Finite Group Databasebeta

← #20275018

Subgroups — #20275018

Order 6784.

376 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 101 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×#11719917C211yesmaximal, characteristic, complemented
2C2×#13133144C212yesmaximal, complemented
3C4×C22⋊Dic53C211yesmaximal, characteristic, complemented
4#16810382C212yesmaximal, complemented
5C53×#525C211yesmaximal, characteristic
6C2×#11719917—24no—
7C2×#11719917—21no—
8C2×#11719917C2212yes—
9C22×#88322—21no—
10C2×C22⋊Dic53—21no—
11C2×C4×Dic53C2212yes—
12#13133144—24no—
13#13133124—21no—
14C2×#41×C53C2211yescharacteristic
15C23×C4×C53C2211yescharacteristic
16#123×C53—21no—
17C22⋊Q8×C53C411yescharacteristic, complemented
18C22⋊Q8×C53C411yescharacteristic, complemented
19#121×C53C411yescharacteristic, complemented
20#121×C53C411yescharacteristic, complemented
21#120×C53—21no—
22C4×D4×C53—21no—
23C4×D4×C53—21no—
24#117×C53C2211yescharacteristic
25C24×C53D411yescharacteristic
26#41×C53—22no—
27#41×C53—22no—
28C2×Q8×C53C2×C411yescharacteristic
29C2×D4×C53—21no—
30C2×D4×C53—21no—
31C2×D4×C53—21no—
32C2×D4×C53C2×C411yescharacteristic
33C22×C4×C53—22no—
34C22×C4×C53—22no—
35C22×C4×C53—21no—
36C22×C4×C53C2×C411yescharacteristic
37C22×C4×C53C2×C411yescharacteristic
38C22×C4×C53D411yescharacteristic
39C22×C4×C53D411yescharacteristic
40C22×C4×C53D411yescharacteristic
41C22×C4×C53C2311yescharacteristic
42C22×Dic53—22no—
43C4⋊C4×C53—21no—
44C4⋊C4×C53—21no—
45C4⋊C4×C53C2×C411yescharacteristic
46C4⋊C4×C53C2×C411yescharacteristic
47C22⋊C4×C53—21no—
48C22⋊C4×C53—21no—
49C22⋊C4×C53—21no—
50C22⋊C4×C53—21no—
51C22⋊C4×C53—21no—
52C42×C53—21no—
53C22⋊Dic53—24no—
54C4×Dic53—22no—
55C4×Dic53—22no—
56#11719917—24no—
57#11719917—22no—
58#11719917D414yescomplemented
59C2×#88322—24no—
60C2×#88322—22no—
61C23×C53—22no—
62C23×C53—22no—
63C23×C53—21no—
64C23×C53—41no—
65C23×C53C22⋊C411yescharacteristic
66C23×C53C22⋊C411yescharacteristic
67C23×C53C2×D411yescharacteristic
68Q8×C53—22no—
69D4×C53—22no—
70D4×C53—42no—
71D4×C53—42no—
72D4×C53—42no—
73C2×C4×C53—22no—
74C2×C4×C53—22no—
75C2×C4×C53—22no—
76C2×C4×C53—21no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—41no—
84C2×C4×C53—41no—
85C2×C4×C53C22⋊C411yescharacteristic
86C2×C4×C53C22⋊C411yescharacteristic
87C2×C4×C53C22×C411yescharacteristic
88C2×C4×C53C2×D412yes—
89C2×C4×C53C2×D411yescharacteristic
90C2×C4×C53#4112yes—
91C2×Dic53—24no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94#88322—24no—
95C22×C53—21no—
96C22×C53—21no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—21no—
102C22×C53—44no—
103C22×C53—41no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53#10412yescomplemented
107C22×C53C2×C22⋊C411yescharacteristic
108C22×C53#12112yes—
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—41no—
112C4×C53—41no—
113C4×C53—41no—
114C4×C53C4×D412yes—
115C4×C53#12011yescharacteristic
116C4×C53#12311yescharacteristic
117Dic53—44no—
118#1871—531nomaximal, Sylow
119C2×C53—22no—
120C2×C53—22no—
121C2×C53—41no—
122C2×C53—81no—
123C2×C53#38611yescharacteristic
124C2×C53C2×#10411yescharacteristic
125C2×C53#42111yescharacteristic
