Finite Group Databasebeta

← #34203440

Subgroups — #34203440

Order 20224.

564 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 123 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#23766112C211yesmaximal, characteristic
2C2×#19098357C212yesmaximal
3C4×#14579076C211yesmaximal, characteristic
4#23761524C212yesmaximal
5C4×#99×C79C211yesmaximal, characteristic
6C2×#14579148C2211yescharacteristic
7C2×C4×Dic158C411yescharacteristic, complemented
8C2×#14579076C2212yes—
9#19098468C411yescharacteristic, complemented
10#19098466C2211yescharacteristic
11#19098451—21no—
12#19098451—21no—
13#19098436—21no—
14#19098436—21no—
15#19098432C411yescharacteristic, complemented
16C4×C22⋊Dic79—21no—
17#19098409C411yescharacteristic, complemented
18#19098357C418yescomplemented
19#19098330—21no—
20C2×C4×D4×C79C2211yescharacteristic
21C2×#99×C79C2212yes—
22C4×C22⋊C4×C79—21no—
23#342×C79—21no—
24C22×Dic158C2×C411yescharacteristic
25#14579148—22no—
26#14579148C2×C414yes—
27C4×Dic158—41no—
28C4×Dic158—41no—
29C2×C4⋊Dic79C2×C411yescharacteristic
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C22⋊Dic79—22no—
36C2×C22⋊Dic79—21no—
37C2×C22⋊Dic79—21no—
38C2×C4×Dic79—21no—
39C2×C4×Dic79—41no—
40C2×C4×Dic79C2×C411yescharacteristic
41C2×C4×Dic79C2×C411yescharacteristic
42#14579076—22no—
43#14579076C2×C414yes—
44C79⋊#95—21no—
45C79⋊#95—21no—
46C79⋊#95—21no—
47C79⋊#95—21no—
48C79⋊#95—41no—
49C22×D4×C79C2311yescharacteristic
50C23×C4×C79D411yescharacteristic
51C23×C4×C79D411yescharacteristic
52C4×D4×C79—41no—
53C4×D4×C79—41no—
54C2×C4⋊C4×C79C2×C411yescharacteristic
55C2×C22⋊C4×C79—22no—
56C2×C22⋊C4×C79D411yescharacteristic
57C2×C22⋊C4×C79D411yescharacteristic
58C2×C42×C79—41no—
59C2×C42×C79C2×C411yescharacteristic
60#99×C79—22no—
61#99×C79C2×C414yes—
62#95×C79—41no—
63C2×Dic158—21no—
64C2×Dic158—41no—
65C2×Dic158—41no—
66C2×Dic158C4212yescomplemented
67C22×Dic79—21no—
68C22×Dic79—21no—
69C22×Dic79—21no—
70C22×Dic79—21no—
71C22×Dic79—42no—
72C4⋊Dic79—41no—
73Dic79⋊C4—41no—
74Dic79⋊C4—41no—
75Dic79⋊C4—81no—
76Dic79⋊C4—81no—
77C22⋊Dic79—24no—
78C22⋊Dic79—24no—
79C22⋊Dic79—24no—
80C22⋊Dic79—42no—
81C22⋊Dic79—81no—
82C22⋊Dic79—81no—
83C4×Dic79—21no—
84C4×Dic79—41no—
85C4×Dic79—42no—
86C4×Dic79—82no—
87C4×Dic79C4212yescomplemented
88C24×C79C2×D411yescharacteristic
89C24×C79C2×D411yescharacteristic
90C2×D4×C79—41no—
91C2×D4×C79—41no—
92C2×D4×C79C22×C412yes—
93C2×D4×C79#4112yes—
94C22×C4×C79—21no—
95C22×C4×C79—21no—
96C22×C4×C79—21no—
97C22×C4×C79—21no—
98C22×C4×C79—42no—
99C22×C4×C79—81no—
100C22×C4×C79—81no—
101C22×C4×C79C22⋊C411yescharacteristic
102C22×C4×C79C22⋊C411yescharacteristic
103C22×C4×C79C22⋊C411yescharacteristic
104C22×C4×C79C22⋊C411yescharacteristic
105C22×C4×C79C22×C411yescharacteristic
106C4⋊C4×C79—41no—
107C22⋊C4×C79—24no—
108C22⋊C4×C79—41no—
109C22⋊C4×C79—41no—
110C22⋊C4×C79—42no—
111C42×C79—21no—
112C42×C79—21no—
113C42×C79—82no—
