Finite Group Databasebeta

← #20290906

Subgroups — #20290906

Order 6784.

324 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 72 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#16811565C211yesmaximal, characteristic
2C4×C22⋊#6157C211yesmaximal, characteristic
3#16811501C212yesmaximal
4#16811501C212yesmaximal
5C2×#13133195C211yesmaximal, characteristic
6C2×C4⋊#6157—21no—
7C2×C22⋊#6157C2212yes—
8C2×C22⋊#6157C2212yes—
9C2×C4×#6157—21no—
10(C2×#6157)⋊C4—21no—
11(C2×#6157)⋊C4—21no—
12C22×Dic106C411yescharacteristic, complemented
13C22×C4×D53C2211yescharacteristic
14#13133195—21no—
15#13133195—21no—
16#13133195—21no—
17#13133195—21no—
18C2×#11719919C411yescharacteristic, complemented
19C2×#11719919C411yescharacteristic, complemented
20C2×#11719921C2211yescharacteristic
21C2×#11719921C2211yescharacteristic
22C2×C4⋊C4×C53C411yescharacteristic, complemented
23C23×D53C2311yescharacteristic
24C22×#6157—22no—
25C22×#6157—22no—
26C22×C4×C53C2×C411yescharacteristic
27C22×C4×C53C2×C411yescharacteristic
28C22×C4×C53C2×C411yescharacteristic
29C22×Dic53C2×C411yescharacteristic
30C22×Dic53C2×C411yescharacteristic
31C22×Dic53C2×C411yescharacteristic
32C2×C4×D53—42no—
33C2×C4×D53D411yescharacteristic
34C2×C4×D53D411yescharacteristic
35C2×C4×D53D411yescharacteristic
36C2×C4×D53D411yescharacteristic
37C2×Dic106—21no—
38C2×Dic106—21no—
39C2×Dic106—42no—
40C22⋊#6157—22no—
41C22⋊#6157—24no—
42C22⋊#6157Q814yescomplemented
43C4⋊#6157—42no—
44C4×#6157—42no—
45C4⋊C4×C53—21no—
46C4⋊C4×C53—21no—
47#11719921—21no—
48#11719921—21no—
49#11719921—21no—
50#11719921—21no—
51#11719919—21no—
52#11719919—21no—
53#11719919—21no—
54#11719919—21no—
55C23×C53C22×C411yescharacteristic
56C22×D53—41no—
57C22×D53—41no—
58C22×D53C2×D411yescharacteristic
59C22×D53C2×D411yescharacteristic
60C22×D53C2×Q812yes—
61C22×D53#4112yes—
62C2×#6157—24no—
63C2×#6157—42no—
64C2×#6157—42no—
65C2×#6157—42no—
66C2×#6157—42no—
67C2×#6157—42no—
68C2×C4×C53—21no—
69C2×C4×C53—21no—
70C2×C4×C53—41no—
71C2×C4×C53—41no—
72C2×C4×C53—41no—
73C2×C4×C53C22⋊C411yescharacteristic
74C2×C4×C53C22⋊C411yescharacteristic
75C2×Dic53—22no—
76C2×Dic53—21no—
77C2×Dic53—21no—
78C2×Dic53—41no—
79C2×Dic53—41no—
80C2×Dic53C22⋊C411yescharacteristic
81C2×Dic53C22⋊C411yescharacteristic
82C4×D53—22no—
83C4×D53—22no—
84C4×D53—41no—
85C4×D53—41no—
86Dic106—42no—
87Dic106—42no—
88C22×C53—21no—
89C22×C53—21no—
90C22×C53C2×C22⋊C411yescharacteristic
91C22×C53C4×Q812yes—
92C2×D53—21no—
93C2×D53—21no—
94C2×D53—21no—
95C2×D53—21no—
96C2×D53—42no—
97C2×D53—42no—
98C2×D53C22⋊Q812yes—
99C2×D53#12411yescharacteristic
100C2×D53C4⋊Q811yescharacteristic
101#6157—44no—
102#6157—82no—
103C4×C53—21no—
104C4×C53—21no—
105C4×C53—41no—
106C4×C53—41no—
107Dic53—21no—
108Dic53—41no—
