Finite Group Databasebeta

← C3×#87879 #11925558

Subgroups — C3×#87879 #11925558

Order 1200.

192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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
1C3×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C3×C5×C22⋊D5C211yesmaximal, characteristic, complemented
3C2×C3×C5×Dic5C211yesmaximal, characteristic, complemented
4C3×#6039C211yesmaximal, characteristic, complemented
5C3×#6038C211yesmaximal, characteristic, complemented
6C3×#6036C211yesmaximal, characteristic, complemented
7C3×D5×Dic5C211yesmaximal, characteristic, complemented
8#87879C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C52C2211yescharacteristic
10C2×C3×C5⋊D5C2211yescharacteristic
11C2×C3×C5×D5C2211yescharacteristic
12C3×#1039C2211yescharacteristic
13C3×C5×Dic5C2211yescharacteristic
14C3×C5×Dic5C2211yescharacteristic
15C3×C5×Dic5C2211yescharacteristic
16C3×#711—51nomaximal
17C3×#709—51nomaximal
18C22⋊(C5⋊D5)C2×C311yescharacteristic, complemented
19C5×C22⋊D5C2×C311yescharacteristic, complemented
20C2×C5×Dic5C2×C311yescharacteristic, complemented
21#6039C2×C311yescharacteristic, complemented
22#6038C2×C311yescharacteristic, complemented
23#6036C2×C311yescharacteristic, complemented
24D5×Dic5C2×C311yescharacteristic, complemented
25C2×C3×C52—21no—
26C2×C3×C52C2311yescharacteristic
27C3×C5⋊D5—21no—
28C3×C5×D5—21no—
29C3×C5×D4—51no—
30C2×C3×C4×C5—51no—
31C3×C22⋊D5—51no—
32C3×C22⋊D5—51no—
33C3×C22⋊D5—51no—
34C3×C22⋊D5—102no—
35C3×C22⋊D5D511yescharacteristic, complemented
36C2×C3×Dic5—51no—
37C2×C3×Dic5D511yescharacteristic, complemented
38C3×D20—51no—
39C3×C4×D5—51no—
40C3×C4×D5—51no—
41C3×C4×D5—51no—
42C3×Dic10—51no—
43C3×Dic10—51no—
44C22×C52C22×C311yescharacteristic
45C2×C5⋊D5C22×C311yescharacteristic
46C2×C5×D5C22×C311yescharacteristic
47#1039C22×C311yescharacteristic
48C5×Dic5C22×C311yescharacteristic
49C5×Dic5C22×C311yescharacteristic
50C5×Dic5C22×C311yescharacteristic
51#711—51no—
52#709—51no—
53C3×C52#4111yescharacteristic, complemented, Hall
54C22×C3×C5—22no—
55C22×C3×C5—51no—
56C22×C3×C5C2×D511yescharacteristic
57C22×C3×C5C2×D511yescharacteristic
58C2×C3×D5—51no—
59C2×C3×D5—51no—
60C2×C3×D5—102no—
61C2×C3×D5C2×D511yescharacteristic
62C3×C4×C5—51no—
63C3×C4×C5—51no—
64C3×C4×C5—51no—
65C3×Dic5—51no—
66C3×Dic5—51no—
67C3×Dic5—102no—
68C3×Dic5C2×D511yescharacteristic
69C3×Dic5C2×D511yescharacteristic
70C3×Dic5C2×D511yescharacteristic
71C2×C52—21no—
72C2×C52C23×C311yescharacteristic
73C5⋊D5—21no—
74C5×D5—21no—
75C3×#41—251noHall
76C5×D4—51no—
77C2×C4×C5—51no—
78C22⋊D5—51no—
79C22⋊D5—51no—
80C22⋊D5—51no—
81C22⋊D5—102no—
82C22⋊D5C3×D511yescharacteristic, complemented
83C2×Dic5—51no—
84C2×Dic5C3×D511yescharacteristic, complemented
85D20—51no—
86C4×D5—51no—
87C4×D5—51no—
88C4×D5—51no—
89Dic10—51no—
90Dic10—51no—
91C2×C3×C5—21no—
92C2×C3×C5—21no—
93C2×C3×C5—22no—
94C2×C3×C5—42no—
95C2×C3×C5—101no—
96C2×C3×C5C22×D511yescharacteristic
97C2×C3×C5C22×D511yescharacteristic
98C3×D5—21no—
99C3×D5—101no—
100C3×D5—101no—
101C3×D5—202no—
102C52C3×#4111yescharacteristic, complemented, Sylow
103C3×Q8—251no—
104C3×D4—51no—
105C3×D4—251no—
106C3×D4—251no—
107C2×C3×C4—51no—
108C2×C3×C4—251no—
109C2×C3×C4—251no—
110C22×C5—22no—
111C22×C5—51no—
112C22×C5C2×C3×D511yescharacteristic
113C22×C5C2×C3×D511yescharacteristic
114C2×D5—51no—
115C2×D5—51no—
116C2×D5—102no—
117C2×D5C2×C3×D511yescharacteristic
118C4×C5—51no—
119C4×C5—51no—
120C4×C5—51no—
121Dic5—51no—
122Dic5—51no—
123Dic5—102no—
124Dic5C2×C3×D511yescharacteristic
125Dic5C2×C3×D511yescharacteristic
126Dic5C2×C3×D511yescharacteristic
127#41—251noSylow
128C3×C5—22no—
129C3×C5#70911yescharacteristic, complemented
130C3×C5#71111yescharacteristic, complemented
131C22×C3—51no—
132C22×C3—251no—
133C22×C3D5211yescharacteristic
134C3×C4—51no—
135C3×C4—51no—
136C3×C4—51no—
137C3×C4—251no—
138C2×C5—21no—
139C2×C5—21no—
140C2×C5—22no—
141C2×C5—42no—
142C2×C5—101no—
143C2×C5C22×C3×D511yescharacteristic
144C2×C5C22×C3×D511yescharacteristic
145D5—21no—
146D5—101no—
147D5—101no—
148D5—202no—
149Q8—251no—
150D4—51no—
151D4—251no—
152D4—251no—
153C2×C4—51no—
154C2×C4—251no—
155C2×C4—251no—
156C2×C3—21no—
157C2×C3—101no—
158C2×C3—501no—
159C2×C3C2×D5211yescharacteristic
160C5—22no—
161C5C3×#70911yescharacteristic, complemented
162C5C3×#71111yescharacteristic, complemented
163C22—51no—
164C22—251no—
165C22C3×D5211yescharacteristic
166C4—51no—
167C4—51no—
168C4—51no—
169C4—251no—
170C3#8787911yescharacteristic, direct product factor, Sylow
171C2—21no—
172C2—101no—
173C2—501no—
174C2C2×C3×D5211yescharacteristic
subgroupG/H
centerC2×C3C2×D52
commutator, cosocleC2×C52C23×C3
FrattiniC2C2×C3×D52
FittingC22×C3×C52C22
socleC2×C3×C52C23
hypercenterC22×C3D52
hypocenterC52C3×#41
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2×C3 ◁ C22×C3

