Finite Group Databasebeta

← C2×#22549668 #30640387

Subgroups — C2×#22549668 #30640387

Order 16256.

368 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C2×#16861205C211yesmaximal, characteristic
2C2×#16861124C211yesmaximal, characteristic, complemented
3#22549668C214yesmaximal, direct product factor
4C2×#99×C127C211yesmaximal, characteristic, complemented
5C22×D508C411yescharacteristic, complemented
6C22×C22⋊D127—21no—
7C22×C22⋊D127—21no—
8#16861205—21no—
9#16861205—21no—
10#16861205C412yescomplemented
11#16861205C2212yes—
12C2×C22⋊Dic127—21no—
13C2×C4×Dic127C411yescharacteristic, complemented
14#16861124—21no—
15#16861124C2212yescomplemented
16C22×D4×C127C2211yescharacteristic
17C2×C22⋊C4×C127—21no—
18#99×C127—21no—
19#99×C127C2212yescomplemented
20C23×D127—21no—
21C2×D508—21no—
22C2×D508—41no—
23C2×D508—41no—
24C2×D508C2×C412yescomplemented
25C2×C22⋊D127—22no—
26C2×C22⋊D127—22no—
27C2×C22⋊D127—22no—
28C2×C22⋊D127—22no—
29C2×C22⋊D127—44no—
30C2×C22⋊D127—44no—
31C22×Dic127—21no—
32C22×Dic127—41no—
33C22⋊Dic127—22no—
34C22⋊Dic127—41no—
35C4×Dic127—21no—
36C4×Dic127C2×C412yescomplemented
37C24×C127D411yescharacteristic
38C24×C127D411yescharacteristic
39C2×D4×C127—41no—
40C2×D4×C127—41no—
41C2×D4×C127C2×C411yescharacteristic
42C2×D4×C127D411yescharacteristic
43C2×D4×C127D411yescharacteristic
44C2×D4×C127C2311yescharacteristic
45C22×C4×C127—41no—
46C22×C4×C127C2×C411yescharacteristic
47C22⋊C4×C127—22no—
48C22⋊C4×C127—41no—
49C22×D127—22no—
50C22×D127—42no—
51C22×D127—42no—
52C22×D127—41no—
53C22×D127—41no—
54D508—42no—
55D508—42no—
56C22⋊D127—44no—
57C22⋊D127—44no—
58C22⋊D127—44no—
59C22⋊D127—44no—
60C2×Dic127—24no—
61C2×Dic127—22no—
62C2×Dic127—42no—
63C2×Dic127—82no—
64C23×C127—21no—
65C23×C127—21no—
66C23×C127—21no—
67C23×C127—21no—
68C23×C127—42no—
69C23×C127—42no—
70C23×C127C22⋊C411yescharacteristic
71C23×C127C22⋊C411yescharacteristic
72C23×C127C22⋊C411yescharacteristic
73C23×C127C2×D411yescharacteristic
74C23×C127C2×D411yescharacteristic
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—41no—
78D4×C127—41no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—42no—
82C2×C4×C127—82no—
83C2×C4×C127C22⋊C411yescharacteristic
84C2×C4×C127C22×C411yescharacteristic
85C22×C127—22no—
86C22×C127—21no—
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—41no—
94C22×C127—42no—
95C22×C127—42no—
96C22×C127—81no—
97C22×C127—81no—
98C22×C127#9911yescharacteristic
99C22×C127#9911yescharacteristic
100C22×C127C2×C22⋊C411yescharacteristic
101C2×D127—42no—
102C2×D127—42no—
103C2×D127—42no—
104C2×D127—82no—
105C2×D127—82no—
106C4×C127—21no—
107C4×C127—21no—
108C4×C127—82no—
109Dic127—24no—
110Dic127—82no—
111C2×C127—21no—
112C2×C127—21no—
113C2×C127—41no—
114C2×C127—41no—
115C2×C127—41no—
116C2×C127—41no—
117C2×C127#35312yescomplemented
118C2×C127C2×#9911yescharacteristic
119D127—82no—
120C2×#353—1271nomaximal, Sylow
121C127C2×#35311yescharacteristic, complemented, Sylow
122C2×#127—1271no—
123C2×#99—1271no—
