Finite Group Databasebeta

← #13104723

Subgroups — #13104723

Order 1600.

616 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C2×C4⋊#1043C211yesmaximal, characteristic
2C2×C4×#1043C211yesmaximal, characteristic
3C4⋊C4×C5⋊D5C211yesmaximal, characteristic
4#11716874C211yesmaximal, characteristic
5#11716874C211yesmaximal, characteristic
6#11716874C211yesmaximal, characteristic
7#11716874C211yesmaximal, characteristic
8C22×#1043C2211yescharacteristic
9C22×#1043C2211yescharacteristic
10C22×#1043C2211yescharacteristic
11C22×#1043C2211yescharacteristic
12C2×C4×C5⋊D5C2211yescharacteristic
13C2×C4×C5⋊D5C2211yescharacteristic
14C2×C4×C5⋊D5C2211yescharacteristic
15C4⋊#1043—22no—
16C4×#1043—22no—
17C52×C4⋊C4C411yescharacteristic, complemented
18C4⋊#1039C411yescharacteristic, complemented
19#87806C411yescharacteristic, complemented
20#87806C411yescharacteristic, complemented
21#65711—56nomaximal
22C22×C5⋊D5C2311yescharacteristic
23C2×#1043—21no—
24C2×#1043—21no—
25C2×#1043—21no—
26C2×#1043—21no—
27C2×#1043—21no—
28C2×#1043—21no—
29C2×#1043—21no—
30C2×#1043—21no—
31C2×#1043—21no—
32C2×#1043—21no—
33C2×#1043D412yes—
34C2×#1043Q812yes—
35C2×C4×C52C2×C411yescharacteristic
36C2×C4×C52C2×C411yescharacteristic
37C2×C4×C52C2×C411yescharacteristic
38C2×#1039C2×C411yescharacteristic
39C2×#1039C2×C411yescharacteristic
40C2×#1039C2×C411yescharacteristic
41C4×C5⋊D5—21no—
42C4×C5⋊D5—21no—
43C4×C5⋊D5—21no—
44C4×C5⋊D5—21no—
45C4×C5⋊D5—21no—
46C4×C5⋊D5—21no—
47C2×C4⋊AGL(1,𝔽5)—56no—
48C2×C4×AGL(1,𝔽5)—56no—
49D5×C4⋊C4—56no—
50(C2×AGL(1,𝔽5))⋊C4—56no—
51(C2×AGL(1,𝔽5))⋊C4—56no—
52(C2×AGL(1,𝔽5))⋊C4—56no—
53(C2×AGL(1,𝔽5))⋊C4—56no—
54C22×C52C22×C411yescharacteristic
55C2×C5⋊D5C2×D411yescharacteristic
56C2×C5⋊D5C2×Q811yescharacteristic
57C2×C5⋊D5#4111yescharacteristic
58C2×C5⋊D5#4111yescharacteristic
59C2×C5⋊D5#4111yescharacteristic
60C2×C5⋊D5#4111yescharacteristic
61#1043—22no—
62#1043—22no—
63#1043—41no—
64#1043—41no—
65C4×C52—21no—
66C4×C52—21no—
67C4×C52—21no—
68#1039—21no—
69#1039—21no—
70#1039—21no—
71C22×AGL(1,𝔽5)—56no—
72C22×AGL(1,𝔽5)—56no—
73C22×AGL(1,𝔽5)—56no—
74C22×AGL(1,𝔽5)—56no—
75C2×C4×D5—56no—
76C2×C4×D5—56no—
77C2×C4×D5—56no—
78C4⋊AGL(1,𝔽5)—1012no—
79C4×AGL(1,𝔽5)—1012no—
80C5×C4⋊C4AGL(1,𝔽5)16yescomplemented
81C4⋊Dic5—56no—
82#683—56no—
83#683—56no—
84#387—251noSylow
85C2×C52#11711yescharacteristic
86C2×C52C4×D411yescharacteristic
87C2×C52C4×Q811yescharacteristic
88C5⋊D5C22⋊Q811yescharacteristic
89C5⋊D5#12311yescharacteristic
90C5⋊D5#12511yescharacteristic
91C5⋊D5#12611yescharacteristic
92C22×D5—56no—
93C2×AGL(1,𝔽5)—512no—
94C2×AGL(1,𝔽5)—512no—
95C2×AGL(1,𝔽5)—106no—
96C2×AGL(1,𝔽5)—106no—
97C2×AGL(1,𝔽5)—106no—
