Finite Group Databasebeta

← #20273717

Subgroups — #20273717

Order 6784.

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

normal subgroups only

This group has 254 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#16811204C211yesmaximal, characteristic
2#16810936C212yesmaximal, complemented
3#16810356C212yesmaximal, complemented
4#16810349C211yesmaximal, characteristic, complemented
5C53×#398C211yesmaximal, characteristic, complemented
6C2×#11719945—21no—
7#13133197—22no—
8#13133192—22no—
9#13133191—21no—
10#13133191C2212yes—
11#13133150—22no—
12#13133137—22no—
13C22⋊#88322C2212yes—
14C2×C4⋊C4×C53—21no—
15C2×C4⋊C4×C53C2211yescharacteristic
16C2×C4⋊C4×C53C2211yescharacteristic
17#95×C53—21no—
18#95×C53C2211yescharacteristic
19C22×C4×C53—21no—
20C22×C4×C53—21no—
21C22×C4×C53D411yescharacteristic
22C22×C4×C53D411yescharacteristic
23C22×C4×C53C2311yescharacteristic
24C2×C22⋊D53—22no—
25#11719945—42no—
26#11719945—42no—
27C2×D212—21no—
28C2×D212—21no—
29C2×D212D412yes—
30C2×C4×D53—22no—
31C2×Dic106—21no—
32C4⋊C4×C53—22no—
33C4⋊C4×C53—22no—
34C4⋊C4×C53—42no—
35#11719921—22no—
36#11719921—22no—
37C4⋊Dic53—21no—
38C4⋊Dic53D412yes—
39C2×#88322—22no—
40C23×C53C2×D411yescharacteristic
41C22×D53—22no—
42C2×C4×C53—21no—
43C2×C4×C53—21no—
44C2×C4×C53—22no—
45C2×C4×C53—21no—
46C2×C4×C53—41no—
47C2×C4×C53—41no—
48C2×C4×C53—41no—
49C2×C4×C53—41no—
50C2×C4×C53—41no—
51C2×C4×C53—41no—
52C2×C4×C53—41no—
53C2×C4×C53D812yes—
54C2×C4×C53C2×D412yes—
55C22⋊D53—42no—
56C22⋊D53—42no—
57C2×Dic53—22no—
58D212—42no—
59D212—42no—
60D212—81no—
61C4×D53—44no—
62Dic106—42no—
63#88322—42no—
64C22×C53—21no—
65C22×C53—21no—
66C22×C53#12011yescharacteristic
67C22×C53C2×D811yescharacteristic
68C22×C53AGL(1,ℤ/8)11yescharacteristic
69C2×D53—42no—
70C2×D53—42no—
71C2×D53—42no—
72C4×C53—22no—
73C4×C53—22no—
74C4×C53—42no—
75C4×C53—41no—
76C4×C53—81no—
77Dic53—42no—
78#1610—531nomaximal, Sylow
79C2×C53—21no—
80C2×C53—21no—
81C2×C53#44711yescharacteristic
82C2×C53#45611yescharacteristic
83C2×C53#45811yescharacteristic
84D53—82no—
85#537—531no—
86#480—532no—
87#398D5311yescharacteristic, complemented
88#327—532no—
89#324—531no—
90C53#161011yescharacteristic, complemented, Sylow
91C2×#41—1061no—
92C22⋊Q8—1062no—
93#121—532no—
94#121—1062no—
95#121—1061no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2×D5311yescharacteristic
98C2×C4⋊C4C2×D5311yescharacteristic
99#107—1062no—
100#102—1062no—
101C22⋊C8—532no—
102#95—21no—
103#95C2×D5311yescharacteristic
104#41—2122no—
105#41—2122no—
106C2×Q8—1061no—
107C2×D4—532no—
108C2×D4—1061no—
109C2×D4—1061no—
110C2×D4—1062no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—1062no—
114C22×C4C22⋊D5311yescharacteristic
115C22×C4C22⋊D5311yescharacteristic
116C22×C4C22×D5311yescharacteristic
117C2×C8—1062no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—532no—
122C4⋊C4—1061no—
123C22⋊C4—1062no—
124C22⋊C4—1062no—
125C23—1062no—
126C23C2×C22⋊D5311yescharacteristic
127Q8—2122no—
128D4—4241no—
129D4—2122no—
130D4—2122no—
131D4—2122no—
132D4—2122no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—1062no—
145C2×C4—2124no—
146C2×C4#1171992212yes—
147C2×C4D4×D5312yes—
148C8—2122no—
149C22—21no—
150C22—21no—
151C22—2122no—
152C22—2122no—
153C22—2122no—
154C22#1313318511yescharacteristic
155C22C2×#1171992211yescharacteristic
156C22#1313324211yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—81no—
162C4—2122no—
163C2—21no—
164C2—21no—
165C2—4242no—
166C2#1681075611yescharacteristic
167C2#1681080511yescharacteristic
168C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#398C2
socleC22×C53#120
hypercenter#398D53
hypocenterC53#1610
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #16810936C2semidirect
3. #16810356C2semidirect
4. #16810349C2semidirect
5. C53×#398C2semidirect
87. #398D53semidirect
90. C53#1610semidirect