Finite Group Databasebeta

← #20279619

Subgroups — #20279619

Order 6784.

604 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#13133194C212yesmaximal, complemented
2C2×#13133182C211yesmaximal, characteristic, complemented
3C2×D4×Dic53C211yesmaximal, characteristic
4#16810555C211yesmaximal, characteristic, complemented
5#16810553C211yesmaximal, characteristic, complemented
6#16810537C211yesmaximal, characteristic
7#16810517C212yesmaximal, complemented
8#16810516C212yesmaximal, complemented
9#16810487C211yesmaximal, characteristic, complemented
10#16810481C212yesmaximal, complemented
11C53×#394C211yesmaximal, characteristic
12C22×Dic106C2211yescharacteristic
13C23×Dic53C2212yes—
14#13133194—28no—
15#13133182—24no—
16D4×Dic53—24no—
17C2×C4⋊Dic53C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×#11719919C2212yes—
20C2×#11719919C2211yescharacteristic
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×C22⋊Dic53C2212yes—
25C2×C22⋊Dic53C2212yes—
26C2×C4×Dic53C2211yescharacteristic
27C2×C4×Dic53C2211yescharacteristic
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2212yes—
32#13133124C2211yescharacteristic
33C22×D4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2211yescharacteristic
35C2×C22⋊C4×C53C2212yes—
36C2×C22⋊C4×C53C2212yes—
37#95×C53C2211yescharacteristic
38C24×C53C2312yes—
39C2×D4×C53—24no—
40C2×D4×C53—22no—
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×Dic53—24no—
46C22×Dic53—24no—
47C22×Dic53—24no—
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C22×Dic53C2311yescharacteristic
54C2×Dic106—22no—
55C2×Dic106—22no—
56C2×Dic106D414yescomplemented
57C4⋊C4×C53—22no—
58C22⋊C4×C53—24no—
59C22⋊C4×C53—24no—
60C22⋊Dic53—24no—
61C22⋊Dic53—24no—
62C22⋊Dic53—24no—
63C4⋊Dic53—22no—
64C4⋊Dic53—24no—
65#11719919—22no—
66#11719919—24no—
67#11719919—22no—
68#11719919—24no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53D414yes—
72C23×C53—22no—
73C23×C53—22no—
74C23×C53—22no—
75C23×C53—22no—
76C23×C53—22no—
77C23×C53—22no—
78C23×C53#4114yes—
79C23×C53C2411yescharacteristic
80D4×C53—28no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53C2×D412yes—
90C2×C4×C53C2×D412yes—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—48no—
103C2×Dic53C2×D412yes—
104C2×Dic53C2×D412yes—
105C2×Dic53C2×D412yes—
106C2×Dic53C2×D412yes—
107C2×Dic53#4112yes—
108Dic106—28no—
109C22×C53—24no—
110C22×C53—24no—
111C22×C53—24no—
112C22×C53—42no—
113C22×C53—42no—
114C22×C53—42no—
115C22×C53—42no—
116C22×C53C22×D411yescharacteristic
117C22×C53C22×D411yescharacteristic
118C22×C53C2×#4111yescharacteristic
119C22×C53C2×#4112yes—
120C22×C532−1+412yes—
121C4×C53—22no—
122C4×C53—22no—
123C4×C53—41no—
124C4×C53—42no—
125Dic53—22no—
126Dic53—22no—
127Dic53—42no—
128Dic53—41no—
129Dic53—42no—
130Dic53#12114yes—
131#2536—531nomaximal, Sylow
132C2×C53—44no—
133C2×C53C2×#12111yescharacteristic
134C2×C53#52511yescharacteristic
135C2×C53#53611yescharacteristic
136C2×C53#54812yes—
137C2×C53#55212yes—
138C2×C22⋊Q8—532no—
139C2×C4×Q8—531no—
140C2×C4×D4—531no—
141#398—531no—
142#398—531no—
143#394—531no—
144#394D5311yescharacteristic
145#388—532no—
146#388—532no—
147#382—531no—
148#380—532no—
149C53#253611yescharacteristic, complemented, Sylow
150C22×Q8—531no—
151C22×D4C2×D5311yescharacteristic
152C23×C4—532no—
153C22⋊Q8—1068no—
154C4×Q8—1064no—
155C4×D4—1064no—
156C2×C4⋊C4—531no—
157C2×C4⋊C4—531no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—532no—
162C2×C4⋊C4C2×D5311yescharacteristic
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4C2×D5312yes—
167C2×C22⋊C4C2×D5312yes—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—531no—
171#95—532no—
172#95—532no—
173#95—532no—
174#95—531no—
175#95C2×D5311yescharacteristic
176C24C22×D5312yes—
177C2×Q8—534no—
178C2×Q8—1062no—
179C2×Q8—1062no—
180C2×D4—24no—
181C2×D4—22no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—532no—
186C22×C4—531no—
187C22×C4—532no—
188C22×C4—1064no—
189C22×C4—1064no—
190C22×C4—1064no—
191C22×C4C22×D5312yes—
192C22×C4C22×D5311yescharacteristic
193C22×C4C22×D5311yescharacteristic
194C22×C4C22×D5311yescharacteristic
195C4⋊C4—22no—
196C4⋊C4—1062no—
197C4⋊C4—1064no—
198C4⋊C4—1064no—
199C4⋊C4—1062no—
200C4⋊C4—1062no—
201C4⋊C4—1064no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—1064no—
205C22⋊C4—1064no—
206C22⋊C4—1064no—
207C42—534no—
208C42—1062no—
209C42—1062no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—22no—
216C23#1171994714yes—
217C23C23×D5311yescharacteristic
218Q8—1068no—
219D4—28no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—532no—
229C2×C4—532no—
230C2×C4—532no—
231C2×C4—532no—
232C2×C4—532no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—1062no—
236C2×C4—1061no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—1062no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—2128no—
245C2×C4D4×D5312yes—
246C2×C4D4×D5312yes—
247C22—24no—
248C22—24no—
249C22—24no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—42no—
254C22C2×D4×D5311yescharacteristic
255C22C2×D4×D5311yescharacteristic
256C22#41×D5311yescharacteristic
257C22C2×#1171994712yes—
258C22#1313326412yes—
259C4—22no—
260C4—22no—
261C4—42no—
262C4—41no—
263C4—534no—
264C4—1062no—
265C4—1062no—
266C4—2122no—
267C4—2122no—
268C4—2121no—
269C2—44no—
270C2#121×D5311yescharacteristic
271C2#1681112911yescharacteristic
272C2#1681119311yescharacteristic
273C2#1681127912yes—
274C2#1681131012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#394C2
hypercenter#394D53
hypocenterC53#2536
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#13133194C2semidirect
2. C2×#13133182C2semidirect
4. #16810555C2semidirect
5. #16810553C2semidirect
7. #16810517C2semidirect
8. #16810516C2semidirect
9. #16810487C2semidirect
10. #16810481C2semidirect
56. C2×Dic106D4semidirect
149. C53#2536semidirect