Finite Group Databasebeta

← C127×#2497 #30635449

Subgroups — C127×#2497 #30635449

Order 16256.

428 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#400×C127C212yesmaximal, complemented
2#396×C127C212yesmaximal
3#389×C127C211yesmaximal, characteristic, complemented
4#389×C127C211yesmaximal, characteristic, complemented
5#388×C127C212yesmaximal
6#387×C127C212yesmaximal, complemented
7#384×C127C211yesmaximal, characteristic, complemented
8#380×C127C211yesmaximal, characteristic
9#380×C127C211yesmaximal, characteristic
10C4×C4⋊C4×C127C211yesmaximal, characteristic, complemented
11C4×C22⋊C4×C127C211yesmaximal, characteristic
12C23×C4×C127C2211yescharacteristic
13C2×C4⋊C4×C127C2211yescharacteristic
14C2×C4⋊C4×C127C2211yescharacteristic
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2212yes—
18C2×C4⋊C4×C127C2212yes—
19C2×C4⋊C4×C127C2212yes—
20C2×C4⋊C4×C127C2212yes—
21C2×C22⋊C4×C127C2211yescharacteristic
22C2×C22⋊C4×C127C2212yes—
23C2×C22⋊C4×C127C2212yes—
24C2×C22⋊C4×C127C2211yescharacteristic
25C2×C42×C127C2211yescharacteristic
26C2×C42×C127C2211yescharacteristic
27C2×C42×C127C2212yes—
28#95×C127C2211yescharacteristic
29#95×C127C2211yescharacteristic
30#95×C127C2212yes—
31#95×C127C2212yes—
32#95×C127C2212yes—
33#95×C127C2212yes—
34#95×C127C2212yes—
35C24×C127C2311yescharacteristic
36C22×C4×C127—22no—
37C22×C4×C127—22no—
38C22×C4×C127—22no—
39C22×C4×C127C2311yescharacteristic
40C22×C4×C127C2311yescharacteristic
41C22×C4×C127C2312yes—
42C22×C4×C127C2312yes—
43C22×C4×C127C2312yes—
44C22×C4×C127C2312yes—
45C22×C4×C127C2311yescharacteristic
46C22×C4×C127C2311yescharacteristic
47C22×C4×C127C2311yescharacteristic
48C22×C4×C127C2311yescharacteristic
49C4⋊C4×C127—24no—
50C4⋊C4×C127—22no—
51C4⋊C4×C127—22no—
52C4⋊C4×C127—22no—
53C4⋊C4×C127—22no—
54C4⋊C4×C127—42no—
55C4⋊C4×C127—42no—
56C4⋊C4×C127—42no—
57C22⋊C4×C127—22no—
58C22⋊C4×C127—42no—
59C22⋊C4×C127—42no—
60C22⋊C4×C127Q814yes—
61C42×C127—22no—
62C42×C127—22no—
63C42×C127—42no—
64C23×C127—21no—
65C23×C127—21no—
66C23×C127—21no—
67C23×C127—21no—
68C23×C127—21no—
69C23×C127—21no—
70C23×C127C2×Q812yes—
71C23×C127C2411yescharacteristic
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—22no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—22no—
95C2×C4×C127—21no—
96C2×C4×C127—44no—
97C2×C4×C127C2×Q812yes—
98C2×C4×C127C2×Q812yes—
99C2×C4×C127#4112yes—
100C2×C4×C127#4112yes—
101C2×C4×C127#4112yes—
102C2×C4×C127#4112yes—
103C22×C127—22no—
104C22×C127—22no—
105C22×C127—22no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127—41no—
109C22×C127—41no—
110C22×C127C22×Q811yescharacteristic
111C22×C127C2×#4111yescharacteristic
112C22×C127C2×#4111yescharacteristic
113C22×C127C2×#4111yescharacteristic
114C22×C127C2×#4111yescharacteristic
115C22×C127H(5,𝔽2)11yescharacteristic
116C22×C127H(5,𝔽2)11yescharacteristic
117C4×C127—22no—
118C4×C127—22no—
119C4×C127—22no—
120C4×C127—41no—
121C4×C127—42no—
122C4×C127—42no—
123C4×C127—42no—
124C4×C127—41no—
125C4×C127—42no—
126C4×C127#12614yes—
127C2×C127—42no—
128C2×C127C2×#12611yescharacteristic
129C2×C127#53311yescharacteristic
130C2×C127#54011yescharacteristic
131C2×C127#55311yescharacteristic
132C2×C127#55411yescharacteristic
133C2×C127#55411yescharacteristic
134C2×C127#55511yescharacteristic
135#2497C12711yesmaximal, characteristic, direct product factor, Sylow
136C127#249711yescharacteristic, direct product factor, Sylow
