Finite Group Databasebeta

← #49137608

Subgroups — #49137608

Order 36300.

160 conjugacy classes of subgroups; only the 19 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 19 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
1C5⋊(C5⋊((C11⋊D11)⋊C3))C211yesmaximal, characteristic, complemented
2C5×C5⋊#10624391C211yesmaximal, characteristic, complemented
3C5×C5⋊#10624391C211yesmaximal, characteristic, complemented
4D552—31nomaximal, Hall
5C52×C112⋊C3C2211yescharacteristic, complemented, Hall
6#10624391⋊D5—51nomaximal
7#10624391⋊D5—51nomaximal
8C5⋊(C5⋊(C11⋊D11))D311yescharacteristic, complemented
9C5×C11×D55—31no—
10C5×C11×D55—31no—
11C5⋊((C11⋊D11)⋊C3)—51no—
12C5⋊((C11⋊D11)⋊C3)—51no—
13C5⋊((C11⋊D11)⋊C3)—102no—
14C5⋊#10624391D511yescharacteristic, complemented
15C5⋊#10624391D511yescharacteristic, complemented
16C5×#10624391—51no—
17C5×#10624391—51no—
18C52×C112C2×D311yescharacteristic, complemented, Hall
19D11×D55—151no—
20D11×D55—151no—
21C5×C112⋊C3—22no—
22C5×C112⋊C3C2×D511yescharacteristic, complemented
23C5×C112⋊C3C2×D511yescharacteristic, complemented
24#10624391⋊C2—251noHall
25C5⋊(C11⋊D11)—51no—
26C5⋊(C11⋊D11)—51no—
27C5⋊(C11⋊D11)—102no—
28C11×D55—31no—
29C11×D55—31no—
30C5×C11×D11—151no—
31C5×C11×D11—151no—
32D5×D55—331no—
33D5×D55—331no—
34(C11⋊D11)⋊C3—251no—
35#10624391—51no—
36#10624391—51no—
37C5×C112—22no—
38C5×C112D3×D511yescharacteristic, complemented
39C5×C112D3×D511yescharacteristic, complemented
40C5⋊D55—331no—
41C5⋊D55—331no—
42C5⋊D55—661no—
43C5×D55—31no—
44C5×D55—31no—
45C5×D5×C11—331no—
46C5×D5×C11—331no—
47D112—751noHall
48C112⋊C3D5211yescharacteristic, complemented, Hall
49D15⋊D5—1211nomaximal, Hall
50C52×C11—31no—
51C52×C11—31no—
52C52×C11—61no—
53C11⋊D11—251no—
54C11×D11—151no—
55C11×D11—151no—
56C2×D55—1651no—
57C2×D55—1651no—
58D5×D11—1651no—
59D5×D11—1651no—
60C5×D15—2421no—
61C5×D15—2421no—
62C3×C5⋊D5—1211no—
63C112D15⋊D511yescharacteristic, complemented, Sylow
64C2×C5×C11—1651no—
65C2×C5×C11—1651no—
66D55—31no—
67D55—31no—
68D55—3301no—
69D55—3301no—
70D55—3301no—
71D55—3301no—
72D55—3301no—
73D55—3302no—
74D55—3302no—
75D55—3304no—
76D55—1651no—
77C5×D11—151no—
78C5×D11—151no—
79D5×C11—331no—
80D5×C11—331no—
81D52—3631noHall
82C3×C52—4841noHall
83D3×D5—6051no—
84D3×D5—6051no—
85C5×C11—31no—
86C5×C11—62no—
87C5×C11—62no—
88C5×C11—64no—
89C5×C11—61no—
90C5×C11—61no—
91C5×C11—61no—
92C5×C11—61no—
93C5×C11—61no—
94C5⋊D5—1211no—
95C5×D5—331no—
96C5×D5—331no—
97C2×D11—8251no—
98C2×D11—8251no—
99D15—2421no—
100D15—2421no—
101C3×D5—6051no—
102C3×D5—6051no—
103C3×D5—12102no—
104C5×D3—12101no—
105C5×D3—12101no—
106C52#10624391⋊C211yescharacteristic, complemented, Sylow
107C2×C11—1651no—
108C2×C11—1651no—
109D11—151no—
110D11—151no—
111D11—8251no—
112D11—16501no—
113D11—16501no—
114C2×D5—18151no—
115C2×D5—18151no—
116C3×C5—4842no—
117C3×C5—4841no—
118C3×C5—4841no—
119C2×D3—30251noHall
120C11—31no—
121C11—61no—
122C11—61no—
123C2×C5—1651no—
124C2×C5—1651no—
125D5—331no—
126D5—331no—
127D5—6051no—
128D5—6051no—
129D5—12102no—
130C2×C3—30251no—
131D3—12101no—
132D3—12101no—
133C5—22no—
134C5#10624391⋊D511yescharacteristic, complemented
135C5#10624391⋊D511yescharacteristic, complemented
136C22—90751noSylow
137C3—4841noSylow
138C2—1651no—
139C2—1651no—
140C2—30251no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, cosocle, hypocenterC52×C112⋊C3C22
Fitting, socleC52×C112C2×D3
LCSG ▷ C52×C112⋊C3
UCSC1

Complements

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

NH ≅ G/Ntype
1. C5⋊(C5⋊((C11⋊D11)⋊C3))C2semidirect
2. C5×C5⋊#10624391C2semidirect
3. C5×C5⋊#10624391C2semidirect
5. C52×C112⋊C3C22semidirect
8. C5⋊(C5⋊(C11⋊D11))D3semidirect
14. C5⋊#10624391D5semidirect
15. C5⋊#10624391D5semidirect
18. C52×C112C2×D3semidirect
22. C5×C112⋊C3C2×D5semidirect
23. C5×C112⋊C3C2×D5semidirect
38. C5×C112D3×D5semidirect
39. C5×C112D3×D5semidirect
48. C112⋊C3D52semidirect
63. C112D15⋊D5semidirect
106. C52#10624391⋊C2semidirect
134. C5#10624391⋊D5semidirect
135. C5#10624391⋊D5semidirect

Maximal subgroups

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

subgroupindex
C5⋊(C5⋊((C11⋊D11)⋊C3))2
C5×C5⋊#106243912
D5523
#10624391⋊D55
D15⋊D5121