Finite Group Databasebeta

← #20275884

Subgroups — #20275884

Order 6784.

244 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 67 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#16810641C211yesmaximal, characteristic
2C2×#13133145C211yesmaximal, characteristic
3#16810596C211yesmaximal, characteristic
4#16810393C211yesmaximal, characteristic, complemented
5#16810383C211yesmaximal, characteristic, complemented
6#16810379C211yesmaximal, characteristic, complemented
7C53×#533C211yesmaximal, characteristic, complemented
8C2×#11719945C2211yescharacteristic
9C2×#11719917C2211yescharacteristic
10C2×#11719917C2211yescharacteristic
11#13133176C2211yescharacteristic
12#13133145—21no—
13#13133145—21no—
14#13133145—22no—
15#13133131—22no—
16C4⋊Q8×C53—21no—
17#125×C53—21no—
18C22⋊Q8×C53—22no—
19C4×Q8×C53—22no—
20#117×C53C2211yescharacteristic
21#117×C53C2211yescharacteristic
22C2×C42×C53C2211yescharacteristic
23C2×Q8×C53—22no—
24C22×C4×C53—21no—
25C22×C4×C53C2311yescharacteristic
26C2×C22⋊D53—21no—
27#11719945—21no—
28#11719945—21no—
29#11719945—21no—
30#11719945—21no—
31C2×D212Q811yescharacteristic, complemented
32C2×C4×D53—21no—
33C2×Dic106Q811yescharacteristic, complemented
34C4⋊C4×C53—24no—
35C4⋊C4×C53—22no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53D412yes—
40C42×C53—21no—
41C42×C53—21no—
42C42×C53D411yescharacteristic
43C42×C53D411yescharacteristic
44C42×C53D412yes—
45C22⋊Dic53—21no—
46C4⋊Dic53—21no—
47C4×Dic53—21no—
48#11719917—21no—
49#11719917—21no—
50#11719917—21no—
51#11719917—21no—
52C2×#88322—21no—
53C2×#88322—21no—
54C23×C53C2×D411yescharacteristic
55C22×D53—21no—
56Q8×C53—44no—
57C2×C4×C53—21no—
58C2×C4×C53—21no—
59C2×C4×C53—22no—
60C2×C4×C53—22no—
61C2×C4×C53—42no—
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×Q811yescharacteristic
65C2×C4×C53#4111yescharacteristic
66C2×C4×C53#4111yescharacteristic
67C2×C4×C53#4111yescharacteristic
68C22⋊D53—41no—
69C22⋊D53—41no—
70C2×Dic53—21no—
71C2×Dic53—41no—
72D212—21no—
73D212—21no—
74C4×D53—41no—
75C4×D53—41no—
76Dic106—21no—
77Dic106—21no—
78#88322—41no—
79#88322—41no—
80C22×C53—41no—
81C22×C53#12011yescharacteristic
82C22×C53C22⋊Q811yescharacteristic
83C22×C53#12411yescharacteristic
84C2×D53—41no—
85C2×D53—81no—
86C4×C53—22no—
87C4×C53—41no—
88C4×C53—42no—
89C4×C53—42no—
90C4×C53C22⋊Q811yescharacteristic
91C4×C53C22⋊Q811yescharacteristic
92C4×C53#12411yescharacteristic
93C4×C53#12411yescharacteristic
94Dic53—41no—
95Dic53—81no—
96#2031—531nomaximal, Sylow
97C2×C53—21no—
98C2×C53—21no—
99C2×C53—21no—
100C2×C53#39311yescharacteristic
101D53—81no—
102#533D5311yescharacteristic, complemented
103#421—531no—
104C2×#104—531no—
105#410—531no—
106#344—531no—
107#339—531no—
108H(3,ℤ/4)—531no—
109C53#203111yescharacteristic, complemented, Sylow
110C2×#41—531no—
111C2×M16—531no—
112C2×M16—531no—
113C4⋊Q8—21no—
114#125—21no—
115C22⋊Q8—22no—
116C4×Q8—22no—
117#117—531no—
118#117C2×D5311yescharacteristic
119#117C2×D5311yescharacteristic
120C2×C42C2×D5311yescharacteristic
121#104—1061no—
122#104—1061no—
123#104—1062no—
124#99—1062no—
125#41—1061no—
126#41—1061no—
127#41—1061no—
128#41—1061no—
129C2×Q8—22no—
130C2×Q8—531no—
131C2×D4—531no—
132C2×D4—1061no—
133C22×C4—21no—
134C22×C4—1061no—
135C22×C4C22×D5311yescharacteristic
136M16—1061no—
137M16—1061no—
138M16—1061no—
139M16—1061no—
140C2×C8—1061no—
141C2×C8—1061no—
142C4⋊C4—24no—
143C4⋊C4—22no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—1061no—
147C22⋊C4—21no—
148C22⋊C4—1061no—
149C22⋊C4C22⋊D5312yes—
150C42—21no—
151C42—21no—
152C42—1061no—
153C42C22⋊D5311yescharacteristic
154C42C22⋊D5311yescharacteristic
155C42C22⋊D5312yes—
156C23—1061no—
157C23C2×C22⋊D5311yescharacteristic
158Q8—44no—
159Q8—1061no—
160Q8—1061no—
161D4—1061no—
162D4—1061no—
163D4—2121no—
164D4—2121no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—1061no—
171C2×C4—2121no—
172C2×C4—2121no—
173C2×C4—2121no—
174C2×C4Q8×D5311yescharacteristic
175C2×C4#1171994911yescharacteristic
176C2×C4#1171994911yescharacteristic
177C2×C4#1171994911yescharacteristic
178C2×C4C2×C22⋊D5311yescharacteristic
179C2×C4C2×C22⋊D5311yescharacteristic
180C8—2121no—
181C8—2121no—
182C22—41no—
183C22—4241no—
184C22—2121no—
185C22#1313318611yescharacteristic
186C22#1313319311yescharacteristic
187C22#1313320611yescharacteristic
188C4—22no—
189C4—41no—
190C4—42no—
191C4—42no—
192C4—4241no—
193C4—2121no—
194C4#1313319311yescharacteristic
195C4#1313319311yescharacteristic
196C4#1313320611yescharacteristic
197C4#1313320611yescharacteristic
198C2—21no—
199C2—21no—
200C2—21no—
201C2—4241no—
202C2#1681053411yescharacteristic
subgroupG/H
centerC4#13133206
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#533C2
socleC2×C53#393
hypercenter#533D53
hypocenterC53#2031
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #533

Complements

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

NH ≅ G/Ntype
4. #16810393C2semidirect
5. #16810383C2semidirect
6. #16810379C2semidirect
7. C53×#533C2semidirect
31. C2×D212Q8semidirect
33. C2×Dic106Q8semidirect
102. #533D53semidirect
109. C53#2031semidirect