Finite Group Databasebeta

← #34180776

Subgroups — #34180776

Order 20224.

754 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 131 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#23763483C211yesmaximal, characteristic, complemented
2#23763477C211yesmaximal, characteristic, complemented
3#23761933C212yesmaximal, complemented
4#23761929C212yesmaximal
5C2×#542×C79C211yesmaximal, characteristic
6#19098527C2212yes—
7#19098426C2212yes—
8#19098332—21no—
9#19098332—21no—
10#19098330—21no—
11#19098330—21no—
12#19098300—24no—
13#19098297—24no—
14#542×C79—24no—
15#542×C79—24no—
16C2×C4⋊Q8×C79C411yescharacteristic, complemented
17C2×#126×C79—21no—
18C2×#124×C79—21no—
19C2×#124×C79C411yescharacteristic, complemented
20C2×#123×C79—21no—
21C2×C22⋊Q8×C79—21no—
22C2×C22⋊Q8×C79C2211yescharacteristic
23C2×#121×C79C2211yescharacteristic
24C2×C4×Q8×C79C411yescharacteristic, complemented
25C2×C4×D4×C79C411yescharacteristic, complemented
26C2×#117×C79C2211yescharacteristic
27C2×#11929608—22no—
28C2×C22⋊Dic79—22no—
29C22⋊(C79⋊C8)—22no—
30C22⋊(C79⋊C8)—22no—
31C79⋊#95—22no—
32C79⋊#95—22no—
33C22×Q8×C79—21no—
34C22×Q8×C79C2×C411yescharacteristic
35C22×D4×C79—21no—
36C22×D4×C79C2×C411yescharacteristic
37C23×C4×C79—21no—
38C23×C4×C79C2311yescharacteristic
39C4⋊Q8×C79—22no—
40C4⋊Q8×C79—22no—
41#126×C79—28no—
42#124×C79—28no—
43#124×C79—22no—
44#124×C79—22no—
45#123×C79—28no—
46C22⋊Q8×C79—28no—
47C22⋊Q8×C79—22no—
48C22⋊Q8×C79D414yes—
49#121×C79—22no—
50#121×C79D414yes—
51C4×Q8×C79—22no—
52C4×Q8×C79—22no—
53C4×D4×C79—22no—
54C4×D4×C79—22no—
55#117×C79—22no—
56#117×C79—22no—
57C2×C4⋊C4×C79—21no—
58C2×C4⋊C4×C79—21no—
59C2×C4⋊C4×C79—21no—
60C2×C4⋊C4×C79C2×C411yescharacteristic
61C2×C4⋊C4×C79C2×C411yescharacteristic
62C2×C4⋊C4×C79D411yescharacteristic
63C2×C4⋊C4×C79D411yescharacteristic
64C2×C22⋊C4×C79—21no—
65C2×C22⋊C4×C79—21no—
66C2×C22⋊C4×C79—21no—
67C2×C22⋊C4×C79—21no—
68C2×C22⋊C4×C79—21no—
69C2×C22⋊C4×C79D411yescharacteristic
70C2×C22⋊C4×C79D411yescharacteristic
71C2×C42×C79—21no—
72C2×C42×C79C2×C411yescharacteristic
73C2×C42×C79C2×C411yescharacteristic
74C22×Dic79—42no—
75#11929608—44no—
76C2×C79⋊C8—42no—
77C22⋊Dic79—44no—
78C24×C79—21no—
79C24×C79C2×D411yescharacteristic
80C2×Q8×C79—24no—
81C2×Q8×C79—21no—
82C2×Q8×C79—42no—
83C2×Q8×C79—82no—
84C2×Q8×C79C22⋊C412yes—
85C2×D4×C79—24no—
86C2×D4×C79—21no—
87C2×D4×C79—42no—
88C2×D4×C79—82no—
89C2×D4×C79C22⋊C412yes—
90C22×C4×C79—21no—
91C22×C4×C79—21no—
92C22×C4×C79—21no—
93C22×C4×C79—24no—
94C22×C4×C79—42no—
95C22×C4×C79—82no—
96C22×C4×C79C22⋊C411yescharacteristic
97C22×C4×C79C22⋊C411yescharacteristic
98C22×C4×C79C22⋊C411yescharacteristic
99C22×C4×C79C22⋊C411yescharacteristic
100C22×C4×C79C22×C411yescharacteristic
101C22×C4×C79C2×D411yescharacteristic
102C22×C4×C79C2×D412yes—
103C22×C4×C79C2×D412yes—
104C4⋊C4×C79—24no—
105C4⋊C4×C79—24no—
106C4⋊C4×C79—21no—
107C4⋊C4×C79—21no—
