Finite Group Databasebeta

← C2×#22550115 #30648302

Subgroups — C2×#22550115 #30648302

Order 16256.

512 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C22×#14523238C211yesmaximal, characteristic, complemented
2C23×C127⋊C8C211yesmaximal, characteristic
3C2×#16861188C211yesmaximal, characteristic
4#22550115C218yesmaximal, direct product factor
5C2×#16861142C211yesmaximal, characteristic, complemented
6C2×#16861133C211yesmaximal, characteristic, complemented
7C2×#16861132C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10C2×#14523238—24no—
11C2×#14523238C2214yescomplemented
12C22×C127⋊C8—24no—
13C22×C127⋊C8C2214yes—
14C22×C127⋊C8C2211yescharacteristic
15C22×C127⋊C8C2211yescharacteristic
16#16861188—22no—
17#16861188C2214yes—
18C2×C4⋊Dic127C2211yescharacteristic
19C2×Dic127⋊C4—21no—
20C2×C22⋊Dic127—21no—
21#16861142C2214yescomplemented
22#16861133C2214yescomplemented
23#16861132C2214yescomplemented
24C22×Q8×C127C2211yescharacteristic
25C23×C4×C127C2211yescharacteristic
26C22⋊Q8×C127—22no—
27C22⋊Q8×C127C2214yes—
28C2×C4⋊C4×C127—21no—
29C2×C4⋊C4×C127C2211yescharacteristic
30C2×C22⋊C4×C127—21no—
31C2×Dic254—21no—
32C2×Dic254—22no—
33C2×Dic254—22no—
34C2×Dic254C2312yes—
35C22×Dic127—21no—
36C22×Dic127—21no—
37#14523238—28no—
38C2×C127⋊C8—28no—
39C2×C127⋊C8—22no—
40C2×C127⋊C8D414yes—
41C2×C127⋊C8C2312yes—
42C2×C127⋊C8C2312yes—
43C4⋊Dic127—21no—
44C4⋊Dic127C2312yes—
45Dic127⋊C4—24no—
46C22⋊Dic127—24no—
47C24×C127D411yescharacteristic
48C2×Q8×C127—21no—
49C2×Q8×C127—22no—
50C2×Q8×C127—22no—
51C2×Q8×C127C2312yes—
52C22×C4×C127—22no—
53C22×C4×C127—22no—
54C22×C4×C127—21no—
55C22×C4×C127—21no—
56C22×C4×C127D412yes—
57C22×C4×C127D411yescharacteristic
58C22×C4×C127C2312yes—
59C22×C4×C127C2311yescharacteristic
60C4⋊C4×C127—24no—
61C4⋊C4×C127—21no—
62C4⋊C4×C127C2312yes—
63C22⋊C4×C127—24no—
64Dic254—22no—
65Dic254—22no—
66Dic254—42no—
67C2×Dic127—22no—
68C2×Dic127—22no—
69C2×Dic127—42no—
70C2×Dic127—42no—
71C127⋊C8—22no—
72C127⋊C8C2×D414yes—
73C23×C127—22no—
74C23×C127—22no—
75C23×C127Q1614yescomplemented
76C23×C127C2×D412yes—
77C23×C127C2×D411yescharacteristic
78Q8×C127—22no—
79Q8×C127—22no—
80Q8×C127—42no—
81C2×C4×C127—22no—
82C2×C4×C127—24no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—22no—
86C2×C4×C127—22no—
87C2×C4×C127—42no—
88C2×C4×C127—42no—
89C2×C4×C127C2×D412yes—
90C2×C4×C127C2×D412yes—
91C2×C4×C127C2×D411yescharacteristic
92C2×C4×C127#4112yes—
93C2×C4×C127C2411yescharacteristic
94C22×C127—22no—
95C22×C127—22no—
96C22×C127—28no—
97C22×C127#12112yes—
98C22×C127C2×Q1612yes—
99C22×C127C2×Q1614yescomplemented
100C22×C127#13512yes—
101C22×C127C22×D411yescharacteristic
102C4×C127—22no—
103C4×C127—42no—
104C4×C127—42no—
105C4×C127#12112yes—
106C4×C127C22×D411yescharacteristic
107C4×C127C2×#4111yescharacteristic
108Dic127—42no—
109Dic127—42no—
110C2×C127—24no—
111C2×C127#46714yescomplemented
112C2×C127C2×#12111yescharacteristic
113C2×C127C22×Q1611yescharacteristic
114C2×C127C2×#13511yescharacteristic
115C2×#467—1271nomaximal, Sylow
116C127C2×#46711yescharacteristic, complemented, Sylow