137#400C2×C12712yescomplemented
138#396C2×C12712yes—
139#389C2×C12711yescharacteristic, complemented
140#389C2×C12711yescharacteristic, complemented
141#388C2×C12712yes—
142#387C2×C12712yescomplemented
143#384C2×C12711yescharacteristic, complemented
144#380C2×C12711yescharacteristic
145#380C2×C12711yescharacteristic
146C4×C4⋊C4C2×C12711yescharacteristic, complemented
147C4×C22⋊C4C2×C12711yescharacteristic
148C23×C4C22×C12711yescharacteristic
149C2×C4⋊C4C22×C12711yescharacteristic
150C2×C4⋊C4C22×C12711yescharacteristic
151C2×C4⋊C4C22×C12711yescharacteristic
152C2×C4⋊C4C22×C12711yescharacteristic
153C2×C4⋊C4C22×C12712yes—
154C2×C4⋊C4C22×C12712yes—
155C2×C4⋊C4C22×C12712yes—
156C2×C4⋊C4C22×C12712yes—
157C2×C22⋊C4C22×C12711yescharacteristic
158C2×C22⋊C4C22×C12712yes—
159C2×C22⋊C4C22×C12712yes—
160C2×C22⋊C4C22×C12711yescharacteristic
161C2×C42C22×C12711yescharacteristic
162C2×C42C22×C12711yescharacteristic
163C2×C42C22×C12712yes—
164#95C22×C12711yescharacteristic
165#95C22×C12711yescharacteristic
166#95C22×C12712yes—
167#95C22×C12712yes—
168#95C22×C12712yes—
169#95C22×C12712yes—
170#95C22×C12712yes—
171C24C23×C12711yescharacteristic
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4C23×C12711yescharacteristic
176C22×C4C23×C12711yescharacteristic
177C22×C4C23×C12712yes—
178C22×C4C23×C12712yes—
179C22×C4C23×C12712yes—
180C22×C4C23×C12712yes—
181C22×C4C23×C12711yescharacteristic
182C22×C4C23×C12711yescharacteristic
183C22×C4C23×C12711yescharacteristic
184C22×C4C23×C12711yescharacteristic
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C22⋊C4—22no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4Q8×C12714yes—
197C42—22no—
198C42—22no—
199C42—42no—
200C23—21no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23C2×Q8×C12712yes—
207C23C24×C12711yescharacteristic
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—44no—
233C2×C4C2×Q8×C12712yes—
234C2×C4C2×Q8×C12712yes—
235C2×C4#41×C12712yes—
236C2×C4#41×C12712yes—
237C2×C4#41×C12712yes—
238C2×C4#41×C12712yes—
239C22—22no—
240C22—22no—
241C22—22no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22C22×Q8×C12711yescharacteristic
247C22C2×#41×C12711yescharacteristic
248C22C2×#41×C12711yescharacteristic
249C22C2×#41×C12711yescharacteristic
250C22C2×#41×C12711yescharacteristic
251C22H(5,𝔽2)×C12711yescharacteristic
252C22H(5,𝔽2)×C12711yescharacteristic
253C4—22no—
254C4—22no—
255C4—22no—
256C4—41no—
257C4—42no—
258C4—42no—
259C4—42no—
260C4—41no—
261C4—42no—
262C4#126×C12714yes—
263C2—42no—
264C2C2×#126×C12711yescharacteristic
265C2#533×C12711yescharacteristic
266C2#540×C12711yescharacteristic
267C2#553×C12711yescharacteristic
268C2#554×C12711yescharacteristic
269C2#554×C12711yescharacteristic
270C2#555×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC24×C127C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. #400×C127C2semidirect
3. #389×C127C2semidirect
4. #389×C127C2semidirect
6. #387×C127C2semidirect
7. #384×C127C2semidirect
10. C4×C4⋊C4×C127C2semidirect
135. #2497C127direct (N × H)
136. C127#2497direct (N × H)
137. #400C2×C127semidirect
139. #389C2×C127semidirect
140. #389C2×C127semidirect
142. #387C2×C127semidirect
143. #384C2×C127semidirect
146. C4×C4⋊C4C2×C127semidirect

Maximal subgroups

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

subgroupindex
#400×C1272
#396×C1272
#389×C1272
#388×C1272
#387×C1272
#384×C1272
#380×C1272
C4×C4⋊C4×C1272
C4×C22⋊C4×C1272
#2497127