Finite Group Databasebeta

← C127×#8317 #44571990

Subgroups — C127×#8317 #44571990

Order 32512.

864 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 864 conjugacy classes of subgroups — too many to draw. Only its 162 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×#3459C211yesmaximal, characteristic
2C127×#1869C213yesmaximal, complemented
3C127×#1459C213yesmaximal, complemented
4C23×C8×C127C2211yescharacteristic
5#543×C127C411yescharacteristic
6#539×C127—23no—
7#538×C127—23no—
8#534×C127C413yes—
9#529×C127—23no—
10#525×C127—23no—
11C2×C4×D4×C127C2213yes—
12C2×C22⋊C8×C127—23no—
13C2×C22⋊C8×C127C2213yes—
14#336×C127—23no—
15#324×C127—23no—
16#323×C127—23no—
17C2×#41×C127—21no—
18C22×D4×C127C2×C413yes—
19C23×C4×C127D413yes—
20C23×C4×C127C2311yescharacteristic
21C22×C8×C127—23no—
22C22×C8×C127—23no—
23C22×C8×C127—42no—
24#126×C127—26no—
25#125×C127—23no—
26#124×C127—23no—
27#123×C127—26no—
28#123×C127—26no—
29C22⋊Q8×C127—23no—
30C22⋊Q8×C127—23no—
31C22⋊Q8×C127—23no—
32#121×C127—23no—
33#121×C127—23no—
34#121×C127—23no—
35#120×C127—26no—
36C4×Q8×C127—21no—
37C4×D4×C127—23no—
38C4×D4×C127—26no—
39C4×D4×C127—26no—
40#117×C127—23no—
41C2×C4⋊C4×C127D413yes—
42C2×C22⋊C4×C127C2×C413yes—
43C2×C22⋊C4×C127D413yes—
44C2×C42×C127D413yes—
45C22⋊C8×C127—26no—
46C22⋊C8×C127—46no—
47C24×C127C22⋊C413yes—
48C24×C127C22×C411yescharacteristic
49#41×C127—81no—
50#41×C127—81no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×D4×C127—26no—
54C2×D4×C127—23no—
55C2×D4×C127—23no—
56C2×D4×C127—26no—
57C2×D4×C127—23no—
58C2×D4×C127—23no—
59C22×C4×C127—23no—
60C22×C4×C127—23no—
61C22×C4×C127—41no—
62C22×C4×C127—41no—
63C22×C4×C127—43no—
64C22×C4×C127—43no—
65C22×C4×C127C22⋊C413yes—
66C22×C4×C127C22⋊C413yes—
67C22×C4×C127C22⋊C413yes—
68C22×C4×C127C2×D413yes—
69C22×C4×C127C2×D413yes—
70C2×C8×C127—26no—
71C2×C8×C127—43no—
72C2×C8×C127—41no—
73C2×C8×C127—43no—
74C2×C8×C127—43no—
75C2×C8×C127—43no—
76C4⋊C4×C127—23no—
77C4⋊C4×C127—23no—
78C4⋊C4×C127—23no—
79C4⋊C4×C127—23no—
80C4⋊C4×C127—26no—
81C22⋊C4×C127—26no—
82C22⋊C4×C127—26no—
83C22⋊C4×C127—26no—
84C22⋊C4×C127—26no—
85C42×C127—23no—
86C42×C127—23no—
87C23×C127—23no—
88C23×C127—23no—
89C23×C127—23no—
90C23×C127—41no—
91C23×C127—41no—
92C23×C127—43no—
93C23×C127—43no—
94C23×C127C2×C22⋊C413yes—
95C23×C127#13111yescharacteristic
96C23×C127#13113yes—
97Q8×C127—81no—
98Q8×C127—81no—
99D4×C127—46no—
100D4×C127—46no—
101D4×C127—83no—
102D4×C127—83no—
103C2×C4×C127—23no—
104C2×C4×C127—23no—
105C2×C4×C127—23no—
106C2×C4×C127—23no—
107C2×C4×C127—23no—
108C2×C4×C127—23no—
109C2×C4×C127—23no—
110C2×C4×C127—43no—
111C2×C4×C127—43no—
112C2×C4×C127—43no—
113C2×C4×C127—43no—
114C2×C4×C127—43no—
115C2×C4×C127—83no—
116C2×C4×C127#12013yes—
117C2×C4×C127#12011yescharacteristic
118C8×C127—42no—
119C8×C127—83no—
120C22×C127—23no—
121C22×C127—23no—
122C22×C127—41no—
123C22×C127—43no—
124C22×C127—46no—
125C22×C127—43no—
126C22×C127—43no—
127C22×C127—43no—
128C22×C127—83no—
129C22×C127#37911yescharacteristic
130C22×C127#40813yes—
131C22×C127#40813yes—
132C4×C127—23no—
133C4×C127—43no—
134C4×C127—43no—
135C4×C127—43no—
136C4×C127B(4,𝔽2)11yescharacteristic
137C4×C127#45811yescharacteristic
138#8317C12711yesmaximal, characteristic, direct product factor, Sylow
