Finite Group Databasebeta

← #20273864

Subgroups — #20273864

Order 6784.

234 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 57 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#16811277C211yesmaximal, characteristic
2#16810973C211yesmaximal, characteristic, complemented
3#16810910C211yesmaximal, characteristic, complemented
4#16810370C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#476C211yesmaximal, characteristic, complemented
8C2×#11719947—21no—
9#13133217C2211yescharacteristic
10#13133201—22no—
11#13133197—21no—
12#13133191—21no—
13C4×D212C2211yescharacteristic
14C2×C4⋊Dic53—21no—
15#13133148—21no—
16C4⋊#88322C2211yescharacteristic
17#13133141—22no—
18#13133137—21no—
19C4×#88322C2211yescharacteristic
20C4⋊Q8×C53C2211yescharacteristic
21C4×D4×C53C2211yescharacteristic
22#106×C53—21no—
23C4⋊C8×C53C2211yescharacteristic
24#102×C53—21no—
25C2×Q8×C53—21no—
26C2×D4×C53D411yescharacteristic
27C22×C4×C53—21no—
28C2×C22⋊D53—22no—
29C22×Dic53—22no—
30#11719947—42no—
31#11719947—42no—
32C2×D212D411yescharacteristic
33C2×C4×D53—21no—
34C2×Dic106—21no—
35C2×C8×C53—21no—
36C4⋊C4×C53—21no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53D411yescharacteristic
39C22⋊C4×C53—21no—
40C42×C53C2311yescharacteristic
41#11719921—22no—
42#11719921—21no—
43C4⋊Dic53—22no—
44C4⋊Dic53—21no—
45C4⋊Dic53D412yes—
46C4⋊Dic53D411yescharacteristic
47C2×#88322—21no—
48C2×#88322—21no—
49C23×C53—21no—
50C22×D53—21no—
51Q8×C53—42no—
52D4×C53—21no—
53D4×C53SD1612yes—
54C2×C4×C53—21no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53C2×D411yescharacteristic
59C2×C4×C53C2×D411yescharacteristic
60C2×C4×C53C2×D411yescharacteristic
61C22⋊D53—44no—
62C2×Dic53—22no—
63C2×Dic53—21no—
64C2×Dic53—44no—
65D212—21no—
66D212—21no—
67C4×D53—41no—
68C4×D53—41no—
69Dic106—41no—
70Dic106—41no—
71C8×C53—41no—
72#88322—41no—
73#88322—41no—
74C22×C53—22no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53#12011yescharacteristic
78C2×D53—41no—
79C2×D53—41no—
80C2×D53—41no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53—81no—
84C4×C53C2×SD1611yescharacteristic
85C4×C53#13511yescharacteristic
86C4×C53AGL(1,ℤ/8)11yescharacteristic
87C4×C53#13711yescharacteristic
88Dic53—42no—
89Dic53—41no—
90#1635—531nomaximal, Sylow
91C2×C53—42no—
92C2×C53#45011yescharacteristic
93C2×C53#45211yescharacteristic
94C2×C53#45411yescharacteristic
95D53—81no—
96#547—531no—
97#487—531no—
98#476—531no—
99#476D5311yescharacteristic, complemented
100#332—531no—
101D4⋊C8—531no—
102D4⋊C8—531no—
103C53#163511yescharacteristic, complemented, Sylow
104C2×#41—1061no—
105C4⋊Q8—531no—
106C4⋊Q8C2×D5311yescharacteristic
107#123—1062no—
108C22⋊Q8—1061no—
109#121—1061no—
110C4×D4—531no—
111C4×D4C2×D5311yescharacteristic
112C2×C4⋊C4—1061no—
113#106—21no—
114#106—1061no—
115C4⋊C8—531no—
116C4⋊C8C2×D5311yescharacteristic
117#103—1062no—
118#102—21no—
119#102—1061no—
120C4×C8—531no—
121#41—2122no—
122#41—2122no—
123C2×Q8—21no—
124C2×Q8—1061no—
125C2×D4—531no—
126C2×D4—1062no—
127C2×D4C22⋊D5311yescharacteristic
128C22×C4—21no—
129C22×C4—1062no—
130C22×C4—1061no—
131C2×C8—21no—
132C2×C8—1061no—
133C2×C8—1061no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4—532no—
137C4⋊C4—531no—
138C4⋊C4—1062no—
139C4⋊C4—1061no—
140C4⋊C4C22⋊D5311yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—1062no—
143C22⋊C4—1061no—
144C42C22×D5311yescharacteristic
145C23—21no—
146C23—1061no—
147Q8—42no—
148Q8—2121no—
149Q8—2121no—
150D4—21no—
151D4—1061no—
152D4—1061no—
153D4—2124no—
154D4#1171992412yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—1061no—
160C2×C4—1062no—
161C2×C4—2124no—
162C2×C4—2121no—
163C2×C4—2121no—
164C2×C4D4×D5311yescharacteristic
165C2×C4D4×D5311yescharacteristic
166C2×C4C2×C22⋊D5311yescharacteristic
167C8—41no—
168C8—2121no—
169C8—2121no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—2121no—
174C22—2121no—
175C22—2121no—
176C22#1313318511yescharacteristic
177C4—21no—
178C4—41no—
179C4—81no—
180C4—2121no—
181C4—2122no—
182C4C2×#1171992411yescharacteristic
183C4#1313323811yescharacteristic
184C4#1313324211yescharacteristic
185C4#1313324611yescharacteristic
186C2—42no—
187C2—4241no—
188C2#1681077311yescharacteristic
189C2#1681078211yescharacteristic
190C2#1681079411yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#476C2
socleC22×C53#120
hypercenter#476D53
hypocenterC53#1635
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #16810973C2semidirect
3. #16810910C2semidirect
4. #16810370C2semidirect
5. D4⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×#476C2semidirect
99. #476D53semidirect
103. C53#1635semidirect