Finite Group Databasebeta

← #20273943

Subgroups — #20273943

Order 6784.

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

normal subgroups only

This group has 212 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#16810973C211yesmaximal, characteristic, complemented
2#16810903C211yesmaximal, characteristic, complemented
3#16810828C211yesmaximal, characteristic, complemented
4#16810372C211yesmaximal, characteristic, complemented
5D4⋊#88322C211yesmaximal, characteristic, complemented
6D4⋊#88322C211yesmaximal, characteristic, complemented
7C53×#555C211yesmaximal, characteristic
8C2×#11719923—21no—
9#13133217C2211yescharacteristic
10#13133150—21no—
11C4⋊#88322C2211yescharacteristic
12C4⋊#88322C2211yescharacteristic
13#13133141—22no—
14#13133140—21no—
15#13133138—21no—
16C4×#88322C2211yescharacteristic
17C4⋊Q8×C53C2211yescharacteristic
18#124×C53—22no—
19#121×C53—21no—
20#121×C53—21no—
21C4×D4×C53C2211yescharacteristic
22C4×D4×C53C2211yescharacteristic
23#117×C53—21no—
24#117×C53—21no—
25C2×Q8×C53D412yes—
26C2×D4×C53—22no—
27C2×D4×C53D411yescharacteristic
28C2×D4×C53D411yescharacteristic
29C22×C4×C53—21no—
30C22×C4×C53—21no—
31C2×Dic106—21no—
32C4⋊C4×C53—21no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53D411yescharacteristic
35C4⋊C4×C53D411yescharacteristic
36C22⋊C4×C53—22no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53—21no—
40C42×C53—22no—
41C42×C53C2311yescharacteristic
42#11719923—42no—
43C4⋊Dic53—21no—
44C4⋊Dic53—21no—
45C2×#88322—21no—
46C2×#88322—21no—
47C2×#88322—21no—
48C23×C53—21no—
49C23×C53—21no—
50Q8×C53—42no—
51D4×C53—21no—
52D4×C53—21no—
53D4×C53—21no—
54D4×C53—21no—
55D4×C53—82no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—41no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×Dic53—41no—
67Dic106—41no—
68Dic106—41no—
69#88322—41no—
70#88322—41no—
71#88322—41no—
72C22×C53—41no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53#12011yescharacteristic
79C4×C53—21no—
80C4×C53—41no—
81C4×C53—41no—
82C4×C53—42no—
83C4×C53#13511yescharacteristic
84C4×C53#13511yescharacteristic
85C4×C53AGL(1,ℤ/8)11yescharacteristic
86C4×C53#13711yescharacteristic
87Dic53—81no—
88#1646—531nomaximal, Sylow
89C2×C53—81no—
90C2×C53—81no—
91C2×C53#44911yescharacteristic
92C2×C53#45211yescharacteristic
93C2×C53#45511yescharacteristic
94#555D5311yescharacteristic
95#487—531no—
96#474—531no—
97#461—531no—
98#333—531no—
99D4⋊C8—531no—
100D4⋊C8—531no—
101C53#164611yescharacteristic, complemented, Sylow
102C2×SD16—1061no—
103C4⋊Q8—531no—
104C4⋊Q8C2×D5311yescharacteristic
105#124—22no—
106#121—21no—
107#121—21no—
108C4×D4C2×D5311yescharacteristic
109C4×D4C2×D5311yescharacteristic
110#117—21no—
111#117—21no—
112#107—1061no—
113C4⋊C8—531no—
114C4⋊C8—531no—
115#103—1062no—
116#103—1061no—
117#102—1061no—
118C4×C8—531no—
119C2×Q8—1061no—
120C2×Q8C22⋊D5312yes—
121C2×D4—22no—
122C2×D4C22⋊D5311yescharacteristic
123C2×D4C22⋊D5311yescharacteristic
124C22×C4—21no—
125C22×C4—21no—
126SD16—2122no—
127C2×C8—1061no—
128C2×C8—1061no—
129C2×C8—1061no—
130C4⋊C4—21no—
131C4⋊C4—21no—
132C4⋊C4—1061no—
133C4⋊C4—1061no—
134C4⋊C4C22⋊D5311yescharacteristic
135C4⋊C4C22⋊D5311yescharacteristic
136C22⋊C4—22no—
137C22⋊C4—22no—
138C22⋊C4—21no—
139C22⋊C4—21no—
140C42—22no—
141C42C22×D5311yescharacteristic
142C23—21no—
143C23—21no—
144Q8—42no—
145Q8—2121no—
146Q8—2121no—
147D4—21no—
148D4—21no—
149D4—21no—
150D4—21no—
151D4—82no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—2121no—
160C2×C4C2×C22⋊D5311yescharacteristic
161C2×C4C2×C22⋊D5311yescharacteristic
162C2×C4C2×C22⋊D5311yescharacteristic
163C8—2121no—
164C8—2121no—
165C8—2121no—
166C22—41no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22—41no—
171C22—41no—
172C22#1313318611yescharacteristic
173C4—21no—
174C4—41no—
175C4—41no—
176C4—42no—
177C4—4241no—
178C4#1313323711yescharacteristic
179C4#1313323711yescharacteristic
180C4#1313324211yescharacteristic
181C4#1313324711yescharacteristic
182C2—81no—
183C2—81no—
184C2#1681076811yescharacteristic
185C2#1681078511yescharacteristic
186C2#1681080111yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#555C2
socleC22×C53#120
hypercenter#555D53
hypocenterC53#1646
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #555

Complements

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

NH ≅ G/Ntype
1. #16810973C2semidirect
2. #16810903C2semidirect
3. #16810828C2semidirect
4. #16810372C2semidirect
5. D4⋊#88322C2semidirect
6. D4⋊#88322C2semidirect
101. C53#1646semidirect