Finite Group Databasebeta

← #20290761

Subgroups — #20290761

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#16811621C212yesmaximal, complemented
3D53⋊#100C211yesmaximal, characteristic, complemented
4#16811490C212yesmaximal, complemented
5#16811186C211yesmaximal, characteristic
6D53⋊M16C2212yes—
7D4×#6157—24no—
8#13133296C2212yes—
9C2×C22⋊#6157—22no—
10#13133281—24no—
11#13133277—22no—
12C2×#11719947C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14#13133213C411yescharacteristic, complemented
15#13133205C411yescharacteristic, complemented
16#13133204C411yescharacteristic, complemented
17#13133189—22no—
18#13133185—22no—
19#13133174C2211yescharacteristic
20#127×C53C411yescharacteristic, complemented
21C23×D53D412yes—
22C22×#6157—24no—
23C2×D4×C53—22no—
24C2×D4×C53C2×C411yescharacteristic
25C2×D4×C53C2×C411yescharacteristic, complemented
26C2×C22⋊D53—21no—
27C2×C22⋊D53—22no—
28C2×C22⋊D53D412yes—
29C22×Dic53—21no—
30#11719947—22no—
31#11719947—22no—
32D4×D53—22no—
33D4×D53—22no—
34C2×D212C2×C411yescharacteristic
35C2×C4×D53C2311yescharacteristic
36C2×Dic106C2×C411yescharacteristic, complemented
37C22⋊#6157—24no—
38C22⋊#6157—22no—
39#11719940—22no—
40C4⋊#6157—22no—
41C4×#6157D414yescomplemented
42#11719936—22no—
43D53⋊C8—22no—
44C42×C53C2×C411yescharacteristic, complemented
45C22⋊Dic53—22no—
46#11719921—22no—
47#11719921—21no—
48#11719919—21no—
49C4×Dic53C2×C411yescharacteristic, complemented
50C23×C53—21no—
51C23×C53C22⋊C412yes—
52C22×D53—22no—
53C22×D53—22no—
54C22×D53—82no—
55C22×D53C22⋊C412yes—
56C22×D53C2×D411yescharacteristic
57C2×#6157—24no—
58C2×#6157—84no—
59D4×C53—22no—
60D4×C53—22no—
61D4×C53—82no—
62C2×C4×C53—21no—
63C2×C4×C53C22×C411yescharacteristic
64C22⋊D53—24no—
65C22⋊D53—41no—
66C22⋊D53—41no—
67C22⋊D53—82no—
68C2×Dic53—21no—
69C2×Dic53—21no—
70C2×Dic53—42no—
71C2×Dic53C2×D411yescharacteristic
72D212—22no—
73C4×D53C2×D412yes—
74C4×D53#4112yes—
75Dic106—22no—
76#88324—42no—
77C22×C53—22no—
78C22×C53—42no—
79C22×C53—42no—
80C22×C53—41no—
81C22×C53C2×C22⋊C411yescharacteristic
82C2×D53—22no—
83C2×D53—41no—
84C2×D53—44no—
85C2×D53—42no—
86C2×D53—82no—
87C2×D53#12011yescharacteristic
88C2×D53#12311yescharacteristic
89#6157—44no—
90C4×C53—41no—
91C4×C53C4×D412yes—
92Dic53—41no—
93Dic53—41no—
94Dic53#12112yes—
95#1911—531nomaximal, Sylow
96C2×C53—21no—
97C2×C53—42no—
98C2×C53—81no—
99C2×C53#38611yescharacteristic
100D53—21no—
101D53—21no—
102D53—42no—
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#615711yescharacteristic, complemented
114#124—531no—
115#124—531no—
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×Q8—531no—
128C2×D4—22no—
129C2×D4—532no—
130C2×D4—531no—
131C2×D4—1062no—
132C2×D4—1062no—
133C2×D4—1062no—
134C2×D4—1061no—
135C2×D4C2×#615711yescharacteristic
136C2×D4C2×#615711yescharacteristic, complemented
137C22×C4—531no—
138C22×C4—1061no—
139C22×C4—1064no—
140M16—1062no—
141M16—1062no—
142C2×C8—1062no—
143C4⋊C4—1061no—
144C4⋊C4—1062no—
145C22⋊C4—1061no—
146C22⋊C4—1062no—
147C22⋊C4—1062no—
148C22⋊C4—1064no—
149C22⋊C4—1062no—
150C42—534no—
151C42—531no—
152C42C2×#615711yescharacteristic, complemented
153C23—21no—
154C23—531no—
155C23—532no—
156C23—1062no—
157C23—1062no—
158C23—4242no—
159C23C22⋊#615712yes—
160Q8—1062no—
161D4—22no—
162D4—22no—
163D4—82no—
164D4—1064no—
165D4—1062no—
166D4—4242no—
167D4—2121no—
168D4—2121no—
169C2×C4—21no—
170C2×C4—531no—
171C2×C4—532no—
172C2×C4—532no—
173C2×C4—1064no—
174C2×C4—1061no—
175C2×C4—1061no—
176C2×C4—4244no—
177C2×C4—2122no—
178C2×C4C22×#615711yescharacteristic
179C8—2122no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—531no—
185C22—531no—
186C22—1062no—
187C22—4242no—
188C22—2121no—
189C22—2122no—
190C22—2124no—
191C22C2×C22⋊#615711yescharacteristic
192C4—41no—
193C4—532no—
194C4—2121no—
195C4—2124no—
196C4—2121no—
197C4D4×#615712yes—
198C2—21no—
199C2—42no—
200C2—81no—
201C2—1061no—
202C2—1061no—
203C2—2122no—
204C2#1681157111yescharacteristic
subgroupG/H
centerC2#16811571
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
Fitting#127×C53C4
socleC2×C53#386
hypercenter#127#6157
hypocenterC53#1911
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #16811690C2semidirect
2. #16811621C2semidirect
3. D53⋊#100C2semidirect
4. #16811490C2semidirect
14. #13133213C4semidirect
15. #13133205C4semidirect
16. #13133204C4semidirect
20. #127×C53C4semidirect
25. C2×D4×C53C2×C4semidirect
36. C2×Dic106C2×C4semidirect
41. C4×#6157D4semidirect
44. C42×C53C2×C4semidirect
49. C4×Dic53C2×C4semidirect
108. C53#1911semidirect
113. #127#6157semidirect
136. C2×D4C2×#6157semidirect
152. C42C2×#6157semidirect