Finite Group Databasebeta

← #20273825

Subgroups — #20273825

Order 6784.

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

normal subgroups only

This group has 216 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#16810971C211yesmaximal, characteristic, complemented
2#16810823C211yesmaximal, characteristic, complemented
3#16810818C211yesmaximal, characteristic
4#16810369C211yesmaximal, characteristic, complemented
5Q8⋊#88322C211yesmaximal, characteristic, complemented
6D4⋊#88322C211yesmaximal, characteristic, complemented
7C53×#556C211yesmaximal, characteristic, complemented
8C2×#11719924—21no—
9C2×#11719922—21no—
10#13133214C2211yescharacteristic
11C4⋊#88322C2211yescharacteristic
12C4⋊#88322C2211yescharacteristic
13#13133137—21no—
14#13133137—21no—
15#13133137—22no—
16C4×#88322C2211yescharacteristic
17C4⋊Q8×C53C2211yescharacteristic
18#126×C53—22no—
19#124×C53—21no—
20C22⋊Q8×C53—21no—
21C4×Q8×C53—21no—
22C4×Q8×C53C2211yescharacteristic
23C4×D4×C53C2211yescharacteristic
24#117×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53D411yescharacteristic
27C2×D4×C53D411yescharacteristic
28C22×C4×C53—21no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212—21no—
32C4⋊C4×C53—22no—
33C4⋊C4×C53—21no—
34C4⋊C4×C53—21no—
35C4⋊C4×C53D411yescharacteristic
36C4⋊C4×C53D411yescharacteristic
37C4⋊C4×C53D412yes—
38C22⋊C4×C53—22no—
39C22⋊C4×C53—22no—
40C22⋊C4×C53—21no—
41C42×C53—22no—
42C42×C53—21no—
43C42×C53C2311yescharacteristic
44#11719924—42no—
45#11719922—42no—
46C2×#88322—21no—
47C2×#88322—21no—
48C2×#88322—21no—
49C23×C53—21no—
50C22×D53—41no—
51Q8×C53—21no—
52Q8×C53—21no—
53Q8×C53—41no—
54Q8×C53—41no—
55D4×C53—21no—
56D4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53—22no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—41no—
62C2×C4×C53—41no—
63C2×C4×C53C2×D411yescharacteristic
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66D212—42no—
67D212—41no—
68D212—41no—
69D212—41no—
70D212—41no—
71#88322—41no—
72#88322—41no—
73#88322—41no—
74C22×C53—41no—
75C22×C53—41no—
76C22×C53—41no—
77C22×C53#12011yescharacteristic
78C2×D53—81no—
79C2×D53—81no—
80C2×D53—81no—
81C4×C53—21no—
82C4×C53—41no—
83C4×C53—41no—
84C4×C53—42no—
85C4×C53—41no—
86C4×C53#13511yescharacteristic
87C4×C53#13511yescharacteristic
88C4×C53AGL(1,ℤ/8)11yescharacteristic
89C4×C53AGL(1,ℤ/8)11yescharacteristic
90#1628—531nomaximal, Sylow
91C2×C53—81no—
92C2×C53#44911yescharacteristic
93C2×C53#44911yescharacteristic
94C2×C53#45411yescharacteristic
95D53—161no—
96#556D5311yescharacteristic, complemented
97#486—531no—
98#460—531no—
99#459—531no—
100#332—531no—
101Q8⋊C8—531no—
102D4⋊C8—531no—
103C53#162811yescharacteristic, complemented, Sylow
104C2×SD16—1061no—
105C2×D8—1061no—
106C4⋊Q8C2×D5311yescharacteristic
107#127—531no—
108#126—22no—
109#124—21no—
110C22⋊Q8—21no—
111C4×Q8—21no—
112C4×Q8C2×D5311yescharacteristic
113C4×D4C2×D5311yescharacteristic
114#117—21no—
115C4⋊C8—531no—
116C4⋊C8—531no—
117#102—1061no—
118#102—1061no—
119#102—1062no—
120C4×C8—531no—
121C2×Q8—21no—
122C2×Q8C22⋊D5311yescharacteristic
123C2×D4—1061no—
124C2×D4—1061no—
125C2×D4—1061no—
126C2×D4C22⋊D5311yescharacteristic
127C22×C4—21no—
128SD16—2122no—
129D8—2122no—
130C2×C8—1061no—
131C2×C8—1061no—
132C2×C8—1061no—
133C4⋊C4—22no—
134C4⋊C4—21no—
135C4⋊C4—21no—
136C4⋊C4C22⋊D5311yescharacteristic
137C4⋊C4C22⋊D5311yescharacteristic
138C4⋊C4C22⋊D5312yes—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—21no—
142C42—22no—
143C42—21no—
144C42C22×D5311yescharacteristic
145C23—21no—
146C23—2121no—
147Q8—21no—
148Q8—21no—
149Q8—41no—
150Q8—41no—
151D4—21no—
152D4—21no—
153D4—2122no—
154D4—2121no—
155D4—2121no—
156D4—2121no—
157D4—2121no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4C2×C22⋊D5311yescharacteristic
165C2×C4C2×C22⋊D5311yescharacteristic
166C2×C4C2×C22⋊D5311yescharacteristic
167C8—2121no—
168C8—2121no—
169C8—2121no—
170C22—41no—
171C22—41no—
172C22—41no—
173C22—4241no—
174C22—4241no—
175C22—4241no—
176C22#1313318611yescharacteristic
177C4—21no—
178C4—41no—
179C4—41no—
180C4—42no—
181C4—41no—
182C4#1313323711yescharacteristic
183C4#1313323711yescharacteristic
184C4#1313324211yescharacteristic
185C4#1313324211yescharacteristic
186C2—81no—
187C2—8481no—
188C2#1681076811yescharacteristic
189C2#1681076811yescharacteristic
190C2#1681079611yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#556C2
socleC22×C53#120
hypercenter#556D53
hypocenterC53#1628
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #16810971C2semidirect
2. #16810823C2semidirect
4. #16810369C2semidirect
5. Q8⋊#88322C2semidirect
6. D4⋊#88322C2semidirect
7. C53×#556C2semidirect
96. #556D53semidirect
103. C53#1628semidirect