Finite Group Databasebeta

← C2×#22549799 #30641951

Subgroups — C2×#22549799 #30641951

Order 16256.

908 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Dic127C211yesmaximal, characteristic, complemented
2C22×Dic127⋊C4C211yesmaximal, characteristic, complemented
3C22×Dic127⋊C4C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic127C211yesmaximal, characteristic
5C22×C22⋊Dic127C211yesmaximal, characteristic
6#22549799C218yesmaximal, direct product factor
7C2×C127⋊#95C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C127C211yesmaximal, characteristic
9C23×Dic127C2211yescharacteristic
10C23×Dic127C2211yescharacteristic
11C23×Dic127C2211yescharacteristic
12C23×Dic127C2211yescharacteristic
13C2×C4⋊Dic127—24no—
14C2×C4⋊Dic127C2214yescomplemented
15C2×Dic127⋊C4—24no—
16C2×Dic127⋊C4—24no—
17C2×Dic127⋊C4C2214yescomplemented
18C2×Dic127⋊C4C2214yescomplemented
19C2×C22⋊Dic127—24no—
20C2×C22⋊Dic127—24no—
21C2×C22⋊Dic127C2214yes—
22C2×C22⋊Dic127C2214yes—
23C127⋊#95C2214yescomplemented
24C25×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic
26C23×C4×C127C2211yescharacteristic
27C2×C22⋊C4×C127—24no—
28C2×C22⋊C4×C127C2214yes—
29C22×Dic127—22no—
30C22×Dic127—22no—
31C22×Dic127—22no—
32C22×Dic127—22no—
33C22×Dic127—22no—
34C22×Dic127—22no—
35C22×Dic127—22no—
36C22×Dic127—22no—
37C22×Dic127—22no—
38C22×Dic127—22no—
39C22×Dic127D414yes—
40C22×Dic127D414yes—
41C22×Dic127C2312yes—
42C22×Dic127C2312yes—
43C22×Dic127C2312yes—
44C22×Dic127C2312yes—
45C4⋊Dic127—28no—
46Dic127⋊C4—28no—
47Dic127⋊C4—28no—
48C22⋊Dic127—28no—
49C22⋊Dic127—28no—
50C24×C127—22no—
51C24×C127—22no—
52C24×C127—22no—
53C24×C127—22no—
54C24×C127—22no—
55C24×C127—22no—
56C24×C127Q814yes—
57C24×C127C2312yes—
58C24×C127C2311yescharacteristic
59C22×C4×C127—22no—
60C22×C4×C127—22no—
61C22×C4×C127—22no—
62C22×C4×C127—22no—
63C22×C4×C127—22no—
64C22×C4×C127D414yes—
65C22×C4×C127C2312yes—
66C22×C4×C127C2312yes—
67C22⋊C4×C127—28no—
68C2×Dic127—28no—
69C2×Dic127—28no—
70C2×Dic127—22no—
71C2×Dic127—22no—
72C2×Dic127—22no—
73C2×Dic127—22no—
74C2×Dic127—22no—
75C2×Dic127—22no—
76C2×Dic127—22no—
77C2×Dic127—22no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—44no—
81C2×Dic127—44no—
82C2×Dic127C2×D414yes—
83C2×Dic127C2×D414yes—
84C23×C127—28no—
85C23×C127—28no—
86C23×C127—28no—
87C23×C127—22no—
88C23×C127—22no—
89C23×C127—22no—
90C23×C127—22no—
91C23×C127—22no—
92C23×C127—22no—
93C23×C127—44no—
94C23×C127—44no—
95C23×C127—44no—
96C23×C127—44no—
97C23×C127C2×D412yes—
98C23×C127C2×D412yes—
99C23×C127C2×D412yes—
100C23×C127C2×Q814yes—
101C23×C127C2×Q812yes—
102C23×C127#4112yes—
103C23×C127#4112yes—
104C23×C127#4112yes—
105C23×C127C2411yescharacteristic
106C2×C4×C127—28no—
107C2×C4×C127—22no—
108C2×C4×C127—22no—
109C2×C4×C127—22no—
110C2×C4×C127—22no—
111C2×C4×C127—22no—
112C2×C4×C127—44no—
113C2×C4×C127C2×D414yes—
114C22×C127—24no—
115C22×C127—24no—
116C22×C127—24no—
117C22×C127—42no—
118C22×C127—42no—
119C22×C127—42no—
120C22×C127—42no—
121C22×C127—416no—
122C22×C127#12114yes—
123C22×C127#12114yes—
124C22×C127#12114yes—
125C22×C127C22⋊Q814yes—
126C22×C127C22⋊Q814yes—
127C22×C127C22⋊Q814yes—
128C22×C127#12614yes—
129C22×C127C22×D411yescharacteristic
130C22×C127C22×D411yescharacteristic
131C22×C127C22×D411yescharacteristic
132C22×C127C22×Q811yescharacteristic
133C22×C127C2×#4111yescharacteristic
134C22×C127C2×#4111yescharacteristic
135C22×C127C2×#4111yescharacteristic
136C4×C127—24no—
137C4×C127—42no—
138Dic127—24no—
139Dic127—24no—
140Dic127—42no—
141Dic127—42no—
142C2×C127—44no—
143C2×C127#39618yescomplemented
144C2×C127C2×#12111yescharacteristic
145C2×C127C2×#12111yescharacteristic
146C2×C127C2×#12111yescharacteristic
147C2×C127C2×C22⋊Q811yescharacteristic
148C2×C127C2×C22⋊Q811yescharacteristic
149C2×C127C2×C22⋊Q811yescharacteristic
150C2×C127C2×#12611yescharacteristic
151C2×#396—1271nomaximal, Sylow
