Finite Group Databasebeta

← #20273780

Subgroups — #20273780

Order 6784.

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

normal subgroups only

This group has 278 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#16810999C211yesmaximal, characteristic, complemented
2#16810823C211yesmaximal, characteristic, complemented
3#16810818C211yesmaximal, characteristic, complemented
4#16810367C211yesmaximal, characteristic, complemented
5Q8⋊#88322C211yesmaximal, characteristic, complemented
6D4⋊#88322C211yesmaximal, characteristic, complemented
7C53×#550C211yesmaximal, characteristic, complemented
8C2×#11719924—21no—
9C2×#11719922—21no—
10C2×#11719922—22no—
11#13133214C2211yescharacteristic
12C4⋊#88322C2211yescharacteristic
13C4⋊#88322C2211yescharacteristic
14#13133137—21no—
15#13133137—21no—
16C4×#88322C2211yescharacteristic
17C2×#41×C53—21no—
18#127×C53—21no—
19#127×C53C2211yescharacteristic
20#121×C53—21no—
21#121×C53—22no—
22C4×Q8×C53C2211yescharacteristic
23C4×D4×C53—21no—
24C4×D4×C53C2211yescharacteristic
25#41×C53—42no—
26#41×C53—42no—
27C2×Q8×C53D411yescharacteristic
28C2×D4×C53—22no—
29C2×D4×C53—22no—
30C2×D4×C53—21no—
31C2×D4×C53—21no—
32C2×D4×C53D411yescharacteristic
33C2×D4×C53D412yes—
34C22×C4×C53—22no—
35C22×C4×C53—21no—
36C2×D212—21no—
37C2×D212—21no—
38C2×D212—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53D411yescharacteristic
41C4⋊C4×C53D411yescharacteristic
42C22⋊C4×C53—22no—
43C22⋊C4×C53—21no—
44C42×C53—21no—
45C42×C53C2311yescharacteristic
46#11719924—42no—
47#11719922—44no—
48#11719922—42no—
49C2×#88322—21no—
50C2×#88322—21no—
51C2×#88322—21no—
52C23×C53—21no—
53C23×C53—22no—
54C22×D53—41no—
55Q8×C53—21no—
56Q8×C53D812yescomplemented
57D4×C53—21no—
58D4×C53—21no—
59D4×C53—42no—
60D4×C53—44no—
61D4×C53—41no—
62D4×C53—41no—
63D4×C53—41no—
64D4×C53—41no—
65D4×C53—82no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—21no—
69C2×C4×C53—44no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53C2×D411yescharacteristic
73C2×C4×C53C2×D411yescharacteristic
74C2×C4×C53C2×D411yescharacteristic
75D212—42no—
76D212—41no—
77D212—41no—
78D212—41no—
79D212—41no—
80#88322—22no—
81#88322—41no—
82#88322—41no—
83C22×C53—42no—
84C22×C53—41no—
85C22×C53—41no—
86C22×C53—42no—
87C22×C53—42no—
88C22×C53—41no—
89C22×C53#12011yescharacteristic
90C2×D53—81no—
91C2×D53—81no—
92C2×D53—81no—
93C4×C53—22no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97C4×C53C2×D811yescharacteristic
98C4×C53#13511yescharacteristic
99C4×C53AGL(1,ℤ/8)11yescharacteristic
100C4×C53AGL(1,ℤ/8)11yescharacteristic
101#1621—531nomaximal, Sylow
102C2×C53—81no—
103C2×C53—82no—
104C2×C53#44711yescharacteristic
105C2×C53#44911yescharacteristic
106C2×C53#45311yescharacteristic
107D53—161no—
108#550D5311yescharacteristic, complemented
109#493—531no—
110#460—531no—
111#459—531no—
112#331—531no—
113Q8⋊C8—531no—
114D4⋊C8—531no—
115C53#162111yescharacteristic, complemented, Sylow
116C2×#41—21no—
117C2×SD16—1061no—
118C2×D8—1061no—
119C2×D8—1062no—
120#127—21no—
121#127—531no—
122#127C2×D5311yescharacteristic
123#121—22no—
124#121—21no—
125C4×Q8C2×D5311yescharacteristic
126C4×D4—21no—
127C4×D4C2×D5311yescharacteristic
128C4⋊C8—531no—
129C4⋊C8—531no—
130#102—1061no—
131#102—1061no—
132C4×C8—531no—
133#41—42no—
134#41—42no—
135C2×Q8C22⋊D5311yescharacteristic
136C2×D4—22no—
137C2×D4—22no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—1061no—
141C2×D4—1061no—
142C2×D4—1061no—
143C2×D4C22⋊D5311yescharacteristic
144C2×D4C22⋊D5312yes—
145C22×C4—22no—
146C22×C4—21no—
147SD16—2122no—
148D8—2124no—
149D8—2122no—
150C2×C8—1061no—
151C2×C8—1061no—
152C2×C8—1061no—
153C4⋊C4—21no—
154C4⋊C4C22⋊D5311yescharacteristic
155C4⋊C4C22⋊D5311yescharacteristic
156C22⋊C4—22no—
157C22⋊C4—21no—
158C42—21no—
159C42C22×D5311yescharacteristic
160C23—21no—
161C23—22no—
162C23—2121no—
163Q8—21no—
164Q8#1171992212yescomplemented
165D4—21no—
166D4—21no—
167D4—42no—
168D4—44no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—82no—
174D4—2122no—
175D4—2121no—
176D4—2121no—
177D4—2121no—
178D4—2121no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—44no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4C2×C22⋊D5311yescharacteristic
186C2×C4C2×C22⋊D5311yescharacteristic
187C2×C4C2×C22⋊D5311yescharacteristic
188C8—1062no—
189C8—2121no—
190C8—2121no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—41no—
197C22—4241no—
198C22—4241no—
199C22—4241no—
200C22#1313318611yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—41no—
205C4C2×#1171992211yescharacteristic
206C4#1313323711yescharacteristic
207C4#1313324211yescharacteristic
208C4#1313324311yescharacteristic
209C2—81no—
210C2—82no—
211C2—8481no—
212C2#1681075711yescharacteristic
213C2#1681076811yescharacteristic
214C2#1681079111yescharacteristic
subgroupG/H
centerC22#13133186
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#550C2
socleC22×C53#120
hypercenter#550D53
hypocenterC53#1621
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #550

Complements

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

NH ≅ G/Ntype
1. #16810999C2semidirect
2. #16810823C2semidirect
3. #16810818C2semidirect
4. #16810367C2semidirect
5. Q8⋊#88322C2semidirect
6. D4⋊#88322C2semidirect
7. C53×#550C2semidirect
56. Q8×C53D8semidirect
108. #550D53semidirect
115. C53#1621semidirect
164. Q8#11719922semidirect