Finite Group Databasebeta

← #34197519

Subgroups — #34197519

Order 20224.

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

normal subgroups only

This group has 756 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#23761721C211yesmaximal, characteristic, complemented
2#23761718C211yesmaximal, characteristic, complemented
3#23761348C212yesmaximal, complemented
4#23761328C212yesmaximal, complemented
5C4×#120×C79C211yesmaximal, characteristic
6#19098616—21no—
7C2×C22⋊(C79⋊C8)—21no—
8#19098289—21no—
9#19098287—21no—
10#19098287C2212yes—
11#19098287C2212yes—
12C24×C4×C79C411yescharacteristic
13C2×#120×C79C411yescharacteristic
14C2×C4×D4×C79—21no—
15C2×C4×D4×C79C2211yescharacteristic
16#386×C79—21no—
17#386×C79C411yescharacteristic
18#385×C79—21no—
19#385×C79C2211yescharacteristic
20#379×C79C2211yescharacteristic
21C4×C22⋊C4×C79—21no—
22C4×C22⋊C4×C79C411yescharacteristic
23C22×C79⋊C8—41no—
24C4⋊(C79⋊C8)—41no—
25C22⋊(C79⋊C8)—22no—
26C22⋊(C79⋊C8)—22no—
27C22⋊(C79⋊C8)—41no—
28C4×C79⋊C8—41no—
29C25×C79C2×C411yescharacteristic
30C22×D4×C79—21no—
31C22×D4×C79C2×C411yescharacteristic
32C23×C4×C79—21no—
33C23×C4×C79—21no—
34C23×C4×C79—44no—
35C23×C4×C79—41no—
36C23×C4×C79—41no—
37C23×C4×C79C2×C411yescharacteristic
38C23×C4×C79C2×C411yescharacteristic
39C23×C4×C79D411yescharacteristic
40#120×C79—22no—
41#120×C79C814yescomplemented
42C4×D4×C79—41no—
43C4×D4×C79—41no—
44C4×D4×C79—44no—
45C2×C4⋊C4×C79—21no—
46C2×C4⋊C4×C79D411yescharacteristic
47C2×C22⋊C4×C79—21no—
48C2×C22⋊C4×C79—21no—
49C2×C22⋊C4×C79—21no—
50C2×C22⋊C4×C79—21no—
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79C2×C411yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79C2311yescharacteristic
55C2×C42×C79—21no—
56C2×C42×C79D411yescharacteristic
57#95×C79—21no—
58#95×C79—21no—
59#95×C79—21no—
60C2×C79⋊C8—42no—
61C2×C79⋊C8—42no—
62C2×C79⋊C8—82no—
63C24×C79—21no—
64C24×C79—21no—
65C24×C79—42no—
66C24×C79—41no—
67C24×C79—42no—
68C24×C79—41no—
69C24×C79C22⋊C411yescharacteristic
70C24×C79C2×C812yescomplemented
71C24×C79C22×C411yescharacteristic
72C2×D4×C79—24no—
73C2×D4×C79—41no—
74C2×D4×C79—41no—
75C2×D4×C79—42no—
76C2×D4×C79—42no—
77C2×D4×C79C2×C812yes—
78C2×D4×C79M1612yes—
79C22×C4×C79—24no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—21no—
85C22×C4×C79—21no—
86C22×C4×C79—21no—
87C22×C4×C79—42no—
88C22×C4×C79—42no—
89C22×C4×C79—42no—
90C22×C4×C79—42no—
91C22×C4×C79—42no—
92C22×C4×C79—42no—
93C22×C4×C79—42no—
94C22×C4×C79—44no—
95C22×C4×C79—44no—
96C22×C4×C79—44no—
97C22×C4×C79—41no—
98C22×C4×C79—41no—
99C22×C4×C79—41no—
100C22×C4×C79—44no—
101C22×C4×C79C22⋊C411yescharacteristic
102C22×C4×C79C22⋊C411yescharacteristic
103C22×C4×C79C22⋊C411yescharacteristic
104C22×C4×C79C2×D411yescharacteristic
105C22×C4×C79C2×D411yescharacteristic
106C4⋊C4×C79—41no—
107C4⋊C4×C79—42no—
108C22⋊C4×C79—21no—
109C22⋊C4×C79—24no—
110C22⋊C4×C79—41no—
111C22⋊C4×C79—41no—
112C22⋊C4×C79—42no—
113C22⋊C4×C79—42no—
114C22⋊C4×C79—42no—
115C22⋊C4×C79—41no—
116C22⋊C4×C79C2×C812yescomplemented
117C42×C79—21no—
118C42×C79—21no—
119C42×C79—42no—
120C23×C79—22no—
121C23×C79—22no—
122C23×C79—22no—
123C23×C79—21no—
124C23×C79—21no—
125C23×C79—21no—