Complements

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

NH ≅ G/Ntype
1. C3×C22⋊(C5⋊D5)C2semidirect
2. C3×C5×C22⋊D5C2semidirect
3. C2×C3×C5×Dic5C2semidirect
4. C3×#6039C2semidirect
5. C3×#6038C2semidirect
6. C3×#6036C2semidirect
7. C3×D5×Dic5C2semidirect
8. #87879C3direct (N × H)
18. C22⋊(C5⋊D5)C2×C3semidirect
19. C5×C22⋊D5C2×C3semidirect
20. C2×C5×Dic5C2×C3semidirect
21. #6039C2×C3semidirect
22. #6038C2×C3semidirect
23. #6036C2×C3semidirect
24. D5×Dic5C2×C3semidirect
35. C3×C22⋊D5D5semidirect
37. C2×C3×Dic5D5semidirect
53. C3×C52#41semidirect
82. C22⋊D5C3×D5semidirect
84. C2×Dic5C3×D5semidirect
102. C52C3×#41semidirect
129. C3×C5#709semidirect
130. C3×C5#711semidirect
161. C5C3×#709semidirect
162. C5C3×#711semidirect
170. C3#87879direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C22⋊(C5⋊D5)2
C3×C5×C22⋊D52
C2×C3×C5×Dic52
C3×#60392
C3×#60382
C3×#60362
C3×D5×Dic52
#878793
C3×#7115
C3×#7095