Finite Group Databasebeta

← C2×#22550533 #30652194

Subgroups — C2×#22550533 #30652194

Order 16256.

1620 conjugacy classes of subgroups; only the 573 normal subgroups are shown.

normal subgroups only

This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 573 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×Q8×D127C211yesmaximal, characteristic, complemented
2#22550533C2116yesmaximal, direct product factor
3C2×#16861196C213yesmaximal, complemented
4C2×#16861188C213yesmaximal, complemented
5C2×#16861180C213yesmaximal, complemented
6C2×Q8×Dic127C211yesmaximal, characteristic, complemented
7C2×C4×C22⋊D127C213yesmaximal, complemented
8C22×#41×C127C211yesmaximal, characteristic, complemented
9C22×Dic254C2213yes—
10C2×Q8×D127—28no—
11C2×Q8×D127C2218yescomplemented
12C22×C22⋊D127C2213yes—
13C22×C4×D127C2213yes—
14#16861196C22124yescomplemented
15#16861188C22124yescomplemented
16#16861180C22124yescomplemented
17Q8×Dic127C2218yescomplemented
18C4×C22⋊D127C22124yescomplemented
19C2×C4⋊Dic127C2213yes—
20C2×Dic127⋊C4C2213yes—
21C2×C22⋊Dic127C2213yes—
22C2×C22⋊Dic127C2216yes—
23C2×(C2×D127)⋊C4C2211yescharacteristic
24C2×C4×Dic127C2213yes—
25C2×#41×C127—28no—
26C2×#41×C127C2218yescomplemented
27C22×Q8×C127C2211yescharacteristic
28C22×D4×C127C2213yes—
29C23×C4×C127C2213yes—
30C23×D127C2311yescharacteristic
31C2×Dic254—212no—
32C2×Dic254C23112yes—
33Q8×D127—232no—
34C2×C22⋊D127—212no—
35C2×C22⋊D127C23112yes—
36C22×Dic127C2313yes—
37C22×Dic127C2313yes—
38C22×Dic127C2311yescharacteristic
39C2×C4×D127—212no—
40C2×C4×D127C23112yes—
41C4⋊Dic127C23112yes—
42Dic127⋊C4C23112yes—
43C22⋊Dic127C23124yes—
44C22⋊Dic127C23112yes—
45(C2×D127)⋊C4C2314yes—
46C4×Dic127C23112yes—
47C24×C127C2313yes—
48#41×C127—232no—
49C2×Q8×C127D418yescomplemented
50C2×Q8×C127C2314yes—
51C2×D4×C127—212no—
52C2×D4×C127C23112yes—
53C22×C4×C127—212no—
54C22×C4×C127C23112yes—
55C22×C4×C127C2311yescharacteristic
56C22×C4×C127C2313yes—
57C22×D127—22no—
58C22×D127—22no—
59C22×D127#4114yescomplemented
60C22×D127C2412yes—
61Dic254—224no—
62C22⋊D127—224no—
63C2×Dic127—26no—
64C2×Dic127—26no—
65C2×Dic127—22no—
66C2×Dic127C2412yes—
67C2×Dic127C2416yes—
68C2×Dic127C2416yes—
69C4×D127—224no—
70C23×C127—26no—
71C23×C127—26no—
72C23×C127—26no—
73C23×C127C2416yes—
74C23×C127C2411yescharacteristic
75Q8×C127C2×D4116yescomplemented
76D4×C127—224no—
77C2×C4×C127—22no—
78C2×C4×C127—224no—
79C2×C4×C127C2×D4112yes—
80C2×C4×C127C2412yes—
81C2×C4×C127C2416yes—
82C22×C127—26no—
83C22×C127—26no—
84C22×C127—26no—
85C22×C127—412no—
86C22×C127C22×D412yes—
87C22×C127C2×#4112yes—
88C22×C1272−1+412yes—
89C22×C127C2511yescharacteristic
90C2×D127—28no—
91C2×D127—22no—
92C2×D127—22no—
93C2×D127C2×#4114yescomplemented
94C4×C127—22no—
95C4×C127C22×D4112yes—
96Dic127—26no—
97Dic127—26no—
98Dic127—22no—
99C2×C127—46no—
100C2×C127#54814yescomplemented
101C2×C127C23×D411yescharacteristic
102C2×C127C22×#4111yescharacteristic
103C2×C127C2×2−1+411yescharacteristic
104D127—24no—
105C2×#548—1271nomaximal, Sylow
106C127C2×#54811yescharacteristic, complemented, Sylow
107C22×#41D12711yescharacteristic, complemented
108C23×Q8—1271no—