152C127C2×#39611yescharacteristic, complemented, Sylow
153C22×C4⋊C4—1271no—
154C22×C4⋊C4—1271no—
155C22×C4⋊C4—1271no—
156C22×C22⋊C4—1271no—
157C22×C22⋊C4—1271no—
158C22×C22⋊C4D12711yescharacteristic
159#396—1278no—
160C2×#95—1271no—
161C25C2×D12711yescharacteristic
162C23×C4—1271no—
163C23×C4—1271no—
164C23×C4—1271no—
165C23×C4—1271no—
166C23×C4C2×D12711yescharacteristic
167C23×C4C2×D12711yescharacteristic
168C2×C4⋊C4—1274no—
169C2×C4⋊C4—1274no—
170C2×C4⋊C4—1274no—
171C2×C4⋊C4—2544no—
172C2×C4⋊C4—2544no—
173C2×C4⋊C4—2544no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—1274no—
176C2×C22⋊C4—1274no—
177C2×C22⋊C4—2544no—
178C2×C22⋊C4—2544no—
179C2×C22⋊C4C2×D12714yes—
180#95—1274no—
181C24—22no—
182C24—22no—
183C24—22no—
184C24—22no—
185C24—22no—
186C24—22no—
187C24Dic25414yes—
188C24C22×D12712yes—
189C24C22×D12711yescharacteristic
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—1274no—
196C22×C4—1274no—
197C22×C4—1272no—
198C22×C4—1272no—
199C22×C4—1272no—
200C22×C4—1272no—
201C22×C4—2542no—
202C22×C4—2542no—
203C22×C4—2542no—
204C22×C4—2542no—
205C22×C4—2542no—
206C22×C4—2542no—
207C22×C4—2542no—
208C22×C4—2542no—
209C22×C4—2542no—
210C22×C4—2542no—
211C22×C4C22⋊D12714yes—
212C22×C4C22×D12712yes—
213C22×C4C22×D12712yes—
214C4⋊C4—2548no—
215C4⋊C4—2548no—
216C4⋊C4—2548no—
217C22⋊C4—28no—
218C22⋊C4—2548no—
219C22⋊C4—2548no—
220C23—28no—
221C23—28no—
222C23—28no—
223C23—22no—
224C23—22no—
225C23—22no—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—44no—
230C23—44no—
231C23—44no—
232C23—44no—
233C23D4×D12712yes—
234C23D4×D12712yes—
235C23C2×C22⋊D12712yes—
236C23C2×Dic25414yes—
237C23C2×Dic25412yes—
238C23#1452324712yes—
239C23#1452324712yes—
240C23#1452324812yes—
241C23C23×D12711yescharacteristic
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—28no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—44no—
249C2×C4—1274no—
250C2×C4—1274no—
251C2×C4—5084no—
252C2×C4—5084no—
253C2×C4—2542no—
254C2×C4—2542no—
255C2×C4—2542no—
256C2×C4—2542no—
257C2×C4—2542no—
258C2×C4—2542no—
259C2×C4—2542no—
260C2×C4—2542no—
261C2×C4—2542no—
262C2×C4—2542no—
263C2×C4—2548no—
264C2×C4—2548no—
265C2×C4C2×C22⋊D12714yes—
266C22—24no—
267C22—24no—
268C22—24no—
269C22—416no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—42no—
274C22#1686118014yes—
275C22#1686118114yes—
276C22#1686118214yes—
277C22#1686118614yes—
278C22#1686118614yes—
279C22#1686118814yes—
280C22#1686120214yes—
281C22C2×D4×D12711yescharacteristic
282C22C2×D4×D12711yescharacteristic
283C22C22×C22⋊D12711yescharacteristic
284C22C22×Dic25411yescharacteristic
285C22C2×#1452324711yescharacteristic
286C22C2×#1452324711yescharacteristic
287C22C2×#1452324811yescharacteristic
288C4—24no—
289C4—42no—
290C4—5082no—
291C4—5082no—
292C4—2544no—
293C4—2544no—
294C2—44no—
295C2#2254979918yesdirect product factor
296C2C2×#1686118011yescharacteristic
297C2C2×#1686118111yescharacteristic
298C2C2×#1686118211yescharacteristic
299C2C2×#1686118611yescharacteristic
300C2C2×#1686118611yescharacteristic
301C2C2×#1686118811yescharacteristic
302C2C2×#1686120211yescharacteristic
subgroupG/H
centerC24C22×D127
commutator, cosocleC23×C127C24
FrattiniC23C23×D127
FittingC22×C22⋊C4×C127C2
socleC24×C127C23
prime radicalC25×C127C22
hypercenterC22×C22⋊C4D127
hypocenterC127C2×#396
LCSG ▷ C23×C127 ▷ C127
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic127C2semidirect
2. C22×Dic127⋊C4C2semidirect
3. C22×Dic127⋊C4C2semidirect
6. #22549799C2direct (N × H)
7. C2×C127⋊#95C2semidirect
14. C2×C4⋊Dic127C22semidirect
17. C2×Dic127⋊C4C22semidirect
18. C2×Dic127⋊C4C22semidirect
23. C127⋊#95C22semidirect
143. C2×C127#396semidirect
152. C127C2×#396semidirect
295. C2#22549799direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1272
C22×Dic127⋊C42
C22×C22⋊Dic1272
#225497992
C2×C127⋊#952
C22×C22⋊C4×C1272
C2×#396127