Finite Group Databasebeta

← C22×D4×#10587739 #31797763

Subgroups — C22×D4×#10587739 #31797763

Order 17536.

2528 conjugacy classes of subgroups; only the 850 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 850 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
1C24×#10587739C212yesmaximal, complemented
2C2×D4×#10587739C2124yesmaximal, direct product factor
3C22×C4⋊#10587739C211yesmaximal, characteristic, complemented
4C22×C22⋊#10587739C212yesmaximal, complemented
5C22×C4×#10587739C211yesmaximal, characteristic, complemented
6C22×D4×D137C211yesmaximal, characteristic
7C23×#10587739—216no—
8C23×#10587739C2211yescharacteristic
9C23×#10587739C2212yes—
10C23×#10587739C2211yescharacteristic
11C23×#10587739C22124yescomplemented
12D4×#10587739C22164yesdirect product factor
13C2×C4⋊#10587739C22112yescomplemented
14C2×C22⋊#10587739C22124yescomplemented
15C2×C4×#10587739C22112yescomplemented
16C24×D137C2212yes—
17C22×D548C411yescharacteristic, complemented
18C22×C22⋊D137C412yescomplemented
19C2×D4×D137C4112yescomplemented
20C2×D4×D137C22112yes—
21C22×C4×D137C2211yescharacteristic
22C22×D4×C137C411yescharacteristic, complemented
23C22×#10587739—296no—
24C22×#10587739—28no—
25C22×#10587739—24no—
26C22×#10587739D418yesdirect product factor
27C22×#10587739C2316yes—
28C22×#10587739C23112yes—
29C22×#10587739C2316yes—
30C22×#10587739C23132yescomplemented
31C4⋊#10587739C23116yescomplemented
32C22⋊#10587739C23132yescomplemented
33C4×#10587739C23116yescomplemented
34C23×D137—28no—
35C23×D137—28no—
36C23×D137C2×C412yescomplemented
37C23×D137C2×C4112yescomplemented
38C23×D137C2311yescharacteristic
39C23×D137C23112yes—
40C2×D548C2×C4112yescomplemented
41C2×C22⋊D137C2×C4124yescomplemented
42D4×D137C2×C4148yescomplemented
43D4×D137C23116yes—
44C22×Dic137C2×C411yescharacteristic, complemented
45C2×C4×D137C2×C416yescomplemented
46C2×C4×D137C2316yes—
47C24×C137C2×C412yescomplemented
48C2×D4×C137C2×C4112yescomplemented
49C22×C4×C137C2×C411yescharacteristic, complemented
50C2×#10587739—224no—
51C2×#10587739—212no—
52C2×#10587739—264no—
53C2×#10587739C2×D4124yesdirect product factor
54C2×#10587739C2414yes—
55C2×#10587739C2418yes—
56C2×#10587739C2414yes—
57C22×D137—28no—
58C22×D137—248no—
59C22×D137—224no—
60C22×D137—224no—
61C22×D137C22×C413yes—
62C22×D137C22×C4124yescomplemented
63C22×D137C22×C4112yescomplemented
64C22×D137C2×D414yes—
65C22×D137#4114yes—
66C22×D137C2418yes—
67C22×D137C2413yes—
68D548C22×C4116yescomplemented
69C22⋊D137C22×C4132yescomplemented
70C2×Dic137C22×C416yescomplemented
71C4×D137C22×C4112yescomplemented
72C4×D137C2414yes—
73C23×C137—28no—
74C23×C137C22×C411yescharacteristic
75C23×C137C22×C4112yescomplemented
76D4×C137C22×C4116yescomplemented
77C2×C4×C137C22×C416yescomplemented
78C22×C137—224no—
79C22×C137C4×D414yescomplemented
80C22×C137C23×C413yes—
81C22×C137C23×C418yescomplemented
82C2×D137—224no—
83C2×D137—248no—
84C2×D137—28no—
85C2×D137—28no—
86C2×D137C4×D4112yescomplemented
87C2×D137C23×C413yes—
88C2×D137C23×C418yescomplemented
89C2×D137C22×D416yes—
90C2×D137C2×#4116yes—
91C2×D137C2511yescharacteristic
92#10587739—28no—
93#10587739—24no—
94#10587739C22×D418yesdirect product factor
95C4×C137C23×C414yescomplemented
96Dic137C23×C414yescomplemented
97C2×C137—28no—
98C2×C137C2×C4×D416yescomplemented
99C2×C137C24×C411yescharacteristic
100D137—28no—
101D137C2×C4×D416yescomplemented
102D137C23×D411yescharacteristic
103D137C22×#4111yescharacteristic
104C137C22×C4×D411yescharacteristic, complemented, Sylow
105C22×C4×D4—1371nomaximal, Sylow
106C23×D4—1371no—
107C24×C4—1372no—
108C2×C4×D4—13724no—
109C22×C4⋊C4—1371no—
110C22×C22⋊C4—1372no—
111C22×C42—1371no—
112C25—1372no—
113C22×D4—1372no—
114C22×D4—13712no—
115C22×D4—13712no—
116C22×D4—1371no—
117C22×D4#1058773911yescharacteristic, direct product factor
118C23×C4—27416no—
119C23×C4—1371no—
120C23×C4—1372no—
121C23×C4—13724no—
122C23×C4—1371no—
123C23×C4—1371no—
124C4×D4—13764no—
125C2×C4⋊C4—13712no—
126C2×C22⋊C4—13724no—
