Finite Group Databasebeta

← C22×#17984153 #31797414

Subgroups — C22×#17984153 #31797414

Order 17536.

856 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 218 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×#14555520C211yesmaximal, characteristic, complemented
2C22×C4×#10587739C211yesmaximal, characteristic, complemented
3C2×#17984153C2112yesmaximal, direct product factor
4C22×#14555511C211yesmaximal, characteristic
5C23×#10587739—21no—
6C2×#14555520—23no—
7C2×#14555520C2216yescomplemented
8C22×#11764047—21no—
9C2×C4×#10587739—23no—
10C2×C4×#10587739C2216yescomplemented
11#17984153C22116yesdirect product factor
12C2×#14555511—23no—
13C2×#14555511—23no—
14C2×#14555511C416yescomplemented
15C2×#14555511C2216yes—
16C22×D548—21no—
17C22×D548C411yescharacteristic, complemented
18C22×C4×D137—21no—
19C22×C4×D137C2211yescharacteristic
20C22×Q8×C137C411yescharacteristic, complemented
21C22×#10587739—28no—
22C22×#10587739—26no—
23#14555520—26no—
24#14555520C2314yescomplemented
25C2×#11764047—26no—
26C4×#10587739—26no—
27C4×#10587739C2314yescomplemented
28C23×D137—21no—
29C23×D137D411yescharacteristic
30#14555511—212no—
31#14555511—26no—
32#14555511—26no—
33#14555511C2×C4112yescomplemented
34#14555511C2314yes—
35C2×D548—26no—
36C2×D548—26no—
37C2×D548—23no—
38C2×D548C2×C416yescomplemented
39C22×Dic137D411yescharacteristic
40C2×C4×D137—26no—
41C2×C4×D137—26no—
42C2×C4×D137C2×C413yes—
43C2×C4×D137D413yes—
44C2×C4×D137D413yes—
45C2×C4×D137C2313yes—
46C2×Q8×C137—23no—
47C2×Q8×C137C2×C416yescomplemented
48C22×C4×C137—21no—
49C22×C4×C137C2×C411yescharacteristic
50C2×#10587739—224no—
51C2×#10587739—24no—
52#11764047—24no—
53C22×D137—24no—
54C22×D137—26no—
55C22×D137—44no—
56C22×D137C22⋊C413yes—
57C22×D137C2×D413yes—
58D548—212no—
59D548—24no—
60D548—26no—
61D548C22×C414yescomplemented
62C2×Dic137C22⋊C413yes—
63C2×Dic137C2×D413yes—
64C4×D137—212no—
65C4×D137—24no—
66C4×D137C22⋊C416yes—
67C4×D137C22×C413yes—
68C4×D137C2×D413yes—
69C4×D137C2×D413yes—
70C4×D137C2411yescharacteristic
71C23×C137C22⋊C411yescharacteristic
72Q8×C137—26no—
73Q8×C137C22×C414yescomplemented
74C2×C4×C137—26no—
75C2×C4×C137C22⋊C413yes—
76C2×C4×C137C22×C413yes—
77C22×C137#10414yescomplemented
78C22×C137C2×C22⋊C413yes—
79C2×D137—26no—
80C2×D137—26no—
81C2×D137—24no—
82C2×D137—412no—
83C2×D137C2×C22⋊C413yes—
84C2×D137C22×D411yescharacteristic
85#10587739—28no—
86C4×C137—24no—
87C4×C137C2×C22⋊C413yes—
88C4×C137C23×C411yescharacteristic
89Dic137C2×C22⋊C413yes—
90Dic137C22×D411yescharacteristic
91C2×C137C2×#10416yescomplemented
92C2×C137C22×C22⋊C411yescharacteristic
93D137—21no—
94D137—23no—
95D137—44no—
96C137C22×#10411yescharacteristic, complemented, Sylow
97C22×#104—1371nomaximal, Sylow
98C22×#41—1371no—
99C22×M16—1371no—
100C22×C42—1371no—
101C2×#104—13712no—
102C2×#41—2743no—
103C2×#41—2743no—
