Finite Group Databasebeta

← C2×#30632428 #44564483

Subgroups — C2×#30632428 #44564483

Order 32128.

1344 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 469 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
1#30632428C2116yesmaximal, direct product factor
2C2×#22548712C211yesmaximal, characteristic, complemented
3C2×#22548709C211yesmaximal, characteristic, complemented
4C2×#22548700C213yesmaximal, complemented
5C2×#22548690C213yesmaximal, complemented
6C2×#22548688C213yesmaximal, complemented
7C2×#22548685C213yesmaximal, complemented
8C2×#126×C251C211yesmaximal, characteristic, complemented
9C24×D251C2211yescharacteristic
10C22×D1004C2213yes—
11C22×D1004C2213yes—
12C22×D1004C2213yescomplemented
13C22×C22⋊D251C2213yes—
14C22×C4×D251C2213yescomplemented
15#22548712C2218yescomplemented
16#22548709C2218yescomplemented
17#22548700C22124yescomplemented
18#22548690C22124yescomplemented
19#22548688C22124yescomplemented
20#22548685C22124yescomplemented
21C2×Dic251⋊C4C2213yescomplemented
22C2×(C2×D251)⋊C4C2213yes—
23C2×(C2×D251)⋊C4C2213yes—
24C2×(C2×D251)⋊C4C2213yescomplemented
25#126×C251C2218yescomplemented
26C2×C4⋊C4×C251C2213yescomplemented
27C2×C22⋊C4×C251C2213yes—
28C2×C42×C251C2211yescharacteristic, complemented
29C23×D251—46no—
30C23×D251C2313yes—
31C23×D251C2311yescharacteristic
32C23×D251C2311yescharacteristic
33C23×D251C2314yes—
34C2×D1004—46no—
35C2×D1004—46no—
36C2×D1004—46no—
37C2×D1004C23112yes—
38C2×D1004C23112yes—
39C2×D1004C23112yescomplemented
40C2×C22⋊D251—46no—
41C2×C22⋊D251C23112yes—
42C22×Dic251C2313yes—
43C2×C4×D251—46no—
44C2×C4×D251C23112yescomplemented
45Dic251⋊C4C23112yescomplemented
46(C2×D251)⋊C4C23112yes—
47(C2×D251)⋊C4C23112yes—
48(C2×D251)⋊C4C23112yescomplemented
49C24×C251C2311yescharacteristic
50C22×C4×C251C2313yes—
51C22×C4×C251C2313yes—
52C4⋊C4×C251C23112yescomplemented
53C22⋊C4×C251C23112yes—
54C42×C251C2314yescomplemented
55C22×D251—26no—
56C22×D251—26no—
57C22×D251—26no—
58C22×D251—26no—
59C22×D251—26no—
60C22×D251—412no—
61C22×D251—416no—
62C22×D251C2412yes—
63C22×D251C2412yes—
64C22×D251C2416yes—
65D1004—412no—
66D1004—412no—
67D1004—412no—
68C22⋊D251—412no—
69C2×Dic251—26no—
70C2×Dic251C2416yes—
71C4×D251—412no—
72C23×C251—26no—
73C23×C251C2412yes—
74C23×C251C2411yescharacteristic
75C2×C4×C251—26no—
76C2×C4×C251—26no—
77C2×C4×C251C2416yes—
78C2×C4×C251C2416yes—
79C22×C251—26no—
80C22×C251—44no—
81C22×C251H(5,𝔽2)16yes—
82C22×C251C2511yescharacteristic
83C2×D251—26no—
84C2×D251—26no—
85C2×D251—26no—
86C2×D251—26no—
87C2×D251—26no—
88C2×D251—416no—
89C2×D251—412no—
90C2×D251—44no—
91C2×D251—44no—
92C4×C251—26no—
93C4×C251—26no—
94Dic251—26no—
95C2×C251—42no—
96C2×C251#56014yescomplemented
97C2×C251C2×H(5,𝔽2)13yes—
98D251—46no—
99D251—42no—
100D251—42no—
101C251C2×#56011yescharacteristic, complemented, Sylow
102C2×#560—2511nomaximal, Sylow
103#560—25116no—
104C2×#127—2511no—
105C2×#126—2511no—
106C2×#126D25111yescharacteristic, complemented
107C2×#123—2513no—
108C2×#121—2513no—
109C2×#121—2513no—
110C2×#120—2513no—
111C25—2511no—
112C22×D4—2513no—
113C22×D4—2513no—
114C22×D4—2513no—
115C22×D4—2513no—
