Finite Group Databasebeta

← #20275982

Subgroups — #20275982

Order 6784.

256 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 69 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
1C22×#11719925C211yesmaximal, characteristic
2#16810650C211yesmaximal, characteristic
3#16810631C211yesmaximal, characteristic
4C2×#13133136C211yesmaximal, characteristic
5C2×#13133135C211yesmaximal, characteristic
6#16810387C211yesmaximal, characteristic
7C53×#419C211yesmaximal, characteristic
8C22×Dic106C2211yescharacteristic
9C2×#11719925—22no—
10C2×#11719925—22no—
11C2×#11719925—22no—
12C2×#11719917C2211yescharacteristic
13C2×#11719917C2211yescharacteristic
14C22×#88322C2211yescharacteristic
15C4⋊#88322—22no—
16#13133140—22no—
17#13133136—22no—
18#13133135—22no—
19C22×Q8×C53C2211yescharacteristic
20C2×M16×C53C2211yescharacteristic
21#117×C53C2211yescharacteristic
22#103×C53—22no—
23C2×Q8×C53—21no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53D412yes—
29C22×C4×C53—21no—
30C22×C4×C53C2311yescharacteristic
31C22×Dic53—21no—
32C2×Dic106—21no—
33C2×Dic106—21no—
34C2×Dic106—21no—
35C2×Dic106—21no—
36C2×Dic106—21no—
37C2×Dic106D412yes—
38M16×C53—21no—
39M16×C53—21no—
40C2×C8×C53—21no—
41C4⋊C4×C53D412yes—
42C22⋊C4×C53—21no—
43C42×C53—21no—
44#11719925—28no—
45#11719917—21no—
46#11719917—21no—
47#11719917—21no—
48#11719917—21no—
49C2×#88322—21no—
50C2×#88322—21no—
51C2×#88322—21no—
52C2×#88322—21no—
53C2×#88322D412yes—
54C23×C53#4111yescharacteristic
55Q8×C53—22no—
56Q8×C53—22no—
57Q8×C53—41no—
58Q8×C53—41no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C2×C4×C53#4111yescharacteristic
69C2×Dic53—22no—
70C2×Dic53—41no—
71C2×Dic53—41no—
72Dic106—22no—
73Dic106—22no—
74Dic106—41no—
75Dic106—41no—
76C8×C53—41no—
77#88322—22no—
78#88322—41no—
79#88322—41no—
80C22×C53—22no—
81C22×C53#12111yescharacteristic
82C22×C53#12311yescharacteristic
83C22×C53#12311yescharacteristic
84C4×C53—42no—
85C4×C53—42no—
86C4×C53#12011yescharacteristic
87C4×C53#12111yescharacteristic
88C4×C53#12111yescharacteristic
89C4×C53#12411yescharacteristic
90Dic53—42no—
91#2047—531nomaximal, Sylow
92C2×C53—21no—
93C2×C53—21no—
94C2×C53#39411yescharacteristic
95C2×C53#47312yes—
96C22×Q16—531no—
97#424—531no—
98#419—531no—
99#419D5311yescharacteristic
100C2×#101—531no—
101C2×#101—531no—
102#341—531no—
103C53#204711yescharacteristic, complemented, Sylow
104C22×Q8—531no—
105C22×Q8C2×D5311yescharacteristic
106C2×Q16—1062no—
107C2×Q16—1062no—
108C2×Q16—1062no—
109C2×M16—531no—
110C2×M16—531no—
111C2×M16C2×D5311yescharacteristic
112C22×C8—531no—
113#117C2×D5311yescharacteristic
114C4⋊C8—1062no—
115#103—22no—
116#103—1062no—
117#101—1062no—
118#101—1062no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8—21no—
124C2×Q8—532no—
125C2×Q8—1061no—
126C2×Q8—1061no—
127C2×Q8—1061no—
128C2×Q8—1061no—
129C2×Q8—1061no—
130C2×Q8D21212yes—
131C22×C4—21no—
132C22×C4—1061no—
133C22×C4C22×D5311yescharacteristic
134Q16—1068no—
135M16—21no—
136M16—21no—
137M16—1061no—
138M16—1061no—
139M16—1061no—
140M16—1061no—
141C2×C8—21no—
142C2×C8—532no—
143C2×C8—1061no—
144C2×C8—1061no—
145C2×C8—1061no—
146C2×C8—1061no—
147C4⋊C4C22⋊D5312yes—
148C22⋊C4—21no—
149C42—21no—
150C23#1171994711yescharacteristic
151Q8—22no—
152Q8—22no—
153Q8—41no—
154Q8—41no—
155Q8—1062no—
156Q8—1062no—
157Q8—2121no—
158Q8—2121no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—1062no—
164C2×C4—2121no—
165C2×C4—2121no—
166C2×C4C2×D21211yescharacteristic
167C2×C4#1171994511yescharacteristic
168C2×C4D4×D5311yescharacteristic
169C2×C4D4×D5311yescharacteristic
170C2×C4#1171994711yescharacteristic
171C2×C4C2×C22⋊D5311yescharacteristic
172C8—41no—
173C8—1062no—
174C8—2121no—
175C8—2121no—
176C22—22no—
177C22#1313318911yescharacteristic
178C22#1313319911yescharacteristic
179C22#1313320111yescharacteristic
180C4—42no—
181C4—42no—
182C4—2122no—
183C4#1313318711yescharacteristic
184C4#1313318811yescharacteristic
185C4#1313319111yescharacteristic
186C4#1313320311yescharacteristic
187C2—21no—
188C2—21no—
189C2#1681053911yescharacteristic
190C2#1681089612yes—
subgroupG/H
centerC22#13133189
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#419C2
socleC22×C53#121
hypercenter#419D53
hypocenterC53#2047
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
103. C53#2047semidirect