Finite Group Databasebeta

← #21423403

Subgroups — #21423403

Order 7552.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#16838144C211yesmaximal, characteristic
2#16837797C211yesmaximal, characteristic, complemented
3#16837762C212yesmaximal, complemented
4#16837141C212yesmaximal, complemented
5C59×#329C211yesmaximal, characteristic, complemented
6C2×#11745672—21no—
7C2×#11745670—21no—
8#14273727C2211yescharacteristic, complemented
9#14273716—21no—
10#14273715C2212yes—
11#14273713—22no—
12#14273697—22no—
13#14273695—21no—
14#14273652—22no—
15#14273649—22no—
16#14273638C2211yescharacteristic, complemented
17#124×C59C2211yescharacteristic
18#97×C59C2212yescomplemented
19C23×D59—21no—
20C2×Q8×C59D411yescharacteristic
21C2×D4×C59D411yescharacteristic
22C2×C22⋊D59—22no—
23C2×D236—21no—
24C2×D236D412yes—
25C2×Dic118—21no—
26C2×Dic118D412yes—
27C2×C8×C59—22no—
28C22⋊C4×C59—22no—
29C42×C59C2311yescharacteristic
30C22⋊Dic59—21no—
31#11745672—42no—
32#11745670—42no—
33(C2×D59)⋊C4—22no—
34(C2×D59)⋊C4—22no—
35(C2×D59)⋊C4—22no—
36C4⋊Dic59—22no—
37C4×Dic59—21no—
38C2×C59⋊C8—21no—
39C23×C59—21no—
40C22×D59—22no—
41C22×D59—81no—
42C22×D59—82no—
43Q8×C59—41no—
44D4×C59—41no—
45C2×C4×C59—21no—
46C2×C4×C59C2×D411yescharacteristic
47C2×C4×C59C2×D412yes—
48C22⋊D59—82no—
49C2×Dic59—22no—
50C2×Dic59—41no—
51D236—42no—
52D236—81no—
53Dic118—42no—
54Dic118—41no—
55Dic118—41no—
56C8×C59—42no—
57C59⋊C8—22no—
58C22×C59—41no—
59C22×C59—42no—
60C22×C59#12011yescharacteristic
61C2×D59—42no—
62C2×D59—42no—
63C2×D59—42no—
64C2×D59—84no—
65C4×C59—21no—
66C4×C59—22no—
67C4×C59—41no—
68Dic59—42no—
69Dic59—81no—
70#1659—591nomaximal, Sylow
71C2×C59—81no—
72C2×C59#45311yescharacteristic
73C2×C59#45512yes—
74D59—82no—
75H(3,𝔽4)—591no—
76#497—591no—
77#489—592no—
78#329—592no—
79#329D5911yescharacteristic, complemented
80C59#165911yescharacteristic, complemented, Sylow
81C2×Q16—1181no—
82C2×SD16—1181no—
83C4⋊Q8—591no—
84#124—592no—
85#124—1182no—
86#124—1181no—
87#124C2×D5911yescharacteristic
88#120—1182no—
89#120—1181no—
90#103—1182no—
91#102—1182no—
92#97—591no—
93#97C2×D5912yescomplemented
94C24—1181no—
95C2×Q8—592no—
96C2×Q8—1181no—
97C2×Q8D23611yescharacteristic
98C2×D4—592no—
99C2×D4—1181no—
100C2×D4—1182no—
101C2×D4D23611yescharacteristic
102Q16—2362no—
103SD16—2362no—
104C2×C8—22no—
105C2×C8—1181no—
106C4⋊C4—1182no—
107C22⋊C4—22no—
108C22⋊C4—1182no—
109C22⋊C4—1181no—
110C22⋊C4—1182no—
111C22⋊C4—1182no—
112C42—1181no—
113C42C22×D5911yescharacteristic
114C23—21no—
115C23—1182no—
116C23—4721no—
117C23—4722no—
118Q8—41no—
119Q8—2361no—
120Q8—2361no—
121Q8—2362no—
122D4—41no—
123D4—4721no—
124D4—4722no—
125D4—2362no—
126C2×C4—21no—
127C2×C4—1182no—
128C2×C4—2361no—
129C2×C4C2×D23611yescharacteristic
130C2×C4D4×D5912yes—
131C8—42no—
132C8—1182no—
133C22—41no—
134C22—42no—
135C22—4724no—
136C22—2362no—
137C22—2362no—
138C22—2362no—
139C22#1427369711yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—4721no—
144C4—2362no—
145C2—81no—
146C2—4722no—
147C2#1683757211yescharacteristic
148C2#1683758112yes—
subgroupG/H
centerC22#14273697
commutatorC42×C59C23
FrattiniC42C22×D59
FittingC59×#329C2
socleC22×C59#120
hypercenter#329D59
hypocenterC59#1659
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #16837797C2semidirect
3. #16837762C2semidirect
4. #16837141C2semidirect
5. C59×#329C2semidirect
8. #14273727C22semidirect
16. #14273638C22semidirect
18. #97×C59C22semidirect
79. #329D59semidirect
80. C59#1659semidirect
93. #97C2×D59semidirect