Finite Group Databasebeta

← C22×#11716862 #72242367

Subgroups — C22×#11716862 #72242367

Order 3200.

2400 conjugacy classes of subgroups; only the 486 normal subgroups are shown.

normal subgroups only

This group has 2400 conjugacy classes of subgroups — too many to draw. Only its 486 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×C4⋊#1043C212yesmaximal
2C22×C8×C5⋊D5C211yesmaximal, characteristic
3C2×#11716862C2112yesmaximal, direct product factor
4C23×#1043—22no—
5C22×C4×C5⋊D5C2211yescharacteristic
6C2×C4⋊#1043—26no—
7C2×C4⋊#1043C22112yes—
8C22×C52×C8C411yescharacteristic, complemented
9C22×#6029C411yescharacteristic, complemented
10C2×C8×C5⋊D5C416yescomplemented
11C2×C8×C5⋊D5C2216yes—
12#11716862C22116yesdirect product factor
13C22×#3994—56nomaximal
14C23×C5⋊D5D411yescharacteristic
15C22×#1043—212no—
16C22×#1043—48no—
17C22×C4×C52C2×C411yescharacteristic
18C22×#1039Q811yescharacteristic
19C2×C4×C5⋊D5C2×C413yes—
20C2×C4×C5⋊D5D413yes—
21C2×C4×C5⋊D5Q813yes—
22C2×C4×C5⋊D5C2313yes—
23C4⋊#1043—212no—
24C4⋊#1043C2318yes—
25C2×C52×C8C2×C416yescomplemented
26C2×#6029C2×C416yescomplemented
27C8×C5⋊D5C2×C4112yescomplemented
28C8×C5⋊D5C2314yes—
29C22×C4⋊AGL(1,𝔽5)—512no—
30C22×C8×D5—56no—
31C2×#3994—572no—
32C23×C52C4⋊C411yescharacteristic
33C22×C5⋊D5C4⋊C413yes—
34C22×C5⋊D5D814yes—
35C22×C5⋊D5Q1614yes—
36C22×C5⋊D5C2×D413yes—
37C2×#1043—28no—
38C2×#1043—424no—
39C2×C4×C52C4⋊C413yes—
40C2×C4×C52C22×C413yes—
41C2×#1039C4⋊C413yes—
42C2×#1039C2×Q813yes—
43C4×C5⋊D5C4⋊C416yes—
44C4×C5⋊D5C22×C413yes—
45C4×C5⋊D5C2×D413yes—
46C4×C5⋊D5C2×Q813yes—
47C4×C5⋊D5C2411yescharacteristic
48C52×C8C22×C414yescomplemented
49#6029C22×C414yescomplemented
50C23×AGL(1,𝔽5)—1012no—
51C22×C4×D5—56no—
52C2×C4⋊AGL(1,𝔽5)—572no—
53C2×C4⋊AGL(1,𝔽5)—1036no—
54C22×C5×C8AGL(1,𝔽5)16yescomplemented
55C22×#168—56no—
56C2×C8×D5—536no—
57C2×C8×D5—536no—
58#3994—596no—
59C22×#107—251noSylow
60C22×C52#10714yescomplemented
61C22×C52C2×C4⋊C413yes—
62C2×C5⋊D5#107112yescomplemented
63C2×C5⋊D5C2×C4⋊C413yes—
64C2×C5⋊D5C2×D816yes—
65C2×C5⋊D5C2×Q1616yes—
66C2×C5⋊D5C22×D411yescharacteristic
67#1043—48no—
68C4×C52C2×C4⋊C413yes—
69C4×C52C23×C411yescharacteristic
70#1039C2×C4⋊C413yes—
71#1039C22×Q811yescharacteristic
72C23×D5—56no—
73C22×AGL(1,𝔽5)—1072no—
74C22×AGL(1,𝔽5)—2048no—
75C22×C4×C5C2×AGL(1,𝔽5)16yes—
76C22×Dic5—56no—
77C2×C4×D5—518no—
78C2×C4×D5—518no—
79C2×C4×D5—518no—
80C2×C4×D5—518no—
81C4⋊AGL(1,𝔽5)—548no—
82C4⋊AGL(1,𝔽5)—1072no—
83C2×C5×C8C2×AGL(1,𝔽5)136yescomplemented
84C2×#168—536no—
85C8×D5—524no—
86C8×D5—572no—
87C23×C8—251no—
88C22×C4⋊C4—252no—
89C2×#107—2512no—
90C2×C52C2×#10716yescomplemented
91C2×C52C22×C4⋊C411yescharacteristic
92C5⋊D5C2×#10716yescomplemented
93C5⋊D5C22×D811yescharacteristic
94C5⋊D5C22×Q1611yescharacteristic
95C23×C5C4⋊AGL(1,𝔽5)16yes—
96C22×D5—518no—
97C22×D5—518no—
98C22×D5—524no—
99C22×D5—524no—
100C2×AGL(1,𝔽5)—1048no—
101C2×AGL(1,𝔽5)—20144no—
102C2×C4×C5C4⋊AGL(1,𝔽5)118yes—
