Finite Group Databasebeta

← #20278358

Subgroups — #20278358

Order 6784.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×Q8×D53C211yesmaximal, characteristic
2C2×#13133217C211yesmaximal, characteristic, complemented
3C2×#13133215C211yesmaximal, characteristic, complemented
4C2×#13133197C211yesmaximal, characteristic
5#16810544C214yesmaximal, complemented
6#16810528C211yesmaximal, characteristic, complemented
7#16810497C214yesmaximal
8C4×#11719921C211yesmaximal, characteristic
9C53×#391C211yesmaximal, characteristic, complemented
10C22×Dic106C2212yes—
11C22×Dic106C2211yescharacteristic
12C22×Dic106C2211yescharacteristic
13C2×Q8×D53—24no—
14C2×Q8×D53—24no—
15C2×Q8×D53—24no—
16C22×C4×D53C2212yes—
17C22×C4×D53C2211yescharacteristic
18#13133217—24no—
19#13133215—24no—
20#13133197—24no—
21C2×C4⋊Dic53C2214yes—
22C2×C4⋊Dic53C2212yes—
23C2×#11719919C2214yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27#13133124C2214yes—
28#13133124C2212yes—
29C22×Q8×C53C2211yescharacteristic
30C2×C4⋊C4×C53C2211yescharacteristic
31C2×C42×C53C2211yescharacteristic
32#95×C53C2214yes—
33C23×D53C2311yescharacteristic
34C2×Q8×C53—22no—
35C2×Q8×C53—22no—
36C2×Q8×C53D414yes—
37C22×C4×C53C2312yes—
38C22×C4×C53C2312yes—
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×Dic53C2312yes—
42C22×Dic53C2311yescharacteristic
43C22×Dic53C2312yes—
44C22×Dic53C2312yes—
45Q8×D53—232no—
46C2×C4×D53—24no—
47C2×C4×D53—24no—
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53—24no—
51C2×C4×D53D414yes—
52C2×Dic106—24no—
53C2×Dic106—22no—
54C2×Dic106—22no—
55C2×Dic106—24no—
56C2×Dic106—24no—
57C2×Dic106—22no—
58C2×Dic106—22no—
59C2×Dic106—22no—
60C2×Dic106D414yes—
61C4⋊C4×C53—22no—
62C42×C53—22no—
63#11719921—24no—
64#11719921Q818yescomplemented
65C4⋊Dic53—24no—
66C4⋊Dic53—28no—
67#11719919—28no—
68C4×Dic53—22no—
69C23×C53C2411yescharacteristic
70C22×D53—21no—
71C22×D53—21no—
72C22×D53—22no—
73C22×D53C2×D412yes—
74C22×D53C2×Q814yes—
75Q8×C53—28no—
76C2×C4×C53—22no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—22no—
80C2×C4×C53—22no—
81C2×C4×C53—24no—
82C2×C4×C53C2×D414yes—
83C2×C4×C53C2×D412yes—
84C2×C4×C53C2×D412yes—
85C2×C4×C53C2×D412yes—
86C2×C4×C53C2×Q814yescomplemented
87C2×Dic53—22no—
88C2×Dic53—21no—
89C2×Dic53—21no—
90C2×Dic53—22no—
91C2×Dic53—22no—
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53C2×D412yes—
97C2×Dic53C2×D414yes—
98C2×Dic53C2×Q814yescomplemented
99C4×D53—216no—
100C4×D53—28no—
101Dic106—216no—
102Dic106—28no—
103Dic106—44no—
104C22×C53C22×D411yescharacteristic
105C22×C53C22×D411yescharacteristic
106C22×C53C22×D411yescharacteristic
107C22×C53C22×Q812yes—
108C22×C532−1+412yes—
109C2×D53—22no—
110C2×D53—22no—
111C2×D53—24no—
112C2×D53—24no—
113C2×D53C4⋊Q814yescomplemented
114C4×C53—24no—
115C4×C53—24no—
116C4×C53—42no—
117C4×C53#12014yes—
118Dic53—22no—
119Dic53—24no—
120Dic53—24no—
121Dic53—42no—
122#2434—531nomaximal, Sylow
123C2×C53C2×#12011yescharacteristic
124C2×C53C2×C4⋊Q811yescharacteristic
