Finite Group Databasebeta

← C127×#15616 #44579289

Subgroups — C127×#15616 #44579289

Order 32512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#396×C127C214yesmaximal
2C2×#389×C127C214yesmaximal, complemented
3C127×#1441C214yesmaximal, complemented
4C127×#1440C211yesmaximal, characteristic
5C127×#1437C212yesmaximal
6C22×C4⋊C4×C127C2214yes—
7C22×C4⋊C4×C127C2214yes—
8C22×C4⋊C4×C127C2214yes—
9C22×C22⋊C4×C127C2214yes—
10C22×C22⋊C4×C127C2212yes—
11C22×C22⋊C4×C127C2211yescharacteristic
12C22×C42×C127C2212yes—
13#396×C127—216no—
14#389×C127—216no—
15C2×#95×C127C2214yes—
16C2×#95×C127C2214yes—
17C2×#95×C127C2214yes—
18C2×#95×C127C2212yes—
19C25×C127C2311yescharacteristic
20C23×C4×C127C2314yes—
21C23×C4×C127C2314yes—
22C23×C4×C127C2312yes—
23C23×C4×C127C2314yes—
24C2×C4⋊C4×C127—28no—
25C2×C4⋊C4×C127—28no—
26C2×C4⋊C4×C127—28no—
27C2×C4⋊C4×C127—28no—
28C2×C4⋊C4×C127—28no—
29C2×C4⋊C4×C127—28no—
30C2×C4⋊C4×C127—28no—
31C2×C4⋊C4×C127—44no—
32C2×C4⋊C4×C127D4116yes—
33C2×C22⋊C4×C127—28no—
34C2×C22⋊C4×C127—28no—
35C2×C22⋊C4×C127—28no—
36C2×C22⋊C4×C127—28no—
37C2×C22⋊C4×C127—42no—
38C2×C22⋊C4×C127—42no—
39C2×C22⋊C4×C127Q814yes—
40C2×C42×C127—28no—
41C2×C42×C127—42no—
42#95×C127—28no—
43#95×C127—28no—
44#95×C127—28no—
45#95×C127—42no—
46C24×C127—24no—
47C24×C127—24no—
48C24×C127—22no—
49C24×C127—24no—
50C24×C127C2×Q812yes—
51C24×C127C2411yescharacteristic
52C22×C4×C127—24no—
53C22×C4×C127—24no—
54C22×C4×C127—24no—
55C22×C4×C127—24no—
56C22×C4×C127—24no—
57C22×C4×C127—24no—
58C22×C4×C127—24no—
59C22×C4×C127—24no—
60C22×C4×C127—24no—
61C22×C4×C127—22no—
62C22×C4×C127—22no—
63C22×C4×C127—24no—
64C22×C4×C127—24no—
65C22×C4×C127—24no—
66C22×C4×C127—24no—
67C22×C4×C127—24no—
68C22×C4×C127—24no—
69C22×C4×C127—24no—
70C22×C4×C127—24no—
71C22×C4×C127—24no—
72C22×C4×C127—24no—
73C22×C4×C127C2×D418yes—
74C22×C4×C127C2×D418yes—
75C22×C4×C127C2×D418yes—
76C22×C4×C127C2×Q814yes—
77C22×C4×C127#4118yes—
78C4⋊C4×C127—232no—
79C4⋊C4×C127—416no—
80C4⋊C4×C127—416no—
81C22⋊C4×C127—44no—
82C22⋊C4×C127—48no—
83C22⋊C4×C127—416no—
84C42×C127—48no—
85C23×C127—24no—
86C23×C127—22no—
87C23×C127—28no—
88C23×C127—44no—
89C23×C127—44no—
90C23×C127—44no—
91C23×C127—42no—
92C23×C127—44no—
93C23×C127—44no—
94C23×C127—44no—
95C23×C127—42no—
96C23×C127C22×D414yes—
97C23×C127C22×Q811yescharacteristic
98C23×C127C2×#4114yes—
99C23×C127H(5,𝔽2)12yes—
100C23×C1272−1+414yes—
101C2×C4×C127—28no—
102C2×C4×C127—28no—
103C2×C4×C127—28no—
104C2×C4×C127—28no—
105C2×C4×C127—28no—
106C2×C4×C127—28no—
107C2×C4×C127—28no—
108C2×C4×C127—24no—
109C2×C4×C127—28no—
110C2×C4×C127—28no—
111C2×C4×C127—28no—
112C2×C4×C127—44no—
113C2×C4×C127—44no—
114C2×C4×C127—44no—
115C2×C4×C127—42no—
116C2×C4×C127—44no—
117C2×C4×C127—44no—
118C2×C4×C127—42no—
119C2×C4×C127—44no—
120C2×C4×C127—44no—
121C2×C4×C127—44no—
122C2×C4×C127—44no—
123C2×C4×C127—44no—
124C2×C4×C127—44no—
125C2×C4×C127—44no—
126C2×C4×C127#121116yes—
127C22×C127—42no—
128C22×C127—44no—
129C22×C127—48no—
130C22×C127—84no—
131C22×C127—84no—
132C22×C127C2×#12114yes—
133C22×C127#53312yes—
134C22×C127#53714yes—
135C22×C127#54114yes—
136C22×C127#54311yescharacteristic
137C22×C127D4212yes—
138C22×C127#54714yes—
139C22×C127#54814yes—
