×#41×C127—44no—
24C2×#41×C127C2×C412yes—
25C22×Q8×C127C2×C411yescharacteristic
26C22×D4×C127—22no—
27C22×D4×C127—22no—
28C22×D4×C127C2×C411yescharacteristic
29C22×D4×C127D412yes—
30C23×C4×C127D412yes—
31C23×C4×C127C2311yescharacteristic
32C22×C8×C127—22no—
33C4⋊Q8×C127—22no—
34#124×C127—22no—
35#123×C127—28no—
36#123×C127—24no—
37#123×C127—24no—
38#123×C127—22no—
39C22⋊Q8×C127—24no—
40C22⋊Q8×C127D414yes—
41#121×C127—28no—
42#121×C127D414yes—
43#120×C127—24no—
44#120×C127—22no—
45C4×D4×C127—24no—
46C4×D4×C127—24no—
47#117×C127C2×C412yes—
48C2×C4⋊C4×C127—24no—
49C2×C22⋊C4×C127—24no—
50C2×C22⋊C4×C127—22no—
51C2×C22⋊C4×C127—22no—
52#99×C127—44no—
53C22⋊C8×C127—24no—
54C22⋊C8×C127—44no—
55#95×C127—24no—
56C24×C127—22no—
57C24×C127C22⋊C412yes—
58C24×C127C2×D411yescharacteristic
59#41×C127—44no—
60#41×C127—416no—
61#41×C127—48no—
62C2×Q8×C127—22no—
63C2×Q8×C127—42no—
64C2×Q8×C127C22⋊C412yes—
65C2×D4×C127—24no—
66C2×D4×C127—24no—
67C2×D4×C127—24no—
68C2×D4×C127—22no—
69C2×D4×C127—24no—
70C2×D4×C127—24no—
71C2×D4×C127—22no—
72C2×D4×C127—42no—
73C2×D4×C127—48no—
74C2×D4×C127—44no—
75C2×D4×C127—44no—
76C2×D4×C127C22⋊C412yes—
77C22×C4×C127—22no—
78C22×C4×C127—24no—
79C22×C4×C127—24no—
80C22×C4×C127—22no—
81C22×C4×C127—24no—
82C22×C4×C127—42no—
83C22×C4×C127—42no—
84C22×C4×C127—42no—
85C22×C4×C127C22⋊C412yes—
86C22×C4×C127C22×C411yescharacteristic
87C22×C4×C127C2×D411yescharacteristic
88C22×C4×C127C2×D414yes—
89C2×C8×C127—44no—
90C2×C8×C127—42no—
91C4⋊C4×C127—28no—
92C4⋊C4×C127—24no—
93C4⋊C4×C127C22⋊C414yes—
94C22⋊C4×C127—24no—
95C22⋊C4×C127—28no—
96C22⋊C4×C127—22no—
97C22⋊C4×C127—24no—
98C22⋊C4×C127—24no—
99C22⋊C4×C127—44no—
100C42×C127—22no—
101C23×C127—22no—
102C23×C127—22no—
103C23×C127—22no—
104C23×C127—22no—
105C23×C127—42no—
106C23×C127—44no—
107C23×C127—44no—
108C23×C127—84no—
109C23×C127#9914yes—
110C23×C127C2×C22⋊C411yescharacteristic
111C23×C127#12012yes—
112Q8×C127—42no—
113Q8×C127—44no—
114D4×C127—28no—
115D4×C127—42no—
116D4×C127—48no—
117D4×C127—48no—
118D4×C127—44no—
119D4×C127—84no—
120C2×C4×C127—22no—
121C2×C4×C127—22no—
122C2×C4×C127—24no—
123C2×C4×C127—22no—
124C2×C4×C127—22no—
125C2×C4×C127—22no—
126C2×C4×C127—22no—
127C2×C4×C127—44no—
128C2×C4×C127—48no—
129C2×C4×C127—44no—
130C2×C4×C127—44no—
131C2×C4×C127—84no—
132C2×C4×C127—82no—
133C2×C4×C127—84no—
134C2×C4×C127C2×C22⋊C412yes—
135C2×C4×C127#12012yes—
136C8×C127—82no—
137C22×C127—22no—
138C22×C127—24no—
139C22×C127—24no—
140C22×C127—22no—
141C22×C127—44no—
142C22×C127—44no—
143C22×C127—44no—
144C22×C127—42no—
145C22×C127—82no—
146C22×C127—82no—
147C22×C127—164no—
148C22×C127#37911yescharacteristic
149C22×C127C2×#9912yes—
150C4×C127—22no—
151C4×C127—22no—
152C4×C127—44no—
153C4×C127—42no—
154C4×C127—42no—
155C4×C127—162no—
156#8462C12711yesmaximal, characteristic, direct product factor, Sylow
157C2×C127—22no—
158C2×C127—42no—
159C2×C127—44no—
160C2×C127—82no—
161C2×C127#178111yescharacteristic
162C2×C127#179412yes—
163#3485C2×C12711yescharacteristic
164#1782C2×C12711yescharacteristic, complemented
165#1780C2×C12711yescharacteristic, complemented
166#1516C2×C12714yescomplemented
167C127#846211yescharacteristic, direct product factor, Sylow
168C2×H(5,𝔽2)C4×C12711yescharacteristic, complemented
169C22×#41C22×C12711yescharacteristic
