Finite Group Databasebeta

← #34197878

Subgroups — #34197878

Order 20224.

534 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 91 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#23761741C211yesmaximal, characteristic, complemented
2#23761718C211yesmaximal, characteristic, complemented
3#23761330C212yesmaximal, complemented
4#23761328C212yesmaximal, complemented
5C79×#2534C211yesmaximal, characteristic
6#19098620—21no—
7C2×C22⋊(C79⋊C8)—21no—
8#19098289—21no—
9#19098287—21no—
10#19098287C2212yes—
11#19098287C2212yes—
12C2×#123×C79C411yescharacteristic
13C2×#120×C79C411yescharacteristic
14C2×C4×D4×C79C2211yescharacteristic
15C22×C22⋊C4×C79C411yescharacteristic
16#396×C79—21no—
17#394×C79—21no—
18#388×C79—21no—
19#386×C79—21no—
20#381×C79C2211yescharacteristic
21#380×C79C411yescharacteristic
22#379×C79C2211yescharacteristic
23C22×C79⋊C8—41no—
24C4⋊(C79⋊C8)—41no—
25C22⋊(C79⋊C8)—22no—
26C22⋊(C79⋊C8)—22no—
27C22⋊(C79⋊C8)—41no—
28#14579072—41no—
29C25×C79C2×C411yescharacteristic
30C22×D4×C79—21no—
31C22×D4×C79C2×C411yescharacteristic
32C23×C4×C79C2×C411yescharacteristic
33C23×C4×C79C2×C411yescharacteristic
34C23×C4×C79D411yescharacteristic
35#123×C79—42no—
36#120×C79—42no—
37C4×D4×C79—41no—
38C4×D4×C79—41no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79D411yescharacteristic
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—21no—
48C2×C22⋊C4×C79—41no—
49C2×C22⋊C4×C79—41no—
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79C2×C411yescharacteristic
52C2×C22⋊C4×C79D411yescharacteristic
53C2×C22⋊C4×C79C2311yescharacteristic
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—22no—
63C24×C79—41no—
64C24×C79—42no—
65C24×C79—41no—
66C24×C79—81no—
67C24×C79C22⋊C411yescharacteristic
68C24×C79C22×C411yescharacteristic
69C2×D4×C79—22no—
70C2×D4×C79—22no—
71C2×D4×C79—41no—
72C2×D4×C79—42no—
73C2×D4×C79—42no—
74C2×D4×C79—81no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—21no—
80C22×C4×C79—21no—
81C22×C4×C79—21no—
82C22×C4×C79—22no—
83C22×C4×C79—41no—
84C22×C4×C79—41no—
85C22×C4×C79—81no—
86C22×C4×C79—81no—
87C22×C4×C79C22⋊C411yescharacteristic
88C22×C4×C79C22⋊C411yescharacteristic
89C22×C4×C79C22⋊C411yescharacteristic
90C22×C4×C79C2×D411yescharacteristic
91C22×C4×C79C2×D411yescharacteristic
92C4⋊C4×C79—41no—
93C4⋊C4×C79—42no—
94C4⋊C4×C79—81no—
95C22⋊C4×C79—41no—
96C22⋊C4×C79—41no—
97C22⋊C4×C79—42no—
98C22⋊C4×C79—42no—
99C22⋊C4×C79—42no—
100C22⋊C4×C79—42no—
101C22⋊C4×C79—41no—
102C22⋊C4×C79—41no—
103C22⋊C4×C79—41no—
104C22⋊C4×C79—41no—
105C22⋊C4×C79—81no—
106C42×C79—21no—
107C42×C79—21no—
108C23×C79—21no—
109C23×C79—21no—
110C23×C79—21no—
111C23×C79—21no—
112C23×C79—22no—
113C23×C79—41no—
114C23×C79—41no—
115C23×C79—41no—
116C23×C79—41no—
117C23×C79—41no—
118C23×C79—41no—
119C23×C79—41no—
120C23×C79—42no—
121C23×C79—42no—
122C23×C79—41no—
123C23×C79—41no—
124C23×C79—41no—
125C23×C79—82no—
126C23×C79—84no—
127C23×C79—84no—
128C23×C79—82no—
129C23×C79#10012yes—
130C23×C79C2×C22⋊C411yescharacteristic
131C23×C79#13112yes—
132D4×C79—44no—
133D4×C79—84no—
134C2×C4×C79—21no—
135C2×C4×C79—21no—
136C2×C4×C79—21no—
137C2×C4×C79—21no—
138C2×C4×C79—21no—
139C2×C4×C79—21no—
140C2×C4×C79—41no—
141C2×C4×C79—42no—
142C2×C4×C79—41no—
143C2×C4×C79—41no—
144C2×C4×C79—41no—
145C2×C4×C79—41no—
146C2×C4×C79—41no—
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—81no—
155C2×C4×C79—81no—
156C2×C4×C79—81no—
157C2×C4×C79—82no—
158C2×C4×C79#9911yescharacteristic
159C2×C4×C79#9911yescharacteristic
160C79⋊C8—82no—
161C79⋊C8—82no—
162C22×C79—21no—
163C22×C79—21no—
164C22×C79—22no—
165C22×C79—22no—
166C22×C79—42no—
167C22×C79—42no—
168C22×C79—42no—
169C22×C79—42no—
170C22×C79—42no—
171C22×C79—41no—
172C22×C79—41no—
173C22×C79—41no—
174C22×C79—41no—
175C22×C79—42no—
176C22×C79—81no—
177C22×C79—81no—
178C22×C79—81no—
179C22×C79—84no—
180C22×C79—84no—
