Finite Group Databasebeta

← C2×#31771346 #46825003

Subgroups — C2×#31771346 #46825003

Order 34432.

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#31771346C2116yesmaximal, direct product factor
2C2×#22575450C211yesmaximal, characteristic, complemented
3C2×#22575447C211yesmaximal, characteristic, complemented
4C2×#22575438C213yesmaximal, complemented
5C2×#22575428C213yesmaximal, complemented
6C2×#22575426C213yesmaximal, complemented
7C2×#22575423C213yesmaximal, complemented
8C2×#126×C269C211yesmaximal, characteristic, complemented
9C24×D269C2211yescharacteristic
10C22×D1076C2213yes—
11C22×D1076C2213yes—
12C22×D1076C2213yescomplemented
13C22×C22⋊D269C2213yes—
14C22×C4×D269C2213yescomplemented
15#22575450C2218yescomplemented
16#22575447C2218yescomplemented
17#22575438C22124yescomplemented
18#22575428C22124yescomplemented
19#22575426C22124yescomplemented
20#22575423C22124yescomplemented
21C2×#16864381C2213yescomplemented
22C2×#16864379C2213yes—
23C2×#16864379C2213yes—
24C2×#16864379C2213yescomplemented
25#126×C269C2218yescomplemented
26C2×C4⋊C4×C269C2213yescomplemented
27C2×C22⋊C4×C269C2213yes—
28C2×C42×C269C2211yescharacteristic, complemented
29C23×D269—46no—
30C23×D269C2311yescharacteristic
31C23×D269C2311yescharacteristic
32C23×D269C2313yes—
33C23×D269C2314yes—
34C2×D1076—46no—
35C2×D1076—46no—
36C2×D1076—46no—
37C2×D1076C23112yes—
38C2×D1076C23112yes—
39C2×D1076C23112yescomplemented
40C2×C22⋊D269—46no—
41C2×C22⋊D269C23112yes—
42C22×Dic269C2313yes—
43C2×C4×D269—46no—
44C2×C4×D269C23112yescomplemented
45#16864381C23112yescomplemented
46#16864379C23112yes—
47#16864379C23112yes—
48#16864379C23112yescomplemented
49C24×C269C2311yescharacteristic
50C22×C4×C269C2313yes—
51C22×C4×C269C2313yes—
52C4⋊C4×C269C23112yescomplemented
53C22⋊C4×C269C23112yes—
54C42×C269C2314yescomplemented
55C22×D269—26no—
56C22×D269—26no—
57C22×D269—26no—
58C22×D269—26no—
59C22×D269—26no—
60C22×D269—412no—
61C22×D269—416no—
62C22×D269C2412yes—
63C22×D269C2412yes—
64C22×D269C2416yes—
65D1076—412no—
66D1076—412no—
67D1076—412no—
68C22⋊D269—412no—
69C2×Dic269—26no—
70C2×Dic269C2416yes—
71C4×D269—412no—
72C23×C269—26no—
73C23×C269C2412yes—
74C23×C269C2411yescharacteristic
75C2×C4×C269—26no—
76C2×C4×C269—26no—
77C2×C4×C269C2416yes—
78C2×C4×C269C2416yes—
79C22×C269—26no—
80C22×C269—44no—
81C22×C269H(5,𝔽2)16yes—
82C22×C269C2511yescharacteristic
83C2×D269—26no—
84C2×D269—26no—
85C2×D269—26no—
86C2×D269—26no—
87C2×D269—26no—
88C2×D269—416no—
89C2×D269—412no—
90C2×D269—44no—
91C2×D269—44no—
92C4×C269—26no—
93C4×C269—26no—
94Dic269—26no—
95C2×C269—42no—
96C2×C269#56014yescomplemented
97C2×C269C2×H(5,𝔽2)13yes—
98D269—46no—
99D269—42no—
100D269—42no—
101C269C2×#56011yescharacteristic, complemented, Sylow
102C2×#560—2691nomaximal, Sylow
103#560—26916no—
104C2×#127—2691no—
105C2×#126—2691no—
106C2×#126D26911yescharacteristic, complemented
107C2×#123—2693no—
108C2×#121—2693no—
109C2×#121—2693no—
110C2×#120—2693no—
111C25—2691no—
112C22×D4—2693no—
113C22×D4—2693no—
114C22×D4—2693no—
115C22×D4—2693no—
116C23×C4—2693no—
117#127—2698no—
118#126—2698no—
