Finite Group Databasebeta

← #20290760

Subgroups — #20290760

Order 6784.

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

normal subgroups only

This group has 362 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#16811690C211yesmaximal, characteristic, complemented
2#16811622C212yesmaximal, complemented
3D53⋊#100C211yesmaximal, characteristic, complemented
4#16811490C212yesmaximal, complemented
5#16811191C211yesmaximal, characteristic
6D53⋊M16C2212yes—
7D4×#6157—24no—
8#13133296C2212yes—
9C2×C22⋊#6157—22no—
10#13133282—24no—
11#13133277—22no—
12C2×#11719949C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14#13133214C411yescharacteristic, complemented
15#13133213C411yescharacteristic, complemented
16#13133206C411yescharacteristic, complemented
17#13133190—22no—
18#13133187—22no—
19#13133174C2211yescharacteristic
20#124×C53C411yescharacteristic, complemented
21C23×D53D412yes—
22C22×#6157—24no—
23C2×Q8×C53C2×C411yescharacteristic, complemented
24C2×D4×C53C2×C411yescharacteristic
25C2×C22⋊D53—22no—
26C2×C22⋊D53D412yes—
27#11719949—22no—
28#11719949—22no—
29D4×D53—22no—
30D4×D53—22no—
31C2×D212—21no—
32C2×D212—22no—
33C2×D212C2×C411yescharacteristic
34C2×D212C2×C411yescharacteristic, complemented
35C2×C4×D53—21no—
36C2×C4×D53C2311yescharacteristic
37C22⋊#6157—24no—
38C22⋊#6157—22no—
39#11719940—22no—
40C4⋊#6157—22no—
41C4×#6157D414yescomplemented
42#11719936—22no—
43D53⋊C8—22no—
44C22⋊C4×C53—22no—
45C42×C53C2×C411yescharacteristic, complemented
46#11719921—22no—
47#11719921—21no—
48#11719919—21no—
49C4×Dic53C2×C411yescharacteristic, complemented
50C23×C53C22⋊C412yes—
51C22×D53—22no—
52C22×D53—22no—
53C22×D53—21no—
54C22×D53—82no—
55C22×D53C22⋊C412yes—
56C22×D53C2×D411yescharacteristic
57C2×#6157—24no—
58C2×#6157—84no—
59Q8×C53—22no—
60D4×C53—22no—
61C2×C4×C53—21no—
62C2×C4×C53—21no—
63C2×C4×C53C22×C411yescharacteristic
64C22⋊D53—24no—
65C22⋊D53—82no—
66C2×Dic53—21no—
67C2×Dic53C2×D411yescharacteristic
68D212—22no—
69D212—22no—
70D212—41no—
71D212—41no—
72D212—82no—
73C4×D53—42no—
74C4×D53C2×D412yes—
75C4×D53#4112yes—
76#88324—42no—
77C22×C53—22no—
78C22×C53—42no—
79C22×C53C2×C22⋊C411yescharacteristic
80C2×D53—22no—
81C2×D53—41no—
82C2×D53—44no—
83C2×D53—42no—
84C2×D53—42no—
85C2×D53—41no—
86C2×D53—82no—
87C2×D53#12011yescharacteristic
88C2×D53#12311yescharacteristic
89#6157—44no—
90C4×C53—41no—
91C4×C53—41no—
92C4×C53C4×D412yes—
93Dic53—41no—
94Dic53#12112yes—
95#1911—531nomaximal, Sylow
96C2×C53—21no—
97C2×C53—42no—
98C2×C53#38611yescharacteristic
99D53—21no—
100D53—21no—
101D53—42no—
102D53—81no—
103#535—531no—
104#518—531no—
105#421—532no—
106C2×#100—531no—
107H(3,ℤ/4)—532no—
108C53#191111yescharacteristic, complemented, Sylow
109C2×#41—531no—
110C22×D4—531no—
111C2×M16—532no—
112#127—531no—
113#127—531no—
114#124—531no—
115#124#615711yescharacteristic, complemented
116#121—1062no—
117#120—1062no—
118C4×D4—1064no—
119#117—532no—
120#117—531no—
121C2×C22⋊C4—1062no—
122#104—1064no—
123#100—1062no—
124C24—532no—
125#41—1062no—
126#41—1062no—
127C2×Q8C2×#615711yescharacteristic, complemented
128C2×D4—532no—
129C2×D4—531no—
130C2×D4—531no—
131C2×D4—1062no—
132C2×D4—1061no—
133C2×D4—1062no—
134C2×D4—1062no—
135C2×D4—1062no—
136C2×D4C2×#615711yescharacteristic
137C22×C4—531no—
138C22×C4—1061no—
139C22×C4—1064no—
140M16—1062no—
141M16—1062no—
142C2×C8—1062no—
143C4⋊C4—1061no—
144C4⋊C4—1062no—
145C22⋊C4—22no—
146C22⋊C4—1061no—
147C22⋊C4—1062no—
148C22⋊C4—1062no—
149C22⋊C4—1064no—
150C42—534no—
151C42—531no—
152C42C2×#615711yescharacteristic, complemented
153C23—531no—
154C23—532no—
155C23—1062no—
156C23—1062no—
157C23—1061no—
158C23—4242no—
159C23C22⋊#615712yes—
160Q8—22no—
161D4—22no—
162D4—1064no—
163D4—1062no—
164D4—1062no—
165D4—4242no—
166D4—4242no—
167D4—2121no—
168D4—2121no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—531no—
172C2×C4—532no—
173C2×C4—532no—
174C2×C4—1064no—
175C2×C4—1061no—
176C2×C4—4244no—
177C2×C4—2122no—
178C2×C4C22×#615711yescharacteristic
179C8—2122no—
180C22—22no—
181C22—42no—
182C22—531no—
183C22—531no—
184C22—1062no—
185C22—4242no—
186C22—2121no—
187C22—2122no—
188C22—2122no—
189C22—2124no—
190C22—2121no—
191C22C2×C22⋊#615711yescharacteristic
192C4—41no—
193C4—41no—
194C4—532no—
195C4—2121no—
196C4—2124no—
197C4D4×#615712yes—
198C2—21no—
199C2—42no—
200C2—1061no—
201C2—1061no—
202C2—4241no—
203C2—2122no—
204C2#1681157111yescharacteristic
subgroupG/H
centerC2#16811571
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#124×C53C4
socleC2×C53#386
hypercenter#124#6157
hypocenterC53#1911
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #16811690C2semidirect
2. #16811622C2semidirect
3. D53⋊#100C2semidirect
4. #16811490C2semidirect
14. #13133214C4semidirect
15. #13133213C4semidirect
16. #13133206C4semidirect
20. #124×C53C4semidirect
23. C2×Q8×C53C2×C4semidirect
34. C2×D212C2×C4semidirect
41. C4×#6157D4semidirect
45. C42×C53C2×C4semidirect
49. C4×Dic53C2×C4semidirect
108. C53#1911semidirect
115. #124#6157semidirect
127. C2×Q8C2×#6157semidirect
152. C42C2×#6157semidirect