Finite Group Databasebeta

← #20290124

Subgroups — #20290124

Order 6784.

274 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 58 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#16811594C211yesmaximal, characteristic
2#16811566C211yesmaximal, characteristic
3#16811477C212yesmaximal, complemented
4#16811476C212yesmaximal, complemented
5C22⋊Q8×D53C211yesmaximal, characteristic
6C2×#11719936—21no—
7C2×C22⋊#6157—21no—
8#13133274C2212yes—
9C22⋊#88324—21no—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4C2212yes—
12C2×Q8×D53C411yescharacteristic, complemented
13C22×C4×D53C2211yescharacteristic
14#13133197C2211yescharacteristic
15#13133196C411yescharacteristic, complemented
16#13133195—21no—
17#13133194—21no—
18#13133193C2211yescharacteristic
19C4⋊C4×D53—21no—
20C4⋊C4×D53C411yescharacteristic, complemented
21C22⋊C4×D53—21no—
22C22⋊Q8×C53C411yescharacteristic, complemented
23C23×D53D411yescharacteristic
24C22×#6157—22no—
25C2×Q8×C53C2×C411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C22×Dic53D411yescharacteristic
28Q8×D53—42no—
29C2×C4×D53—21no—
30C2×C4×D53—21no—
31C2×C4×D53—42no—
32C2×C4×D53C2×C411yescharacteristic
33C2×C4×D53D411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53C2311yescharacteristic
36C2×Dic106—21no—
37C2×Dic106C2×C411yescharacteristic
38C22⋊#6157—42no—
39C2×#88324—22no—
40#11719936—42no—
41C4⋊C4×C53—21no—
42C4⋊C4×C53C2×C411yescharacteristic
43C22⋊C4×C53—21no—
44C22⋊Dic53—21no—
45#11719921—21no—
46#11719921—21no—
47C4⋊Dic53—21no—
48C4⋊Dic53C2×C411yescharacteristic
49#11719919—21no—
50#11719919—21no—
51#11719919—21no—
52C23×C53C22⋊C411yescharacteristic
53C22×D53—22no—
54C22×D53—22no—
55C22×D53—41no—
56C22×D53C22⋊C411yescharacteristic
57C22×D53C2×D411yescharacteristic
58C2×#6157—42no—
59C2×#6157—42no—
60C2×#6157—42no—
61Q8×C53—41no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—41no—
65C2×C4×C53C22⋊C411yescharacteristic
66C2×C4×C53C22×C411yescharacteristic
67C2×Dic53—21no—
68C2×Dic53—21no—
69C2×Dic53—41no—
70C2×Dic53C22⋊C411yescharacteristic
71C2×Dic53C2×D411yescharacteristic
72C4×D53—22no—
73C4×D53—22no—
74C4×D53—41no—
75C4×D53—41no—
76C4×D53—42no—
77C4×D53—42no—
78Dic106—42no—
79Dic106—41no—
80#88324—42no—
81C22×C53—21no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53C2×C22⋊C411yescharacteristic
85C2×D53—21no—
86C2×D53—21no—
87C2×D53—21no—
88C2×D53—21no—
89C2×D53—44no—
90C2×D53#9912yes—
91C2×D53#13712yes—
92#6157—82no—
93C4×C53—21no—
94C4×C53—21no—
95C4×C53—41no—
96C4×C53—41no—
97Dic53—21no—
98Dic53—21no—
99Dic53—41no—
100Dic53—41no—
101#1511—531nomaximal, Sylow
102C2×C53—41no—
103C2×C53C2×#9911yescharacteristic
104C2×C53#41911yescharacteristic
105C2×C53#42111yescharacteristic
106D53—22no—
107D53—22no—
108C2×C22⋊Q8—531no—
109#407—531no—
110#381—531no—
111#328—532no—
112#328—532no—
113C53#151111yescharacteristic, complemented, Sylow
114C22×Q8—531no—
115C23×C4—531no—
116C2×M16—1061no—
117C22⋊Q8—531no—
118C22⋊Q8—531no—
119C22⋊Q8—531no—
120C22⋊Q8—1061no—
121C22⋊Q8—1061no—
122C22⋊Q8#615711yescharacteristic, complemented
123C2×C4⋊C4—531no—
124C2×C4⋊C4—1061no—
125C2×C22⋊C4—1061no—
126C2×C22⋊C4—1061no—
127C22⋊C8—532no—
128C22⋊C8—1061no—
129#95—532no—
130#95—1061no—
131C24—531no—
132C2×Q8—531no—
133C2×Q8—1061no—
134C2×Q8—2122no—
135C2×Q8C2×#615711yescharacteristic
136C22×C4—531no—
137C22×C4—531no—
138C22×C4—531no—
139C22×C4—531no—
140C22×C4—531no—
141C22×C4—1061no—
142C22×C4—1061no—
143C22×C4—1062no—
144C22×C4—2122no—
145C22×C4C2×#615711yescharacteristic
146M16—2122no—
147C2×C8—1062no—
148C4⋊C4—21no—
149C4⋊C4—531no—
150C4⋊C4—1061no—
151C4⋊C4—1061no—
152C4⋊C4—1061no—
153C4⋊C4—1061no—
154C4⋊C4C2×#615711yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—1061no—
157C22⋊C4—1061no—
158C22⋊C4—1061no—
159C22⋊C4—2122no—
160C23—531no—
161C23—531no—
162C23—1062no—
163C23—1062no—
164C23—2121no—
165C23C22⋊#615711yescharacteristic
166Q8—41no—
167Q8—2121no—
168Q8—2122no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—531no—
173C2×C4—531no—
174C2×C4—1062no—
175C2×C4—1062no—
176C2×C4—1061no—
177C2×C4—1061no—
178C2×C4—2121no—
179C2×C4—2121no—
180C2×C4—2122no—
181C2×C4—2122no—
182C2×C4—2121no—
183C2×C4—2122no—
184C2×C4—2122no—
185C2×C4—2122no—
186C2×C4C22⋊#615711yescharacteristic
187C2×C4C22×#615711yescharacteristic
188C8—2122no—
189C22—21no—
190C22—21no—
191C22—21no—
192C22—532no—
193C22—532no—
194C22—1061no—
195C22—1061no—
196C22—1061no—
197C22—1061no—
198C22—2124no—
199C22C2×C22⋊#615711yescharacteristic
200C4—21no—
201C4—21no—
202C4—41no—
203C4—41no—
204C4—1061no—
205C4—1061no—
206C4—4242no—
207C4—2121no—
208C4—2121no—
209C2—41no—
210C2—1062no—
211C2—1062no—
212C2D53⋊#9911yescharacteristic
213C2#1681161611yescharacteristic
214C2#1681162011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC22⋊Q8×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC22⋊Q8#6157
hypocenterC53#1511
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #16811477C2semidirect
4. #16811476C2semidirect
12. C2×Q8×D53C4semidirect
15. #13133196C4semidirect
20. C4⋊C4×D53C4semidirect
22. C22⋊Q8×C53C4semidirect
113. C53#1511semidirect
122. C22⋊Q8#6157semidirect