Finite Group Databasebeta

← #20279656

Subgroups — #20279656

Order 6784.

948 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 171 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×#13133213C212yesmaximal, complemented
2C2×#13133213C211yesmaximal, characteristic, complemented
3C2×#13133190C212yesmaximal, complemented
4C2×#13133189C212yesmaximal, complemented
5C2×#13133177C211yesmaximal, characteristic, complemented
6#16810533C211yesmaximal, characteristic, complemented
7#16810532C211yesmaximal, characteristic, complemented
8#16810531C211yesmaximal, characteristic, complemented
9#16810523C212yesmaximal, complemented
10#16810474C211yesmaximal, characteristic, complemented
11C53×#392C211yesmaximal, characteristic, complemented
12C22×D212C2212yes—
13C22×D212C2211yescharacteristic
14C22×C22⋊D53C2212yes—
15C22×C22⋊D53C2212yes—
16C22×C22⋊D53C2212yes—
17C22×C22⋊D53C2212yes—
18C22×C22⋊D53C2211yescharacteristic
19C23×Dic53C2211yescharacteristic
20C22×C4×D53C2211yescharacteristic
21#13133213—24no—
22#13133213—28no—
23#13133190—28no—
24#13133189—28no—
25#13133177—24no—
26C2×#11719919C2211yescharacteristic
27C2×#11719919C2212yes—
28C2×C22⋊Dic53C2212yes—
29C2×#11719921C2212yes—
30C2×#11719921C2212yes—
31C2×#11719921C2211yescharacteristic
32C2×C4×Dic53C2212yes—
33C2×C4×Dic53C2211yescharacteristic
34#13133124C2211yescharacteristic
35C22×D4×C53C2212yes—
36C22×D4×C53C2211yescharacteristic
37C2×C22⋊C4×C53C2212yes—
38C2×C22⋊C4×C53C2211yescharacteristic
39#95×C53C2211yescharacteristic
40C24×C53C2312yes—
41C24×C53C2311yescharacteristic
42C23×D53C2312yes—
43C23×D53C2311yescharacteristic
44C2×D4×C53—24no—
45C2×D4×C53—24no—
46C2×D4×C53—22no—
47C2×D4×C53—42no—
48C2×D4×C53—42no—
49C22×C4×C53C2311yescharacteristic
50C22×C4×C53C2311yescharacteristic
51C22×C4×C53C2312yes—
52C2×C22⋊D53—24no—
53C2×C22⋊D53—24no—
54C2×C22⋊D53—22no—
55C2×C22⋊D53—24no—
56C2×C22⋊D53—24no—
57C2×C22⋊D53—24no—
58C2×C22⋊D53—24no—
59C2×C22⋊D53—24no—
60C2×C22⋊D53—24no—
61C2×C22⋊D53—24no—
62C2×C22⋊D53—22no—
63C2×C22⋊D53—24no—
64C2×C22⋊D53—42no—
65C2×C22⋊D53—42no—
66C2×C22⋊D53D414yescomplemented
67C22×Dic53—22no—
68C22×Dic53—24no—
69C22×Dic53C2311yescharacteristic
70C22×Dic53C2312yes—
71C22×Dic53C2311yescharacteristic
72C22×Dic53C2311yescharacteristic
73C2×D212—24no—
74C2×D212—24no—
75C2×D212—22no—
76C2×D212—42no—
77C2×D212—42no—
78C2×C4×D53—22no—
79C2×C4×D53—24no—
80C22⋊C4×C53—22no—
81C22⋊C4×C53—24no—
82C22⋊Dic53—24no—
83#11719921—22no—
84#11719921—24no—
85#11719921—24no—
86#11719919—22no—
87#11719919D418yescomplemented
88C4×Dic53—24no—
89C4×Dic53D414yescomplemented
90C23×C53—22no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—22no—
101C23×C53C2×D412yes—
102C23×C53C2411yescharacteristic
103C22×D53—22no—
104C22×D53—21no—
105C22×D53—21no—
106C22×D53—22no—
107C22×D53—22no—
108C22×D53—22no—
109C22×D53—22no—
110C22×D53—22no—
111C22×D53—22no—
112C22×D53—22no—
113C22×D53—22no—
114C22×D53C2×D412yes—
115D4×C53—48no—
116D4×C53—44no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—22no—
120C2×C4×C53—21no—
121C2×C4×C53—22no—
122C2×C4×C53C2×D412yes—
123C2×C4×C53C2×D414yes—
124C22⋊D53—28no—
125C22⋊D53—48no—
126C22⋊D53—48no—
127C22⋊D53—48no—
128C22⋊D53—48no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—21no—
132C2×Dic53—22no—
133C2×Dic53—44no—
134C2×Dic53C2×D414yes—
135C2×Dic53C2×D412yes—
136C2×Dic53C2×D412yes—
137C2×Dic53C2×D412yes—
138C2×Dic53C2×D414yes—
139C2×Dic53#4112yes—
140D212—48no—
141D212—44no—
142C4×D53—44no—
143C22×C53—22no—
144C22×C53—24no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—42no—
153C22×C53—42no—
154C22×C53—42no—
155C22×C53C22×D411yescharacteristic
156C22×C53C22×D411yescharacteristic
157C22×C53C22×D412yes—
158C22×C53C2×#4111yescharacteristic
159C22×C53H(5,𝔽2)12yes—
160C2×D53—22no—
161C2×D53—24no—
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—42no—
165C2×D53—42no—
166C2×D53—42no—
167C2×D53—42no—
168C2×D53—42no—
169C2×D53—42no—
170C2×D53—42no—
171C2×D53—42no—
172C4×C53—22no—
173C4×C53—24no—
174C4×C53—41no—
175Dic53—24no—
176Dic53—41no—