119#126C2×D26918yescomplemented
120#123—26924no—
121#121—26924no—
122#121—26924no—
123#120—26924no—
124C2×C4⋊C4—2693no—
125C2×C4⋊C4C2×D26913yescomplemented
126C2×C22⋊C4—2693no—
127C2×C22⋊C4—2693no—
128C2×C22⋊C4—2693no—
129C2×C22⋊C4C2×D26913yes—
130C2×C42C2×D26911yescharacteristic, complemented
131C24—2691no—
132C24—2691no—
133C24—2693no—
134C24—2694no—
135C24—10766no—
136C24C22×D26911yescharacteristic
137C2×D4—26912no—
138C2×D4—26912no—
139C2×D4—26912no—
140C2×D4—26912no—
141C2×D4—10766no—
142C2×D4—10766no—
143C2×D4—10766no—
144C2×D4—10766no—
145C22×C4—2693no—
146C22×C4—26912no—
147C22×C4—10766no—
148C22×C4C22×D26913yes—
149C22×C4C22×D26913yes—
150C4⋊C4—26912no—
151C4⋊C4C22×D269112yescomplemented
152C22⋊C4—26912no—
153C22⋊C4—26912no—
154C22⋊C4—26912no—
155C22⋊C4C22×D269112yes—
156C42C22×D26914yescomplemented
157C23—26no—
158C23—2692no—
159C23—2692no—
160C23—2696no—
161C23—5386no—
162C23—5386no—
163C23—5386no—
164C23—5386no—
165C23—5386no—
166C23—107612no—
167C23—107616no—
168C23C23×D26912yes—
169C23C23×D26911yescharacteristic
170D4—107612no—
171D4—107612no—
172D4—107612no—
173D4—107612no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—2696no—
177C2×C4—5386no—
178C2×C4—107612no—
179C2×C4C23×D26916yes—
180C2×C4C23×D26916yes—
181C22—26no—
182C22—44no—
183C22—5386no—
184C22—5386no—
185C22—5386no—
186C22—5386no—
187C22—5386no—
188C22—107616no—
189C22—10764no—
190C22—10764no—
191C22—107612no—
192C22#2257549916yes—
193C22C24×D26911yescharacteristic
194C4—26no—
195C4—26no—
196C4—5386no—
197C2—42no—
198C2—10762no—
199C2—10762no—
200C2—10766no—
201C2#3177134614yesdirect product factor
202C2C2×#2257549913yes—
subgroupG/H
centerC23C23×D269
commutator, cosocleC22×C269C25
FrattiniC22C24×D269
FittingC2×#126×C269C2
socleC23×C269C24
hypercenterC2×#126D269
hypocenterC269C2×#560
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #31771346C2direct (N × H)
2. C2×#22575450C2semidirect
3. C2×#22575447C2semidirect
4. C2×#22575438C2semidirect
5. C2×#22575428C2semidirect
6. C2×#22575426C2semidirect
7. C2×#22575423C2semidirect
8. C2×#126×C269C2semidirect
12. C22×D1076C22semidirect
14. C22×C4×D269C22semidirect
15. #22575450C22semidirect
16. #22575447C22semidirect
17. #22575438C22semidirect
18. #22575428C22semidirect
19. #22575426C22semidirect
20. #22575423C22semidirect
21. C2×#16864381C22semidirect
24. C2×#16864379C22semidirect
25. #126×C269C22semidirect
26. C2×C4⋊C4×C269C22semidirect
28. C2×C42×C269C22semidirect
39. C2×D1076C23semidirect
44. C2×C4×D269C23semidirect
45. #16864381C23semidirect
48. #16864379C23semidirect
52. C4⋊C4×C269C23semidirect
54. C42×C269C23semidirect
96. C2×C269#560semidirect
101. C269C2×#560semidirect
106. C2×#126D269semidirect
119. #126C2×D269semidirect
125. C2×C4⋊C4C2×D269semidirect
130. C2×C42C2×D269semidirect
151. C4⋊C4C22×D269semidirect
156. C42C22×D269semidirect
201. C2#31771346direct (N × H)

Maximal subgroups

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

subgroupindex
#317713462
C2×#225754502
C2×#225754472
C2×#225754382
C2×#225754282
C2×#225754262
C2×#225754232
C2×#126×C2692
C2×#560269