124C2×#99D12711yescharacteristic, complemented
125#353—1274no—
126C22×D4—1271no—
127C22×D4—2541no—
128C22×D4—2541no—
129C22×D4C2×D12711yescharacteristic
130#127—1272no—
131#127—1272no—
132#127—2541no—
133#127—2541no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—2541no—
136C2×C42—1271no—
137#99—21no—
138#99—1272no—
139#99—2541no—
140#99C2×D12712yescomplemented
141C24—2541no—
142C24C22⋊D12711yescharacteristic
143C24D50811yescharacteristic
144C2×D4—41no—
145C2×D4—41no—
146C2×D4—1272no—
147C2×D4—5081no—
148C2×D4—5081no—
149C2×D4—5084no—
150C2×D4—5084no—
151C2×D4—2542no—
152C2×D4—2542no—
153C2×D4—2542no—
154C2×D4—2542no—
155C2×D4—2541no—
156C2×D4C4×D12711yescharacteristic
157C2×D4C22⋊D12711yescharacteristic
158C2×D4D50811yescharacteristic
159C2×D4C22×D12711yescharacteristic
160C22×C4—41no—
161C22×C4—5081no—
162C22×C4—2541no—
163C22×C4C4×D12711yescharacteristic
164C22⋊C4—22no—
165C22⋊C4—41no—
166C22⋊C4—5081no—
167C22⋊C4—2542no—
168C42—1272no—
169C42—2541no—
170C23—21no—
171C23—21no—
172C23—21no—
173C23—21no—
174C23—42no—
175C23—42no—
176C23—5082no—
177C23—5082no—
178C23—5081no—
179C23—5081no—
180C23—2542no—
181C23(C2×D127)⋊C411yescharacteristic
182C23(C2×D127)⋊C411yescharacteristic
183C23(C2×D127)⋊C411yescharacteristic
184C23C2×C22⋊D12711yescharacteristic
185C23C2×D50811yescharacteristic
186D4—41no—
187D4—41no—
188D4—41no—
189D4—41no—
190D4—5082no—
191D4—5082no—
192D4—5084no—
193D4—5084no—
194D4—5084no—
195D4—5084no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—42no—
199C2×C4—82no—
200C2×C4—10162no—
201C2×C4—5082no—
202C2×C4—2544no—
203C2×C4—2542no—
204C2×C4(C2×D127)⋊C411yescharacteristic
205C2×C4C2×C4×D12711yescharacteristic
206C22—22no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—42no—
217C22—81no—
218C22—81no—
219C22—10162no—
220C22—10162no—
221C22—5082no—
222C22—5082no—
223C22—5082no—
224C22#1686112311yescharacteristic
225C22#1686112311yescharacteristic
226C22C2×(C2×D127)⋊C411yescharacteristic
227C4—21no—
228C4—21no—
229C4—82no—
230C4—10162no—
231C4—2544no—
232C2—21no—
233C2—21no—
234C2—41no—
235C2—41no—
236C2—41no—
237C2—41no—
238C2—10162no—
239C2#2254966812yesdirect product factor
240C2C2×#1686112311yescharacteristic
subgroupG/H
centerC22#16861123
commutatorC2×C4×C127C22×C4
FrattiniC2×D4C22×D127
FittingC2×#99×C127C2
socleC22×C127#99
prime radicalC2×#16861205C2
cosocleC2×D4×C127C23
hypercenterC2×#99D127
hypocenterC127C2×#353
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16861124C2semidirect
3. #22549668C2direct (N × H)
4. C2×#99×C127C2semidirect
5. C22×D508C4semidirect
10. #16861205C4semidirect
13. C2×C4×Dic127C4semidirect
15. #16861124C22semidirect
19. #99×C127C22semidirect
24. C2×D508C2×C4semidirect
36. C4×Dic127C2×C4semidirect
117. C2×C127#353semidirect
121. C127C2×#353semidirect
124. C2×#99D127semidirect
140. #99C2×D127semidirect
239. C2#22549668direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612052
C2×#168611242
#225496682
C2×#99×C1272
C2×#353127