126C23×C79—21no—
127C23×C79—41no—
128C23×C79—44no—
129C23×C79—42no—
130C23×C79—41no—
131C23×C79—41no—
132C23×C79—41no—
133C23×C79—44no—
134C23×C79—42no—
135C23×C79—41no—
136C23×C79—41no—
137C23×C79—42no—
138C23×C79—42no—
139C23×C79—42no—
140C23×C79—48no—
141C23×C79—42no—
142C23×C79—41no—
143C23×C79—41no—
144C23×C79C22⋊C812yes—
145C23×C79C2×C22⋊C411yescharacteristic
146C23×C79C22×C811yescharacteristic
147C23×C79C2×M1611yescharacteristic
148D4×C79—42no—
149D4×C79—42no—
150D4×C79—48no—
151C2×C4×C79—21no—
152C2×C4×C79—21no—
153C2×C4×C79—22no—
154C2×C4×C79—22no—
155C2×C4×C79—22no—
156C2×C4×C79—21no—
157C2×C4×C79—21no—
158C2×C4×C79—21no—
159C2×C4×C79—44no—
160C2×C4×C79—42no—
161C2×C4×C79—41no—
162C2×C4×C79—41no—
163C2×C4×C79—41no—
164C2×C4×C79—48no—
165C2×C4×C79—42no—
166C2×C4×C79—41no—
167C2×C4×C79—42no—
168C2×C4×C79—44no—
169C2×C4×C79—42no—
170C2×C4×C79—41no—
171C2×C4×C79—41no—
172C2×C4×C79—41no—
173C2×C4×C79—42no—
174C2×C4×C79—41no—
175C2×C4×C79—41no—
176C2×C4×C79—41no—
177C2×C4×C79—81no—
178C2×C4×C79C22⋊C812yes—
179C2×C4×C79#9911yescharacteristic
180C2×C4×C79#9911yescharacteristic
181C79⋊C8—82no—
182C79⋊C8—82no—
183C22×C79—21no—
184C22×C79—21no—
185C22×C79—24no—
186C22×C79—21no—
187C22×C79—21no—
188C22×C79—22no—
189C22×C79—42no—
190C22×C79—41no—
191C22×C79—41no—
192C22×C79—42no—
193C22×C79—42no—
194C22×C79—42no—
195C22×C79—42no—
196C22×C79—44no—
197C22×C79—44no—
198C22×C79—44no—
199C22×C79—44no—
200C22×C79—42no—
201C22×C79—41no—
202C22×C79—41no—
203C22×C79—41no—
204C22×C79—42no—
205C22×C79—41no—
206C22×C79—42no—
207C22×C79—81no—
208C22×C79C2×C22⋊C811yescharacteristic
209C22×C79C2×#9911yescharacteristic
210C22×C79#41311yescharacteristic
211C4×C79—21no—
212C4×C79—22no—
213C4×C79—42no—
214C4×C79—42no—
215C4×C79—41no—
216C4×C79—41no—
217C4×C79—41no—
218C4×C79—81no—
219C4×C79AΣL(1,𝔽16)12yes—
220#11471—791nomaximal, Sylow
221C2×C79—21no—
222C2×C79—21no—
223C2×C79—44no—
224C2×C79—41no—
225C2×C79—41no—
226C2×C79—42no—
227C2×C79#145911yescharacteristic
228C2×C79C2×AΣL(1,𝔽16)11yescharacteristic
229C2×C79#212011yescharacteristic
230C4×#120D7911yescharacteristic
231#1460—791no—
232#1459—791no—
233#1326—792no—
234#1316—792no—
235C79#1147111yescharacteristic, complemented, Sylow
236C24×C4Dic7911yescharacteristic
237C2×#120Dic7911yescharacteristic
238C2×C4×D4—21no—
239C2×C4×D4C2×D7911yescharacteristic
240#431—1581no—
241C2×C22⋊C8—1581no—
242#386—21no—
243#386Dic7911yescharacteristic
244#385—21no—
245#385C2×D7911yescharacteristic
246#379C2×D7911yescharacteristic
247C4×C22⋊C4—21no—
248C4×C22⋊C4Dic7911yescharacteristic
249#324—1581no—
250#323—792no—
251#323—792no—
252#323—1581no—
253C25C2×Dic7911yescharacteristic
254C22×D4—21no—
255C22×D4C2×Dic7911yescharacteristic
256C23×C4—21no—
257C23×C4—21no—
258C23×C4—44no—
259C23×C4—41no—
260C23×C4—41no—
261C23×C4C2×Dic7911yescharacteristic
262C23×C4C2×Dic7911yescharacteristic
263C23×C4C22⋊D7911yescharacteristic
264C22×C8—3161no—
265#120—22no—
266#120C79⋊C814yescomplemented
267C4×D4—41no—
268C4×D4—41no—
269C4×D4—44no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22⋊D7911yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4C2×Dic7911yescharacteristic
277C2×C22⋊C4C2×Dic7911yescharacteristic
