Finite Group Databasebeta

← #21423055

Subgroups — #21423055

Order 7552.

256 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 53 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#16838028C211yesmaximal, characteristic
2#16837560C212yesmaximal, complemented
3#16837314C211yesmaximal, characteristic
4#16837140C212yesmaximal, complemented
5C59×#323C211yesmaximal, characteristic
6C22×Dic118C2211yescharacteristic
7C2×Dic236—22no—
8#14273706—21no—
9#14273706C2212yes—
10#14273704—22no—
11#14273704—22no—
12C2×C22⋊Dic59—21no—
13C2×C22⋊Dic59—21no—
14#14273652—22no—
15C59⋊#95—21no—
16C59⋊#95C2211yescharacteristic
17C2×C22⋊C4×C59C2211yescharacteristic
18C22⋊C8×C59C2212yes—
19C24×C59D411yescharacteristic
20C22×C4×C59D411yescharacteristic
21C22×C4×C59C2311yescharacteristic
22C22×Dic59—21no—
23C22×Dic59—21no—
24C22×Dic59—21no—
25C22×Dic59—21no—
26C2×Dic118—22no—
27C2×Dic118—21no—
28C2×Dic118—42no—
29C2×Dic118D412yes—
30C2×C8×C59—22no—
31C22⋊C4×C59—22no—
32C22⋊Dic59—22no—
33C22⋊Dic59—22no—
34C22⋊Dic59—42no—
35C4⋊Dic59—21no—
36C4⋊Dic59D412yes—
37Dic59⋊C4—22no—
38Dic59⋊C4—22no—
39Dic236—44no—
40C23×C59—21no—
41C23×C59—21no—
42C23×C59—41no—
43C23×C59—41no—
44C23×C59Q1612yescomplemented
45C23×C59C2×D411yescharacteristic
46C2×C4×C59—21no—
47C2×C4×C59—41no—
48C2×C4×C59—41no—
49C2×C4×C59C2×D412yes—
50C2×Dic59—22no—
51C2×Dic59—22no—
52C2×Dic59—22no—
53C2×Dic59—41no—
54C2×Dic59—41no—
55C2×Dic59—41no—
56C2×Dic59—41no—
57C2×Dic59—41no—
58C2×Dic59—81no—
59C2×Dic59—81no—
60Dic118—24no—
61Dic118—42no—
62Dic118—42no—
63Dic118—81no—
64C8×C59—42no—
65C22×C59—21no—
66C22×C59—21no—
67C22×C59—22no—
68C22×C59—42no—
69C22×C59—41no—
70C22×C59—41no—
71C22×C59—81no—
72C22×C59#12011yescharacteristic
73C22×C59C2×Q1611yescharacteristic
74C22×C59#13711yescharacteristic
75C4×C59—22no—
76C4×C59—41no—
77Dic59—42no—
78Dic59—42no—
79Dic59—42no—
80Dic59—81no—
81#1602—591nomaximal, Sylow
82C2×C59—21no—
83C2×C59—21no—
84C2×C59—42no—
85C2×C59#45111yescharacteristic
86C2×C59#45511yescharacteristic
87C2×C59B(4,𝔽2)11yescharacteristic
88#543—591no—
89#451—592no—
90#393—591no—
91#328—592no—
92#323D5911yescharacteristic
93C59#160211yescharacteristic, complemented, Sylow
94C22×Q8—591no—
95C2×Q16—1182no—
96C22⋊Q8—592no—
97C22⋊Q8—1182no—
98C22⋊Q8—1182no—
99C22⋊Q8—1181no—
100C2×C22⋊C4—1181no—
101C2×C22⋊C4—1181no—
102C2×C22⋊C4C2×D5911yescharacteristic
103#103—1182no—
104C22⋊C8C2×D5912yes—
105#95—591no—
106#95—1181no—
107C24D23611yescharacteristic
108C2×Q8—592no—
109C2×Q8—1181no—
110C2×Q8—1182no—
111C2×Q8—2362no—
112C22×C4—1181no—
113C22×C4—1181no—
114C22×C4—1181no—
115C22×C4—1181no—
116C22×C4D23611yescharacteristic
117C22×C4C22×D5911yescharacteristic
118Q16—2364no—
119C2×C8—22no—
120C4⋊C4—592no—
121C4⋊C4—1182no—
122C4⋊C4—1181no—
123C4⋊C4—1182no—
124C22⋊C4—22no—
125C22⋊C4—1182no—
126C22⋊C4—1182no—
127C22⋊C4—2362no—
128C23—21no—
129C23—21no—
130C23—41no—
131C23—41no—
132C23Dic23612yescomplemented
133C23C2×D23611yescharacteristic
134Q8—1184no—
135Q8—4721no—
136Q8—2362no—
137Q8—2362no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—1182no—
142C2×C4—1182no—
143C2×C4—1182no—
144C2×C4—4721no—
145C2×C4—4721no—
146C2×C4—2361no—
147C2×C4—2361no—
148C2×C4—2361no—
149C2×C4—2361no—
150C2×C4—2361no—
151C2×C4D4×D5912yes—
152C8—42no—
153C22—21no—
154C22—21no—
155C22—22no—
156C22—42no—
157C22—41no—
158C22—41no—
159C22—81no—
160C22#1427369711yescharacteristic
161C22C2×Dic23611yescharacteristic
162C22#1427376011yescharacteristic
163C4—22no—
164C4—41no—
165C4—4721no—
166C4—2362no—
167C4—2362no—
168C4—2362no—
169C2—21no—
170C2—21no—
171C2—42no—
172C2#1683756011yescharacteristic
173C2#1683758211yescharacteristic
174C2#1683759011yescharacteristic
subgroupG/H
centerC22#14273697
commutatorC22×C4×C59C23
FrattiniC22×C4C22×D59
FittingC59×#323C2
socleC22×C59#120
hypercenter#323D59
hypocenterC59#1602
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #16837560C2semidirect
4. #16837140C2semidirect
44. C23×C59Q16semidirect
93. C59#1602semidirect
132. C23Dic236semidirect