139C2×C127—23no—
140C2×C127—23no—
141C2×C127—83no—
142C2×C127#178011yescharacteristic
143C2×C127#179011yescharacteristic
144C2×C127#179111yescharacteristic
145#3459C2×C12711yescharacteristic
146#1869C2×C12713yescomplemented
147#1459C2×C12713yescomplemented
148C127#831711yescharacteristic, direct product factor, Sylow
149C23×C8C22×C12711yescharacteristic
150#543C4×C12711yescharacteristic
151#539—23no—
152#538—23no—
153#534C4×C12713yes—
154#529—23no—
155#525—23no—
156C2×C4×D4C22×C12713yes—
157C2×C22⋊C8—23no—
158C2×C22⋊C8C22×C12713yes—
159#336—23no—
160#324—23no—
161#323—23no—
162C2×#41—21no—
163C22×D4C2×C4×C12713yes—
164C23×C4D4×C12713yes—
165C23×C4C23×C12711yescharacteristic
166C22×C8—23no—
167C22×C8—23no—
168C22×C8—42no—
169#126—26no—
170#125—23no—
171#124—23no—
172#123—26no—
173#123—26no—
174C22⋊Q8—23no—
175C22⋊Q8—23no—
176C22⋊Q8—23no—
177#121—23no—
178#121—23no—
179#121—23no—
180#120—26no—
181C4×Q8—21no—
182C4×D4—23no—
183C4×D4—26no—
184C4×D4—26no—
185#117—23no—
186C2×C4⋊C4D4×C12713yes—
187C2×C22⋊C4C2×C4×C12713yes—
188C2×C22⋊C4D4×C12713yes—
189C2×C42D4×C12713yes—
190C22⋊C8—26no—
191C22⋊C8—46no—
192C24C22⋊C4×C12713yes—
193C24C22×C4×C12711yescharacteristic
194#41—81no—
195#41—81no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×D4—26no—
199C2×D4—23no—
200C2×D4—23no—
201C2×D4—26no—
202C2×D4—23no—
203C2×D4—23no—
204C22×C4—23no—
205C22×C4—23no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—43no—
209C22×C4—43no—
210C22×C4C22⋊C4×C12713yes—
211C22×C4C22⋊C4×C12713yes—
212C22×C4C22⋊C4×C12713yes—
213C22×C4C2×D4×C12713yes—
214C22×C4C2×D4×C12713yes—
215C2×C8—26no—
216C2×C8—43no—
217C2×C8—41no—
218C2×C8—43no—
219C2×C8—43no—
220C2×C8—43no—
221C4⋊C4—23no—
222C4⋊C4—23no—
223C4⋊C4—23no—
224C4⋊C4—23no—
225C4⋊C4—26no—
226C22⋊C4—26no—
227C22⋊C4—26no—
228C22⋊C4—26no—
229C22⋊C4—26no—
230C42—23no—
231C42—23no—
232C23—23no—
233C23—23no—
234C23—23no—
235C23—41no—
236C23—41no—
237C23—43no—
238C23—43no—
239C23C2×C22⋊C4×C12713yes—
240C23#131×C12711yescharacteristic
241C23#131×C12713yes—
242Q8—81no—
243Q8—81no—
244D4—46no—
245D4—46no—
246D4—83no—
247D4—83no—
248C2×C4—23no—
249C2×C4—23no—
250C2×C4—23no—
251C2×C4—23no—
252C2×C4—23no—
253C2×C4—23no—
254C2×C4—23no—
255C2×C4—43no—
256C2×C4—43no—
257C2×C4—43no—
258C2×C4—43no—
259C2×C4—43no—
260C2×C4—83no—
261C2×C4#120×C12713yes—
262C2×C4#120×C12711yescharacteristic
263C8—42no—
264C8—83no—
265C22—23no—
266C22—23no—
267C22—41no—
268C22—43no—
269C22—46no—
270C22—43no—
271C22—43no—
272C22—43no—
273C22—83no—
274C22#379×C12711yescharacteristic
275C22#408×C12713yes—
276C22#408×C12713yes—
277C4—23no—
278C4—43no—
279C4—43no—
280C4—43no—
281C4B(4,𝔽2)×C12711yescharacteristic
282C4#458×C12711yescharacteristic
283C2—23no—
284C2—23no—
285C2—83no—
286C2C127×#178011yescharacteristic
287C2C127×#179011yescharacteristic
288C2C127×#179111yescharacteristic
subgroupG/H
centerC2×C4×C127#120
commutatorC24C22×C4×C127
Frattini, cosocleC23×C4C23×C127
socleC22×C127#379
prime radicalC127×#3459C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1869C2semidirect
3. C127×#1459C2semidirect
138. #8317C127direct (N × H)
146. #1869C2×C127semidirect
147. #1459C2×C127semidirect
148. C127#8317direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34592
C127×#18692
C127×#14592
#8317127