116C23×C4—2513no—
117#127—2518no—
118#126—2518no—
119#126C2×D25118yescomplemented
120#123—25124no—
121#121—25124no—
122#121—25124no—
123#120—25124no—
124C2×C4⋊C4—2513no—
125C2×C4⋊C4C2×D25113yescomplemented
126C2×C22⋊C4—2513no—
127C2×C22⋊C4—2513no—
128C2×C22⋊C4—2513no—
129C2×C22⋊C4C2×D25113yes—
130C2×C42C2×D25111yescharacteristic, complemented
131C24—10046no—
132C24—2514no—
133C24—2511no—
134C24—2511no—
135C24—2513no—
136C24C22×D25111yescharacteristic
137C2×D4—10046no—
138C2×D4—10046no—
139C2×D4—10046no—
140C2×D4—10046no—
141C2×D4—25112no—
142C2×D4—25112no—
143C2×D4—25112no—
144C2×D4—25112no—
145C22×C4—10046no—
146C22×C4—25112no—
147C22×C4—2513no—
148C22×C4C22×D25113yes—
149C22×C4C22×D25113yes—
150C4⋊C4—25112no—
151C4⋊C4C22×D251112yescomplemented
152C22⋊C4—25112no—
153C22⋊C4—25112no—
154C22⋊C4—25112no—
155C22⋊C4C22×D251112yes—
156C42C22×D25114yescomplemented
157C23—26no—
158C23—100412no—
159C23—100416no—
160C23—5026no—
161C23—5026no—
162C23—5026no—
163C23—5026no—
164C23—5026no—
165C23—2512no—
166C23—2512no—
167C23—2516no—
168C23C23×D25112yes—
169C23C23×D25111yescharacteristic
170D4—100412no—
171D4—100412no—
172D4—100412no—
173D4—100412no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—100412no—
177C2×C4—5026no—
178C2×C4—2516no—
179C2×C4C23×D25116yes—
180C2×C4C23×D25116yes—
181C22—26no—
182C22—44no—
183C22—100416no—
184C22—10044no—
185C22—10044no—
186C22—100412no—
187C22—5026no—
188C22—5026no—
189C22—5026no—
190C22—5026no—
191C22—5026no—
192C22#2254876116yes—
193C22C24×D25111yescharacteristic
194C4—26no—
195C4—26no—
196C4—5026no—
197C2—42no—
198C2—10042no—
199C2—10042no—
200C2—10046no—
201C2#3063242814yesdirect product factor
202C2C2×#2254876113yes—
subgroupG/H
centerC23C23×D251
commutator, cosocleC22×C251C25
FrattiniC22C24×D251
FittingC2×#126×C251C2
socleC23×C251C24
hypercenterC2×#126D251
hypocenterC251C2×#560
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #30632428C2direct (N × H)
2. C2×#22548712C2semidirect
3. C2×#22548709C2semidirect
4. C2×#22548700C2semidirect
5. C2×#22548690C2semidirect
6. C2×#22548688C2semidirect
7. C2×#22548685C2semidirect
8. C2×#126×C251C2semidirect
12. C22×D1004C22semidirect
14. C22×C4×D251C22semidirect
15. #22548712C22semidirect
16. #22548709C22semidirect
17. #22548700C22semidirect
18. #22548690C22semidirect
19. #22548688C22semidirect
20. #22548685C22semidirect
21. C2×Dic251⋊C4C22semidirect
24. C2×(C2×D251)⋊C4C22semidirect
25. #126×C251C22semidirect
26. C2×C4⋊C4×C251C22semidirect
28. C2×C42×C251C22semidirect
39. C2×D1004C23semidirect
44. C2×C4×D251C23semidirect
45. Dic251⋊C4C23semidirect
48. (C2×D251)⋊C4C23semidirect
52. C4⋊C4×C251C23semidirect
54. C42×C251C23semidirect
96. C2×C251#560semidirect
101. C251C2×#560semidirect
106. C2×#126D251semidirect
119. #126C2×D251semidirect
125. C2×C4⋊C4C2×D251semidirect
130. C2×C42C2×D251semidirect
151. C4⋊C4C22×D251semidirect
156. C42C22×D251semidirect
201. C2#30632428direct (N × H)

Maximal subgroups

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

subgroupindex
#306324282
C2×#225487122
C2×#225487092
C2×#225487002
C2×#225486902
C2×#225486882
C2×#225486852
C2×#126×C2512
C2×#560251