140C22×C127D4×Q814yes—
141C22×C127#55314yes—
142C22×C127#56312yes—
143C4×C127—216no—
144C4×C127—48no—
145C4×C127—44no—
146C4×C127—84no—
147#15616C12711yesmaximal, characteristic, direct product factor, Sylow
148C2×C127—82no—
149C2×C127#241614yes—
150C2×C127#242914yes—
151C2×C127#260811yescharacteristic
152C2×C127#268312yes—
153C2×C127#272514yes—
154C2×#396C2×C12714yes—
155C2×#389C2×C12714yescomplemented
156#1441C2×C12714yescomplemented
157#1440C2×C12711yescharacteristic
158#1437C2×C12712yes—
159C127#1561611yescharacteristic, direct product factor, Sylow
160C22×C4⋊C4C22×C12714yes—
161C22×C4⋊C4C22×C12714yes—
162C22×C4⋊C4C22×C12714yes—
163C22×C22⋊C4C22×C12714yes—
164C22×C22⋊C4C22×C12712yes—
165C22×C22⋊C4C22×C12711yescharacteristic
166C22×C42C22×C12712yes—
167#396—216no—
168#389—216no—
169C2×#95C22×C12714yes—
170C2×#95C22×C12714yes—
171C2×#95C22×C12714yes—
172C2×#95C22×C12712yes—
173C25C23×C12711yescharacteristic
174C23×C4C23×C12714yes—
175C23×C4C23×C12714yes—
176C23×C4C23×C12712yes—
177C23×C4C23×C12714yes—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4D4×C127116yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4Q8×C12714yes—
194C2×C42—28no—
195C2×C42—42no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—42no—
200C24—24no—
201C24—24no—
202C24—22no—
203C24—24no—
204C24C2×Q8×C12712yes—
205C24C24×C12711yescharacteristic
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4C2×D4×C12718yes—
228C22×C4C2×D4×C12718yes—
229C22×C4C2×D4×C12718yes—
230C22×C4C2×Q8×C12714yes—
231C22×C4#41×C12718yes—
232C4⋊C4—232no—
233C4⋊C4—416no—
234C4⋊C4—416no—
235C22⋊C4—44no—
236C22⋊C4—48no—
237C22⋊C4—416no—
238C42—48no—
239C23—24no—
240C23—22no—
241C23—28no—
242C23—44no—
243C23—44no—
244C23—44no—
245C23—42no—
246C23—44no—
247C23—44no—
248C23—44no—
249C23—42no—
250C23C22×D4×C12714yes—
251C23C22×Q8×C12711yescharacteristic
252C23C2×#41×C12714yes—
253C23H(5,𝔽2)×C12712yes—
254C232−1+4×C12714yes—
255C2×C4—28no—
256C2×C4—28no—
257C2×C4—28no—
258C2×C4—28no—
259C2×C4—28no—
260C2×C4—28no—
261C2×C4—28no—
262C2×C4—24no—
263C2×C4—28no—
264C2×C4—28no—
265C2×C4—28no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—44no—
269C2×C4—42no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—44no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4#121×C127116yes—
281C22—42no—
282C22—44no—
283C22—48no—
284C22—84no—
285C22—84no—
286C22C2×#121×C12714yes—
287C22#533×C12712yes—
288C22#537×C12714yes—
289C22#541×C12714yes—
290C22#543×C12711yescharacteristic
291C22D42×C12712yes—
292C22#547×C12714yes—
293C22#548×C12714yes—
294C22D4×Q8×C12714yes—
295C22#553×C12714yes—
296C22#563×C12712yes—
297C4—216no—
298C4—48no—
299C4—44no—
300C4—84no—
301C2—82no—
302C2C127×#241614yes—
303C2C127×#242914yes—
304C2C127×#260811yescharacteristic
305C2C127×#268312yes—
306C2C127×#272514yes—
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
prime radicalC25×C127C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#389×C127C2semidirect
3. C127×#1441C2semidirect
147. #15616C127direct (N × H)
155. C2×#389C2×C127semidirect
156. #1441C2×C127semidirect
159. C127#15616direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#396×C1272
C2×#389×C1272
C127×#14412
C127×#14402
C127×#14372
#15616127