108C4⋊C4×C79—24no—
109C4⋊C4×C79—22no—
110C4⋊C4×C79—22no—
111C4⋊C4×C79C22⋊C412yes—
112C4⋊C4×C79C22⋊C412yes—
113C22⋊C4×C79—24no—
114C22⋊C4×C79—24no—
115C22⋊C4×C79—24no—
116C22⋊C4×C79—24no—
117C22⋊C4×C79—22no—
118C22⋊C4×C79—22no—
119C22⋊C4×C79—24no—
120C42×C79—24no—
121C42×C79—21no—
122C42×C79—21no—
123C42×C79—21no—
124C42×C79—21no—
125C23×C79—22no—
126C23×C79—22no—
127C23×C79—41no—
128C23×C79—41no—
129C23×C79—42no—
130C23×C79—42no—
131C23×C79—42no—
132C23×C79C2×C22⋊C411yescharacteristic
133C23×C79#12012yes—
134Q8×C79—42no—
135Q8×C79—42no—
136Q8×C79—84no—
137D4×C79—42no—
138D4×C79—42no—
139D4×C79—84no—
140C2×C4×C79—22no—
141C2×C4×C79—22no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—22no—
145C2×C4×C79—21no—
146C2×C4×C79—21no—
147C2×C4×C79—22no—
148C2×C4×C79—22no—
149C2×C4×C79—42no—
150C2×C4×C79—42no—
151C2×C4×C79—42no—
152C2×C4×C79—42no—
153C2×C4×C79—42no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—84no—
159C2×C4×C79C2×C22⋊C411yescharacteristic
160C2×C4×C79C2×C22⋊C411yescharacteristic
161C2×C4×C79#12012yes—
162C2×Dic79—82no—
163C2×Dic79—82no—
164C2×Dic79—82no—
165C79⋊C8—82no—
166C22×C79—21no—
167C22×C79—21no—
168C22×C79—22no—
169C22×C79—22no—
170C22×C79—22no—
171C22×C79—42no—
172C22×C79—42no—
173C22×C79—42no—
174C22×C79—82no—
175C22×C79—84no—
176C22×C79#37911yescharacteristic
177C22×C79#41012yes—
178C4×C79—21no—
179C4×C79—21no—
180C4×C79—22no—
181C4×C79—42no—
182C4×C79—42no—
183C4×C79—41no—
184C4×C79—41no—
185C4×C79—41no—
186C4×C79—41no—
187C4×C79—42no—
188Dic79—162no—
189#8519—791nomaximal, Sylow
190C2×C79—22no—
191C2×C79—42no—
192C2×C79—82no—
193C2×C79#178211yescharacteristic
194C2×C79#179511yescharacteristic
195C2×C79#179611yescharacteristic
196C2×#542D7911yescharacteristic
197#1822—791no—
198#1820—791no—
199#1511—792no—
200#1510—792no—
201C79#851911yescharacteristic, complemented, Sylow
202#542—24no—
203#542—24no—
204C2×C4⋊Q8Dic7911yescharacteristic, complemented
205C2×#126—21no—
206C2×#124—21no—
207C2×#124Dic7911yescharacteristic, complemented
208C2×#123—21no—
209C2×C22⋊Q8—21no—
210C2×C22⋊Q8C2×D7911yescharacteristic
211C2×#121C2×D7911yescharacteristic
212C2×C4×Q8Dic7911yescharacteristic, complemented
213C2×C4×D4Dic7911yescharacteristic, complemented
214C2×#117C2×D7911yescharacteristic
215#407—792no—
216#381—792no—
217#343—1581no—
218#343—1581no—
219#342—1581no—
220#342—1581no—
221#328—1584no—
222#327—1584no—
223C22×Q8—21no—
224C22×Q8C2×Dic7911yescharacteristic
225C22×D4—21no—
226C22×D4C2×Dic7911yescharacteristic
227C23×C4—21no—
228C23×C4C22×D7911yescharacteristic
229C2×M16—1582no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232#126—28no—
233#124—28no—
234#124—22no—
235#124—22no—
236#123—28no—
237C22⋊Q8—28no—
238C22⋊Q8—22no—
239C22⋊Q8C22⋊D7914yes—
240#121—22no—
241#121C22⋊D7914yes—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×D4—22no—
245C4×D4—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×Dic7911yescharacteristic
