Finite Group Databasebeta

← #34197529

Subgroups — #34197529

Order 20224.

548 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 119 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
5C79×#2321C211yesmaximal, characteristic
6#19098616—21no—
7C2×C22⋊(C79⋊C8)—21no—
8#19098289—21no—
9#19098287—21no—
10#19098287C2212yes—
11#19098287C2212yes—
12C2×#120×C79C411yescharacteristic
13C2×C4×D4×C79C2211yescharacteristic
14C22×C22⋊C4×C79C411yescharacteristic
15#390×C79—21no—
16#388×C79—21no—
17#388×C79—21no—
18#386×C79—21no—
19#386×C79C411yescharacteristic
20#385×C79C2211yescharacteristic
21#379×C79C2211yescharacteristic
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×C79C2×C411yescharacteristic
33C23×C4×C79C2×C411yescharacteristic
34C23×C4×C79D411yescharacteristic
35#120×C79—22no—
36#120×C79C814yescomplemented
37C4×D4×C79—41no—
38C4×D4×C79—41no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79D411yescharacteristic
41C2×C22⋊C4×C79—21no—
42C2×C22⋊C4×C79—21no—
43C2×C22⋊C4×C79—21no—
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79—21no—
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79—41no—
48C2×C22⋊C4×C79—41no—
49C2×C22⋊C4×C79C2×C411yescharacteristic
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79C2311yescharacteristic
53C2×C42×C79—21no—
54C2×C42×C79D411yescharacteristic
55#95×C79—21no—
56#95×C79—21no—
57#95×C79—21no—
58C2×C79⋊C8—42no—
59C2×C79⋊C8—42no—
60C2×C79⋊C8—82no—
61C24×C79—21no—
62C24×C79—21no—
63C24×C79—41no—
64C24×C79—41no—
65C24×C79—81no—
66C24×C79—81no—
67C24×C79C22⋊C411yescharacteristic
68C24×C79C2×C812yescomplemented
69C24×C79C22×C411yescharacteristic
70C2×D4×C79—24no—
71C2×D4×C79—41no—
72C2×D4×C79—41no—
73C2×D4×C79—81no—
74C2×D4×C79—81no—
75C2×D4×C79C2×C812yes—
76C2×D4×C79M1612yes—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—21no—
83C22×C4×C79—21no—
84C22×C4×C79—41no—
85C22×C4×C79—81no—
86C22×C4×C79—81no—
87C22×C4×C79—81no—
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C22⋊C411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C22×C4×C79C2×D411yescharacteristic
93C4⋊C4×C79—41no—
94C4⋊C4×C79—81no—
95C22⋊C4×C79—21no—
96C22⋊C4×C79—24no—
97C22⋊C4×C79—41no—
98C22⋊C4×C79—41no—
99C22⋊C4×C79—42no—
100C22⋊C4×C79—41no—
101C22⋊C4×C79—41no—
102C22⋊C4×C79—41no—
103C22⋊C4×C79—81no—
104C22⋊C4×C79—81no—
105C22⋊C4×C79—81no—
106C22⋊C4×C79C2×C812yescomplemented
107C42×C79—21no—
108C42×C79—21no—
109C42×C79—81no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79—21no—
113C23×C79—21no—
114C23×C79—22no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—41no—
121C23×C79—41no—
122C23×C79—41no—
123C23×C79—42no—
124C23×C79—42no—
125C23×C79—41no—
126C23×C79—41no—
127C23×C79—82no—
128C23×C79—82no—
129C23×C79—82no—
130C23×C79—82no—
131C23×C79—84no—
132C23×C79C22⋊C812yes—
133C23×C79C2×C22⋊C411yescharacteristic
134C23×C79C22×C811yescharacteristic
135C23×C79C2×M1611yescharacteristic
136D4×C79—42no—
137D4×C79—42no—
138D4×C79—84no—
139C2×C4×C79—21no—
140C2×C4×C79—21no—
141C2×C4×C79—22no—
142C2×C4×C79—21no—
143C2×C4×C79—21no—
144C2×C4×C79—21no—
145C2×C4×C79—41no—
146C2×C4×C79—42no—
147C2×C4×C79—41no—
148C2×C4×C79—41no—
149C2×C4×C79—41no—
150C2×C4×C79—41no—
151C2×C4×C79—41no—
152C2×C4×C79—41no—
153C2×C4×C79—41no—
154C2×C4×C79—41no—
155C2×C4×C79—41no—
156C2×C4×C79—41no—
157C2×C4×C79—41no—
158C2×C4×C79—81no—
159C2×C4×C79—81no—
160C2×C4×C79—81no—
161C2×C4×C79—82no—
162C2×C4×C79C22⋊C812yes—
163C2×C4×C79#9911yescharacteristic
164C2×C4×C79#9911yescharacteristic
165C79⋊C8—82no—
166C79⋊C8—82no—
167C22×C79—21no—
168C22×C79—21no—
169C22×C79—21no—
170C22×C79—21no—
171C22×C79—22no—
172C22×C79—42no—
173C22×C79—41no—
174C22×C79—41no—
175C22×C79—42no—
176C22×C79—42no—
177C22×C79—42no—
178C22×C79—41no—
179C22×C79—41no—
180C22×C79—41no—
181C22×C79—41no—
182C22×C79—42no—
183C22×C79—81no—
184C22×C79—81no—
185C22×C79—81no—
186C22×C79—84no—
187C22×C79—84no—
188C22×C79—81no—
189C22×C79—81no—
190C22×C79C2×C22⋊C811yescharacteristic
191C22×C79C2×#9911yescharacteristic
192C22×C79#41311yescharacteristic
