Finite Group Databasebeta

← #20273734

Subgroups — #20273734

Order 6784.

288 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 53 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#16811185C211yesmaximal, characteristic
2#16810941C212yesmaximal, complemented
3#16810356C212yesmaximal, complemented
4#16810347C211yesmaximal, characteristic, complemented
5C53×#399C211yesmaximal, characteristic, complemented
6C22×C22⋊D53—21no—
7#13133201—22no—
8#13133191—21no—
9#13133191C2212yes—
10#13133187—22no—
11#13133148—22no—
12#13133137—22no—
13C22⋊#88322C2212yescomplemented
14C2×C4⋊C4×C53—21no—
15C2×C4⋊C4×C53C2211yescharacteristic, complemented
16C2×C22⋊C4×C53—21no—
17C2×C22⋊C4×C53C2211yescharacteristic
18#95×C53C2211yescharacteristic, complemented
19C24×C53D411yescharacteristic
20C23×D53—21no—
21C22×C4×C53—21no—
22C22×C4×C53—21no—
23C22×C4×C53D411yescharacteristic
24C22×C4×C53C2311yescharacteristic
25C2×C22⋊D53—22no—
26C2×C22⋊D53—22no—
27C2×C22⋊D53—42no—
28C2×C22⋊D53—42no—
29C22×Dic53—21no—
30C2×D212—21no—
31C2×D212D412yes—
32C4⋊C4×C53—22no—
33C4⋊C4×C53—42no—
34C22⋊C4×C53—22no—
35C22⋊C4×C53—42no—
36#11719921—22no—
37#11719921—22no—
38C4⋊Dic53—21no—
39C4⋊Dic53D412yes—
40C2×#88322—22no—
41C23×C53—21no—
42C23×C53—21no—
43C23×C53—41no—
44C23×C53—41no—
45C23×C53SD1612yes—
46C23×C53C2×D411yescharacteristic
47C22×D53—22no—
48C22×D53—42no—
49C22×D53—81no—
50C22×D53—81no—
51C2×C4×C53—22no—
52C2×C4×C53—21no—
53C2×C4×C53—41no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53C2×D412yes—
61C22⋊D53—44no—
62C22⋊D53—42no—
63C22⋊D53—42no—
64C2×Dic53—22no—
65C2×Dic53—42no—
66D212—42no—
67D212—81no—
68#88322—42no—
69C22×C53—21no—
70C22×C53—21no—
71C22×C53—22no—
72C22×C53—42no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53—81no—
76C22×C53#12011yescharacteristic
77C22×C53C2×SD1611yescharacteristic
78C22×C53AGL(1,ℤ/8)11yescharacteristic
79C2×D53—42no—
80C2×D53—42no—
81C2×D53—42no—
82C2×D53—84no—
83C4×C53—22no—
84C4×C53—42no—
85C4×C53—41no—
86C4×C53—81no—
87Dic53—42no—
88#1613—531nomaximal, Sylow
89C2×C53—21no—
90C2×C53—21no—
91C2×C53—42no—
92C2×C53#45011yescharacteristic
93C2×C53#45311yescharacteristic
94C2×C53#45811yescharacteristic
95D53—82no—
96#534—531no—
97#481—532no—
98#399D5311yescharacteristic, complemented
99#327—532no—
100#323—531no—
101C53#161311yescharacteristic, complemented, Sylow
102C22×D4—1061no—
103#123—1062no—
104#121—532no—
105#121—1061no—
106#120—1062no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2×D5311yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C2×D5311yescharacteristic
111#106—1062no—
112#102—1062no—
113C22⋊C8—532no—
114#95C2×D5311yescharacteristic, complemented
115C24—1061no—
116C24C22⋊D5311yescharacteristic
117C2×D4—532no—
118C2×D4—1061no—
119C2×D4—1062no—
120C2×D4—1062no—
121C2×D4—2122no—
122C2×D4—2122no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—1061no—
126C22×C4C22⋊D5311yescharacteristic
127C22×C4C22×D5311yescharacteristic
128C2×C8—1062no—
129C4⋊C4—22no—
130C4⋊C4—42no—
131C4⋊C4—532no—
132C4⋊C4—1061no—
133C22⋊C4—22no—
134C22⋊C4—42no—
135C22⋊C4—1062no—
136C22⋊C4—1062no—
137C23—21no—
138C23—21no—
139C23—41no—
140C23—41no—
141C23—1062no—
142C23—4241no—
143C23—4241no—
144C23—2122no—
145C23#1171992412yes—
146C23C2×C22⋊D5311yescharacteristic
147D4—4241no—
148D4—2122no—
149D4—2124no—
150D4—2122no—
151D4—2122no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—1062no—
162C2×C4—2122no—
163C2×C4D4×D5312yes—
164C8—2122no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—81no—
172C22—4244no—
173C22—2122no—
174C22—2122no—
175C22—2122no—
176C22#1313318511yescharacteristic
177C22C2×#1171992411yescharacteristic
178C22#1313324211yescharacteristic
179C4—22no—
180C4—42no—
181C4—41no—
182C4—81no—
183C4—2122no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—4242no—
188C2#1681077311yescharacteristic
189C2#1681079011yescharacteristic
190C2#1681081211yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC22×C4×C53C23
FrattiniC22×C4C22×D53
FittingC53×#399C2
socleC22×C53#120
hypercenter#399D53
hypocenterC53#1613
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #399

Complements

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

NH ≅ G/Ntype
2. #16810941C2semidirect
3. #16810356C2semidirect
4. #16810347C2semidirect
5. C53×#399C2semidirect
13. C22⋊#88322C22semidirect
15. C2×C4⋊C4×C53C22semidirect
18. #95×C53C22semidirect
98. #399D53semidirect
101. C53#1613semidirect
108. C2×C4⋊C4C2×D53semidirect
114. #95C2×D53semidirect