278C2×C22⋊C4C22⋊D7911yescharacteristic
279C2×C22⋊C4C22×D7911yescharacteristic
280C2×C42—21no—
281C2×C42C22⋊D7911yescharacteristic
282C4⋊C8—3161no—
283C22⋊C8—3161no—
284C22⋊C8—1582no—
285C22⋊C8—1582no—
286C4×C8—3161no—
287#95—21no—
288#95—21no—
289#95—21no—
290C24—21no—
291C24—21no—
292C24—42no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24C22⋊Dic7911yescharacteristic
297C24C2×C79⋊C812yescomplemented
298C24C22×Dic7911yescharacteristic
299C2×D4—24no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4C2×C79⋊C812yes—
305C2×D4#1192960812yes—
306C22×C4—24no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4C22⋊Dic7911yescharacteristic
329C22×C4C22⋊Dic7911yescharacteristic
330C22×C4C22⋊Dic7911yescharacteristic
331C22×C4C2×C22⋊D7911yescharacteristic
332C22×C4C2×C22⋊D7911yescharacteristic
333C2×C8—3162no—
334C2×C8—3162no—
335C2×C8—6322no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C22⋊C4—21no—
339C22⋊C4—24no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—41no—
346C22⋊C4C2×C79⋊C812yescomplemented
347C42—21no—
348C42—21no—
349C42—42no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—44no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—44no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—48no—
371C23—42no—
372C23—41no—
373C23—41no—
374C23C22⋊(C79⋊C8)12yes—
375C23C2×C22⋊Dic7911yescharacteristic
376C23C22×C79⋊C811yescharacteristic
377C23C2×#1192960811yescharacteristic
378D4—42no—
379D4—42no—
380D4—48no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—81no—
408C2×C4C22⋊(C79⋊C8)12yes—
409C2×C4#1457907611yescharacteristic
410C2×C4#1457907611yescharacteristic
411C8—6322no—
412C8—6322no—
413C22—21no—
414C22—21no—
415C22—24no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—42no—
420C22—41no—
421C22—41no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—42no—
431C22—41no—
432C22—41no—
433C22—42no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—81no—
438C22C2×C22⋊(C79⋊C8)11yescharacteristic
439C22C2×#1457907611yescharacteristic
440C22#1909854711yescharacteristic
441C4—21no—
442C4—22no—
443C4—42no—
444C4—41no—
445C4—42no—
446C4—41no—
447C4—41no—
448C4—81no—
449C4#1909835012yes—
450C2—21no—
451C2—21no—
452C2—44no—
453C2—41no—
454C2—41no—
455C2—42no—
456C2#2376171811yescharacteristic
457C2C2×#1909835011yescharacteristic
458C2#2376504111yescharacteristic
subgroupG/H
centerC2×C4#14579076
commutatorC23×C79C22×C8
FrattiniC2×C22⋊C4C22×D79
FittingC4×#120×C79C2
socleC22×C79C2×#99
prime radicalC2×#120×C79C4
cosocleC2×C22⋊C4×C79C23
hypercenterC4×#120D79
hypocenterC79#11471
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#120

Complements

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

NH ≅ G/Ntype
1. #23761721C2semidirect
2. #23761718C2semidirect
3. #23761348C2semidirect
4. #23761328C2semidirect
41. #120×C79C8semidirect
70. C24×C79C2×C8semidirect
116. C22⋊C4×C79C2×C8semidirect
235. C79#11471semidirect
266. #120C79⋊C8semidirect
297. C24C2×C79⋊C8semidirect
346. C22⋊C4C2×C79⋊C8semidirect

Maximal subgroups

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

subgroupindex
#237617212
#237617182
#237613482
#237613282
C4×#120×C792
#1147179