Finite Group Databasebeta

← #11764921

Subgroups — #11764921

Order 1120.

264 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 66 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×D5×D7C211yesmaximal, characteristic
2C2×C4×D35C211yesmaximal, characteristic, complemented
3C2×#63304C211yesmaximal, characteristic, complemented
4C7×#685C211yesmaximal, characteristic, complemented
5C5×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
6#10587855C211yesmaximal, characteristic, complemented
7(C2×D7)⋊Dic5C211yesmaximal, characteristic, complemented
8C22×D35C2211yescharacteristic
9C22×C7×D5C2211yescharacteristic
10C22×C5×D7C2211yescharacteristic
11C2×D5×D7—22no—
12C2×D5×D7—22no—
13C2×D5×D7C414yescomplemented
14C2×C4×C5×C7C2211yescharacteristic, complemented
15C2×Dic35C2211yescharacteristic, complemented
16C4×D35—22no—
17C2×C7×Dic5C2211yescharacteristic, complemented
18C2×C5×Dic7C2211yescharacteristic, complemented
19#63304—22no—
20D7×C22⋊C4—51nomaximal, Hall
21D5×C22⋊C4—71nomaximal, Hall
22C22×C5×C7C2311yescharacteristic
23C2×D35C2×C412yes—
24C2×D35D412yes—
25C2×D35D412yes—
26C2×C7×D5—21no—
27C2×C7×D5—21no—
28C2×C7×D5C2×C412yescomplemented
29C2×C5×D7—21no—
30C2×C5×D7—21no—
31C2×C5×D7C2×C412yescomplemented
32D5×D7—28no—
33C4×C5×C7—21no—
34Dic35—21no—
35C5×Dic7—21no—
36C7×Dic5—21no—
37C23×D7—51no—
38C2×C4×D7—51no—
39C2×C4×D7—51no—
40C7×C22⋊C4—51no—
41C22⋊Dic7—51no—
42(C2×D7)⋊C4—51no—
43(C2×D7)⋊C4D511yescharacteristic, complemented
44C23×D5—71no—
45C2×C4×D5—71no—
46C2×C4×D5—71no—
47C5×C22⋊C4—71no—
48C22⋊Dic5—71no—
49#685—71no—
50#685D711yescharacteristic, complemented
51C2×C5×C7C22×C411yescharacteristic
52C2×C5×C7C2×D411yescharacteristic
53C2×C5×C7C2×D411yescharacteristic
54D35C22⋊C414yescomplemented
55C5×D7—22no—
56C7×D5—22no—
57C23×C7—51no—
58C22×D7—51no—
59C22×D7—54no—
60C22×D7—102no—
61C22×D7—102no—
62C22×D7C2×D511yescharacteristic
63C2×C4×C7—51no—
64C2×C4×C7C2×D511yescharacteristic, complemented
65C2×Dic7—51no—
66C2×Dic7C2×D511yescharacteristic, complemented
67C4×D7—102no—
68C4×D7—102no—
69C23×C5—71no—
70C22×D5—71no—
71C22×D5—74no—
72C22×D5—142no—
73C22×D5—142no—
74C22×D5C2×D711yescharacteristic
75C2×C4×C5—71no—
76C2×C4×C5C2×D711yescharacteristic, complemented
77C2×Dic5—71no—
78C2×Dic5C2×D711yescharacteristic, complemented
79C4×D5—142no—
80C4×D5—142no—
81C5×C7C2×C22⋊C411yescharacteristic, complemented, Hall
82C2×C22⋊C4—351noSylow
83C22×C7—52no—
84C22×C7—101no—
85C22×C7—101no—
86C22×C7C22×D511yescharacteristic
87C2×D7—21no—
88C2×D7—21no—
89C2×D7—52no—
90C2×D7—52no—
91C2×D7—52no—
92C2×D7—108no—
93C2×D7C4×D512yescomplemented
94C4×C7—21no—
95C4×C7—101no—
96Dic7—21no—
97Dic7—101no—
98C22×C5—72no—
99C22×C5—141no—
100C22×C5—141no—
101C22×C5C22×D711yescharacteristic
102C2×D5—21no—
103C2×D5—21no—
104C2×D5—72no—
105C2×D5—72no—
106C2×D5—72no—
107C2×D5—148no—
108C2×D5C4×D712yescomplemented
109C4×C5—21no—
110C4×C5—141no—
111Dic5—21no—
112Dic5—141no—
113C24—351no—
114C22×C4—351no—
115C22×C4—351no—
116C22⋊C4—51no—
117C22⋊C4—71no—
118C22⋊C4—351no—
119C22⋊C4—351no—
120C2×C7—102no—
121C2×C7C2×C4×D511yescharacteristic
122C2×C7D4×D511yescharacteristic
123C2×C7D4×D511yescharacteristic
124D7—22no—
125D7—54no—
126C2×C5—142no—
127C2×C5C2×C4×D711yescharacteristic
128C2×C5D4×D711yescharacteristic
129C2×C5D4×D711yescharacteristic
130D5—22no—
131D5—74no—
132C23—51no—
133C23—71no—
134C23—354no—
135C23—351no—
136C23—702no—
137C23—702no—
138C2×C4—51no—
139C2×C4—71no—
140C2×C4—351no—
141C2×C4—702no—
142C2×C4—702no—
143C2×C4D5×D711yescharacteristic, complemented
144C7D5×C22⋊C411yescharacteristic, complemented, Sylow
145C5D7×C22⋊C411yescharacteristic, complemented, Sylow
146C22—52no—
147C22—72no—
148C22—101no—
149C22—101no—
150C22—141no—
151C22—141no—
152C22—352no—
153C22—352no—
154C22—352no—
155C22—708no—
156C22C2×D5×D711yescharacteristic
157C4—21no—
158C4—101no—
159C4—141no—
160C4—701no—
161C2—102no—
162C2—142no—
163C2—354no—
164C2C4×D5×D711yescharacteristic
165C2#1058795011yescharacteristic
166C2#1058796311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×D7
commutatorC2×C5×C7C22×C4
FittingC2×C4×C5×C7C22
socle, cosocleC22×C5×C7C23
prime radicalC22×D5×D7C2
hypercenterC2×C4D5×D7
hypocenterC5×C7C2×C22⋊C4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D35C2semidirect
3. C2×#63304C2semidirect
4. C7×#685C2semidirect
5. C5×(C2×D7)⋊C4C2semidirect
6. #10587855C2semidirect
7. (C2×D7)⋊Dic5C2semidirect
13. C2×D5×D7C4semidirect
14. C2×C4×C5×C7C22semidirect
15. C2×Dic35C22semidirect
17. C2×C7×Dic5C22semidirect
18. C2×C5×Dic7C22semidirect
28. C2×C7×D5C2×C4semidirect
31. C2×C5×D7C2×C4semidirect
43. (C2×D7)⋊C4D5semidirect
50. #685D7semidirect
54. D35C22⋊C4semidirect
64. C2×C4×C7C2×D5semidirect
66. C2×Dic7C2×D5semidirect
76. C2×C4×C5C2×D7semidirect
78. C2×Dic5C2×D7semidirect
81. C5×C7C2×C22⋊C4semidirect
93. C2×D7C4×D5semidirect
108. C2×D5C4×D7semidirect
143. C2×C4D5×D7semidirect
144. C7D5×C22⋊C4semidirect
145. C5D7×C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×D5×D72
C2×C4×D352
C2×#633042
C7×#6852
C5×(C2×D7)⋊C42
#105878552
(C2×D7)⋊Dic52
D7×C22⋊C45
D5×C22⋊C47