114C23×C79—21no—
115C23×C79—21no—
116C23×C79—21no—
117C23×C79—21no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79C2×C22⋊C411yescharacteristic
123C23×C79C4×D412yes—
124C23×C79C4×D412yes—
125D4×C79—42no—
126D4×C79—42no—
127C2×C4×C79—21no—
128C2×C4×C79—21no—
129C2×C4×C79—21no—
130C2×C4×C79—21no—
131C2×C4×C79—21no—
132C2×C4×C79—41no—
133C2×C4×C79—42no—
134C2×C4×C79—42no—
135C2×C4×C79—41no—
136C2×C4×C79—41no—
137C2×C4×C79—41no—
138C2×C4×C79—44no—
139C2×C4×C79—81no—
140C2×C4×C79—81no—
141C2×C4×C79—82no—
142C2×C4×C79—82no—
143C2×C4×C79#9911yescharacteristic
144C2×C4×C79#9911yescharacteristic
145C2×C4×C79C2×C4211yescharacteristic
146C2×C4×C79#11711yescharacteristic
147C2×Dic79—22no—
148C2×Dic79—22no—
149C2×Dic79—41no—
150C2×Dic79—41no—
151C2×Dic79—41no—
152C2×Dic79—41no—
153C2×Dic79—44no—
154C2×Dic79—41no—
155C2×Dic79—41no—
156C2×Dic79—41no—
157C2×Dic79—41no—
158C2×Dic79—41no—
159C2×Dic79—41no—
160C2×Dic79—82no—
161C2×Dic79—82no—
162Dic158—42no—
163Dic158—42no—
164C22×C79—21no—
165C22×C79—21no—
166C22×C79—22no—
167C22×C79—22no—
168C22×C79—42no—
169C22×C79—42no—
170C22×C79—41no—
171C22×C79—41no—
172C22×C79—41no—
173C22×C79—41no—
174C22×C79—81no—
175C22×C79—81no—
176C22×C79C4×C22⋊C411yescharacteristic
177C22×C79C2×#9911yescharacteristic
178C22×C79#41011yescharacteristic
179C4×C79—21no—
180C4×C79—21no—
181C4×C79—21no—
182C4×C79—21no—
183C4×C79—41no—
184C4×C79—41no—
185C4×C79—41no—
186C4×C79—84no—
187Dic79—42no—
188Dic79—42no—
189Dic79—81no—
190Dic79—84no—
191Dic79—81no—
192#12627—791nomaximal, Sylow
193C2×C79—21no—
194C2×C79—21no—
195C2×C79—42no—
196C2×C79—42no—
197C2×C79C4×#9911yescharacteristic
198C2×C79#212711yescharacteristic
199C2×C79#212911yescharacteristic
200#2326—791no—
201C2×#354—792no—
202C4×#99—791no—
203C4×#99D7911yescharacteristic
204#1393—792no—
205C79#1262711yescharacteristic, complemented, Sylow
206C2×#124—791no—
207C2×C4×Q8—791no—
208C2×C4×D4C2×D7911yescharacteristic
209C2×#99—792no—
210C2×#99C2×D7912yes—
211#391—791no—
212#390—791no—
213#388—1581no—
214#388—1581no—
215#385—1581no—
216#385—1581no—
217#384—791no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—1581no—
220#376—791no—
221#354—798no—
222#342—21no—
223#342—1581no—
224C22×Q8—791no—
225C22×D4C22×D7911yescharacteristic
226C23×C4D31611yescharacteristic
227C23×C4C22⋊D7911yescharacteristic
228#124—794no—
229#124—1582no—
230C4×Q8—3161no—
231C4×Q8—3161no—
232C4×D4—41no—
233C4×D4—41no—
234C2×C4⋊C4—791no—
235C2×C4⋊C4—1581no—
236C2×C4⋊C4—1581no—
237C2×C4⋊C4—1581no—
238C2×C4⋊C4C4×D7911yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—1582no—
241C2×C22⋊C4—1581no—
242C2×C22⋊C4—1581no—
243C2×C22⋊C4—1581no—
244C2×C22⋊C4—1581no—
245C2×C22⋊C4D31611yescharacteristic
246C2×C22⋊C4C22⋊D7911yescharacteristic
247C2×C42—41no—
248C2×C42—3161no—
249C2×C42—791no—
250C2×C42—791no—
251C2×C42—1581no—
252C2×C42C4×D7911yescharacteristic
253#99—22no—
254#99—794no—
