Finite Group Databasebeta

← #20275373

Subgroups — #20275373

Order 6784.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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#16810666C211yesmaximal, characteristic
2C2×#13133141C211yesmaximal, characteristic
3C2×#13133136C211yesmaximal, characteristic
4#16810579C211yesmaximal, characteristic
5#16810391C211yesmaximal, characteristic
6#16810387C211yesmaximal, characteristic
7C53×#419C211yesmaximal, characteristic
8C2×#11719917C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C22×#88322C2211yescharacteristic
11C2×C4⋊Dic53C2211yescharacteristic
12#13133150—21no—
13#13133148—21no—
14#13133141—21no—
15#13133141—21no—
16#13133136C414yescomplemented
17#13133128—21no—
18C4×#88322—21no—
19C22×Q8×C53C2211yescharacteristic
20C2×M16×C53C2211yescharacteristic
21#117×C53C2211yescharacteristic
22#103×C53—21no—
23#103×C53—21no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53—21no—
27C2×Q8×C53—21no—
28C2×Q8×C53—21no—
29C2×Q8×C53C2×C412yes—
30C22×C4×C53—21no—
31C22×C4×C53C2311yescharacteristic
32C22×Dic53—21no—
33M16×C53—21no—
34M16×C53—21no—
35C2×C8×C53—21no—
36C4⋊C4×C53D411yescharacteristic
37C4⋊C4×C53D411yescharacteristic
38C22⋊C4×C53—21no—
39C42×C53—21no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42#11719917—21no—
43#11719917—21no—
44#11719917C2×C412yes—
45#11719917C2×C412yes—
46C2×#88322—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C2×#88322—21no—
50C2×#88322D411yescharacteristic
51C2×#88322D411yescharacteristic
52C23×C53#4111yescharacteristic
53Q8×C53—22no—
54Q8×C53—22no—
55Q8×C53—41no—
56Q8×C53—41no—
57C2×C4×C53—22no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53C22×C411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C2×C4×C53#4111yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C2×Dic53—41no—
69C2×Dic53—81no—
70C8×C53—41no—
71#88322—22no—
72#88322—22no—
73#88322—21no—
74#88322—21no—
75C22×C53—21no—
76C22×C53—21no—
77C22×C53#11711yescharacteristic
78C22×C53#12111yescharacteristic
79C22×C53#12611yescharacteristic
80C4×C53—42no—
81C4×C53—41no—
82C4×C53—41no—
83C4×C53C4×D411yescharacteristic
84C4×C53C4×D411yescharacteristic
85C4×C53#12311yescharacteristic
86C4×C53#12411yescharacteristic
87Dic53—81no—
88#1930—531nomaximal, Sylow
89C2×C53—21no—
90C2×C53—21no—
91C2×C53#38811yescharacteristic
92C2×C53#47111yescharacteristic
93C2×C53#47311yescharacteristic
94#428—531no—
95#419D5311yescharacteristic
96C2×#103—531no—
97C2×#101—531no—
98#405—531no—
99#343—531no—
100#341—531no—
101C53#193011yescharacteristic, complemented, Sylow
102C22×Q8C2×D5311yescharacteristic
103C2×M16—531no—
104C2×M16—531no—
105C2×M16C2×D5311yescharacteristic
106C22×C8—531no—
107#117C2×D5311yescharacteristic
108C2×C4⋊C4—531no—
109#107—1061no—
110#106—1061no—
111#103—21no—
112#103—21no—
113#103—1061no—
114#103—1061no—
115#101—534no—
116#97—1061no—
117C4×C8—1061no—
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—21no—
121C2×Q8—21no—
122C2×Q8—21no—
123C2×Q8C4×D5312yes—
124C22×C4—21no—
125C22×C4—1061no—
126C22×C4C22×D5311yescharacteristic
127M16—21no—
128M16—21no—
129M16—532no—
130M16—532no—
131M16—1061no—
132M16—1061no—
133C2×C8—21no—
134C2×C8—531no—
135C2×C8—531no—
136C2×C8—1061no—
137C2×C8—1061no—
138C2×C8—1061no—
139C2×C8—1061no—
140C4⋊C4—1061no—
141C4⋊C4—1061no—
142C4⋊C4C22⋊D5311yescharacteristic
143C4⋊C4C22⋊D5311yescharacteristic
144C22⋊C4—21no—
145C42—21no—
146C23#1171994711yescharacteristic
147Q8—22no—
148Q8—22no—
149Q8—41no—
150Q8—41no—
151C2×C4—22no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—4241no—
157C2×C4—2121no—
158C2×C4C2×C4×D5311yescharacteristic
159C2×C4#1171994511yescharacteristic
160C2×C4D4×D5311yescharacteristic
161C2×C4#1171994711yescharacteristic
162C2×C4#1171994711yescharacteristic
163C2×C4C2×C22⋊D5311yescharacteristic
164C8—41no—
165C8—1062no—
166C8—1062no—
167C8—1061no—
168C8—1061no—
169C22—21no—
170C22—21no—
171C22#1313317311yescharacteristic
172C22#1313318911yescharacteristic
173C22#1313321011yescharacteristic
174C4—42no—
175C4—41no—
176C4—41no—
177C4—4241no—
178C4#1313317711yescharacteristic
179C4C4×C22⋊D5311yescharacteristic
180C4#1313319911yescharacteristic
181C4#1313320411yescharacteristic
182C2—21no—
183C2—21no—
184C2#1681051711yescharacteristic
185C2#1681088811yescharacteristic
186C2#1681089611yescharacteristic
subgroupG/H
centerC22#13133189
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#419C2
socleC22×C53#121
hypercenter#419D53
hypocenterC53#1930
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
16. #13133136C4semidirect
101. C53#1930semidirect