126C22×M16—531no—
127#525D5311yescharacteristic
128C2×#104—532no—
129C4×C22⋊C4—531no—
130#339—532no—
131C53#187111yescharacteristic, complemented, Sylow
132C2×#41C2×D5311yescharacteristic
133C23×C4C2×D5311yescharacteristic
134C2×M16—532no—
135C2×M16—1064no—
136C2×M16—1061no—
137C22×C8—1061no—
138#123—21no—
139C22⋊Q8Dic5311yescharacteristic, complemented
140C22⋊Q8Dic5311yescharacteristic, complemented
141#121Dic5311yescharacteristic, complemented
142#121Dic5311yescharacteristic, complemented
143#120—21no—
144C4×D4—21no—
145C4×D4—21no—
146#117C2×D5311yescharacteristic
147C2×C22⋊C4—1061no—
148C2×C42—532no—
149#104—1064no—
150#95—1061no—
151C24C22⋊D5311yescharacteristic
152#41—22no—
153#41—22no—
154C2×Q8C2×Dic5311yescharacteristic
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4C2×Dic5311yescharacteristic
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—1062no—
163C22×C4C2×Dic5311yescharacteristic
164C22×C4C2×Dic5311yescharacteristic
165C22×C4C22⋊D5311yescharacteristic
166C22×C4C22⋊D5311yescharacteristic
167C22×C4C22⋊D5311yescharacteristic
168C22×C4C22×D5311yescharacteristic
169M16—534no—
170M16—1064no—
171M16—1062no—
172C2×C8—1064no—
173C2×C8—1062no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4C2×Dic5311yescharacteristic
177C4⋊C4C2×Dic5311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—1064no—
184C42—21no—
185C42—1062no—
186C42—1062no—
187C23—22no—
188C23—22no—
189C23—21no—
190C23—41no—
191C23C22⋊Dic5311yescharacteristic
192C23C22⋊Dic5311yescharacteristic
193C23C2×C22⋊D5311yescharacteristic
194Q8—22no—
195D4—22no—
196D4—42no—
197D4—42no—
198D4—42no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—1064no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4C22⋊Dic5311yescharacteristic
215C2×C4C22⋊Dic5311yescharacteristic
216C2×C4D4×D5312yes—
217C2×C4#1171994712yes—
218C2×C4C22×Dic5311yescharacteristic
219C2×C4C2×C22⋊D5311yescharacteristic
220C8—1064no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—21no—
227C22—21no—
228C22—44no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22#1313314412yescomplemented
233C22C2×C22⋊Dic5311yescharacteristic
234C22#1313318912yes—
235C4—21no—
236C4—21no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—2124no—
241C4D4×Dic5312yes—
242C4#1313318511yescharacteristic
243C4#1313319811yescharacteristic
244C2—22no—
245C2—22no—
246C2—41no—
247C2—81no—
248C2#1681050311yescharacteristic
249C2C2×#1313314411yescharacteristic
250C2#1681064011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D53
commutatorC2×C4×C53C22×C4
FrattiniC22×C4C22×D53
FittingC53×#525C2
socleC22×C53C2×C22⋊C4
hypercenter#525D53
hypocenterC53#1871
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×#11719917C2semidirect
2. C2×#13133144C2semidirect
3. C4×C22⋊Dic53C2semidirect
4. #16810382C2semidirect
17. C22⋊Q8×C53C4semidirect
18. C22⋊Q8×C53C4semidirect
19. #121×C53C4semidirect
20. #121×C53C4semidirect
58. #11719917D4semidirect
106. C22×C53#104semidirect
131. C53#1871semidirect
139. C22⋊Q8Dic53semidirect
140. C22⋊Q8Dic53semidirect
141. #121Dic53semidirect
142. #121Dic53semidirect
232. C22#13133144semidirect