Finite Group Databasebeta

← #20273777

Subgroups — #20273777

Order 6784.

278 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 55 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#16811293C211yesmaximal, characteristic, complemented
2#16810820C211yesmaximal, characteristic, complemented
3#16810815C211yesmaximal, characteristic, complemented
4#16810367C211yesmaximal, characteristic, complemented
5#16810355C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×#493C211yesmaximal, characteristic, complemented
8C2×#11719947—21no—
9C2×#11719923—21no—
10C2×#11719922—21no—
11#13133213—21no—
12#13133189—22no—
13#13133188—21no—
14C4×Dic106C2211yescharacteristic
15C4×D212C2211yescharacteristic
16D4×Dic53—21no—
17C4⋊#88322C2211yescharacteristic
18C4⋊#88322C2211yescharacteristic
19#13133138—21no—
20#13133138—21no—
21C2×D8×C53—22no—
22#127×C53C2211yescharacteristic
23#127×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×D4×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29C2×D4×C53—21no—
30C2×D4×C53D412yes—
31C2×C22⋊D53—22no—
32C2×C22⋊D53—22no—
33C22×Dic53—22no—
34#11719947—42no—
35#11719947—42no—
36C2×D212D411yescharacteristic
37C2×C4×D53—21no—
38C2×Dic106D411yescharacteristic
39D8×C53—44no—
40C2×C8×C53—21no—
41C42×C53C2311yescharacteristic
42C22⋊Dic53—22no—
43#11719923—42no—
44#11719922—42no—
45#11719921—21no—
46C4⋊Dic53D411yescharacteristic
47C4⋊Dic53D411yescharacteristic
48#11719919—21no—
49C4×Dic53—21no—
50C2×#88322—21no—
51C2×#88322—21no—
52C23×C53—22no—
53C23×C53—41no—
54C22×D53—21no—
55D4×C53—42no—
56D4×C53—42no—
57D4×C53—41no—
58D4×C53—41no—
59D4×C53—41no—
60D4×C53—41no—
61D4×C53—41no—
62D4×C53—41no—
63D4×C53—41no—
64D4×C53—41no—
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×C4×C53C2×D411yescharacteristic
68C22⋊D53—44no—
69C22⋊D53—82no—
70C2×Dic53—21no—
71C2×Dic53—21no—
72C2×Dic53—21no—
73C2×Dic53—44no—
74D212—21no—
75D212—21no—
76C4×D53—41no—
77C4×D53—41no—
78Dic106—21no—
79Dic106D812yescomplemented
80C8×C53—22no—
81#88322—41no—
82#88322—41no—
83C22×C53—42no—
84C22×C53—42no—
85C22×C53—42no—
86C22×C53—81no—
87C22×C53—81no—
88C22×C53—81no—
89C22×C53#12011yescharacteristic
90C2×D53—41no—
91C2×D53—41no—
92C2×D53—41no—
93C4×C53—21no—
94C4×C53C2×D811yescharacteristic
95C4×C53#13511yescharacteristic
96C4×C53AGL(1,ℤ/8)11yescharacteristic
97C4×C53AGL(1,ℤ/8)11yescharacteristic
98Dic53—22no—
99Dic53—41no—
100Dic53—41no—
101#1621—531nomaximal, Sylow
102C2×C53—82no—
103C2×C53—161no—
104C2×C53#44711yescharacteristic
105C2×C53#44911yescharacteristic
106C2×C53#45311yescharacteristic
107D53—81no—
108#550—531no—
109#493D5311yescharacteristic, complemented
110#460—531no—
111#459—531no—
112#331—531no—
113Q8⋊C8—531no—
114D4⋊C8—531no—
115C53#162111yescharacteristic, complemented, Sylow
116C2×#41—1061no—
117C2×SD16—1061no—
118C2×D8—22no—
119C2×D8—1061no—
120#127—1061no—
121#127C2×D5311yescharacteristic
122#127C2×D5311yescharacteristic
123#121—1061no—
124#121—1062no—
125C4×Q8—531no—
126C4×D4—531no—
127C4×D4—1061no—
128C4⋊C8—531no—
129C4⋊C8—531no—
130#102—1061no—
131#102—1061no—
132C4×C8C2×D5311yescharacteristic
133#41—2122no—
134#41—2122no—
135C2×Q8—531no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—531no—
142C2×D4—1062no—
143C2×D4—1062no—
144C2×D4C22⋊D5312yes—
145C22×C4—1062no—
146C22×C4—1061no—
147SD16—2122no—
148D8—44no—
149D8—2122no—
150C2×C8—21no—
151C2×C8—1061no—
152C2×C8—1061no—
153C4⋊C4—531no—
154C4⋊C4—531no—
155C4⋊C4—1061no—
156C22⋊C4—1062no—
157C22⋊C4—1061no—
158C42—1061no—
159C42C22×D5311yescharacteristic
160C23—22no—
161C23—41no—
162C23—1061no—
163Q8—532no—
164Q8—1061no—
165D4—42no—
166D4—42no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—41no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—1061no—
176D4—1061no—
177D4—4242no—
178D4—2124no—
179C2×C4—1061no—
180C2×C4—1061no—
181C2×C4—1061no—
182C2×C4—2124no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4D4×D5311yescharacteristic
186C2×C4D4×D5311yescharacteristic
187C2×C4C2×C22⋊D5311yescharacteristic
188C8—22no—
189C8—2121no—
190C8—2121no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22—81no—
196C22—81no—
197C22—2121no—
198C22—2121no—
199C22—2121no—
200C22#1313318511yescharacteristic
201C4—21no—
202C4—1062no—
203C4—2121no—
204C4—2121no—
205C4D8×D5311yescharacteristic
206C4#1313323611yescharacteristic
207C4#1313324111yescharacteristic
208C4#1313324111yescharacteristic
209C2—82no—
210C2—161no—
211C2—4241no—
212C2#1681075511yescharacteristic
213C2#1681076511yescharacteristic
214C2#1681079011yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#493C2
socleC22×C53#120
hypercenter#493D53
hypocenterC53#1621
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #16811293C2semidirect
2. #16810820C2semidirect
3. #16810815C2semidirect
4. #16810367C2semidirect
5. #16810355C2semidirect
6. #16810352C2semidirect
7. C53×#493C2semidirect
79. Dic106D8semidirect
109. #493D53semidirect
115. C53#1621semidirect