Finite Group Databasebeta

← #20273718

Subgroups — #20273718

Order 6784.

254 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 57 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#16810934C212yesmaximal, complemented
2#16810553C211yesmaximal, characteristic, complemented
3#16810357C212yesmaximal, complemented
4#16810349C211yesmaximal, characteristic, complemented
5C53×#537C211yesmaximal, characteristic
6C2×C4⋊Dic53C2211yescharacteristic
7C2×#11719919—21no—
8#13133150—22no—
9#13133138—22no—
10C22⋊#88322C2212yes—
11#13133124—21no—
12#13133124C2211yescharacteristic
13C2×#41×C53—21no—
14C22⋊Q8×C53—22no—
15#121×C53—22no—
16#121×C53—21no—
17#121×C53C2212yes—
18C2×C4⋊C4×C53C2211yescharacteristic
19#41×C53—42no—
20#41×C53—42no—
21C2×Q8×C53—21no—
22C2×D4×C53—21no—
23C2×D4×C53—22no—
24C2×D4×C53—21no—
25C2×D4×C53D412yes—
26C22×C4×C53—22no—
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D411yescharacteristic
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—22no—
34C4⋊C4×C53D412yes—
35C22⋊C4×C53—22no—
36C22⋊C4×C53—22no—
37C4⋊Dic53—22no—
38#11719919—42no—
39C2×#88322—22no—
40C23×C53—22no—
41C23×C53C2×D411yescharacteristic
42Q8×C53—42no—
43D4×C53—42no—
44D4×C53—42no—
45D4×C53—42no—
46D4×C53—42no—
47D4×C53—81no—
48C2×C4×C53—21no—
49C2×C4×C53—21no—
50C2×C4×C53—22no—
51C2×C4×C53—21no—
52C2×C4×C53—41no—
53C2×C4×C53—44no—
54C2×C4×C53—41no—
55C2×C4×C53D812yes—
56C2×C4×C53C2×D412yes—
57C2×Dic53—22no—
58C2×Dic53—41no—
59C2×Dic53—41no—
60C2×Dic53—41no—
61C2×Dic53—41no—
62C2×Dic53—41no—
63#88322—42no—
64C22×C53—21no—
65C22×C53—21no—
66C22×C53—42no—
67C22×C53—42no—
68C22×C53—42no—
69C22×C53#12011yescharacteristic
70C22×C53C2×D811yescharacteristic
71C22×C53AGL(1,ℤ/8)11yescharacteristic
72C4×C53—22no—
73C4×C53—22no—
74C4×C53—42no—
75C4×C53—41no—
76Dic53—42no—
77Dic53—81no—
78#1610—531nomaximal, Sylow
79C2×C53—21no—
80C2×C53—21no—
81C2×C53—82no—
82C2×C53#44711yescharacteristic
83C2×C53#45611yescharacteristic
84C2×C53#45811yescharacteristic
85#537D5311yescharacteristic
86#480—532no—
87#398—531no—
88#327—532no—
89#324—531no—
90C53#161011yescharacteristic, complemented, Sylow
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C2×D5312yes—
96C2×C4⋊C4—531no—
97C2×C4⋊C4—1061no—
98C2×C4⋊C4C2×D5311yescharacteristic
99#107—1062no—
100#102—1062no—
101C22⋊C8—532no—
102#95—531no—
103#95—1061no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C22⋊D5312yes—
111C22×C4—22no—
112C22×C4—1061no—
113C22×C4—1061no—
114C22×C4C22⋊D5311yescharacteristic
115C22×C4C22⋊D5311yescharacteristic
116C22×C4C22×D5311yescharacteristic
117C2×C8—1062no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—1062no—
121C4⋊C4—2122no—
122C4⋊C4C22⋊D5312yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×C22⋊D5311yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—44no—
139C2×C4—41no—
140C2×C4—1062no—
141C2×C4—2121no—
142C2×C4—2121no—
143C2×C4—2121no—
144C2×C4—2121no—
145C2×C4—2121no—
146C2×C4#1171992212yes—
147C2×C4C2×C22⋊D5312yes—
148C8—2122no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#1313318611yescharacteristic
155C22C2×#1171992211yescharacteristic
156C22#1313324311yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—4241no—
162C4—2122no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#1681075711yescharacteristic
167C2#1681080611yescharacteristic
168C2#1681081311yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#537C2
socleC22×C53#120
hypercenter#537D53
hypocenterC53#1610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #16810934C2semidirect
2. #16810553C2semidirect
3. #16810357C2semidirect
4. #16810349C2semidirect
90. C53#1610semidirect