Finite Group Databasebeta

← C22×C3×C5×A5 #72219085

Subgroups — C22×C3×C5×A5 #72219085

Order 3600.

334 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 40 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×C3×C5×A5C213yesmaximal, direct product factor
2C22×C5×A5C311yesmaximal, characteristic, direct product factor
3C3×C5×A5C2211yescharacteristic, direct product factor
4C22×C3×C5×A4—51nomaximal
5C22×C3×A5C511yesmaximal, characteristic, direct product factor
6C22×C3×C5×D5—61nomaximal
7C2×C5×A5C2×C313yesdirect product factor
8C22×C3×C5×D3—101nomaximal
9C2×C3×C5×A4—53no—
10C2×C3×A5C2×C513yesdirect product factor
11C22×C3×C52—61no—
12C2×C3×C5×D5—63no—
13C2×C3×C5×D5—63no—
14C5×A5C22×C311yescharacteristic, direct product factor
15C24×C3×C5—51no—
16C22×C5×A4—51no—
17C22×C5×A4—52no—
18C22×A5C3×C511yescharacteristic, direct product factor
19C22×C5×D5—61no—
20C22×C32×C5—101no—
21C2×C3×C5×D3—103no—
22C2×C3×C5×D3—103no—
23C3×C5×A4—51no—
24C3×A5C22×C511yescharacteristic, direct product factor
25C2×C3×C52—63no—
26C3×C5×D5—61no—
27C3×C5×D5—63no—
28C22×C3×A4—51noHall
29C23×C3×C5—53no—
30C23×C3×C5—151no—
31C23×C3×C5—153no—
32C22×C5×D3—101no—
33C22×C3×D5—61no—
34C2×C5×A4—53no—
35C2×C5×A4—56no—
36C2×A5C2×C3×C513yesdirect product factor
37C22×C52—61no—
38C2×C5×D5—63no—
39C2×C5×D5—63no—
40C2×C32×C5—103no—
41C3×C5×D3—103no—
42C3×C5×D3—101no—
43C24×C5—51no—
44C3×C52—61no—
45C22×C3×D3—101no—
46C2×C3×A4—53no—
47C22×C3×C5—51no—
48C22×C3×C5—61no—
49C22×C3×C5—101no—
50C22×C3×C5—122no—
51C22×C3×C5—153no—
52C22×C3×C5—152no—
53C22×C3×C5—153no—
54C22×C3×C5—153no—
55C22×C3×C5—201no—
56C22×C3×C5A511yesmaximal normal, characteristic, direct product factor
57C2×C5×D3—103no—
58C2×C5×D3—103no—
59C2×C3×D5—63no—
60C2×C3×D5—63no—
61C5×A4—51no—
62C5×A4—52no—
63A5C22×C3×C511yescharacteristic, direct product factor
64C2×C52—63no—
65C5×D5—63no—
66C5×D5—61no—
67C24×C3—51no—
68C22×A4—51no—
69C22×A4—52no—
70C32×C5—101no—
71C23×C5—53no—
72C23×C5—153no—
73C23×C5—151no—
74C22×D5—61no—
75C22×C32—101no—
76C2×C3×D3—103no—
77C2×C3×D3—103no—
78C3×A4—51no—
79C2×C3×C5—63no—
80C2×C3×C5—103no—
81C2×C3×C5—126no—
82C2×C3×C5—151no—
83C2×C3×C5—153no—
84C2×C3×C5—203no—
85C2×C3×C5C2×A513yesdirect product factor
86C3×D5—61no—
87C3×D5—63no—
88C5×D3—103no—
89C5×D3—101no—
90C52—61noSylow
91C23×C3—53no—
92C23×C3—153no—
93C23×C3—151no—
94C22×D3—101no—
95C2×A4—53no—
96C2×A4—56no—
97C22×C5—51no—
98C22×C5—61no—
99C22×C5—122no—
100C22×C5—153no—
101C22×C5—152no—
102C22×C5—153no—
103C22×C5—153no—
104C22×C5C3×A511yescharacteristic, direct product factor
105C2×D5—63no—
106C2×D5—63no—
107C2×C32—103no—
108C3×D3—101no—
109C3×D3—103no—
110C24—51noSylow
111C3×C5—61no—
112C3×C5—101no—
113C3×C5—122no—
114C3×C5—201no—
115C3×C5C22×A511yescharacteristic, direct product factor
116C22×C3—51no—
117C22×C3—101no—
118C22×C3—153no—
119C22×C3—152no—
120C22×C3—153no—
121C22×C3—153no—
122C22×C3—201no—
123C22×C3C5×A511yescharacteristic, direct product factor
124C2×D3—103no—
125C2×D3—103no—
126A4—51no—
127A4—52no—
128C2×C5—63no—
129C2×C5—126no—
130C2×C5—151no—
131C2×C5—153no—
132C2×C5C2×C3×A513yesdirect product factor
133D5—61no—
134D5—63no—
135C32—101noSylow
136C23—53no—
137C23—153no—
138C23—151no—
139C2×C3—103no—
140C2×C3—151no—
141C2×C3—153no—
142C2×C3—203no—
143C2×C3C2×C5×A513yesdirect product factor
144D3—101no—
145D3—103no—
146C5—61no—
147C5—122no—
148C5C22×C3×A511yescharacteristic, direct product factor
149C22—51no—
150C22—153no—
151C22—152no—
152C22—153no—
153C22—153no—
154C22C3×C5×A511yescharacteristic, direct product factor
155C3—101no—
156C3—201no—
157C3C22×C5×A511yescharacteristic, direct product factor
158C2—151no—
159C2—153no—
160C2C2×C3×C5×A513yesdirect product factor
subgroupG/H
center, Fitting, hypercenterC22×C3×C5A5
commutator, hypocenterA5C22×C3×C5
FrattiniC1G
LCSG ▷ A5
UCSC1 ◁ C22×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×A5C2direct (N × H)
2. C22×C5×A5C3direct (N × H)
3. C3×C5×A5C22direct (N × H)
5. C22×C3×A5C5direct (N × H)
7. C2×C5×A5C2×C3direct (N × H)
10. C2×C3×A5C2×C5direct (N × H)
14. C5×A5C22×C3direct (N × H)
18. C22×A5C3×C5direct (N × H)
24. C3×A5C22×C5direct (N × H)
36. C2×A5C2×C3×C5direct (N × H)
56. C22×C3×C5A5direct (N × H)
63. A5C22×C3×C5direct (N × H)
85. C2×C3×C5C2×A5direct (N × H)
104. C22×C5C3×A5direct (N × H)
115. C3×C5C22×A5direct (N × H)
123. C22×C3C5×A5direct (N × H)
132. C2×C5C2×C3×A5direct (N × H)
143. C2×C3C2×C5×A5direct (N × H)
148. C5C22×C3×A5direct (N × H)
154. C22C3×C5×A5direct (N × H)
157. C3C22×C5×A5direct (N × H)
160. C2C2×C3×C5×A5direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×A52
C22×C5×A53
C22×C3×C5×A45
C22×C3×A55
C22×C3×C5×D56
C22×C3×C5×D310