Finite Group Databasebeta

← #20273778

Subgroups — #20273778

Order 6784.

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

normal subgroups only

This group has 278 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#16811295C211yesmaximal, characteristic, complemented
2#16810999C211yesmaximal, characteristic, complemented
3#16810815C211yesmaximal, characteristic, complemented
4#16810366C211yesmaximal, characteristic, complemented
5Q8⋊#88322C211yesmaximal, characteristic, complemented
6#16810350C211yesmaximal, characteristic, complemented
7C53×#460C211yesmaximal, characteristic, complemented
8C2×#11719949—21no—
9C2×#11719922—21no—
10C2×#11719922—22no—
11#13133214—21no—
12#13133214C2211yescharacteristic
13#13133192—21no—
14#13133192—22no—
15C4×D212—21no—
16C4×D212C2211yescharacteristic
17C4⋊#88322C2211yescharacteristic
18#13133138—21no—
19C4×#88322C2211yescharacteristic
20C2×SD16×C53—21no—
21#127×C53C2211yescharacteristic
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24#102×C53—21no—
25C2×Q8×C53D411yescharacteristic
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—21no—
29#11719949—42no—
30#11719949—42no—
31C2×D212—22no—
32C2×D212—21no—
33C2×D212—21no—
34C2×D212—22no—
35C2×D212D411yescharacteristic
36C2×D212D412yes—
37C2×C4×D53—22no—
38C2×C4×D53—21no—
39SD16×C53—42no—
40C2×C8×C53—21no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53D411yescharacteristic
43C42×C53—21no—
44C42×C53C2311yescharacteristic
45#11719922—44no—
46#11719922—42no—
47#11719921—22no—
48#11719921—21no—
49C4⋊Dic53D411yescharacteristic
50C2×#88322—21no—
51C2×#88322—21no—
52C23×C53—41no—
53C22×D53—21no—
54C22×D53—22no—
55Q8×C53—21no—
56Q8×C53D812yescomplemented
57D4×C53—42no—
58D4×C53—41no—
59D4×C53—41no—
60D4×C53—41no—
61D4×C53—41no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53C2×D411yescharacteristic
65C2×C4×C53C2×D411yescharacteristic
66C2×C4×C53C2×D411yescharacteristic
67C2×Dic53—21no—
68D212—21no—
69D212—21no—
70D212—42no—
71D212—44no—
72D212—41no—
73D212—41no—
74D212—41no—
75D212—41no—
76D212—82no—
77C4×D53—44no—
78C4×D53—41no—
79C4×D53—41no—
80C8×C53—41no—
81#88322—22no—
82#88322—41no—
83C22×C53—81no—
84C22×C53—81no—
85C22×C53—81no—
86C22×C53#12011yescharacteristic
87C2×D53—42no—
88C2×D53—41no—
89C2×D53—41no—
90C2×D53—42no—
91C2×D53—42no—
92C2×D53—41no—
93C4×C53—22no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53C2×D811yescharacteristic
97C4×C53#13511yescharacteristic
98C4×C53AGL(1,ℤ/8)11yescharacteristic
99C4×C53AGL(1,ℤ/8)11yescharacteristic
100Dic53—41no—
101#1621—531nomaximal, Sylow
102C2×C53—161no—
103C2×C53#44711yescharacteristic
104C2×C53#44911yescharacteristic
105C2×C53#45311yescharacteristic
106D53—81no—
107D53—82no—
108#550—531no—
109#493—531no—
110#460D5311yescharacteristic, complemented
111#459—531no—
112#331—531no—
113Q8⋊C8—531no—
114D4⋊C8—531no—
115C53#162111yescharacteristic, complemented, Sylow
116C2×#41—1061no—
117C2×SD16—21no—
118C2×D8—1061no—
119C2×D8—1062no—
120#127—531no—
121#127—1061no—
122#127C2×D5311yescharacteristic
123#121—1062no—
124#121—1061no—
125C4×Q8C2×D5311yescharacteristic
126C4×D4—531no—
127C4×D4—1061no—
128C4⋊C8—531no—
129C4⋊C8C2×D5311yescharacteristic
130#102—21no—
131#102—1061no—
132C4×C8—531no—
133#41—2122no—
134#41—2122no—
135C2×Q8C22⋊D5311yescharacteristic
136C2×D4—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—531no—
140C2×D4—532no—
141C2×D4—1062no—
142C2×D4—1062no—
143C2×D4—1061no—
144C2×D4—1061no—
145C22×C4—1062no—
146C22×C4—1061no—
147SD16—42no—
148D8—2124no—
149D8—2122no—
150C2×C8—21no—
151C2×C8—1061no—
152C2×C8—1061no—
153C4⋊C4—21no—
154C4⋊C4—531no—
155C4⋊C4C22⋊D5311yescharacteristic
156C22⋊C4—1062no—
157C22⋊C4—1061no—
158C42—21no—
159C42C22×D5311yescharacteristic
160C23—41no—
161C23—1061no—
162C23—1062no—
163Q8—21no—
164Q8#1171992212yescomplemented
165D4—42no—
166D4—41no—
167D4—41no—
168D4—41no—
169D4—41no—
170D4—1061no—
171D4—1061no—
172D4—4242no—
173D4—2122no—
174D4—2124no—
175D4—2121no—
176D4—2121no—
177D4—2121no—
178D4—2121no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—1061no—
182C2×C4—2124no—
183C2×C4—2121no—
184C2×C4—2121no—
185C2×C4D4×D5311yescharacteristic
186C2×C4D4×D5311yescharacteristic
187C2×C4C2×C22⋊D5311yescharacteristic
188C8—41no—
189C8—1062no—
190C8—2121no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—2122no—
195C22—2121no—
196C22—2121no—
197C22—2122no—
198C22—2121no—
199C22—2122no—
200C22#1313318511yescharacteristic
201C4—22no—
202C4—21no—
203C4—41no—
204C4—2121no—
205C4C2×#1171992211yescharacteristic
206C4#1313323811yescharacteristic
207C4#1313324211yescharacteristic
208C4#1313324411yescharacteristic
209C2—161no—
210C2—4241no—
211C2—4242no—
212C2#1681075611yescharacteristic
213C2#1681076711yescharacteristic
214C2#1681078911yescharacteristic
subgroupG/H
centerC22#13133185
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×#460C2
socleC22×C53#120
hypercenter#460D53
hypocenterC53#1621
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #16811295C2semidirect
2. #16810999C2semidirect
3. #16810815C2semidirect
4. #16810366C2semidirect
5. Q8⋊#88322C2semidirect
6. #16810350C2semidirect
7. C53×#460C2semidirect
56. Q8×C53D8semidirect
110. #460D53semidirect
115. C53#1621semidirect
164. Q8#11719922semidirect