Finite Group Databasebeta

← #20273868

Subgroups — #20273868

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#16810974C211yesmaximal, characteristic, complemented
3#16810914C211yesmaximal, characteristic, complemented
4#16810913C211yesmaximal, characteristic, complemented
5#16810368C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×D4⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719947—21no—
9#13133217C2211yescharacteristic
10#13133217C2211yescharacteristic
11#13133201—22no—
12#13133197—21no—
13#13133191—21no—
14C4×D212C2211yescharacteristic
15C2×C4⋊Dic53—21no—
16#13133149—21no—
17#13133148—21no—
18C4⋊#88322C2211yescharacteristic
19#13133142—22no—
20#13133139—21no—
21#13133138—21no—
22C4×D4×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53D411yescharacteristic
26C22×C4×C53—21no—
27C2×C22⋊D53—22no—
28C22×Dic53—22no—
29#11719947—42no—
30#11719947—42no—
31C2×D212D411yescharacteristic
32C2×C4×D53—21no—
33C2×Dic106—21no—
34C2×Dic106—21no—
35C2×C8×C53—21no—
36C2×C8×C53—21no—
37C4⋊C4×C53D411yescharacteristic
38C22⋊C4×C53—21no—
39C42×C53C2311yescharacteristic
40#11719921—22no—
41#11719921—21no—
42C4⋊Dic53—22no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C4⋊Dic53—21no—
46C4⋊Dic53D412yes—
47C4⋊Dic53D411yescharacteristic
48C2×#88322—21no—
49C23×C53—21no—
50C22×D53—21no—
51D4×C53—21no—
52D4×C53SD1612yes—
53C2×C4×C53—21no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C22⋊D53—44no—
60C2×Dic53—22no—
61C2×Dic53—21no—
62C2×Dic53—44no—
63C2×Dic53—41no—
64D212—21no—
65D212—21no—
66C4×D53—41no—
67C4×D53—41no—
68Dic106—41no—
69Dic106—41no—
70Dic106—42no—
71C8×C53—41no—
72C8×C53—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×C53C2×SD1611yescharacteristic
84C4×C53#13511yescharacteristic
85C4×C53AGL(1,ℤ/8)11yescharacteristic
86C4×C53#13711yescharacteristic
87Dic53—42no—
88Dic53—41no—
89Dic53—81no—
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#476—531no—
100#332—531no—
101D4⋊C8—531no—
102D4⋊C8D5311yescharacteristic, complemented
103C53#163511yescharacteristic, complemented, Sylow
104C2×#41—1061no—
105C4⋊Q8—531no—
106C4⋊Q8—531no—
107#123—1062no—
108C22⋊Q8—1061no—
109#121—1061no—
110C4×D4—531no—
111C4×D4C2×D5311yescharacteristic
112C2×C4⋊C4—1061no—
113#106—1061no—
114#106—1061no—
115C4⋊C8—531no—
116C4⋊C8C2×D5311yescharacteristic
117#103—1062no—
118#102—1061no—
119#102—1061no—
120C4×C8C2×D5311yescharacteristic
121#41—2122no—
122#41—2122no—
123C2×Q8—1061no—
124C2×Q8—1061no—
125C2×D4—531no—
126C2×D4—1062no—
127C2×D4D21211yescharacteristic
128C22×C4—21no—
129C22×C4—1062no—
130C22×C4—1061no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—1061no—
134C4⋊C4—532no—
135C4⋊C4—531no—
136C4⋊C4—1062no—
137C4⋊C4—1061no—
138C4⋊C4—1061no—
139C4⋊C4—1061no—
140C4⋊C4D21211yescharacteristic
141C22⋊C4—21no—
142C22⋊C4—1062no—
143C22⋊C4—1061no—
144C42C22×D5311yescharacteristic
145C23—21no—
146C23—1061no—
147Q8—2122no—
148Q8—2121no—
149Q8—2121no—
150D4—21no—
151D4—1061no—
152D4—1061no—
153D4—2124no—
154D4#1171991312yes—
155C2×C4—21no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—1061no—
159C2×C4—1062no—
160C2×C4—2121no—
161C2×C4—2124no—
162C2×C4—2121no—
163C2×C4—2121no—
164C2×C4C2×D21211yescharacteristic
165C2×C4D4×D5311yescharacteristic
166C2×C4D4×D5311yescharacteristic
167C8—41no—
168C8—41no—
169C8—2121no—
170C22—22no—
171C22—41no—
172C22—41no—
173C22—2121no—
174C22—2121no—
175C22—2121no—
176C22#1313318711yescharacteristic
177C4—21no—
178C4—41no—
179C4—4241no—
180C4—2121no—
181C4—2122no—
182C4C2×#1171991311yescharacteristic
183C4#1313323611yescharacteristic
184C4#1313324511yescharacteristic
185C4#1313324611yescharacteristic
186C2—42no—
187C2—4241no—
188C2#1681077611yescharacteristic
189C2#1681078611yescharacteristic
190C2#1681079711yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×D4⋊C8C2
socleC22×C53#120
hypercenterD4⋊C8D53
hypocenterC53#1635
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #16810974C2semidirect
3. #16810914C2semidirect
4. #16810913C2semidirect
5. #16810368C2semidirect
6. #16810352C2semidirect
7. C53×D4⋊C8C2semidirect
102. D4⋊C8D53semidirect
103. C53#1635semidirect