Finite Group Databasebeta

← #34198758

Subgroups — #34198758

Order 20224.

436 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 83 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#23769098C211yesmaximal, characteristic
2#23761734C211yesmaximal, characteristic
3#23761423C212yesmaximal
4#23761399C212yesmaximal
5C79×#1740C211yesmaximal, characteristic
6C2×#14579153C411yescharacteristic, complemented
7#19098620—21no—
8#19098502C411yescharacteristic, complemented
9#19098499C2211yescharacteristic
10#19098493C411yescharacteristic, complemented
11#19098488C2211yescharacteristic
12#19098458—21no—
13#19098457—21no—
14#19098447—21no—
15#19098436—21no—
16#19098409C411yescharacteristic, complemented
17#19098289—21no—
18#19098287C2212yes—
19C2×#117×C79—21no—
20C2×#117×C79C2211yescharacteristic
21#342×C79—21no—
22#342×C79C2212yes—
23#14579153—41no—
24#14579153—41no—
25C2×C4⋊Dic79C2×C411yescharacteristic
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×C4×Dic79—21no—
34C4⋊(C79⋊C8)—41no—
35C22⋊(C79⋊C8)—22no—
36#14579072—41no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45#117×C79—24no—
46#117×C79—41no—
47#117×C79—41no—
48#117×C79—42no—
49C2×C4⋊C4×C79—41no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79—22no—
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C22⋊C4×C79C2311yescharacteristic
55C2×C42×C79—41no—
56C2×C42×C79C2×C411yescharacteristic
57C2×C42×C79D411yescharacteristic
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C2×C79⋊C8—42no—
63C4⋊Dic79—41no—
64C4⋊Dic79—41no—
65Dic79⋊C4—42no—
66Dic79⋊C4—42no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69Dic79⋊C4—42no—
70C22⋊Dic79—42no—
71C22⋊Dic79—41no—
72C22⋊Dic79—41no—
73C4×Dic79—42no—
74C24×C79C2×D411yescharacteristic
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—22no—
80C22×C4×C79—42no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22⋊C411yescharacteristic
84C22×C4×C79C22⋊C411yescharacteristic
85C22×C4×C79C22×C411yescharacteristic
86C22×C4×C79C2×D411yescharacteristic
87C4⋊C4×C79—41no—
88C4⋊C4×C79—41no—
89C4⋊C4×C79—44no—
90C22⋊C4×C79—24no—
91C22⋊C4×C79—41no—
92C22⋊C4×C79—41no—
93C22⋊C4×C79—42no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—44no—
99C23×C79—21no—
100C23×C79—21no—
101C23×C79—41no—
102C23×C79—41no—
103C23×C79C2×C22⋊C411yescharacteristic
104C23×C79AGL(1,ℤ/8)11yescharacteristic
105C23×C79#13711yescharacteristic
106C2×C4×C79—22no—
107C2×C4×C79—22no—
108C2×C4×C79—22no—
109C2×C4×C79—22no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—21no—
114C2×C4×C79—21no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—44no—
123C2×C4×C79—82no—
124C2×C4×C79—82no—
125C2×C4×C79#9911yescharacteristic
126C2×C4×C79#9911yescharacteristic
127C2×C4×C79#10412yes—
128C2×Dic79—22no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C2×Dic79—41no—
133C2×Dic79—41no—
134C2×Dic79—41no—
135C2×Dic79—41no—
136C2×Dic79—41no—
137C2×Dic79—41no—
138C2×Dic79—41no—
139C2×Dic79—41no—
140C79⋊C8—82no—
141C22×C79—21no—
142C22×C79—21no—
143C22×C79—21no—
144C22×C79—21no—
145C22×C79—41no—
146C22×C79—41no—
147C22×C79—41no—
148C22×C79—41no—
149C22×C79—81no—
150C22×C79C2×#9911yescharacteristic
151C22×C79#41711yescharacteristic
152C22×C79C2×#10411yescharacteristic
153C4×C79—21no—
154C4×C79—21no—
155C4×C79—22no—
156C4×C79—41no—
157C4×C79—41no—
158C4×C79—41no—
159C4×C79—41no—
160C4×C79—84no—
161Dic79—42no—
162Dic79—81no—
163Dic79—81no—
164Dic79—81no—
165#11760—791nomaximal, Sylow
166C2×C79—21no—
167C2×C79—21no—
168C2×C79—41no—
169C2×C79—41no—
