Finite Group Databasebeta

← C2×#31774638 #46850482

Subgroups — C2×#31774638 #46850482

Order 34688.

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#31774638C2116yesmaximal, direct product factor
2C2×#22576157C211yesmaximal, characteristic, complemented
3C2×#22576154C211yesmaximal, characteristic, complemented
4C2×#22576145C213yesmaximal, complemented
5C2×#22576135C213yesmaximal, complemented
6C2×#22576133C213yesmaximal, complemented
7C2×#22576130C213yesmaximal, complemented
8C2×#126×C271C211yesmaximal, characteristic, complemented
9C24×D271C2211yescharacteristic
10C22×D1084C2213yes—
11C22×D1084C2213yes—
12C22×D1084C2213yescomplemented
13C22×C22⋊D271C2213yes—
14C22×C4×D271C2213yescomplemented
15#22576157C2218yescomplemented
16#22576154C2218yescomplemented
17#22576145C22124yescomplemented
18#22576135C22124yescomplemented
19#22576133C22124yescomplemented
20#22576130C22124yescomplemented
21C2×Dic271⋊C4C2213yescomplemented
22C2×(C2×D271)⋊C4C2213yes—
23C2×(C2×D271)⋊C4C2213yes—
24C2×(C2×D271)⋊C4C2213yescomplemented
25#126×C271C2218yescomplemented
26C2×C4⋊C4×C271C2213yescomplemented
27C2×C22⋊C4×C271C2213yes—
28C2×C42×C271C2211yescharacteristic, complemented
29C23×D271—46no—
30C23×D271C2313yes—
31C23×D271C2311yescharacteristic
32C23×D271C2311yescharacteristic
33C23×D271C2314yes—
34C2×D1084—46no—
35C2×D1084—46no—
36C2×D1084—46no—
37C2×D1084C23112yes—
38C2×D1084C23112yes—
39C2×D1084C23112yescomplemented
40C2×C22⋊D271—46no—
41C2×C22⋊D271C23112yes—
42C22×Dic271C2313yes—
43C2×C4×D271—46no—
44C2×C4×D271C23112yescomplemented
45Dic271⋊C4C23112yescomplemented
46(C2×D271)⋊C4C23112yes—
47(C2×D271)⋊C4C23112yes—
48(C2×D271)⋊C4C23112yescomplemented
49C24×C271C2311yescharacteristic
50C22×C4×C271C2313yes—
51C22×C4×C271C2313yes—
52C4⋊C4×C271C23112yescomplemented
53C22⋊C4×C271C23112yes—
54C42×C271C2314yescomplemented
55C22×D271—26no—
56C22×D271—26no—
57C22×D271—26no—
58C22×D271—26no—
59C22×D271—26no—
60C22×D271—412no—
61C22×D271—416no—
62C22×D271C2412yes—
63C22×D271C2412yes—
64C22×D271C2416yes—
65D1084—412no—
66D1084—412no—
67D1084—412no—
68C22⋊D271—412no—
69C2×Dic271—26no—
70C2×Dic271C2416yes—
71C4×D271—412no—
72C23×C271—26no—
73C23×C271C2412yes—
74C23×C271C2411yescharacteristic
75C2×C4×C271—26no—
76C2×C4×C271—26no—
77C2×C4×C271C2416yes—
78C2×C4×C271C2416yes—
79C22×C271—26no—
80C22×C271—44no—
81C22×C271H(5,𝔽2)16yes—
82C22×C271C2511yescharacteristic
83C2×D271—26no—
84C2×D271—26no—
85C2×D271—26no—
86C2×D271—26no—
87C2×D271—26no—
88C2×D271—416no—
89C2×D271—412no—
90C2×D271—44no—
91C2×D271—44no—
92C4×C271—26no—
93C4×C271—26no—
94Dic271—26no—
95C2×C271—42no—
96C2×C271#56014yescomplemented
97C2×C271C2×H(5,𝔽2)13yes—
98D271—46no—
99D271—42no—
100D271—42no—
101C271C2×#56011yescharacteristic, complemented, Sylow
102C2×#560—2711nomaximal, Sylow
103#560—27116no—
104C2×#127—2711no—
105C2×#126—2711no—
106C2×#126D27111yescharacteristic, complemented
107C2×#123—2713no—
108C2×#121—2713no—
109C2×#121—2713no—
110C2×#120—2713no—
111C25—2711no—
112C22×D4—2713no—
113C22×D4—2713no—
114C22×D4—2713no—
115C22×D4—2713no—
116C23×C4—2713no—
117#127—2718no—