125C2×C53#53611yescharacteristic
126C2×C53D4×Q812yes—
127C2×C53D4×Q812yes—
128D53—24no—
129C23×Q8—531no—
130C2×C4⋊Q8—531no—
131C2×C4⋊Q8—531no—
132C2×C22⋊Q8—531no—
133#395—534no—
134#391—531no—
135#391D5311yescharacteristic, complemented
136#385—534no—
137C4×C22⋊C4—531no—
138C53#243411yescharacteristic, complemented, Sylow
139C22×Q8—531no—
140C22×Q8—531no—
141C22×Q8—532no—
142C22×Q8—1064no—
143C22×Q8—1064no—
144C22×Q8—1064no—
145C22×Q8C2×D5311yescharacteristic
146C23×C4—531no—
147C23×C4—532no—
148C4⋊Q8—1064no—
149C4⋊Q8—1064no—
150C22⋊Q8—1064no—
151C2×C4⋊C4—532no—
152C2×C4⋊C4—534no—
153C2×C4⋊C4—534no—
154C2×C4⋊C4C2×D5311yescharacteristic
155C2×C22⋊C4—532no—
156C2×C22⋊C4—532no—
157C2×C42—531no—
158C2×C42C2×D5311yescharacteristic
159#95—532no—
160#95—534no—
161#95C2×D5314yes—
162C24—531no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—534no—
166C2×Q8—1064no—
167C2×Q8—1064no—
168C2×Q8—10632no—
169C2×Q8—1064no—
170C2×Q8—1062no—
171C2×Q8—1062no—
172C2×Q8—1062no—
173C2×Q8—1062no—
174C2×Q8—1062no—
175C2×Q8D21214yes—
176C22×C4—532no—
177C22×C4—531no—
178C22×C4—532no—
179C22×C4—534no—
180C22×C4—532no—
181C22×C4—1064no—
182C22×C4—1064no—
183C22×C4—1064no—
184C22×C4—1062no—
185C22×C4—1062no—
186C22×C4C22×D5312yes—
187C22×C4C22×D5312yes—
188C22×C4C22×D5312yes—
189C22×C4C22×D5311yescharacteristic
190C4⋊C4—22no—
191C4⋊C4—1064no—
192C4⋊C4—1068no—
193C4⋊C4—1068no—
194C22⋊C4—538no—
195C22⋊C4—1064no—
196C42—22no—
197C42—1062no—
198C23—532no—
199C23—534no—
200C23—1061no—
201C23—1061no—
202C23—1062no—
203C23C23×D5311yescharacteristic
204Q8—28no—
205Q8—10616no—
206Q8—1068no—
207Q8—2124no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—24no—
214C2×C4—532no—
215C2×C4—534no—
216C2×C4—534no—
217C2×C4—10616no—
218C2×C4—1061no—
219C2×C4—1062no—
220C2×C4—1062no—
221C2×C4—1061no—
222C2×C4—1062no—
223C2×C4—1062no—
224C2×C4—1068no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—1062no—
228C2×C4C2×D21214yes—
229C2×C4C2×D21212yes—
230C2×C4D4×D5312yes—
231C2×C4D4×D5312yes—
232C2×C4Q8×D5314yescomplemented
233C22—534no—
234C22—1064no—
235C22—1064no—
236C22—1062no—
237C22—1062no—
238C22C2×D4×D5311yescharacteristic
239C22C2×D4×D5311yescharacteristic
240C22C22×D21211yescharacteristic
241C22C2×Q8×D5312yes—
242C22#1313326412yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—1064no—
247C4—1062no—
248C4—1064no—
249C4—2122no—
250C4#1313318714yes—
251C2—1064no—
252C2C2×#1313318711yescharacteristic
253C2C4⋊Q8×D5311yescharacteristic
254C2#1681119311yescharacteristic
255C2#1681128612yes—
256C2Q8×D21212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#391C2
hypercenter#391D53
hypocenterC53#2434
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
2. C2×#13133217C2semidirect
3. C2×#13133215C2semidirect
5. #16810544C2semidirect
6. #16810528C2semidirect
9. C53×#391C2semidirect
64. #11719921Q8semidirect
86. C2×C4×C53C2×Q8semidirect
98. C2×Dic53C2×Q8semidirect
113. C2×D53C4⋊Q8semidirect
135. #391D53semidirect
138. C53#2434semidirect
232. C2×C4Q8×D53semidirect