Finite Group Databasebeta

← #20290717

Subgroups — #20290717

Order 6784.

332 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 66 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#16811703C211yesmaximal, characteristic, complemented
2#16811622C212yesmaximal, complemented
3#16811604C211yesmaximal, characteristic, complemented
4#16811494C212yesmaximal, complemented
5#16811174C211yesmaximal, characteristic
6#13133305—24no—
7D53⋊M16—22no—
8D53⋊M16C2212yes—
9#13133296C2212yes—
10#13133282—24no—
11#13133276—22no—
12C2×#11719949C2211yescharacteristic
13C2×#11719949C2211yescharacteristic
14#13133214C411yescharacteristic, complemented
15#13133206C411yescharacteristic, complemented
16#13133206C411yescharacteristic, complemented
17#13133192—22no—
18#13133179—22no—
19C42×D53C2211yescharacteristic
20C4⋊Q8×C53C411yescharacteristic, complemented
21C2×Q8×C53C2×C411yescharacteristic
22C2×Q8×C53C2×C411yescharacteristic
23#11719949—22no—
24#11719949—22no—
25#11719949—22no—
26#11719949—22no—
27C2×D212—22no—
28C2×D212—21no—
29C2×D212C2×C411yescharacteristic
30C2×D212C2×C411yescharacteristic
31C2×D212D412yes—
32C2×C4×D53—21no—
33C2×C4×D53—21no—
34C2×C4×D53D412yes—
35C2×C4×D53C2311yescharacteristic
36C22⋊#6157—22no—
37#11719940—22no—
38C4⋊#6157—22no—
39C4×#6157—22no—
40#11719936—24no—
41#11719936D414yescomplemented
42D53⋊C8—24no—
43D53⋊C8—22no—
44C4⋊C4×C53—22no—
45C42×C53C2×C411yescharacteristic
46#11719921—22no—
47#11719921—22no—
48C4×Dic53—21no—
49C4×Dic53C2×C411yescharacteristic
50C22×D53—21no—
51C22×D53C22⋊C412yes—
52C22×D53C2×D411yescharacteristic
53C2×#6157—42no—
54Q8×C53—22no—
55Q8×C53—22no—
56C2×C4×C53—21no—
57C2×C4×C53—21no—
58C2×C4×C53C22⋊C412yes—
59C2×C4×C53C22×C411yescharacteristic
60C2×Dic53—21no—
61C2×Dic53C2×D411yescharacteristic
62D212—24no—
63D212—22no—
64D212—22no—
65D212—44no—
66D212—42no—
67C4×D53—22no—
68C4×D53—22no—
69C4×D53—42no—
70C4×D53—41no—
71C4×D53—41no—
72C4×D53C2×D412yes—
73C4×D53#4112yes—
74#88324—24no—
75C22×C53C2×C22⋊C411yescharacteristic
76C2×D53—22no—
77C2×D53—41no—
78C2×D53—42no—
79C2×D53—42no—
80C2×D53—41no—
81C2×D53#12112yes—
82#6157—82no—
83C4×C53—22no—
84C4×C53—22no—
85C4×C53—41no—
86C4×C53C4×D412yes—
87Dic53—41no—
88Dic53#12011yescharacteristic
89Dic53#12311yescharacteristic
90#1884—531nomaximal, Sylow
91C2×C53—21no—
92C2×C53#38611yescharacteristic
93D53—22no—
94D53—42no—
95D53—81no—
96#568—531no—
97#532—531no—
98#421—532no—
99#410—531no—
100#339—532no—
101C53#188411yescharacteristic, complemented, Sylow
102C2×#41—531no—
103C2×#41—531no—
104#131—1064no—
105C2×M16—532no—
106C2×M16—1062no—
107C4⋊Q8#615711yescharacteristic, complemented
108#127—531no—
109#124—531no—
110#124—531no—
111#121—1062no—
112C4×D4—1062no—
113#117—532no—
114C2×C42—531no—
115#104—1064no—
116#99—1062no—
117#41—1062no—
118#41—1062no—
119#41—1062no—
120#41—1062no—
121C2×Q8C2×#615711yescharacteristic
122C2×Q8C2×#615711yescharacteristic
123C2×D4—532no—
124C2×D4—531no—
125C2×D4—531no—
126C2×D4—1062no—
127C2×D4—1061no—
128C22×C4—531no—
129C22×C4—532no—
130C22×C4—1061no—
131C22×C4—1061no—
132M16—534no—
133M16—1062no—
134M16—1064no—
135C2×C8—1062no—
136C2×C8—1064no—
137C4⋊C4—22no—
138C4⋊C4—1062no—
139C22⋊C4—1062no—
140C22⋊C4—1062no—
141C22⋊C4—1062no—
142C42—531no—
143C42—1061no—
144C42—1062no—
145C42C2×#615711yescharacteristic
146C23—531no—
147C23—532no—
148C23—1061no—
149Q8—22no—
150Q8—22no—
151D4—1064no—
152D4—1062no—
153D4—1062no—
154D4—2124no—
155D4—2122no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—531no—
159C2×C4—532no—
160C2×C4—532no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—1061no—
164C2×C4—2122no—
165C2×C4—2121no—
166C2×C4—2121no—
167C2×C4—2122no—
168C2×C4C22⋊#615712yes—
169C2×C4C22×#615711yescharacteristic
170C8—1064no—
171C22—532no—
172C22—1062no—
173C22—2121no—
174C22—2122no—
175C22—2122no—
176C22—2121no—
177C22C2×C22⋊#615711yescharacteristic
178C4—22no—
179C4—22no—
180C4—41no—
181C4—531no—
182C4—531no—
183C4—4242no—
184C4—2121no—
185C4D4×#615712yes—
186C2—21no—
187C2—1062no—
188C2—4241no—
189C2—2122no—
190C2#1681157111yescharacteristic
subgroupG/H
centerC2#16811571
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC4⋊Q8×C53C4
socleC2×C53#386
hypercenterC4⋊Q8#6157
hypocenterC53#1884
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. #16811703C2semidirect
2. #16811622C2semidirect
3. #16811604C2semidirect
4. #16811494C2semidirect
14. #13133214C4semidirect
15. #13133206C4semidirect
16. #13133206C4semidirect
20. C4⋊Q8×C53C4semidirect
41. #11719936D4semidirect
101. C53#1884semidirect
107. C4⋊Q8#6157semidirect