Finite Group Databasebeta

← #20275706

Subgroups — #20275706

Order 6784.

228 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#13133208C211yesmaximal, characteristic
2#16810646C211yesmaximal, characteristic
3M16×Dic53C211yesmaximal, characteristic
4#16810392C211yesmaximal, characteristic
5#16810391C211yesmaximal, characteristic
6#16810391C211yesmaximal, characteristic
7C53×#343C211yesmaximal, characteristic
8C2×#11719917C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10#13133208—21no—
11#13133208—21no—
12#13133208—21no—
13#13133208—21no—
14C2×C4⋊Dic53C412yes—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C412yes—
17C2×C4×Dic53C2211yescharacteristic
18#13133146—21no—
19#13133146—21no—
20#13133127—21no—
21C8×Dic53—21no—
22C2×M16×C53C2211yescharacteristic
23C2×M16×C53C2211yescharacteristic
24C2×C4⋊C4×C53C2211yescharacteristic
25C22×C4×C53C2×C412yes—
26C22×C4×C53C2311yescharacteristic
27C22×Dic53C2×C412yes—
28C22×Dic53C2×C411yescharacteristic
29C22×Dic53C2×C411yescharacteristic
30M16×C53—21no—
31M16×C53—21no—
32M16×C53—21no—
33M16×C53Q812yes—
34C2×C8×C53—21no—
35C2×C8×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊Dic53—21no—
39C4⋊Dic53—21no—
40C4⋊Dic53—22no—
41C4⋊Dic53—22no—
42#11719919—22no—
43#11719919—22no—
44C4×Dic53D411yescharacteristic
45C4×Dic53D411yescharacteristic
46C4×Dic53D411yescharacteristic
47C4×Dic53D411yescharacteristic
48#11719917—21no—
49#11719917—21no—
50#11719917—21no—
51#11719917Q812yes—
52C2×#88322—21no—
53C2×#88322—21no—
54C23×C53C22×C411yescharacteristic
55C2×C4×C53—22no—
56C2×C4×C53—42no—
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53C2×Q811yescharacteristic
60C2×C4×C53C2×Q811yescharacteristic
61C2×C4×C53#4111yescharacteristic
62C2×C4×C53#4111yescharacteristic
63C2×Dic53—22no—
64C2×Dic53—41no—
65C2×Dic53—41no—
66C2×Dic53—42no—
67C2×Dic53C22⋊C411yescharacteristic
68C2×Dic53C22⋊C411yescharacteristic
69C2×Dic53C22⋊C411yescharacteristic
70C2×Dic53C22⋊C411yescharacteristic
71C8×C53—22no—
72C8×C53—41no—
73#88322—22no—
74#88322—41no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53C2×C22⋊C411yescharacteristic
78C22×C53C4×Q811yescharacteristic
79C22×C53C4×Q811yescharacteristic
80C4×C53—42no—
81C4×C53C22⋊Q811yescharacteristic
82C4×C53C22⋊Q811yescharacteristic
83C4×C53#12411yescharacteristic
84C4×C53C4⋊Q811yescharacteristic
85Dic53—21no—
86Dic53—21no—
87Dic53—21no—
88Dic53—21no—
89Dic53—42no—
90#1998—531nomaximal, Sylow
91C2×C53—21no—
92C2×C53—21no—
93C2×C53#39111yescharacteristic
94C2×C53#41311yescharacteristic
95C2×C53#41311yescharacteristic
96C2×#125—531no—
97#423—531no—
98C4×M16—531no—
99#343—531no—
100#343—531no—
101#343—531no—
102#343D5311yescharacteristic
103C53#199811yescharacteristic, complemented, Sylow
104C2×M16—531no—
105C2×M16—531no—
106C2×M16C2×D5311yescharacteristic
107C2×M16C2×D5311yescharacteristic
108#125—1061no—
109#125—1061no—
110#125—1061no—
111#125—1061no—
112C2×C4⋊C4—532no—
113C2×C4⋊C4—532no—
114C2×C4⋊C4—531no—
115C2×C4⋊C4C2×D5311yescharacteristic
116C2×C42—531no—
117C4⋊C8—1061no—
118C4⋊C8—1061no—
119#97—1061no—
120C4×C8—1061no—
121C22×C4—531no—
122C22×C4—531no—
123C22×C4—532no—
124C22×C4C4×D5312yes—
125C22×C4C22×D5311yescharacteristic
126M16—21no—
127M16—21no—
128M16—21no—
129M16—532no—
130M16—1061no—
131M16—1061no—
132M16—1061no—
133M16Dic10612yes—
134C2×C8—21no—
135C2×C8—21no—
136C2×C8—1061no—
137C2×C8—1061no—
138C4⋊C4—21no—
139C4⋊C4—21no—
140C4⋊C4—1062no—
141C4⋊C4—1062no—
142C4⋊C4—1062no—
143C4⋊C4—1062no—
144C4⋊C4—1061no—
145C4⋊C4—1061no—
146C42—531no—
147C42—531no—
148C42—531no—
149C42—531no—
150C23C2×C4×D5311yescharacteristic
151C2×C4—22no—
152C2×C4—42no—
153C2×C4—531no—
154C2×C4—531no—
155C2×C4—531no—
156C2×C4—531no—
157C2×C4—1062no—
158C2×C4—2121no—
159C2×C4—2121no—
160C2×C4—2122no—
161C2×C4C2×Dic10611yescharacteristic
162C2×C4#1171994511yescharacteristic
163C2×C4D4×D5311yescharacteristic
164C2×C4D4×D5311yescharacteristic
165C2×C4#1171994711yescharacteristic
166C2×C4Q8×D5311yescharacteristic
167C8—22no—
168C8—41no—
169C8—1062no—
170C8—2121no—
171C22—21no—
172C22—21no—
173C22C22⋊C4×D5311yescharacteristic
174C22#1313318211yescharacteristic
175C22C4×Dic10611yescharacteristic
176C4—42no—
177C4—1061no—
178C4—1061no—
179C4—1061no—
180C4—1061no—
181C4—2122no—
182C4#1313319411yescharacteristic
183C4#1313319511yescharacteristic
184C4#1313320311yescharacteristic
185C4#1313321611yescharacteristic
186C2—21no—
187C2—21no—
188C2#1681052711yescharacteristic
189C2#1681060411yescharacteristic
190C2#1681060611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D53
commutatorC23×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#343C2
socleC22×C53C2×C22⋊C4
hypercenter#343D53
hypocenterC53#1998
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
103. C53#1998semidirect