Finite Group Databasebeta

← #20273849

Subgroups — #20273849

Order 6784.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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#16811002C211yesmaximal, characteristic, complemented
2#16810918C211yesmaximal, characteristic, complemented
3#16810913C211yesmaximal, characteristic
4#16810372C211yesmaximal, characteristic, complemented
5Q8⋊#88322C211yesmaximal, characteristic, complemented
6D4⋊#88322C211yesmaximal, characteristic
7C53×#554C211yesmaximal, characteristic
8C2×#11719925—22no—
9#13133217C2211yescharacteristic
10#13133150—21no—
11#13133148—21no—
12C4⋊#88322C2211yescharacteristic
13C4⋊#88322C2211yescharacteristic
14#13133141—21no—
15#13133138—21no—
16C4×#88322C2211yescharacteristic
17C4⋊Q8×C53C2211yescharacteristic
18#125×C53—21no—
19C22⋊Q8×C53—22no—
20C22⋊Q8×C53—21no—
21C4×Q8×C53C2211yescharacteristic
22C4×D4×C53—21no—
23C4×D4×C53C2211yescharacteristic
24C2×C4⋊C4×C53—21no—
25C2×Q8×C53D411yescharacteristic
26C2×Q8×C53D412yes—
27C2×D4×C53D411yescharacteristic
28C22×C4×C53—22no—
29C22×C4×C53—21no—
30C2×Dic106—21no—
31C4⋊C4×C53—22no—
32C4⋊C4×C53—22no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53D411yescharacteristic
37C4⋊C4×C53D411yescharacteristic
38C22⋊C4×C53—22no—
39C22⋊C4×C53—21no—
40C42×C53—21no—
41C42×C53C2311yescharacteristic
42#11719925—44no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C2×#88322—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C23×C53—21no—
49Q8×C53—21no—
50Q8×C53—21no—
51Q8×C53—42no—
52D4×C53—21no—
53D4×C53Q1612yescomplemented
54C2×C4×C53—21no—
55C2×C4×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53—82no—
61C2×C4×C53C2×D411yescharacteristic
62C2×C4×C53C2×D411yescharacteristic
63C2×C4×C53C2×D411yescharacteristic
64C2×Dic53—41no—
65Dic106—42no—
66#88322—22no—
67#88322—41no—
68#88322—41no—
69C22×C53—22no—
70C22×C53—41no—
71C22×C53—41no—
72C22×C53#12011yescharacteristic
73C4×C53—21no—
74C4×C53—41no—
75C4×C53—41no—
76C4×C53—41no—
77C4×C53—42no—
78C4×C53C2×Q1611yescharacteristic
79C4×C53#13511yescharacteristic
80C4×C53#13711yescharacteristic
81C4×C53#13711yescharacteristic
82Dic53—81no—
83#1632—531nomaximal, Sylow
84C2×C53—42no—
85C2×C53#45111yescharacteristic
86C2×C53#45211yescharacteristic
87C2×C53#45311yescharacteristic
88#554D5311yescharacteristic
89#494—531no—
90#477—531no—
91#476—531no—
92#333—531no—
93Q8⋊C8—531no—
94D4⋊C8—531no—
95C53#163211yescharacteristic, complemented, Sylow
96C2×Q16—1062no—
97C4⋊Q8—531no—
98C4⋊Q8C2×D5311yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C2×D5311yescharacteristic
103C4×D4—21no—
104C4×D4C2×D5311yescharacteristic
105C2×C4⋊C4—21no—
106#107—1061no—
107#106—1061no—
108C4⋊C8—531no—
109C4⋊C8—531no—
110#103—1061no—
111#102—1061no—
112C4×C8—531no—
113C2×Q8—1061no—
114C2×Q8C22⋊D5311yescharacteristic
115C2×Q8C22⋊D5312yes—
116C2×D4C22⋊D5311yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—2124no—
120C2×C8—1061no—
121C2×C8—1061no—
122C2×C8—1061no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—1061no—
129C4⋊C4—1061no—
130C4⋊C4C22⋊D5311yescharacteristic
131C4⋊C4C22⋊D5311yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C22×D5311yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—2122no—
141D4—21no—
142D4#1171992512yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—2121no—
151C2×C4C2×C22⋊D5311yescharacteristic
152C2×C4C2×C22⋊D5311yescharacteristic
153C2×C4C2×C22⋊D5311yescharacteristic
154C8—1062no—
155C8—2121no—
156C8—2121no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1313318611yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—4241no—
167C4C2×#1171992511yescharacteristic
168C4#1313323711yescharacteristic
169C4#1313324711yescharacteristic
170C4#1313324811yescharacteristic
171C2—42no—
172C2#1681077911yescharacteristic
173C2#1681078511yescharacteristic
174C2#1681079111yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#554C2
socleC22×C53#120
hypercenter#554D53
hypocenterC53#1632
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #16811002C2semidirect
2. #16810918C2semidirect
4. #16810372C2semidirect
5. Q8⋊#88322C2semidirect
53. D4×C53Q16semidirect
95. C53#1632semidirect
142. D4#11719925semidirect