255#99—1582no—
256#99C4×D7914yes—
257#95—41no—
258#95—3161no—
259#95—1581no—
260#95—1581no—
261#95—1581no—
262#95—1581no—
263C24C2×C22⋊D7911yescharacteristic
264C24C2×D31611yescharacteristic
265C2×Q8—3161no—
266C2×Q8—3161no—
267C2×Q8—792no—
268C2×Q8—1581no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4C2×C4×D7912yes—
272C2×D4#1192962412yes—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—3162no—
281C22×C4—1581no—
282C22×C4—1581no—
283C22×C4—1581no—
284C22×C4—1581no—
285C22×C4(C2×D79)⋊C411yescharacteristic
286C22×C4(C2×D79)⋊C411yescharacteristic
287C22×C4(C2×D79)⋊C411yescharacteristic
288C22×C4(C2×D79)⋊C411yescharacteristic
289C22×C4C2×C4×D7911yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—3161no—
292C4⋊C4—3161no—
293C4⋊C4—3161no—
294C4⋊C4—6321no—
295C4⋊C4—6321no—
296C22⋊C4—24no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—42no—
300C22⋊C4—3162no—
301C22⋊C4—6321no—
302C22⋊C4—6321no—
303C22⋊C4—1584no—
304C22⋊C4—1584no—
305C22⋊C4—1584no—
306C42—21no—
307C42—21no—
308C42—82no—
309C42—3161no—
310C42—3162no—
311C42—792no—
312C42—6322no—
313C42—1581no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C2×(C2×D79)⋊C411yescharacteristic
323C23C4×C22⋊D7912yes—
324C23C4×D31612yes—
325Q8—3162no—
326Q8—3162no—
327D4—42no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—44no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—3161no—
346C2×C4—3161no—
347C2×C4—3161no—
348C2×C4—3161no—
349C2×C4—3164no—
350C2×C4—3161no—
351C2×C4—3161no—
352C2×C4—3161no—
353C2×C4—3161no—
354C2×C4—3161no—
355C2×C4—3161no—
356C2×C4—6322no—
357C2×C4—6322no—
358C2×C4—1582no—
359C2×C4—1582no—
360C2×C4#1457907511yescharacteristic
361C2×C4#1457907511yescharacteristic
362C2×C4C42×D7911yescharacteristic
363C2×C4#1457911811yescharacteristic
364C22—21no—
365C22—21no—
366C22—22no—
367C22—22no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22—81no—
376C22C4×(C2×D79)⋊C411yescharacteristic
377C22C2×#1457907511yescharacteristic
378C22#1909853611yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—84no—
387C4—3162no—
388C4—3162no—
389C4—6321no—
390C4—6324no—
391C4—6321no—
392C2—21no—
393C2—21no—
394C2—42no—
395C2—42no—
396C2C4×#1457907511yescharacteristic
397C2#2376506711yescharacteristic
398C2#2376507411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC2×C4×C79C2×C42
FrattiniC22×D4C22×D79
FittingC4×#99×C79C2
socleC22×C79C2×#99
prime radical, cosocleC22×D4×C79C23
hypercenterC4×#99D79
hypocenterC79#12627
LCSG ▷ C2×C4×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
7. C2×C4×Dic158C4semidirect
9. #19098468C4semidirect
15. #19098432C4semidirect
17. #19098409C4semidirect
18. #19098357C4semidirect
66. C2×Dic158C42semidirect
87. C4×Dic79C42semidirect
205. C79#12627semidirect

Maximal subgroups

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

subgroupindex
#237661122
C2×#190983572
C4×#145790762
#237615242
C4×#99×C792
#1262779