Finite Group Databasebeta

← #20273758

Subgroups — #20273758

Order 6784.

202 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 51 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#16811252C211yesmaximal, characteristic
2#16810959C211yesmaximal, characteristic
3#16810956C211yesmaximal, characteristic
4#16810563C211yesmaximal, characteristic
5#16810359C211yesmaximal, characteristic
6#16810348C211yesmaximal, characteristic
7C53×#328C211yesmaximal, characteristic
8#13133216—21no—
9#13133215—21no—
10#13133207—21no—
11#13133207—21no—
12#13133196C2211yescharacteristic
13#13133194—21no—
14#13133173—21no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2211yescharacteristic
17#13133151—21no—
18#13133150—21no—
19#13133142—21no—
20#13133141—21no—
21C22⋊#88322C2211yescharacteristic
22#13133124—21no—
23#13133124—21no—
24C22⋊Q8×C53C2211yescharacteristic
25C22⋊C8×C53C2211yescharacteristic
26#95×C53C2211yescharacteristic
27C2×Q8×C53D411yescharacteristic
28C22×C4×C53—21no—
29C22×C4×C53C2311yescharacteristic
30C22×Dic53—21no—
31C22×Dic53D411yescharacteristic
32C22×Dic53D411yescharacteristic
33C2×Dic106—21no—
34C2×Dic106D411yescharacteristic
35C2×C8×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53D411yescharacteristic
38C22⋊C4×C53—21no—
39C22⋊Dic53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53—21no—
43C4⋊Dic53D411yescharacteristic
44#11719919—21no—
45#11719919—21no—
46#11719919—21no—
47#11719919—21no—
48#11719919—21no—
49#11719919—21no—
50#11719919—21no—
51C4×Dic53—21no—
52C4×Dic53—21no—
53C2×#88322—21no—
54C23×C53C2×D411yescharacteristic
55Q8×C53—41no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×Dic53—21no—
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—41no—
70C2×Dic53—42no—
71C2×Dic53—41no—
72C2×Dic53Q1612yes—
73Dic106—41no—
74Dic106—81no—
75C8×C53—41no—
76#88322—41no—
77C22×C53—21no—
78C22×C53—21no—
79C22×C53#12011yescharacteristic
80C22×C53C2×Q1611yescharacteristic
81C22×C53#13711yescharacteristic
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—41no—
85C4×C53—41no—
86C4×C53—81no—
87Dic53—22no—
88Dic53—41no—
89Dic53—41no—
90Dic53—41no—
91Dic53—81no—
92#1617—531nomaximal, Sylow
93C2×C53—21no—
94C2×C53—21no—
95C2×C53#45111yescharacteristic
96C2×C53#45411yescharacteristic
97C2×C53#45811yescharacteristic
98#544—531no—
99#484—531no—
100#484—531no—
101#400—531no—
102#328—531no—
103#328D5311yescharacteristic
104#324—531no—
105C53#161711yescharacteristic, complemented, Sylow
106C4⋊Q8—1061no—
107C4⋊Q8—1061no—
108#125—1061no—
109#125—1061no—
110C22⋊Q8—531no—
111C22⋊Q8—1061no—
112C22⋊Q8C2×D5311yescharacteristic
113#117—1061no—
114C2×C4⋊C4—531no—
115C2×C4⋊C4—531no—
116#107—1061no—
117#107—1061no—
118#103—1061no—
119#103—1061no—
120C22⋊C8—531no—
121C22⋊C8C2×D5311yescharacteristic
122#95—1061no—
123#95—1061no—
124#95C2×D5311yescharacteristic
125C2×Q8—531no—
126C2×Q8—1061no—
127C2×Q8D21211yescharacteristic
128C22×C4—21no—
129C22×C4—531no—
130C22×C4—531no—
131C22×C4—1061no—
132C22×C4C22×D5311yescharacteristic
133C2×C8—21no—
134C2×C8—1061no—
135C4⋊C4—21no—
136C4⋊C4—531no—
137C4⋊C4—1061no—
138C4⋊C4—1061no—
139C4⋊C4—1061no—
140C4⋊C4—1061no—
141C4⋊C4—1061no—
142C4⋊C4—1061no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4—1061no—
146C4⋊C4—1061no—
147C4⋊C4D21211yescharacteristic
148C22⋊C4—21no—
149C22⋊C4—1061no—
150C42—1061no—
151C42—1061no—
152C23D4×D5311yescharacteristic
153Q8—41no—
154Q8—4241no—
155Q8—2121no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—532no—
163C2×C4—1061no—
164C2×C4—1061no—
165C2×C4—1061no—
166C2×C4—1061no—
167C2×C4—1061no—
168C2×C4—2122no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4C2×D21211yescharacteristic
172C2×C4D4×D5311yescharacteristic
173C8—41no—
174C8—2121no—
175C22—21no—
176C22—21no—
177C22#1313318711yescharacteristic
178C22Q16×D5311yescharacteristic
179C22#1313324611yescharacteristic
180C4—21no—
181C4—21no—
182C4—41no—
183C4—41no—
184C4—81no—
185C4—1062no—
186C4—4241no—
187C4—2121no—
188C4—2121no—
189C4—2121no—
190C2—21no—
191C2—21no—
192C2#1681078011yescharacteristic
193C2#1681079711yescharacteristic
194C2#1681081411yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#328C2
socleC22×C53#120
hypercenter#328D53
hypocenterC53#1617
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
105. C53#1617semidirect