118#126—2718no—
119#126C2×D27118yescomplemented
120#123—27124no—
121#121—27124no—
122#121—27124no—
123#120—27124no—
124C2×C4⋊C4—2713no—
125C2×C4⋊C4C2×D27113yescomplemented
126C2×C22⋊C4—2713no—
127C2×C22⋊C4—2713no—
128C2×C22⋊C4—2713no—
129C2×C22⋊C4C2×D27113yes—
130C2×C42C2×D27111yescharacteristic, complemented
131C24—2711no—
132C24—2711no—
133C24—2713no—
134C24—2714no—
135C24—10846no—
136C24C22×D27111yescharacteristic
137C2×D4—27112no—
138C2×D4—27112no—
139C2×D4—27112no—
140C2×D4—27112no—
141C2×D4—10846no—
142C2×D4—10846no—
143C2×D4—10846no—
144C2×D4—10846no—
145C22×C4—2713no—
146C22×C4—27112no—
147C22×C4—10846no—
148C22×C4C22×D27113yes—
149C22×C4C22×D27113yes—
150C4⋊C4—27112no—
151C4⋊C4C22×D271112yescomplemented
152C22⋊C4—27112no—
153C22⋊C4—27112no—
154C22⋊C4—27112no—
155C22⋊C4C22×D271112yes—
156C42C22×D27114yescomplemented
157C23—26no—
158C23—2712no—
159C23—2712no—
160C23—2716no—
161C23—5426no—
162C23—5426no—
163C23—5426no—
164C23—5426no—
165C23—5426no—
166C23—108412no—
167C23—108416no—
168C23C23×D27112yes—
169C23C23×D27111yescharacteristic
170D4—108412no—
171D4—108412no—
172D4—108412no—
173D4—108412no—
174C2×C4—26no—
175C2×C4—26no—
176C2×C4—2716no—
177C2×C4—5426no—
178C2×C4—108412no—
179C2×C4C23×D27116yes—
180C2×C4C23×D27116yes—
181C22—26no—
182C22—44no—
183C22—5426no—
184C22—5426no—
185C22—5426no—
186C22—5426no—
187C22—5426no—
188C22—108416no—
189C22—10844no—
190C22—10844no—
191C22—108412no—
192C22#2257620616yes—
193C22C24×D27111yescharacteristic
194C4—26no—
195C4—26no—
196C4—5426no—
197C2—42no—
198C2—10842no—
199C2—10842no—
200C2—10846no—
201C2#3177463814yesdirect product factor
202C2C2×#2257620613yes—
subgroupG/H
centerC23C23×D271
commutator, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC2×#126×C271C2
socleC23×C271C24
hypercenterC2×#126D271
hypocenterC271C2×#560
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #31774638C2direct (N × H)
2. C2×#22576157C2semidirect
3. C2×#22576154C2semidirect
4. C2×#22576145C2semidirect
5. C2×#22576135C2semidirect
6. C2×#22576133C2semidirect
7. C2×#22576130C2semidirect
8. C2×#126×C271C2semidirect
12. C22×D1084C22semidirect
14. C22×C4×D271C22semidirect
15. #22576157C22semidirect
16. #22576154C22semidirect
17. #22576145C22semidirect
18. #22576135C22semidirect
19. #22576133C22semidirect
20. #22576130C22semidirect
21. C2×Dic271⋊C4C22semidirect
24. C2×(C2×D271)⋊C4C22semidirect
25. #126×C271C22semidirect
26. C2×C4⋊C4×C271C22semidirect
28. C2×C42×C271C22semidirect
39. C2×D1084C23semidirect
44. C2×C4×D271C23semidirect
45. Dic271⋊C4C23semidirect
48. (C2×D271)⋊C4C23semidirect
52. C4⋊C4×C271C23semidirect
54. C42×C271C23semidirect
96. C2×C271#560semidirect
101. C271C2×#560semidirect
106. C2×#126D271semidirect
119. #126C2×D271semidirect
125. C2×C4⋊C4C2×D271semidirect
130. C2×C42C2×D271semidirect
151. C4⋊C4C22×D271semidirect
156. C42C22×D271semidirect
201. C2#31774638direct (N × H)

Maximal subgroups

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

subgroupindex
#317746382
C2×#225761572
C2×#225761542
C2×#225761452
C2×#225761352
C2×#225761332
C2×#225761302
C2×#126×C2712
C2×#560271