109#548—12716no—
110C2×#124—1273no—
111C2×C22⋊Q8—1273no—
112C2×#121—1273no—
113C2×C4×Q8—1271no—
114C2×C4×D4—1273no—
115C2×#41—28no—
116C2×#41C2×D12718yescomplemented
117C22×Q8—1278no—
118C22×Q8—1273no—
119C22×Q8—2548no—
120C22×Q8C2×D12711yescharacteristic
121C22×D4—1273no—
122C22×D4C2×D12713yes—
123C23×C4—1273no—
124C23×C4C2×D12713yes—
125#124—12724no—
126C22⋊Q8—12724no—
127#121—12724no—
128C4×Q8—1278no—
129C4×D4—12724no—
130C2×C4⋊C4—1273no—
131C2×C4⋊C4—1273no—
132C2×C22⋊C4—1273no—
133C2×C22⋊C4—1276no—
134C2×C22⋊C4—1271no—
135C2×C42—1273no—
136C24—1271no—
137C24C22×D12713yes—
138#41—232no—
139C2×Q8—12712no—
140C2×Q8—25432no—
141C2×Q8—25412no—
142C2×Q8C22⋊D12718yescomplemented
143C2×Q8C22×D12714yes—
144C2×D4—212no—
145C2×D4—12712no—
146C2×D4—25412no—
147C2×D4C22×D127112yes—
148C22×C4—212no—
149C22×C4—1271no—
150C22×C4—1273no—
151C22×C4—1273no—
152C22×C4—12712no—
153C22×C4—25412no—
154C22×C4C22×D127112yes—
155C22×C4C22×D12711yescharacteristic
156C22×C4C22×D12713yes—
157C4⋊C4—12712no—
158C4⋊C4—12712no—
159C22⋊C4—12712no—
160C22⋊C4—12724no—
161C22⋊C4—1274no—
162C42—12712no—
163C23—26no—
164C23—26no—
165C23—26no—
166C23—1274no—
167C23—1272no—
168C23—2542no—
169C23—2542no—
170C23C23×D12716yes—
171C23C23×D12711yescharacteristic
172Q8—25424no—
173Q8C2×C22⋊D127116yescomplemented
174D4—224no—
175D4—25424no—
176C2×C4—22no—
177C2×C4—224no—
178C2×C4—1272no—
179C2×C4—1276no—
180C2×C4—1276no—
181C2×C4—2546no—
182C2×C4—2546no—
183C2×C4—2542no—
184C2×C4—25424no—
185C2×C4C2×C22⋊D127112yes—
186C2×C4C23×D12712yes—
187C2×C4C23×D12716yes—
188C22—26no—
189C22—26no—
190C22—26no—
191C22—412no—
192C22—1274no—
193C22—2548no—
194C22—2542no—
195C22—2542no—
196C22C22×C22⋊D12712yes—
197C22#41×D12712yes—
198C22#1686125612yes—
199C22C24×D12711yescharacteristic
200C4—22no—
201C4—2546no—
202C4—2546no—
203C4—2542no—
204C4C22×C22⋊D127112yes—
205C2—46no—
206C2—2544no—
207C2#2255053314yesdirect product factor
208C2C23×C22⋊D12711yescharacteristic
209C2C2×#41×D12711yescharacteristic
210C2C2×#1686125611yescharacteristic
subgroupG/H
centerC23C23×D127
commutator, cosocleC22×C127C25
FrattiniC22C24×D127
FittingC22×#41×C127C2
socleC23×C127C24
hypercenterC22×#41D127
hypocenterC127C2×#548
LCSG ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D127C2semidirect
2. #22550533C2direct (N × H)
3. C2×#16861196C2semidirect
4. C2×#16861188C2semidirect
5. C2×#16861180C2semidirect
6. C2×Q8×Dic127C2semidirect
7. C2×C4×C22⋊D127C2semidirect
8. C22×#41×C127C2semidirect
11. C2×Q8×D127C22semidirect
14. #16861196C22semidirect
15. #16861188C22semidirect
16. #16861180C22semidirect
17. Q8×Dic127C22semidirect
18. C4×C22⋊D127C22semidirect
26. C2×#41×C127C22semidirect
49. C2×Q8×C127D4semidirect
59. C22×D127#41semidirect
75. Q8×C127C2×D4semidirect
93. C2×D127C2×#41semidirect
100. C2×C127#548semidirect
106. C127C2×#548semidirect
107. C22×#41D127semidirect
116. C2×#41C2×D127semidirect
142. C2×Q8C22⋊D127semidirect
173. Q8C2×C22⋊D127semidirect
207. C2#22550533direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Q8×D1272
#225505332
C2×#168611962
C2×#168611882
C2×#168611802
C2×Q8×Dic1272
C2×C4×C22⋊D1272
C22×#41×C1272
C2×#548127