Finite Group Databasebeta

← #22626533

Subgroups — #22626533

Order 9344.

716 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 198 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
1#17989655C212yesmaximal, complemented
2#17989567C216yesmaximal, complemented
3#17989566C216yesmaximal, complemented
4C2×#14556638C211yesmaximal, characteristic
5Q8×#64408—28no—
6#14556670C22112yes—
7#14556650C2214yes—
8#14556649C22112yes—
9#14556638—28no—
10C2×#11924028C2213yes—
11C2×#11924027C413yescomplemented
12C2×#11924027C2213yes—
13C22×Dic146C413yescomplemented
14C2×Q8×D73C411yescharacteristic, complemented
15C2×Q8×D73C2211yescharacteristic
16C2×#41×C73C411yescharacteristic, complemented
17C4⋊#64408—212no—
18C22⋊#64408C2318yes—
19C4×#64408—212no—
20#11924028—212no—
21#11924027—212no—
22#11924027—212no—
23C2×Dic146—212no—
24C2×Dic146C2×C413yes—
25C2×Dic146C2×C413yes—
26C2×Dic146C2×C413yes—
27Q8×D73—24no—
28Q8×D73D418yes—
29C2×D292C2311yescharacteristic
30C2×C22⋊D73C2×C413yes—
31C2×C22⋊D73C2313yes—
32C22×Dic73C2×C416yes—
33C2×C4×D73C2×C413yes—
34C2×C4×D73C2313yes—
35#41×C73—24no—
36C2×Q8×C73C2×C411yescharacteristic
37C2×D4×C73C2×C413yes—
38C22×C4×C73C2×C413yes—
39C23×C73C22×C413yes—
40C22×D73C22×C411yescharacteristic
41C22×D73C2411yescharacteristic
42C2×#64408—28no—
43Q8×C73C22⋊C414yes—
44D4×C73—26no—
45C2×C4×C73—26no—
46C2×C4×C73C22×C411yescharacteristic
47C2×C4×C73C22×C413yes—
48C22⋊D73—26no—
49C22⋊D73—26no—
50C2×Dic73—212no—
51C2×Dic73C22×C413yes—
52C2×Dic73C22×C413yes—
53D292—22no—
54C4×D73—26no—
55C4×D73C2×D4112yes—
56Dic146—26no—
57Dic146—26no—
58Dic146C22⋊C4112yes—
59C22×C73—23no—
60C22×C73—43no—
61C22×C73C23×C411yescharacteristic
62C2×D73—21no—
63C2×D73—41no—
64C2×D73—41no—
65C2×D73C22×D412yes—
66#64408—48no—
67C4×C73—21no—
68C4×C73C2×C22⋊C416yes—
69Dic73—23no—
70Dic73C2×C22⋊C416yes—
71C2×C73—21no—
72C2×C73—43no—
73C2×C73C22×C22⋊C411yescharacteristic
74D73—22no—
75D73—41no—
76#2884—731nomaximal, Sylow
77C73#288411yescharacteristic, complemented, Sylow
78C2×2−1+4—731no—
79#519—732no—
80#410—736no—
81#410—736no—
822−1+4—1468no—
83C2×#41—733no—
84C2×#41—733no—
85C2×#41—733no—
86C2×#41#6440811yescharacteristic, complemented
87C22×Q8—731no—
88C22×Q8—731no—
89C22×Q8—733no—
90C4×Q8—1468no—
91#117—7312no—
92#99—7312no—
93#99—734no—
94#41—24no—
95#41—14612no—
96#41—14612no—
97#41—14612no—
98C2×Q8—738no—
99C2×Q8—733no—
100C2×Q8—733no—
101C2×Q8—733no—
102C2×Q8—14612no—
103C2×Q8—1464no—
104C2×Q8C2×#6440811yescharacteristic
105C2×D4—733no—
106C2×D4—733no—
107C2×D4—731no—
108C2×D4C2×#6440813yes—
109C22×C4—733no—
110C22×C4—736no—
111C22×C4—733no—
112C22×C4C2×#6440813yes—
113C4⋊C4—14612no—
114C22⋊C4—738no—
115C42—14612no—
116C23—731no—
117C23—731no—
118C23C22×#6440813yes—
119Q8—7312no—
120Q8—1466no—
121Q8—1466no—
122Q8C22⋊#6440814yes—
123D4—26no—
124D4—1462no—
125D4—1466no—
126D4—1466no—
127C2×C4—26no—
128C2×C4—733no—
129C2×C4—7312no—
130C2×C4—733no—
131C2×C4—1468no—
132C2×C4—14612no—
133C2×C4—1466no—
134C2×C4C22×#6440811yescharacteristic
135C2×C4C22×#6440813yes—
136C22—23no—
137C22—43no—
138C22—2921no—
139C22—2921no—
140C22—732no—
141C22—1461no—
142C22C23×#6440811yescharacteristic
143C4—21no—
144C4—2928no—
145C4—736no—
146C4—1463no—
147C4C2×C22⋊#6440816yes—
148C2—21no—
149C2—43no—
150C2—2921no—
151C2—1462no—
152C2C22×C22⋊#6440811yescharacteristic
subgroupG/H
centerC2C22×C22⋊#64408
commutatorC22×C73C23×C4
FrattiniC22C23×#64408
FittingC2×#41×C73C4
socleC2×C73C22×C22⋊C4
hypercenterC2×#41#64408
hypocenterC73#2884
LCSG ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×Q8 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. #17989655C2semidirect
2. #17989567C2semidirect
3. #17989566C2semidirect
11. C2×#11924027C4semidirect
13. C22×Dic146C4semidirect
14. C2×Q8×D73C4semidirect
16. C2×#41×C73C4semidirect
77. C73#2884semidirect
86. C2×#41#64408semidirect