103C2×C4×C5C22×AGL(1,𝔽5)118yes—
104C2×Dic5—518no—
105C2×Dic5—518no—
106C4×D5—536no—
107C4×D5—518no—
108C4×D5—518no—
109C4×D5—518no—
110C4×D5—56no—
111C5×C8C22×AGL(1,𝔽5)124yescomplemented
112#168—524no—
113C23×C4—251no—
114C23×C4—502no—
115C22×C8—251no—
116C22×C8—256no—
117C22×C8—256no—
118C22×C8#104311yescharacteristic, complemented
119C2×C4⋊C4—2512no—
120C2×C4⋊C4—506no—
121#107—2516no—
122C52C22×#10711yescharacteristic, complemented, Sylow
123C22×C5#3994124yescomplemented
124C22×C5C2×C4⋊AGL(1,𝔽5)118yes—
125C2×D5—572no—
126C2×D5—536no—
127C2×D5—536no—
128C2×D5—518no—
129C2×D5—56no—
130AGL(1,𝔽5)—2048no—
131C4×C5C2×C4⋊AGL(1,𝔽5)118yes—
132C4×C5C23×AGL(1,𝔽5)16yes—
133Dic5—518no—
134Dic5—56no—
135C24—251no—
136C22×C4—251no—
137C22×C4—253no—
138C22×C4—253no—
139C22×C4—253no—
140C22×C4—253no—
141C22×C4—5012no—
142C22×C4—1008no—
143C22×C4C2×#104311yescharacteristic
144C2×C8—256no—
145C2×C8—254no—
146C2×C8—2512no—
147C2×C8C2×#104316yescomplemented
148C4⋊C4—258no—
149C4⋊C4—5012no—
150C2×C5C2×#3994136yescomplemented
151C2×C5C22×C4⋊AGL(1,𝔽5)16yes—
152D5—56no—
153D5—56no—
154D5—536no—
155C23—253no—
156C23—253no—
157C23—254no—
158C23—254no—
159C23C4⋊#104311yescharacteristic
160C2×C4—253no—
161C2×C4—253no—
162C2×C4—253no—
163C2×C4—256no—
164C2×C4—253no—
165C2×C4—251no—
166C2×C4—253no—
167C2×C4—508no—
168C2×C4—10024no—
169C2×C4C4⋊#104313yes—
170C2×C4C22×#104313yes—
171C8—254no—
172C8C22×#104314yescomplemented
173C5C22×#399416yescomplemented
174C22—251no—
175C22—253no—
176C22—256no—
177C22—256no—
178C22—2512no—
179C22#1171686214yesdirect product factor
180C22C2×C4⋊#104313yes—
181C4—253no—
182C4—251no—
183C4—1008no—
184C4C2×C4⋊#104313yes—
185C4C23×#104311yescharacteristic
186C2—251no—
187C2—251no—
188C2—256no—
189C2C2×#1171686216yesdirect product factor
190C2C22×C4⋊#104311yescharacteristic
subgroupG/H
centerC23C4⋊#1043
commutatorC4×C52C23×C4
FrattiniC4C23×#1043
FittingC22×C52×C8C4
socleC23×C52C4⋊C4
prime radicalC23×C5⋊D5D4
cosocleC4×C5⋊D5C24
hypercenterC22×C8#1043
hypocenterC52C22×#107
LCSG ▷ C4×C52 ▷ C2×C52 ▷ C52
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
3. C2×#11716862C2direct (N × H)
8. C22×C52×C8C4semidirect
9. C22×#6029C4semidirect
10. C2×C8×C5⋊D5C4semidirect
12. #11716862C22direct (N × H)
25. C2×C52×C8C2×C4semidirect
26. C2×#6029C2×C4semidirect
27. C8×C5⋊D5C2×C4semidirect
48. C52×C8C22×C4semidirect
49. #6029C22×C4semidirect
54. C22×C5×C8AGL(1,𝔽5)semidirect
60. C22×C52#107semidirect
62. C2×C5⋊D5#107semidirect
83. C2×C5×C8C2×AGL(1,𝔽5)semidirect
90. C2×C52C2×#107semidirect
92. C5⋊D5C2×#107semidirect
111. C5×C8C22×AGL(1,𝔽5)semidirect
118. C22×C8#1043semidirect
122. C52C22×#107semidirect
123. C22×C5#3994semidirect
147. C2×C8C2×#1043semidirect
150. C2×C5C2×#3994semidirect
172. C8C22×#1043semidirect
173. C5C22×#3994semidirect
179. C22#11716862direct (N × H)
189. C2C2×#11716862direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊#10432
C22×C8×C5⋊D52
C2×#117168622
C22×#39945