109Dic53—41no—
110Dic53#9912yes—
111#1996—531nomaximal, Sylow
112C2×C53—22no—
113C2×C53#39111yescharacteristic
114C2×C53C2×#9911yescharacteristic
115C2×C53#41011yescharacteristic
116D53—22no—
117D53—21no—
118D53—21no—
119C2×C22⋊Q8—531no—
120#380—531no—
121C4×C22⋊C4—531no—
122#342—532no—
123#342—532no—
124C53#199611yescharacteristic, complemented, Sylow
125C22×Q8—531no—
126C23×C4—531no—
127C22⋊Q8—1061no—
128C22⋊Q8—1061no—
129C22⋊Q8—1061no—
130C22⋊Q8—1061no—
131C2×C4⋊C4—531no—
132C2×C4⋊C4—531no—
133C2×C4⋊C4—1061no—
134C2×C4⋊C4#615711yescharacteristic, complemented
135C2×C22⋊C4—532no—
136C2×C22⋊C4—531no—
137C2×C22⋊C4—531no—
138C2×C22⋊C4—532no—
139C2×C42—1061no—
140#95—1061no—
141#95—1061no—
142C24—531no—
143C2×Q8—1061no—
144C2×Q8—1061no—
145C2×Q8—2122no—
146C22×C4—531no—
147C22×C4—531no—
148C22×C4—531no—
149C22×C4—531no—
150C22×C4—531no—
151C22×C4—531no—
152C22×C4—531no—
153C22×C4—1062no—
154C22×C4—1062no—
155C22×C4—2122no—
156C22×C4C2×#615711yescharacteristic
157C22×C4C2×#615711yescharacteristic
158C22×C4C2×#615711yescharacteristic
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—1061no—
162C4⋊C4—1061no—
163C4⋊C4—1061no—
164C4⋊C4—1061no—
165C4⋊C4—2122no—
166C22⋊C4—534no—
167C22⋊C4—1061no—
168C22⋊C4—1061no—
169C22⋊C4—1061no—
170C22⋊C4—1061no—
171C22⋊C4—1062no—
172C22⋊C4—1064no—
173C42—2122no—
174C23—531no—
175C23—531no—
176C23—532no—
177C23—532no—
178C23—2121no—
179C23—2121no—
180C23C22×#615711yescharacteristic
181Q8—2122no—
182Q8—2122no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—531no—
189C2×C4—531no—
190C2×C4—1062no—
191C2×C4—1062no—
192C2×C4—1062no—
193C2×C4—1064no—
194C2×C4—1061no—
195C2×C4—1061no—
196C2×C4—2121no—
197C2×C4—2121no—
198C2×C4—2122no—
199C2×C4—2122no—
200C2×C4—2122no—
201C2×C4—2122no—
202C2×C4—2122no—
203C2×C4—2121no—
204C2×C4—2121no—
205C2×C4C22⋊#615711yescharacteristic
206C2×C4C22⋊#615711yescharacteristic
207C22—21no—
208C22—21no—
209C22—532no—
210C22—531no—
211C22—531no—
212C22—1061no—
213C22—1061no—
214C22—1061no—
215C22—1061no—
216C22—2122no—
217C22—2122no—
218C22C2×C22⋊#615711yescharacteristic
219C22Q8×#615712yes—
220C4—21no—
221C4—21no—
222C4—41no—
223C4—41no—
224C4—532no—
225C4—1061no—
226C4—4242no—
227C4—2124no—
228C4—2121no—
229C4—2121no—
230C2—22no—
231C2—1061no—
232C2—1062no—
233C2—1061no—
234C2(C2×#6157)⋊Q811yescharacteristic
235C2D53⋊#9911yescharacteristic
236C2#1681160211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C4⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C4⋊C4#6157
hypocenterC53#1996
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
12. C22×Dic106C4semidirect
18. C2×#11719919C4semidirect
19. C2×#11719919C4semidirect
22. C2×C4⋊C4×C53C4semidirect
42. C22⋊#6157Q8semidirect
124. C53#1996semidirect
134. C2×C4⋊C4#6157semidirect