170C2×C79#150411yescharacteristic
171C2×C79#212311yescharacteristic
172C2×C79#212611yescharacteristic
173#2611—791no—
174#1740D7911yescharacteristic
175#1464—791no—
176#1352—792no—
177#1344—792no—
178C79#1176011yescharacteristic, complemented, Sylow
179C2×#125—791no—
180C2×#117—21no—
181C2×#117C2×D7911yescharacteristic
182#432—1581no—
183#400—791no—
184#399—791no—
185#398—791no—
186#397—791no—
187#389—1581no—
188#388—1581no—
189#387—1581no—
190#385—1581no—
191#376—791no—
192#342—21no—
193#342C2×D7912yes—
194#324—1581no—
195#323—792no—
196C23×C4C22⋊D7911yescharacteristic
197#125—3161no—
198#125—3161no—
199#117—24no—
200#117—41no—
201#117—41no—
202#117—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—791no—
205C2×C4⋊C4—791no—
206C2×C4⋊C4—1581no—
207C2×C4⋊C4—1581no—
208C2×C4⋊C4—1581no—
209C2×C4⋊C4—1581no—
210C2×C4⋊C4C4×D7911yescharacteristic
211C2×C4⋊C4D31611yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1581no—
214C2×C22⋊C4—1581no—
215C2×C22⋊C4C22⋊D7911yescharacteristic
216C2×C22⋊C4C22×D7911yescharacteristic
217C2×C42—41no—
218C2×C42—1581no—
219C2×C42C4×D7911yescharacteristic
220C2×C42D31611yescharacteristic
221C4⋊C8—3161no—
222C22⋊C8—1582no—
223#97—3161no—
224#95—791no—
225#95—791no—
226#95—1581no—
227#95—1581no—
228#95—1581no—
229#95—1581no—
230#95—1581no—
231C24C2×C22⋊D7911yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—1581no—
239C22×C4—1581no—
240C22×C4—1581no—
241C22×C4—1581no—
242C22×C4(C2×D79)⋊C411yescharacteristic
243C22×C4(C2×D79)⋊C411yescharacteristic
244C22×C4(C2×D79)⋊C411yescharacteristic
245C22×C4(C2×D79)⋊C411yescharacteristic
246C22×C4C2×C4×D7911yescharacteristic
247C22×C4C2×D31611yescharacteristic
248C2×C8—3162no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—44no—
252C4⋊C4—3161no—
253C4⋊C4—3161no—
254C4⋊C4—3162no—
255C4⋊C4—3162no—
256C4⋊C4—3161no—
257C4⋊C4—3161no—
258C4⋊C4—3162no—
259C22⋊C4—24no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—3161no—
264C22⋊C4—3161no—
265C22⋊C4—3162no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—44no—
271C42—3162no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×(C2×D79)⋊C411yescharacteristic
277C23#1457918711yescharacteristic
278C23#1457919211yescharacteristic
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—3161no—
299C2×C4—3161no—
300C2×C4—3161no—
301C2×C4—3161no—
302C2×C4—3161no—
303C2×C4—3161no—
304C2×C4—3161no—
305C2×C4—3161no—
306C2×C4—3161no—
307C2×C4—3161no—
308C2×C4—3161no—
309C2×C4—1582no—
310C2×C4#1457907511yescharacteristic
311C2×C4#1457907511yescharacteristic
312C2×C4#1457908912yes—
313C8—6322no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1457907511yescharacteristic
324C22#1909856211yescharacteristic
325C22C2×#1457908911yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—84no—
334C4—3162no—
335C4—6321no—
336C4—6321no—
337C4—6321no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#2376190211yescharacteristic
343C2#2376505311yescharacteristic
344C2#2376506311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1740C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1740D79
hypocenterC79#11760
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1740

Complements

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

NH ≅ G/Ntype
6. C2×#14579153C4semidirect
8. #19098502C4semidirect
10. #19098493C4semidirect
16. #19098409C4semidirect
178. C79#11760semidirect

Maximal subgroups

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

subgroupindex
#237690982
#237617342
#237614232
#237613992
C79×#17402
#1176079