177Dic53#12114yescomplemented
178Dic53#12714yescomplemented
179#2541—531nomaximal, Sylow
180C2×C53—42no—
181C2×C53—82no—
182C2×C53C2×#12111yescharacteristic
183C2×C53C2×#12711yescharacteristic
184C2×C53#53511yescharacteristic
185C2×C53D4212yes—
186C2×C53#55012yes—
187D53—42no—
188D53—82no—
189C2×#127—531no—
190C2×#127—532no—
191C2×#121—532no—
192C2×#121—532no—
193C2×C4×D4—531no—
194#392—531no—
195#392—531no—
196#392—531no—
197#392D5311yescharacteristic, complemented
198#390—532no—
199C4×C4⋊C4—531no—
200C53#254111yescharacteristic, complemented, Sylow
201C22×D4—531no—
202C22×D4—532no—
203C22×D4—532no—
204C22×D4—532no—
205C22×D4—532no—
206C22×D4—532no—
207C22×D4—531no—
208C22×D4C2×D5312yes—
209C22×D4C2×D5311yescharacteristic
210C23×C4—531no—
211C23×C4—531no—
212#127—1064no—
213#127—1068no—
214#121—1068no—
215#121—1068no—
216C4×D4—1064no—
217C2×C4⋊C4—532no—
218C2×C4⋊C4—531no—
219C2×C22⋊C4—532no—
220C2×C22⋊C4—531no—
221C2×C22⋊C4—532no—
222C2×C22⋊C4—532no—
223C2×C22⋊C4C2×D5311yescharacteristic
224C2×C22⋊C4C2×D5312yes—
225C2×C42—531no—
226C2×C42—532no—
227#95—531no—
228#95C2×D5311yescharacteristic
229C24—531no—
230C24—532no—
231C24C22×D5312yes—
232C24C22×D5311yescharacteristic
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—534no—
239C2×D4—1064no—
240C2×D4—1062no—
241C2×D4—1062no—
242C2×D4—1064no—
243C2×D4—1064no—
244C2×D4—1064no—
245C2×D4—1064no—
246C2×D4—1064no—
247C2×D4—1064no—
248C2×D4—1064no—
249C2×D4—1064no—
250C2×D4—1064no—
251C2×D4—1064no—
252C2×D4—1064no—
253C2×D4—1062no—
254C2×D4—2122no—
255C2×D4—2122no—
256C2×D4—2122no—
257C2×D4—2122no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—532no—
261C22×C4—531no—
262C22×C4—1062no—
263C22×C4—1064no—
264C22×C4—1062no—
265C22×C4—1064no—
266C22×C4C22×D5311yescharacteristic
267C22×C4C22×D5311yescharacteristic
268C22×C4C22×D5312yes—
269C4⋊C4—538no—
270C4⋊C4—1062no—
271C22⋊C4—22no—
272C22⋊C4—24no—
273C22⋊C4—1064no—
274C22⋊C4—1064no—
275C22⋊C4—1062no—
276C22⋊C4—1064no—
277C42—534no—
278C42—1064no—
279C23—21no—
280C23—22no—
281C23—21no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—532no—
291C23—1061no—
292C23—1062no—
293C23—1061no—
294C23—1062no—
295C23—1062no—
296C23—1062no—
297C23—1062no—
298C23—1062no—
299C23—1062no—
300C23—1062no—
301C23—1062no—
302C23D4×D5312yes—
303C23C23×D5311yescharacteristic
304D4—48no—
305D4—44no—
306D4—1068no—
307D4—2128no—
308D4—2124no—
309D4—2128no—
310D4—2128no—
311D4—2128no—
312D4—2128no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—532no—
319C2×C4—532no—
320C2×C4—532no—
321C2×C4—534no—
322C2×C4—532no—
323C2×C4—534no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—1062no—
327C2×C4—1061no—
328C2×C4—2124no—
329C2×C4—2124no—
330C2×C4D4×D5312yes—
331C2×C4D4×D5314yes—
332C22—22no—
333C22—24no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—1062no—
345C22—1064no—
346C22—2122no—
347C22—2122no—
348C22—2122no—
349C22—2122no—
350C22—2122no—
351C22—2122no—
352C22—2122no—
353C22—2122no—
354C22—2121no—
355C22—2121no—
356C22C2×D4×D5312yes—
357C22C2×D4×D5311yescharacteristic
358C22C2×D4×D5311yescharacteristic
359C22#41×D5311yescharacteristic
360C22#1313326212yes—
361C4—22no—
362C4—24no—
363C4—41no—
364C4—534no—
365C4—534no—
366C4—1064no—
367C4—2121no—
368C2—42no—
369C2—82no—
370C2—4242no—
371C2—2122no—
372C2#121×D5311yescharacteristic
373C2#127×D5311yescharacteristic
374C2#1681118811yescharacteristic
375C2#1681125512yes—
376C2#1681129112yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#392C2
hypercenter#392D53
hypocenterC53#2541
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#13133213C2semidirect
2. C2×#13133213C2semidirect
3. C2×#13133190C2semidirect
4. C2×#13133189C2semidirect
5. C2×#13133177C2semidirect
6. #16810533C2semidirect
7. #16810532C2semidirect
8. #16810531C2semidirect
9. #16810523C2semidirect
10. #16810474C2semidirect
11. C53×#392C2semidirect
66. C2×C22⋊D53D4semidirect
87. #11719919D4semidirect
89. C4×Dic53D4semidirect
177. Dic53#121semidirect
178. Dic53#127semidirect
197. #392D53semidirect
200. C53#2541semidirect