Finite Group Databasebeta

← #20273847

Subgroups — #20273847

Order 6784.

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

normal subgroups only

This group has 214 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#16811330C211yesmaximal, characteristic
2#16811002C211yesmaximal, characteristic, complemented
3#16810915C211yesmaximal, characteristic, complemented
4#16810371C211yesmaximal, characteristic, complemented
5#16810353C211yesmaximal, characteristic, complemented
6D4⋊#88322C211yesmaximal, characteristic
7C53×#476C211yesmaximal, characteristic
8C2×#11719925—22no—
9#13133217C2211yescharacteristic
10#13133208—21no—
11#13133196—21no—
12#13133194—22no—
13C4×Dic106C2211yescharacteristic
14D4×Dic53—21no—
15C2×C4⋊Dic53—21no—
16#13133150—21no—
17C4⋊#88322C2211yescharacteristic
18#13133140—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—
28C22×Dic53—22no—
29C2×Dic106D411yescharacteristic
30C2×Dic106D412yes—
31C2×C8×C53—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53D411yescharacteristic
35C22⋊C4×C53—21no—
36C42×C53C2311yescharacteristic
37C22⋊Dic53—22no—
38#11719925—44no—
39C4⋊Dic53—22no—
40C4⋊Dic53—21no—
41C4⋊Dic53—21no—
42C4⋊Dic53D411yescharacteristic
43#11719919—22no—
44#11719919—21no—
45C4×Dic53—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C23×C53—21no—
49Q8×C53—42no—
50D4×C53—21no—
51D4×C53Q1612yescomplemented
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×Dic53—22no—
60C2×Dic53—21no—
61C2×Dic53—21no—
62C2×Dic53—82no—
63Dic106—21no—
64Dic106—21no—
65Dic106—42no—
66C8×C53—41no—
67#88322—22no—
68#88322—41no—
69C22×C53—22no—
70C22×C53—41no—
71C22×C53—41no—
72C22×C53#12011yescharacteristic
73C4×C53—21no—
74C4×C53—41no—
75C4×C53—81no—
76C4×C53C2×Q1611yescharacteristic
77C4×C53#13511yescharacteristic
78C4×C53#13711yescharacteristic
79C4×C53#13711yescharacteristic
80Dic53—41no—
81Dic53—42no—
82Dic53—41no—
83#1632—531nomaximal, Sylow
84C2×C53—42no—
85C2×C53#45111yescharacteristic
86C2×C53#45211yescharacteristic
87C2×C53#45311yescharacteristic
88#554—531no—
89#494—531no—
90#477—531no—
91#476D5311yescharacteristic
92#333—531no—
93Q8⋊C8—531no—
94D4⋊C8—531no—
95C53#163211yescharacteristic, complemented, Sylow
96C2×Q16—1062no—
97C4⋊Q8—531no—
98C4⋊Q8C2×D5311yescharacteristic
99#125—1061no—
100C22⋊Q8—1062no—
101C22⋊Q8—1061no—
102C4×Q8—531no—
103C4×D4—1061no—
104C4×D4C2×D5311yescharacteristic
105C2×C4⋊C4—1061no—
106#107—1061no—
107#106—21no—
108C4⋊C8—531no—
109C4⋊C8C2×D5311yescharacteristic
110#103—1061no—
111#102—21no—
112C4×C8—531no—
113C2×Q8—21no—
114C2×Q8—531no—
115C2×Q8—532no—
116C2×D4C22⋊D5311yescharacteristic
117C22×C4—21no—
118C22×C4—1062no—
119Q16—2124no—
120C2×C8—21no—
121C2×C8—1061no—
122C2×C8—1061no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—531no—
126C4⋊C4—1062no—
127C4⋊C4—1062no—
128C4⋊C4—1061no—
129C4⋊C4—1061no—
130C4⋊C4—1061no—
131C4⋊C4C22⋊D5311yescharacteristic
132C22⋊C4—21no—
133C22⋊C4—1062no—
134C42—1061no—
135C42C22×D5311yescharacteristic
136C23—21no—
137Q8—42no—
138Q8—1061no—
139Q8—1061no—
140Q8—2122no—
141D4—21no—
142D4#1171992512yescomplemented
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—1061no—
148C2×C4—1061no—
149C2×C4—1062no—
150C2×C4—4242no—
151C2×C4D4×D5311yescharacteristic
152C2×C4D4×D5311yescharacteristic
153C2×C4C2×C22⋊D5311yescharacteristic
154C8—41no—
155C8—1062no—
156C8—2121no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1313318511yescharacteristic
161C4—21no—
162C4—41no—
163C4—81no—
164C4—2121no—
165C4—2121no—
166C4—2122no—
167C4C2×#1171992511yescharacteristic
168C4#1313323811yescharacteristic
169C4#1313324611yescharacteristic
170C4#1313324711yescharacteristic
171C2—42no—
172C2#1681077811yescharacteristic
173C2#1681078211yescharacteristic
174C2#1681078911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#476C2
socleC22×C53#120
hypercenter#476D53
hypocenterC53#1632
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. #16811002C2semidirect
3. #16810915C2semidirect
4. #16810371C2semidirect
5. #16810353C2semidirect
51. D4×C53Q16semidirect
95. C53#1632semidirect
142. D4#11719925semidirect