127C2×C42—13712no—
128C24—2748no—
129C24—2748no—
130C24—1372no—
131C24—13712no—
132C24—13712no—
133C24—1371no—
134C24C2×#1058773912yescomplemented
135C2×D4—13724no—
136C2×D4—13748no—
137C2×D4—13716no—
138C2×D4—13712no—
139C2×D4C2×#10587739112yesdirect product factor
140C22×C4—2748no—
141C22×C4—2744no—
142C22×C4—27496no—
143C22×C4—1376no—
144C22×C4—13712no—
145C22×C4—13732no—
146C22×C4—1376no—
147C22×C4—1378no—
148C22×C4—1371no—
149C22×C4—1376no—
150C22×C4—1376no—
151C22×C4C2×#1058773911yescharacteristic, complemented
152C4⋊C4—13716no—
153C22⋊C4—13732no—
154C42—13716no—
155C23—28no—
156C23—27424no—
157C23—27424no—
158C23—27448no—
159C23—2748no—
160C23—13712no—
161C23—1378no—
162C23—13724no—
163C23—1374no—
164C23—1374no—
165C23—1373no—
166C23—1373no—
167C23C22×#1058773911yescharacteristic
168C23C22×#10587739112yescomplemented
169D4—13732no—
170D4—13716no—
171D4C22×#10587739116yesdirect product factor
172C2×C4—27424no—
173C2×C4—27412no—
174C2×C4—27464no—
175C2×C4—1374no—
176C2×C4—1378no—
177C2×C4—1374no—
178C2×C4—13724no—
179C2×C4—1376no—
180C2×C4—1374no—
181C2×C4—13712no—
182C2×C4C22×#1058773916yescomplemented
183C22—224no—
184C22—2748no—
185C22—2748no—
186C22—27448no—
187C22—27424no—
188C22—1378no—
189C22—1376no—
190C22—1376no—
191C22—13712no—
192C22—1371no—
193C22—1373no—
194C22D4×#1058773914yesdirect product factor
195C22C23×#1058773913yes—
196C22C23×#1058773918yescomplemented
197C4—2748no—
198C4—2744no—
199C4—1378no—
200C4—1374no—
201C4C23×#1058773914yescomplemented
202C2—28no—
203C2—2748no—
204C2—1371no—
205C2—1371no—
206C2—1376no—
207C2C2×D4×#1058773916yesdirect product factor
208C2C24×#1058773911yescharacteristic
subgroupG/H
centerC23C22×#10587739
commutatorC2×C137C24×C4
FrattiniC2C24×#10587739
FittingC22×D4×C137C4
socleC23×C137C22×C4
prime radicalC22×D4×D137C2
cosocleC2×D137C25
hypercenterC22×D4#10587739
hypocenterC137C22×C4×D4
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C24×#10587739C2semidirect
2. C2×D4×#10587739C2direct (N × H)
3. C22×C4⋊#10587739C2semidirect
4. C22×C22⋊#10587739C2semidirect
5. C22×C4×#10587739C2semidirect
11. C23×#10587739C22semidirect
12. D4×#10587739C22direct (N × H)
13. C2×C4⋊#10587739C22semidirect
14. C2×C22⋊#10587739C22semidirect
15. C2×C4×#10587739C22semidirect
17. C22×D548C4semidirect
18. C22×C22⋊D137C4semidirect
19. C2×D4×D137C4semidirect
22. C22×D4×C137C4semidirect
26. C22×#10587739D4direct (N × H)
30. C22×#10587739C23semidirect
31. C4⋊#10587739C23semidirect
32. C22⋊#10587739C23semidirect
33. C4×#10587739C23semidirect
36. C23×D137C2×C4semidirect
37. C23×D137C2×C4semidirect
40. C2×D548C2×C4semidirect
41. C2×C22⋊D137C2×C4semidirect
42. D4×D137C2×C4semidirect
44. C22×Dic137C2×C4semidirect
45. C2×C4×D137C2×C4semidirect
47. C24×C137C2×C4semidirect
48. C2×D4×C137C2×C4semidirect
49. C22×C4×C137C2×C4semidirect
53. C2×#10587739C2×D4direct (N × H)
62. C22×D137C22×C4semidirect
63. C22×D137C22×C4semidirect
68. D548C22×C4semidirect
69. C22⋊D137C22×C4semidirect
70. C2×Dic137C22×C4semidirect
71. C4×D137C22×C4semidirect
75. C23×C137C22×C4semidirect
76. D4×C137C22×C4semidirect
77. C2×C4×C137C22×C4semidirect
79. C22×C137C4×D4semidirect
81. C22×C137C23×C4semidirect
86. C2×D137C4×D4semidirect
88. C2×D137C23×C4semidirect
94. #10587739C22×D4direct (N × H)
95. C4×C137C23×C4semidirect
96. Dic137C23×C4semidirect
98. C2×C137C2×C4×D4semidirect
101. D137C2×C4×D4semidirect
104. C137C22×C4×D4semidirect
117. C22×D4#10587739direct (N × H)
134. C24C2×#10587739semidirect
139. C2×D4C2×#10587739direct (N × H)
151. C22×C4C2×#10587739semidirect
168. C23C22×#10587739semidirect
171. D4C22×#10587739direct (N × H)
182. C2×C4C22×#10587739semidirect
194. C22D4×#10587739direct (N × H)
196. C22C23×#10587739semidirect
201. C4C23×#10587739semidirect
207. C2C2×D4×#10587739direct (N × H)

Maximal subgroups

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

subgroupindex
C24×#105877392
C2×D4×#105877392
C22×C4⋊#105877392
C22×C22⋊#105877392
C22×C4×#105877392
C22×D4×D1372
C22×C4×D4137