104C2×#41—1376no—
105C2×#41—1376no—
106C22×Q8#1058773911yescharacteristic, complemented
107C22×D4—2741no—
108C22×D4—1371no—
109C23×C4—2741no—
110C23×C4—2741no—
111C23×C4—1371no—
112C2×M16—2743no—
113C2×M16—1376no—
114C22×C8—2741no—
115C2×C42—2743no—
116C2×C42—1376no—
117#104—13716no—
118C24—2741no—
119C24—1371no—
120#41—2746no—
121#41—2746no—
122#41—27412no—
123#41—1374no—
124#41—13712no—
125C2×Q8—23no—
126C2×Q8C2×#1058773916yescomplemented
127C2×D4—2746no—
128C2×D4—2746no—
129C2×D4—2743no—
130C2×D4—1376no—
131C22×C4—21no—
132C22×C4—2746no—
133C22×C4—2746no—
134C22×C4—2748no—
135C22×C4—2746no—
136C22×C4—1371no—
137C22×C4—1373no—
138C22×C4—1373no—
139C22×C4—1373no—
140C22×C4—1373no—
141C22×C4C2×#1058773911yescharacteristic
142M16—2746no—
143M16—1374no—
144C2×C8—2746no—
145C42—2746no—
146C42—1374no—
147C23—2744no—
148C23—2746no—
149C23—5484no—
150C23—1373no—
151C23—1373no—
152C23C22⋊#1058773911yescharacteristic
153Q8—26no—
154Q8C22×#1058773914yescomplemented
155D4—2744no—
156D4—27412no—
157D4—2746no—
158D4—1374no—
159C2×C4—26no—
160C2×C4—2744no—
161C2×C4—27412no—
162C2×C4—27424no—
163C2×C4—2744no—
164C2×C4—1373no—
165C2×C4—1373no—
166C2×C4—1373no—
167C2×C4—1373no—
168C2×C4—1376no—
169C2×C4—1371no—
170C2×C4—1373no—
171C2×C4C22⋊#1058773913yes—
172C2×C4C22×#1058773913yes—
173C8—2744no—
174C22—2746no—
175C22—2746no—
176C22—2744no—
177C22—54812no—
178C22—1371no—
179C22—1373no—
180C22#1798415314yesdirect product factor
181C22C2×C22⋊#1058773913yes—
182C4—24no—
183C4—2748no—
184C4—1371no—
185C4—1373no—
186C4C2×C22⋊#1058773913yes—
187C4C23×#1058773911yescharacteristic
188C2—2741no—
189C2—2743no—
190C2—5484no—
191C2C2×#1798415316yesdirect product factor
192C2C22×C22⋊#1058773911yescharacteristic
subgroupG/H
centerC23C22⋊#10587739
commutatorC4×C137C23×C4
FrattiniC4C23×#10587739
FittingC22×Q8×C137C4
socleC23×C137C22⋊C4
prime radicalC22×#14555511C2
cosocleC4×D137C24
hypercenterC22×Q8#10587739
hypocenterC137C22×#104
LCSG ▷ C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14555520C2semidirect
2. C22×C4×#10587739C2semidirect
3. C2×#17984153C2direct (N × H)
7. C2×#14555520C22semidirect
10. C2×C4×#10587739C22semidirect
11. #17984153C22direct (N × H)
14. C2×#14555511C4semidirect
17. C22×D548C4semidirect
20. C22×Q8×C137C4semidirect
24. #14555520C23semidirect
27. C4×#10587739C23semidirect
33. #14555511C2×C4semidirect
38. C2×D548C2×C4semidirect
47. C2×Q8×C137C2×C4semidirect
61. D548C22×C4semidirect
73. Q8×C137C22×C4semidirect
77. C22×C137#104semidirect
91. C2×C137C2×#104semidirect
96. C137C22×#104semidirect
106. C22×Q8#10587739semidirect
126. C2×Q8C2×#10587739semidirect
154. Q8C22×#10587739semidirect
180. C22#17984153direct (N × H)
191. C2C2×#17984153direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145555202
C22×C4×#105877392
C2×#179841532
C22×#145555112
C22×#104137