170#547—24no—
171#546—24no—
172#536C4×C12712yescomplemented
173#534—22no—
174#525C22×C12712yes—
175C2×#123—22no—
176#518C4×C12711yescharacteristic, complemented
177C2×#99—22no—
178C2×C22⋊C8—22no—
179C2×C22⋊C8C22×C12712yescomplemented
180#381C22×C12712yescomplemented
181#328—24no—
182#327—24no—
183H(5,𝔽2)—44no—
184C2×#41—22no—
185C2×#41—24no—
186C2×#41—44no—
187C2×#41C2×C4×C12712yes—
188C22×Q8C2×C4×C12711yescharacteristic
189C22×D4—22no—
190C22×D4—22no—
191C22×D4C2×C4×C12711yescharacteristic
192C22×D4D4×C12712yes—
193C23×C4D4×C12712yes—
194C23×C4C23×C12711yescharacteristic
195C22×C8—22no—
196C4⋊Q8—22no—
197#124—22no—
198#123—28no—
199#123—24no—
200#123—24no—
201#123—22no—
202C22⋊Q8—24no—
203C22⋊Q8D4×C12714yes—
204#121—28no—
205#121D4×C12714yes—
206#120—24no—
207#120—22no—
208C4×D4—24no—
209C4×D4—24no—
210#117C2×C4×C12712yes—
211C2×C4⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215#99—44no—
216C22⋊C8—24no—
217C22⋊C8—44no—
218#95—24no—
219C24—22no—
220C24C22⋊C4×C12712yes—
221C24C2×D4×C12711yescharacteristic
222#41—44no—
223#41—416no—
224#41—48no—
225C2×Q8—22no—
226C2×Q8—42no—
227C2×Q8C22⋊C4×C12712yes—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—22no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—42no—
236C2×D4—48no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4C22⋊C4×C12712yes—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4C22⋊C4×C12712yes—
249C22×C4C22×C4×C12711yescharacteristic
250C22×C4C2×D4×C12711yescharacteristic
251C22×C4C2×D4×C12714yes—
252C2×C8—44no—
253C2×C8—42no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4C22⋊C4×C12714yes—
257C22⋊C4—24no—
258C22⋊C4—28no—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—44no—
263C42—22no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—44no—
270C23—44no—
271C23—84no—
272C23#99×C12714yes—
273C23C2×C22⋊C4×C12711yescharacteristic
274C23#120×C12712yes—
275Q8—42no—
276Q8—44no—
277D4—28no—
278D4—42no—
279D4—48no—
280D4—48no—
281D4—44no—
282D4—84no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—84no—
297C2×C4C2×C22⋊C4×C12712yes—
298C2×C4#120×C12712yes—
299C8—82no—
300C22—22no—
301C22—24no—
302C22—22no—
303C22—24no—
304C22—44no—
305C22—44no—
306C22—42no—
307C22—44no—
308C22—82no—
309C22—82no—
310C22—164no—
311C22#379×C12711yescharacteristic
312C22C2×#99×C12712yes—
313C4—22no—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—44no—
318C4—162no—
319C2—22no—
320C2—42no—
321C2—44no—
322C2—82no—
323C2C127×#178111yescharacteristic
324C2C127×#179412yes—
subgroupG/H
center, socleC22×C127#379
commutatorC22×C4C22×C4×C127
Frattini, cosocleC23×C4C23×C127
prime radicalC127×#3485C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1782C2semidirect
3. C127×#1780C2semidirect
4. C127×#1516C2semidirect
5. C2×H(5,𝔽2)×C127C4semidirect
9. #536×C127C4semidirect
13. #518×C127C4semidirect
16. C2×C22⋊C8×C127C22semidirect
17. #381×C127C22semidirect
156. #8462C127direct (N × H)
164. #1782C2×C127semidirect
165. #1780C2×C127semidirect
166. #1516C2×C127semidirect
167. C127#8462direct (N × H)
168. C2×H(5,𝔽2)C4×C127semidirect
172. #536C4×C127semidirect
176. #518C4×C127semidirect
179. C2×C22⋊C8C22×C127semidirect
180. #381C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#34852
C127×#17822
C127×#17802
C127×#15162
#8462127