193C4×C79—21no—
194C4×C79—21no—
195C4×C79—22no—
196C4×C79—42no—
197C4×C79—41no—
198C4×C79—41no—
199C4×C79—41no—
200C4×C79—81no—
201C4×C79—81no—
202#11473—791nomaximal, Sylow
203C2×C79—21no—
204C2×C79—21no—
205C2×C79—41no—
206C2×C79—41no—
207C2×C79—42no—
208C2×C79—82no—
209C2×C79#145911yescharacteristic
210C2×C79#212111yescharacteristic
211C2×C79#212211yescharacteristic
212#2321D7911yescharacteristic
213#1460—791no—
214#1459—791no—
215#1326—792no—
216#1316—792no—
217C79#1147311yescharacteristic, complemented, Sylow
218C2×#120Dic7911yescharacteristic
219C2×C4×D4C2×D7911yescharacteristic
220C22×C22⋊C4Dic7911yescharacteristic
221#431—1581no—
222C2×C22⋊C8—1581no—
223#390—21no—
224#388—21no—
225#388—21no—
226#386—21no—
227#386Dic7911yescharacteristic
228#385C2×D7911yescharacteristic
229#379C2×D7911yescharacteristic
230C4×C22⋊C4Dic7911yescharacteristic
231#324—1581no—
232#323—792no—
233#323—792no—
234#323—1581no—
235C25C2×Dic7911yescharacteristic
236C22×D4—21no—
237C22×D4C2×Dic7911yescharacteristic
238C23×C4C2×Dic7911yescharacteristic
239C23×C4C2×Dic7911yescharacteristic
240C23×C4C22⋊D7911yescharacteristic
241C22×C8—3161no—
242#120—22no—
243#120C79⋊C814yescomplemented
244C4×D4—41no—
245C4×D4—41no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4C22⋊D7911yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—41no—
255C2×C22⋊C4—41no—
256C2×C22⋊C4C2×Dic7911yescharacteristic
257C2×C22⋊C4C2×Dic7911yescharacteristic
258C2×C22⋊C4C22⋊D7911yescharacteristic
259C2×C22⋊C4C22×D7911yescharacteristic
260C2×C42—21no—
261C2×C42C22⋊D7911yescharacteristic
262C4⋊C8—3161no—
263C22⋊C8—3161no—
264C22⋊C8—1582no—
265C22⋊C8—1582no—
266C4×C8—3161no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C24—21no—
272C24—41no—
273C24—41no—
274C24—81no—
275C24—81no—
276C24C22⋊Dic7911yescharacteristic
277C24C2×C79⋊C812yescomplemented
278C24C22×Dic7911yescharacteristic
279C2×D4—24no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—81no—
283C2×D4—81no—
284C2×D4C2×C79⋊C812yes—
285C2×D4#1192960812yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4C22⋊Dic7911yescharacteristic
298C22×C4C22⋊Dic7911yescharacteristic
299C22×C4C22⋊Dic7911yescharacteristic
300C22×C4C2×C22⋊D7911yescharacteristic
301C22×C4C2×C22⋊D7911yescharacteristic
302C2×C8—3162no—
303C2×C8—3162no—
304C2×C8—6322no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C22⋊C4—21no—
308C22⋊C4—24no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—81no—
316C22⋊C4—81no—
317C22⋊C4—81no—
318C22⋊C4C2×C79⋊C812yescomplemented
319C42—21no—
320C42—21no—
321C42—81no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23—84no—
344C23C22⋊(C79⋊C8)12yes—
345C23C2×C22⋊Dic7911yescharacteristic
346C23C22×C79⋊C811yescharacteristic
347C23C2×#1192960811yescharacteristic
348D4—42no—
349D4—42no—
350D4—84no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4C22⋊(C79⋊C8)12yes—
375C2×C4#1457907611yescharacteristic
376C2×C4#1457907611yescharacteristic
377C8—6322no—
378C8—6322no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—84no—
399C22—84no—
400C22—81no—
401C22—81no—
402C22C2×C22⋊(C79⋊C8)11yescharacteristic
403C22C2×#1457907611yescharacteristic
404C22#1909854711yescharacteristic
405C4—22no—
406C4—21no—
407C4—21no—
408C4—42no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2—42no—
419C2—82no—
420C2#2376171811yescharacteristic
421C2#2376504611yescharacteristic
422C2#2376505011yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC23×C79C22×C8
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2321C2
socleC22×C79C2×#99
prime radicalC2×#120×C79C4
cosocleC2×C22⋊C4×C79C23
hypercenter#2321D79
hypocenterC79#11473
LCSG ▷ C23×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
1. #23761721C2semidirect
2. #23761718C2semidirect
3. #23761348C2semidirect
4. #23761328C2semidirect
36. #120×C79C8semidirect
68. C24×C79C2×C8semidirect
106. C22⋊C4×C79C2×C8semidirect
217. C79#11473semidirect
243. #120C79⋊C8semidirect
277. C24C2×C79⋊C8semidirect
318. 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
C79×#23212
#1147379