117C22×Q16—1271no—
118C23×C8—1271no—
119C2×C22⋊Q8—1271no—
120C2×C22⋊Q8D12711yescharacteristic
121#467—1278no—
122C2×#107—1271no—
123C2×#103—1271no—
124C2×#103—1271no—
125C22×Q8—1271no—
126C22×Q8C2×D12711yescharacteristic
127C23×C4C2×D12711yescharacteristic
128C2×Q16—1274no—
129C2×Q16—2544no—
130C22×C8—1271no—
131C22×C8—1271no—
132C22×C8—1274no—
133C22×C8—2544no—
134C22⋊Q8—22no—
135C22⋊Q8—1274no—
136C22⋊Q8—2542no—
137C22⋊Q8C2×D12714yes—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—1271no—
140C2×C4⋊C4—2541no—
141C2×C4⋊C4C2×D12711yescharacteristic
142C2×C22⋊C4—21no—
143C2×C22⋊C4—2541no—
144#107—1274no—
145#103—1274no—
146#103—1274no—
147C24C22⋊D12711yescharacteristic
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—1272no—
152C2×Q8—2542no—
153C2×Q8—2542no—
154C2×Q8—2541no—
155C2×Q8C22×D12712yes—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2541no—
161C22×C4—2541no—
162C22×C4C22⋊D12712yes—
163C22×C4C22⋊D12711yescharacteristic
164C22×C4C22×D12712yes—
165C22×C4C22×D12711yescharacteristic
166Q16—2548no—
167C2×C8—1274no—
168C2×C8—1272no—
169C2×C8—1272no—
170C2×C8—2548no—
171C2×C8—2542no—
172C4⋊C4—24no—
173C4⋊C4—21no—
174C4⋊C4—1272no—
175C4⋊C4—2544no—
176C4⋊C4—2541no—
177C4⋊C4C22×D12712yes—
178C22⋊C4—24no—
179C22⋊C4—2544no—
180C23—22no—
181C23—22no—
182C23#1452323814yescomplemented
183C23C2×C22⋊D12712yes—
184C23C2×C22⋊D12711yescharacteristic
185Q8—22no—
186Q8—22no—
187Q8—42no—
188Q8—5082no—
189Q8—2542no—
190Q8—2542no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—5082no—
200C2×C4—5082no—
201C2×C4—2542no—
202C2×C4—2542no—
203C2×C4D4×D12712yes—
204C2×C4C2×C22⋊D12712yes—
205C2×C4C2×C22⋊D12711yescharacteristic
206C2×C4#1452324712yes—
207C2×C4C23×D12711yescharacteristic
208C8—1274no—
209C8—2542no—
210C22—22no—
211C22—28no—
212C22—22no—
213C22#1686118112yes—
214C22C2×#1452323812yes—
215C22C2×#1452323814yescomplemented
216C22#1686122912yes—
217C22C22×C22⋊D12711yescharacteristic
218C4—22no—
219C4—42no—
220C4—42no—
221C4—5082no—
222C4—5082no—
223C4#1686118112yes—
224C4C2×D4×D12711yescharacteristic
225C4C2×#1452324711yescharacteristic
226C2—24no—
227C2#2255011514yesdirect product factor
228C2C2×#1686118111yescharacteristic
229C2C22×#1452323811yescharacteristic
230C2C2×#1686122911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×C22⋊Q8×C127C2
socleC23×C127C2×D4
prime radicalC24×C127D4
hypercenterC2×C22⋊Q8D127
hypocenterC127C2×#467
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
4. #22550115C2direct (N × H)
5. C2×#16861142C2semidirect
6. C2×#16861133C2semidirect
7. C2×#16861132C2semidirect
11. C2×#14523238C22semidirect
21. #16861142C22semidirect
22. #16861133C22semidirect
23. #16861132C22semidirect
75. C23×C127Q16semidirect
99. C22×C127C2×Q16semidirect
111. C2×C127#467semidirect
116. C127C2×#467semidirect
182. C23#14523238semidirect
215. C22C2×#14523238semidirect
227. C2#22550115direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145232382
C23×C127⋊C82
C2×#168611882
#225501152
C2×#168611422
C2×#168611332
C2×#168611322
C2×C22⋊Q8×C1272
C2×#467127