98C2×AGL(1,𝔽5)—106no—
99C2×AGL(1,𝔽5)—106no—
100C2×AGL(1,𝔽5)—106no—
101C2×AGL(1,𝔽5)—106no—
102C2×AGL(1,𝔽5)—106no—
103C2×AGL(1,𝔽5)—106no—
104C2×AGL(1,𝔽5)—106no—
105C2×C4×C5C2×AGL(1,𝔽5)16yes—
106C2×C4×C5C2×AGL(1,𝔽5)16yes—
107C2×C4×C5C2×AGL(1,𝔽5)16yes—
108C2×Dic5—56no—
109C2×Dic5—56no—
110C2×Dic5—56no—
111C4×D5—106no—
112C4×D5—106no—
113C4×D5—106no—
114C4×D5—106no—
115C4×D5—106no—
116C4×D5—106no—
117C2×C4⋊C4—251no—
118C2×C4⋊C4—251no—
119C2×C42—251no—
120#95—251no—
121#95—251no—
122#95—251no—
123#95—251no—
124C52#38711yescharacteristic, complemented, Sylow
125C22×C5C22×AGL(1,𝔽5)16yes—
126C2×D5—56no—
127C2×D5—56no—
128C2×D5—56no—
129C2×D5—56no—
130C2×D5—56no—
131C2×D5—56no—
132AGL(1,𝔽5)—1012no—
133AGL(1,𝔽5)—1012no—
134AGL(1,𝔽5)—206no—
135AGL(1,𝔽5)—206no—
136C4×C5—26no—
137C4×C5—26no—
138C4×C5—26no—
139Dic5—106no—
140Dic5—106no—
141Dic5—106no—
142C22×C4—251no—
143C22×C4—251no—
144C22×C4—251no—
145C22×C4—251no—
146C22×C4—251no—
147C22×C4—251no—
148C22×C4—251no—
149C4⋊C4—251no—
150C4⋊C4—251no—
151C4⋊C4—251no—
152C4⋊C4—502no—
153C4⋊C4#104311yescharacteristic, complemented
154C42—502no—
155C2×C5#413016yes—
156C2×C5D4×AGL(1,𝔽5)16yes—
157C2×C5Q8×AGL(1,𝔽5)16yes—
158D5—56no—
159D5—56no—
160D5—56no—
161D5—56no—
162C23—251no—
163C2×C4—252no—
164C2×C4—252no—
165C2×C4—251no—
166C2×C4—251no—
167C2×C4—251no—
168C2×C4—501no—
169C2×C4—501no—
170C2×C4—501no—
171C2×C4—501no—
172C2×C4—501no—
173C2×C4—501no—
174C2×C4—501no—
175C2×C4—501no—
176C2×C4—501no—
177C2×C4—501no—
178C2×C4—501no—
179C2×C4—501no—
180C2×C4—501no—
181C2×C4—501no—
182C2×C4—501no—
183C2×C4—501no—
184C2×C4C2×#104311yescharacteristic
185C2×C4C2×#104311yescharacteristic
186C2×C4C2×#104311yescharacteristic
187C5#6571116yescomplemented
188C22—251no—
189C22—251no—
190C22—251no—
191C22—251no—
192C22—251no—
193C22—251no—
194C22C22×#104311yescharacteristic
195C4—21no—
196C4—21no—
197C4—21no—
198C4—502no—
199C4—502no—
200C4—501no—
201C4—501no—
202C4—501no—
203C4—1001no—
204C4—1001no—
205C2—251no—
206C2—251no—
207C2—251no—
208C2—251no—
209C2#1171743811yescharacteristic
210C2D4×#104311yescharacteristic
211C2Q8×#104311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1043
commutator, socleC22×C52C22×C4
FittingC52×C4⋊C4C4
prime radical, cosocleC22×C5⋊D5C23
hypercenterC4⋊C4#1043
hypocenterC52#387
LCSG ▷ C22×C52 ▷ C52
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
17. C52×C4⋊C4C4semidirect
18. C4⋊#1039C4semidirect
19. #87806C4semidirect
20. #87806C4semidirect
80. C5×C4⋊C4AGL(1,𝔽5)semidirect
124. C52#387semidirect
153. C4⋊C4#1043semidirect
187. C5#65711semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10432
C2×C4×#10432
C4⋊C4×C5⋊D52
#117168742
#657115