Finite Group Databasebeta

← C127×#15158 #44578831

Subgroups — C127×#15158 #44578831

Order 32512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×Q8×C127C211yesmaximal, characteristic
2C2×#395×C127C212yesmaximal
3C2×#389×C127C213yesmaximal
4C127×#1441C216yesmaximal
5C127×C4⋊#95C213yesmaximal
6C23×Q8×C127C2211yescharacteristic
7C2×C4×Q8×C127—212no—
8C22×C4⋊C4×C127C2216yes—
9C22×C4⋊C4×C127C2213yes—
10C22×C4⋊C4×C127C2211yescharacteristic
11C22×C4⋊C4×C127C2213yes—
12C22×C42×C127C2213yes—
13#395×C127—28no—
14#389×C127—212no—
15C2×#95×C127C2216yes—
16C2×#95×C127C2216yes—
17C2×#95×C127C2216yes—
18C22×Q8×C127—26no—
19C22×Q8×C127—26no—
20C22×Q8×C127D414yes—
21C23×C4×C127C2312yes—
22C23×C4×C127C2313yes—
23C23×C4×C127C2316yes—
24C23×C4×C127C2313yes—
25C23×C4×C127C2311yescharacteristic
26C4×Q8×C127—232no—
27C2×C4⋊C4×C127—212no—
28C2×C4⋊C4×C127—26no—
29C2×C4⋊C4×C127—26no—
30C2×C4⋊C4×C127—26no—
31C2×C4⋊C4×C127—26no—
32C2×C4⋊C4×C127—212no—
33C2×C4⋊C4×C127—212no—
34C2×C4⋊C4×C127—212no—
35C2×C4⋊C4×C127Q8112yescomplemented
36C2×C42×C127—212no—
37C2×C42×C127D4112yes—
38#95×C127—212no—
39#95×C127—212no—
40#95×C127—46no—
41C24×C127C2411yescharacteristic
42C2×Q8×C127—28no—
43C2×Q8×C127—224no—
44C2×Q8×C127—224no—
45C22×C4×C127—26no—
46C22×C4×C127—26no—
47C22×C4×C127—26no—
48C22×C4×C127—26no—
49C22×C4×C127—26no—
50C22×C4×C127—26no—
51C22×C4×C127—26no—
52C22×C4×C127—23no—
53C22×C4×C127—26no—
54C22×C4×C127—26no—
55C22×C4×C127—23no—
56C22×C4×C127—26no—
57C22×C4×C127—26no—
58C22×C4×C127—22no—
59C22×C4×C127—26no—
60C22×C4×C127C2×D416yes—
61C22×C4×C127C2×D416yes—
62C22×C4×C127C2×D416yes—
63C22×C4×C127C2×D416yes—
64C22×C4×C127C2×Q8112yes—
65C22×C4×C127C2×Q816yes—
66C22×C4×C127#4116yes—
67C22×C4×C127#4112yes—
68C4⋊C4×C127—224no—
69C4⋊C4×C127—224no—
70C4⋊C4×C127—44no—
71C4⋊C4×C127—424no—
72C42×C127—224no—
73C23×C127C22×D413yes—
74C23×C127C22×D411yescharacteristic
75C23×C127C22×Q813yes—
76C23×C127C2×#4113yes—
77C23×C127C2×#4111yescharacteristic
78C23×C127H(5,𝔽2)11yescharacteristic
79C23×C1272−1+413yes—
80Q8×C127—232no—
81C2×C4×C127—26no—
82C2×C4×C127—212no—
83C2×C4×C127—26no—
84C2×C4×C127—212no—
85C2×C4×C127—26no—
86C2×C4×C127—212no—
87C2×C4×C127—26no—
88C2×C4×C127—26no—
89C2×C4×C127—212no—
90C2×C4×C127—212no—
91C2×C4×C127—46no—
92C2×C4×C127—46no—
93C2×C4×C127—46no—
94C2×C4×C127—46no—
95C2×C4×C127—46no—
96C2×C4×C127—46no—
97C2×C4×C127—42no—
98C2×C4×C127—46no—
99C2×C4×C127—46no—
100C2×C4×C127#12014yes—
101C2×C4×C127#121112yes—
102C2×C4×C127C22⋊Q8112yes—
103C2×C4×C127C4⋊Q8112yes—
104C22×C127C2×#12011yescharacteristic
105C22×C127C2×#12113yes—
106C22×C127C2×C22⋊Q813yes—
107C22×C127#52513yes—
108C22×C127C2×C4⋊Q813yes—
109C22×C127#53613yes—
110C22×C127#53713yes—
111C22×C127D4×Q813yes—
112C22×C127#55011yescharacteristic
113C22×C127#55413yes—
114C22×C127#55613yes—
115C22×C127#55716yes—
116C4×C127—212no—
117C4×C127—46no—
118C4×C127—412no—
119C4×C127—82no—
120C4×C127#39518yescomplemented
121#15158C12711yesmaximal, characteristic, direct product factor, Sylow
122C2×C127C2×#39511yescharacteristic
123C2×C127#253813yes—
124C2×C127#254613yes—
125C2×C127#258113yes—
126C2×C127#258413yes—
127C2×C127#261611yescharacteristic
