Finite Group Databasebeta

← #20273910

Subgroups — #20273910

Order 6784.

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

normal subgroups only

This group has 250 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
2#16810970C211yesmaximal, characteristic, complemented
3#16810904C211yesmaximal, characteristic, complemented
4#16810833C211yesmaximal, characteristic, complemented
5#16810366C211yesmaximal, characteristic, complemented
6#16810352C211yesmaximal, characteristic, complemented
7C53×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×#11719949—21no—
9C2×#11719925—21no—
10#13133217C2211yescharacteristic
11#13133214—21no—
12#13133214C2211yescharacteristic
13#13133192—21no—
14#13133192—22no—
15C4×D212—21no—
16C4×D212C2211yescharacteristic
17#13133151—21no—
18C4⋊#88322C2211yescharacteristic
19#13133140—21no—
20#13133139—21no—
21#13133139—22no—
22C4×Q8×C53C2211yescharacteristic
23C4⋊C8×C53C2211yescharacteristic
24C4×C8×C53C2211yescharacteristic
25C2×Q8×C53D411yescharacteristic
26#11719949—42no—
27#11719949—42no—
28C2×D212—22no—
29C2×D212—21no—
30C2×D212—21no—
31C2×D212—22no—
32C2×D212D411yescharacteristic
33C2×D212D412yes—
34C2×C4×D53—22no—
35C2×C4×D53—21no—
36C2×Dic106—21no—
37C2×C8×C53—21no—
38C2×C8×C53—21no—
39C4⋊C4×C53—21no—
40C4⋊C4×C53D411yescharacteristic
41C42×C53—21no—
42C42×C53C2311yescharacteristic
43#11719925—42no—
44#11719921—22no—
45#11719921—21no—
46C4⋊Dic53—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53D411yescharacteristic
49C2×#88322—21no—
50C22×D53—21no—
51C22×D53—22no—
52Q8×C53—21no—
53Q8×C53D812yes—
54C2×C4×C53—21no—
55C2×C4×C53—21no—
56C2×C4×C53C2×D411yescharacteristic
57C2×C4×C53C2×D411yescharacteristic
58C2×C4×C53C2×D411yescharacteristic
59C2×Dic53—21no—
60C2×Dic53—41no—
61D212—21no—
62D212—21no—
63D212—42no—
64D212—44no—
65D212—41no—
66D212—41no—
67D212—41no—
68D212—41no—
69D212—82no—
70C4×D53—44no—
71C4×D53—41no—
72C4×D53—41no—
73Dic106—41no—
74Dic106—41no—
75C8×C53—41no—
76C8×C53—41no—
77#88322—41no—
78C22×C53#12011yescharacteristic
79C2×D53—42no—
80C2×D53—41no—
81C2×D53—41no—
82C2×D53—42no—
83C2×D53—42no—
84C2×D53—41no—
85C4×C53—22no—
86C4×C53—21no—
87C4×C53—41no—
88C4×C53C2×D811yescharacteristic
89C4×C53#13511yescharacteristic
90C4×C53AGL(1,ℤ/8)11yescharacteristic
91C4×C53#13711yescharacteristic
92Dic53—41no—
93Dic53—81no—
94#1641—531nomaximal, Sylow
95C2×C53#44711yescharacteristic
96C2×C53#45211yescharacteristic
97C2×C53#45511yescharacteristic
98D53—81no—
99D53—82no—
100#550—531no—
101#486—531no—
102#474—531no—
103#462—531no—
104#331—531no—
105Q8⋊C8D5311yescharacteristic, complemented
106D4⋊C8—531no—
107C53#164111yescharacteristic, complemented, Sylow
108C2×#41—1061no—
109C2×Q16—1061no—
110C4⋊Q8—531no—
111#127—531no—
112#127—1061no—
113#121—1062no—
114#121—1061no—
115C4×Q8C2×D5311yescharacteristic
116C4×D4—531no—
117C4×D4—1061no—
118#107—1061no—
119C4⋊C8—531no—
120C4⋊C8C2×D5311yescharacteristic
121#103—1061no—
122#102—1061no—
123#102—1062no—
124C4×C8C2×D5311yescharacteristic
125#41—2122no—
126#41—2122no—
127C2×Q8—1061no—
128C2×Q8D21211yescharacteristic
129C2×D4—531no—
130C2×D4—532no—
131C2×D4—1062no—
132C2×D4—1062no—
133C2×D4—1061no—
134C2×D4—1061no—
135C22×C4—1062no—
136C22×C4—1061no—
137Q16—2122no—
138C2×C8—21no—
139C2×C8—21no—
140C2×C8—1061no—
141C4⋊C4—21no—
142C4⋊C4—531no—
143C4⋊C4—1061no—
144C4⋊C4—1061no—
145C4⋊C4D21211yescharacteristic
146C22⋊C4—1062no—
147C22⋊C4—1061no—
148C42—21no—
149C42C22×D5311yescharacteristic
150C23—1061no—
151C23—1062no—
152Q8—21no—
153Q8—2121no—
154Q8—2121no—
155Q8D42412yes—
156D4—1061no—
157D4—1061no—
158D4—4242no—
159D4—2122no—
160D4—2124no—
161D4—2121no—
162D4—2121no—
163D4—2121no—
164D4—2121no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—1061no—
168C2×C4—2124no—
169C2×C4—2121no—
170C2×C4—2121no—
171C2×C4—2121no—
172C2×C4C2×D21211yescharacteristic
173C2×C4D4×D5311yescharacteristic
174C2×C4D4×D5311yescharacteristic
175C8—41no—
176C8—41no—
177C8—2121no—
178C22—2122no—
179C22—2121no—
180C22—2121no—
181C22—2122no—
182C22—2121no—
183C22—2122no—
184C22#1313318711yescharacteristic
185C4—22no—
186C4—21no—
187C4—41no—
188C4—4241no—
189C4—2121no—
190C4C2×D42411yescharacteristic
191C4#1313323811yescharacteristic
192C4#1313324511yescharacteristic
193C4#1313324911yescharacteristic
194C2—4241no—
195C2—4242no—
196C2#1681075911yescharacteristic
197C2#1681078811yescharacteristic
198C2#1681080211yescharacteristic
subgroupG/H
centerC22#13133187
commutatorC42×C53C23
FrattiniC42C22×D53
FittingC53×Q8⋊C8C2
socleC22×C53#120
hypercenterQ8⋊C8D53
hypocenterC53#1641
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #16810970C2semidirect
3. #16810904C2semidirect
4. #16810833C2semidirect
5. #16810366C2semidirect
6. #16810352C2semidirect
7. C53×Q8⋊C8C2semidirect
105. Q8⋊C8D53semidirect
107. C53#1641semidirect