Finite Group Databasebeta

← #1790×C211 #41009410

Subgroups — #1790×C211 #41009410

Order 27008.

670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 136 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×H(5,𝔽2)×C211C211yesmaximal, characteristic
2C2×#99×C211C213yesmaximal, complemented
3#386×C211C213yesmaximal, complemented
4H(5,𝔽2)×C211—26no—
5H(5,𝔽2)×C211C414yescomplemented
6C2×#41×C211—23no—
7C22×D4×C211—23no—
8C22×D4×C211C2213yes—
9C23×C4×C211C2211yescharacteristic, complemented
10C2×C22⋊C4×C211—23no—
11C2×C22⋊C4×C211C2213yescomplemented
12#99×C211—26no—
13#95×C211—23no—
14C24×C211—21no—
15C24×C211D413yes—
16C24×C211C2311yescharacteristic
17#41×C211—212no—
18#41×C211—212no—
19C2×Q8×C211—21no—
20C2×D4×C211—23no—
21C2×D4×C211—212no—
22C2×D4×C211—212no—
23C2×D4×C211—26no—
24C2×D4×C211—26no—
25C2×D4×C211C2×C416yes—
26C2×D4×C211D416yes—
27C22×C4×C211—23no—
28C22×C4×C211—23no—
29C22×C4×C211—23no—
30C22×C4×C211—42no—
31C22×C4×C211D413yes—
32C22⋊C4×C211—26no—
33C22⋊C4×C211—46no—
34C23×C211—24no—
35C23×C211—23no—
36C23×C211—23no—
37C23×C211—41no—
38C23×C211—41no—
39C23×C211—43no—
40C23×C211—43no—
41C23×C211—81no—
42C23×C211—81no—
43C23×C211C22⋊C416yescomplemented
44C23×C211C22×C411yescharacteristic
45C23×C211C2×D413yes—
46C23×C211C2×D413yes—
47Q8×C211—24no—
48D4×C211—212no—
49D4×C211—212no—
50D4×C211—212no—
51C2×C4×C211—212no—
52C2×C4×C211—23no—
53C2×C4×C211—23no—
54C2×C4×C211—26no—
55C2×C4×C211—43no—
56C2×C4×C211—41no—
57C2×C4×C211—43no—
58C2×C4×C211—43no—
59C2×C4×C211—43no—
60C2×C4×C211C22⋊C416yescomplemented
61C22×C211—23no—
62C22×C211—23no—
63C22×C211—26no—
64C22×C211—41no—
65C22×C211—43no—
66C22×C211—46no—
67C22×C211—43no—
68C22×C211—43no—
69C22×C211—83no—
70C22×C211—84no—
71C22×C211C2×C22⋊C413yes—
72C22×C211#12013yes—
73C22×C211#12011yescharacteristic
74C4×C211—26no—
75C4×C211—42no—
76C4×C211—83no—
77C2×C211—23no—
78C2×C211—23no—
79C2×C211—46no—
80C2×C211#37911yescharacteristic
81C2×C211B(4,𝔽2)11yescharacteristic
82C2×C211#45811yescharacteristic
83C211#179011yescharacteristic, direct product factor, Sylow
84#1790C21111yesmaximal, characteristic, direct product factor, Sylow
85C2×H(5,𝔽2)C2×C21111yescharacteristic
86C2×#99C2×C21113yescomplemented
87#386C2×C21113yescomplemented
88H(5,𝔽2)—26no—
89H(5,𝔽2)C4×C21114yescomplemented
90C2×#41—23no—
91C22×D4—23no—
92C22×D4C22×C21113yes—
93C23×C4C22×C21111yescharacteristic, complemented
94C2×C22⋊C4—23no—
95C2×C22⋊C4C22×C21113yescomplemented
96#99—26no—
97#95—23no—
98C24—21no—
99C24D4×C21113yes—
100C24C23×C21111yescharacteristic
101#41—212no—
102#41—212no—
103C2×Q8—21no—
104C2×D4—23no—
105C2×D4—212no—
106C2×D4—212no—
107C2×D4—26no—
108C2×D4—26no—
109C2×D4C2×C4×C21116yes—
110C2×D4D4×C21116yes—
111C22×C4—23no—
112C22×C4—23no—
113C22×C4—23no—
114C22×C4—42no—
115C22×C4D4×C21113yes—
116C22⋊C4—26no—
117C22⋊C4—46no—
118C23—24no—
119C23—23no—
120C23—23no—
121C23—41no—
122C23—41no—
123C23—43no—
124C23—43no—
125C23—81no—
126C23—81no—
127C23C22⋊C4×C21116yescomplemented
128C23C22×C4×C21111yescharacteristic
129C23C2×D4×C21113yes—
130C23C2×D4×C21113yes—
131Q8—24no—
132D4—212no—
133D4—212no—
134D4—212no—
135C2×C4—212no—
136C2×C4—23no—
137C2×C4—23no—
138C2×C4—26no—
139C2×C4—43no—
140C2×C4—41no—
141C2×C4—43no—
142C2×C4—43no—
143C2×C4—43no—
144C2×C4C22⋊C4×C21116yescomplemented
145C22—23no—
146C22—23no—
147C22—26no—
148C22—41no—
149C22—43no—
150C22—46no—
151C22—43no—
152C22—43no—
153C22—83no—
154C22—84no—
155C22C2×C22⋊C4×C21113yes—
156C22#120×C21113yes—
157C22#120×C21111yescharacteristic
158C4—26no—
159C4—42no—
160C4—83no—
161C2—23no—
162C2—23no—
163C2—46no—
164C2#379×C21111yescharacteristic
165C2B(4,𝔽2)×C21111yescharacteristic
166C2#458×C21111yescharacteristic
subgroupG/H
center, socleC22×C211#120
commutatorC23C22×C4×C211
Frattini, cosocleC24C23×C211
prime radicalC2×H(5,𝔽2)×C211C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C211 ◁ C24×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#99×C211C2semidirect
3. #386×C211C2semidirect
5. H(5,𝔽2)×C211C4semidirect
9. C23×C4×C211C22semidirect
11. C2×C22⋊C4×C211C22semidirect
43. C23×C211C22⋊C4semidirect
60. C2×C4×C211C22⋊C4semidirect
83. C211#1790direct (N × H)
84. #1790C211direct (N × H)
86. C2×#99C2×C211semidirect
87. #386C2×C211semidirect
89. H(5,𝔽2)C4×C211semidirect
93. C23×C4C22×C211semidirect
95. C2×C22⋊C4C22×C211semidirect
127. C23C22⋊C4×C211semidirect
144. C2×C4C22⋊C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×H(5,𝔽2)×C2112
C2×#99×C2112
#386×C2112
#1790211