252C2×C4⋊C4C2×Dic7911yescharacteristic
253C2×C4⋊C4C22⋊D7911yescharacteristic
254C2×C4⋊C4C22⋊D7911yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—1582no—
261C2×C22⋊C4C22⋊D7911yescharacteristic
262C2×C22⋊C4C22⋊D7911yescharacteristic
263C2×C42—21no—
264C2×C42C2×Dic7911yescharacteristic
265C2×C42C2×Dic7911yescharacteristic
266C22⋊C8—1582no—
267C22⋊C8—1582no—
268#95—1582no—
269#95—1582no—
270C24—21no—
271C24C2×C22⋊D7911yescharacteristic
272C2×Q8—24no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—82no—
276C2×Q8C22⋊Dic7912yes—
277C2×D4—24no—
278C2×D4—21no—
279C2×D4—42no—
280C2×D4—82no—
281C2×D4C22⋊Dic7912yes—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—82no—
288C22×C4—3162no—
289C22×C4C22⋊Dic7911yescharacteristic
290C22×C4C22⋊Dic7911yescharacteristic
291C22×C4C22⋊Dic7911yescharacteristic
292C22×C4C22⋊Dic7911yescharacteristic
293C22×C4C22×Dic7911yescharacteristic
294C22×C4C2×C22⋊D7911yescharacteristic
295C22×C4C2×C22⋊D7912yes—
296C22×C4C2×C22⋊D7912yes—
297M16—3164no—
298C2×C8—3162no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—24no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4C22⋊Dic7912yes—
307C4⋊C4C22⋊Dic7912yes—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—24no—
311C22⋊C4—24no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—24no—
315C22⋊C4—3164no—
316C42—24no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C23—22no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23C2×C22⋊Dic7911yescharacteristic
329C23#1457913012yes—
330Q8—42no—
331Q8—42no—
332Q8—84no—
333D4—42no—
334D4—42no—
335D4—84no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4—6322no—
356C2×C4—6322no—
357C2×C4—6322no—
358C2×C4C2×C22⋊Dic7911yescharacteristic
359C2×C4C2×C22⋊Dic7911yescharacteristic
360C2×C4#1457913012yes—
361C8—6322no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—82no—
371C22—84no—
372C22#1909841911yescharacteristic
373C22#1909853712yes—
374C4—22no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—12642no—
385C2—22no—
386C2—42no—
387C2—82no—
388C2#2376331811yescharacteristic
389C2#2376337011yescharacteristic
390C2#2376337511yescharacteristic
subgroupG/H
centerC22#19098419
commutatorC22×C4×C79C22×C4
FrattiniC23×C4C22×D79
FittingC2×#542×C79C2
socleC22×C79#379
prime radicalC2×#121×C79C22
cosocleC23×C4×C79C23
hypercenterC2×#542D79
hypocenterC79#8519
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. #23763483C2semidirect
2. #23763477C2semidirect
3. #23761933C2semidirect
16. C2×C4⋊Q8×C79C4semidirect
19. C2×#124×C79C4semidirect
24. C2×C4×Q8×C79C4semidirect
25. C2×C4×D4×C79C4semidirect
201. C79#8519semidirect
204. C2×C4⋊Q8Dic79semidirect
207. C2×#124Dic79semidirect
212. C2×C4×Q8Dic79semidirect
213. C2×C4×D4Dic79semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237634832
#237634772
#237619332
#237619292
C2×#542×C792
#851979