128C2×C127#284711yescharacteristic
129C22×C4×Q8C2×C12711yescharacteristic
130C2×#395C2×C12712yes—
131C2×#389C2×C12713yes—
132#1441C2×C12716yes—
133C4⋊#95C2×C12713yes—
134C127#1515811yescharacteristic, direct product factor, Sylow
135C23×Q8C22×C12711yescharacteristic
136C2×C4×Q8—212no—
137C22×C4⋊C4C22×C12716yes—
138C22×C4⋊C4C22×C12713yes—
139C22×C4⋊C4C22×C12711yescharacteristic
140C22×C4⋊C4C22×C12713yes—
141C22×C42C22×C12713yes—
142#395—28no—
143#389—212no—
144C2×#95C22×C12716yes—
145C2×#95C22×C12716yes—
146C2×#95C22×C12716yes—
147C22×Q8—26no—
148C22×Q8—26no—
149C22×Q8D4×C12714yes—
150C23×C4C23×C12712yes—
151C23×C4C23×C12713yes—
152C23×C4C23×C12716yes—
153C23×C4C23×C12713yes—
154C23×C4C23×C12711yescharacteristic
155C4×Q8—232no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—26no—
160C2×C4⋊C4—26no—
161C2×C4⋊C4—212no—
162C2×C4⋊C4—212no—
163C2×C4⋊C4—212no—
164C2×C4⋊C4Q8×C127112yescomplemented
165C2×C42—212no—
166C2×C42D4×C127112yes—
167#95—212no—
168#95—212no—
169#95—46no—
170C24C24×C12711yescharacteristic
171C2×Q8—28no—
172C2×Q8—224no—
173C2×Q8—224no—
174C22×C4—26no—
175C22×C4—26no—
176C22×C4—26no—
177C22×C4—26no—
178C22×C4—26no—
179C22×C4—26no—
180C22×C4—26no—
181C22×C4—23no—
182C22×C4—26no—
183C22×C4—26no—
184C22×C4—23no—
185C22×C4—26no—
186C22×C4—26no—
187C22×C4—22no—
188C22×C4—26no—
189C22×C4C2×D4×C12716yes—
190C22×C4C2×D4×C12716yes—
191C22×C4C2×D4×C12716yes—
192C22×C4C2×D4×C12716yes—
193C22×C4C2×Q8×C127112yes—
194C22×C4C2×Q8×C12716yes—
195C22×C4#41×C12716yes—
196C22×C4#41×C12712yes—
197C4⋊C4—224no—
198C4⋊C4—224no—
199C4⋊C4—44no—
200C4⋊C4—424no—
201C42—224no—
202C23C22×D4×C12713yes—
203C23C22×D4×C12711yescharacteristic
204C23C22×Q8×C12713yes—
205C23C2×#41×C12713yes—
206C23C2×#41×C12711yescharacteristic
207C23H(5,𝔽2)×C12711yescharacteristic
208C232−1+4×C12713yes—
209Q8—232no—
210C2×C4—26no—
211C2×C4—212no—
212C2×C4—26no—
213C2×C4—212no—
214C2×C4—26no—
215C2×C4—212no—
216C2×C4—26no—
217C2×C4—26no—
218C2×C4—212no—
219C2×C4—212no—
220C2×C4—46no—
221C2×C4—46no—
222C2×C4—46no—
223C2×C4—46no—
224C2×C4—46no—
225C2×C4—46no—
226C2×C4—42no—
227C2×C4—46no—
228C2×C4—46no—
229C2×C4#120×C12714yes—
230C2×C4#121×C127112yes—
231C2×C4C22⋊Q8×C127112yes—
232C2×C4C4⋊Q8×C127112yes—
233C22C2×#120×C12711yescharacteristic
234C22C2×#121×C12713yes—
235C22C2×C22⋊Q8×C12713yes—
236C22#525×C12713yes—
237C22C2×C4⋊Q8×C12713yes—
238C22#536×C12713yes—
239C22#537×C12713yes—
240C22D4×Q8×C12713yes—
241C22#550×C12711yescharacteristic
242C22#554×C12713yes—
243C22#556×C12713yes—
244C22#557×C12716yes—
245C4—212no—
246C4—46no—
247C4—412no—
248C4—82no—
249C4#395×C12718yescomplemented
250C2C2×#395×C12711yescharacteristic
251C2C127×#253813yes—
252C2C127×#254613yes—
253C2C127×#258113yes—
254C2C127×#258413yes—
255C2C127×#261611yescharacteristic
256C2C127×#284711yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
35. C2×C4⋊C4×C127Q8semidirect
120. C4×C127#395semidirect
121. #15158C127direct (N × H)
134. C127#15158direct (N × H)
164. C2×C4⋊C4Q8×C127semidirect
249. C4#395×C127semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Q8×C1272
C2×#395×C1272
C2×#389×C1272
C127×#14412
C127×C4⋊#952
#15158127