Finite Group Databasebeta

← #20273742

Subgroups — #20273742

Order 6784.

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

normal subgroups only

This group has 236 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#16811249C211yesmaximal, characteristic
2#16810959C212yesmaximal, complemented
3#16810559C211yesmaximal, characteristic
4#16810361C212yesmaximal, complemented
5C53×#323C211yesmaximal, characteristic
6#13133196—21no—
7#13133196C2212yes—
8#13133194—22no—
9#13133194—22no—
10C2×C4⋊Dic53C2211yescharacteristic
11C2×#11719919—21no—
12C2×C22⋊Dic53—21no—
13C2×C22⋊Dic53—21no—
14#13133151—22no—
15#13133142—22no—
16#13133124C2211yescharacteristic
17C2×C22⋊C4×C53C2211yescharacteristic
18C22⋊C8×C53C2212yes—
19C24×C53D411yescharacteristic
20C22×C4×C53D411yescharacteristic
21C22×C4×C53C2311yescharacteristic
22C22×Dic53—21no—
23C22×Dic53—21no—
24C22×Dic53—21no—
25C22×Dic53—21no—
26C2×Dic106—21no—
27C2×Dic106D412yes—
28C2×C8×C53—22no—
29C22⋊C4×C53—22no—
30C22⋊Dic53—22no—
31C22⋊Dic53—22no—
32C22⋊Dic53—42no—
33C4⋊Dic53—21no—
34C4⋊Dic53—22no—
35C4⋊Dic53D412yes—
36#11719919—22no—
37#11719919—22no—
38#11719919—42no—
39C23×C53—21no—
40C23×C53—21no—
41C23×C53—41no—
42C23×C53—41no—
43C23×C53Q1612yes—
44C23×C53C2×D411yescharacteristic
45C2×C4×C53—21no—
46C2×C4×C53—41no—
47C2×C4×C53—41no—
48C2×C4×C53C2×D412yes—
49C2×Dic53—22no—
50C2×Dic53—22no—
51C2×Dic53—22no—
52C2×Dic53—41no—
53C2×Dic53—41no—
54C2×Dic53—41no—
55C2×Dic53—41no—
56C2×Dic53—41no—
57C2×Dic53—81no—
58C2×Dic53—81no—
59Dic106—42no—
60Dic106—81no—
61C8×C53—42no—
62C22×C53—21no—
63C22×C53—21no—
64C22×C53—22no—
65C22×C53—42no—
66C22×C53—41no—
67C22×C53—41no—
68C22×C53—81no—
69C22×C53#12011yescharacteristic
70C22×C53C2×Q1611yescharacteristic
71C22×C53#13711yescharacteristic
72C4×C53—22no—
73C4×C53—41no—
74Dic53—42no—
75Dic53—42no—
76Dic53—42no—
77Dic53—81no—
78#1614—531nomaximal, Sylow
79C2×C53—21no—
80C2×C53—21no—
81C2×C53—42no—
82C2×C53#45111yescharacteristic
83C2×C53#45311yescharacteristic
84C2×C53#45811yescharacteristic
85#543—531no—
86#484—532no—
87#399—531no—
88#328—532no—
89#323D5311yescharacteristic
90C53#161411yescharacteristic, complemented, Sylow
91C22⋊Q8—532no—
92C22⋊Q8—1062no—
93C22⋊Q8—1062no—
94C22⋊Q8—1061no—
95C2×C4⋊C4—531no—
96C2×C4⋊C4—1061no—
97C2×C22⋊C4—1061no—
98C2×C22⋊C4—1061no—
99C2×C22⋊C4C2×D5311yescharacteristic
100#107—1062no—
101#103—1062no—
102C22⋊C8C2×D5312yes—
103#95—531no—
104C24D21211yescharacteristic
105C2×Q8—532no—
106C2×Q8—1061no—
107C22×C4—1061no—
108C22×C4—1061no—
109C22×C4—1061no—
110C22×C4—1061no—
111C22×C4D21211yescharacteristic
112C22×C4C22×D5311yescharacteristic
113C2×C8—22no—
114C4⋊C4—532no—
115C4⋊C4—1062no—
116C4⋊C4—1061no—
117C4⋊C4—1062no—
118C4⋊C4—1062no—
119C4⋊C4—2122no—
120C22⋊C4—22no—
121C22⋊C4—1062no—
122C22⋊C4—1062no—
123C22⋊C4—2122no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic21212yes—
129C23C2×D21211yescharacteristic
130Q8—4241no—
131Q8—2122no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—1062no—
136C2×C4—1062no—
137C2×C4—1062no—
138C2×C4—4241no—
139C2×C4—4241no—
140C2×C4—2121no—
141C2×C4—2121no—
142C2×C4—2121no—
143C2×C4—2121no—
144C2×C4—2121no—
145C2×C4D4×D5312yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#1313318711yescharacteristic
155C22C2×Dic21211yescharacteristic
156C22#1313325011yescharacteristic
157C4—22no—
158C4—41no—
159C4—4241no—
160C4—2122no—
161C4—2122no—
162C4—2122no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#1681078111yescharacteristic
167C2#1681079311yescharacteristic
168C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#323C2
socleC22×C53#120
hypercenter#323D53
hypocenterC53#1614
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #16810959C2semidirect
4. #16810361C2semidirect
90. C53#1614semidirect