181C22×C79—81no—
182C22×C79—81no—
183C22×C79#40811yescharacteristic
184C22×C79C2×#9911yescharacteristic
185C22×C79C2×#10011yescharacteristic
186C4×C79—21no—
187C4×C79—21no—
188C4×C79—41no—
189C4×C79—41no—
190C4×C79—41no—
191C4×C79—41no—
192C4×C79—81no—
193C4×C79—81no—
194C4×C79—81no—
195#11560—791nomaximal, Sylow
196C2×C79—21no—
197C2×C79—21no—
198C2×C79—42no—
199C2×C79—42no—
200C2×C79—82no—
201C2×C79#147111yescharacteristic
202C2×C79#212011yescharacteristic
203C2×C79#212111yescharacteristic
204#2534D7911yescharacteristic
205#1466—791no—
206#1459—791no—
207#1317—792no—
208#1316—792no—
209C79#1156011yescharacteristic, complemented, Sylow
210C2×#123Dic7911yescharacteristic
211C2×#120Dic7911yescharacteristic
212C2×C4×D4C2×D7911yescharacteristic
213C22×C22⋊C4Dic7911yescharacteristic
214#432—1581no—
215C2×C22⋊C8—1581no—
216#396—21no—
217#394—21no—
218#388—21no—
219#386—21no—
220#381C2×D7911yescharacteristic
221#380Dic7911yescharacteristic
222#379C2×D7911yescharacteristic
223#324—1581no—
224#323—792no—
225#323—792no—
226#323—1581no—
227C25C2×Dic7911yescharacteristic
228C22×D4—21no—
229C22×D4C2×Dic7911yescharacteristic
230C23×C4C2×Dic7911yescharacteristic
231C23×C4C2×Dic7911yescharacteristic
232C23×C4C22⋊D7911yescharacteristic
233C22×C8—3161no—
234#123—42no—
235#120—42no—
236C4×D4—41no—
237C4×D4—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C22⋊D7911yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—41no—
248C2×C22⋊C4—41no—
249C2×C22⋊C4C2×Dic7911yescharacteristic
250C2×C22⋊C4C2×Dic7911yescharacteristic
251C2×C22⋊C4C22⋊D7911yescharacteristic
252C2×C22⋊C4C22×D7911yescharacteristic
253C2×C42C22⋊D7911yescharacteristic
254C4⋊C8—3161no—
255C22⋊C8—3161no—
256C22⋊C8—1582no—
257C22⋊C8—1582no—
258#97—3161no—
259#95—21no—
260#95—21no—
261#95—21no—
262C24—21no—
263C24—22no—
264C24—41no—
265C24—42no—
266C24—41no—
267C24—81no—
268C24C22⋊Dic7911yescharacteristic
269C24C22×Dic7911yescharacteristic
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—81no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4C22⋊Dic7911yescharacteristic
289C22×C4C22⋊Dic7911yescharacteristic
290C22×C4C22⋊Dic7911yescharacteristic
291C22×C4C2×C22⋊D7911yescharacteristic
292C22×C4C2×C22⋊D7911yescharacteristic
293C2×C8—3162no—
294C2×C8—3162no—
295C2×C8—6322no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—81no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—81no—
310C42—21no—
311C42—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—22no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—82no—
330C23—84no—
331C23—84no—
332C23—82no—
333C23#1457907812yes—
334C23C2×C22⋊Dic7911yescharacteristic
335C23#1457916812yes—
336D4—44no—
337D4—84no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4#1457907611yescharacteristic
363C2×C4#1457907611yescharacteristic
364C8—6322no—
365C8—6322no—
366C22—21no—
367C22—21no—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—84no—
384C22—84no—
385C22—81no—
386C22—81no—
387C22#1909853011yescharacteristic
388C22C2×#1457907611yescharacteristic
389C22C2×#1457907811yescharacteristic
390C4—21no—
391C4—21no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—42no—
402C2—42no—
403C2—82no—
404C2#2376176011yescharacteristic
405C2#2376504111yescharacteristic
406C2#2376504611yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC24×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2534C2
socleC22×C79C2×#99
prime radicalC2×#120×C79C4
cosocleC2×C22⋊C4×C79C23
hypercenter#2534D79
hypocenterC79#11560
LCSG ▷ C24×C79 ▷ C22×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. #23761741C2semidirect
2. #23761718C2semidirect
3. #23761330C2semidirect
4. #23761328C2semidirect
209. C79#11560semidirect

Maximal subgroups

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

subgroupindex
#237617412
#237617182
#237613302
#237613282
C79×#25342
#1156079