Finite Group Databasebeta

← #34198760

Subgroups — #34198760

Order 20224.

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

normal subgroups only

This group has 436 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#23763133C211yesmaximal, characteristic
2#23761734C211yesmaximal, characteristic
3#23761424C212yesmaximal
4#23761400C212yesmaximal
5C79×#2611C211yesmaximal, characteristic
6C2×#14579120—21no—
7#19098620—21no—
8#19098330—21no—
9#19098330C2212yes—
10#19098289—21no—
11#19098287C2212yes—
12C2×#125×C79C411yescharacteristic, complemented
13C2×#117×C79C2211yescharacteristic
14#400×C79C411yescharacteristic, complemented
15#399×C79C2211yescharacteristic
16#398×C79C411yescharacteristic, complemented
17#397×C79C2211yescharacteristic
18#389×C79—21no—
19#388×C79—21no—
20#387×C79—21no—
21#385×C79—21no—
22#376×C79C411yescharacteristic, complemented
23#14579120—24no—
24#14579120—42no—
25C2×C4⋊Dic79—41no—
26C2×C22⋊Dic79—22no—
27C2×C4×Dic79—41no—
28C4⋊(C79⋊C8)—41no—
29C22⋊(C79⋊C8)—22no—
30#14579072—41no—
31C23×C4×C79D411yescharacteristic
32#125×C79—41no—
33#125×C79—41no—
34#117×C79—41no—
35#117×C79—41no—
36C2×C4⋊C4×C79—21no—
37C2×C4⋊C4×C79—21no—
38C2×C4⋊C4×C79—21no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79C2×C411yescharacteristic
41C2×C4⋊C4×C79C2×C411yescharacteristic
42C2×C4⋊C4×C79C2×C411yescharacteristic
43C2×C4⋊C4×C79D411yescharacteristic
44C2×C22⋊C4×C79—21no—
45C2×C22⋊C4×C79—21no—
46C2×C22⋊C4×C79D411yescharacteristic
47C2×C22⋊C4×C79C2311yescharacteristic
48C2×C42×C79—21no—
49C2×C42×C79C2×C411yescharacteristic
50C2×C42×C79D411yescharacteristic
51#95×C79—21no—
52#95×C79—21no—
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56#95×C79C2×C411yescharacteristic
57#95×C79C2×C411yescharacteristic
58C22×Dic79—42no—
59C2×C79⋊C8—42no—
60C4⋊Dic79—44no—
61C22⋊Dic79—24no—
62C22⋊Dic79—42no—
63C4×Dic79—44no—
64C24×C79C2×D411yescharacteristic
65C22×C4×C79—21no—
66C22×C4×C79—21no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—22no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79C22⋊C411yescharacteristic
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C4⋊C4×C79—41no—
81C4⋊C4×C79—41no—
82C4⋊C4×C79—42no—
83C4⋊C4×C79—42no—
84C4⋊C4×C79—41no—
85C4⋊C4×C79—41no—
86C4⋊C4×C79—41no—
87C4⋊C4×C79—41no—
88C4⋊C4×C79—42no—
89C22⋊C4×C79—41no—
90C22⋊C4×C79—41no—
91C22⋊C4×C79—42no—
92C22⋊C4×C79—41no—
93C22⋊C4×C79—41no—
94C42×C79—21no—
95C42×C79—21no—
96C42×C79—21no—
97C42×C79—21no—
98C42×C79—42no—
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—22no—
115C2×C4×C79—21no—
116C2×C4×C79—21no—
117C2×C4×C79—21no—
118C2×C4×C79—21no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79—41no—
131C2×C4×C79—41no—
132C2×C4×C79—41no—
133C2×C4×C79—41no—
134C2×C4×C79#9911yescharacteristic
135C2×C4×C79#9911yescharacteristic
136C2×C4×C79#10412yes—
137C2×Dic79—44no—
138C2×Dic79—82no—
139C2×Dic79—82no—
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—42no—
157C4×C79—41no—
158C4×C79—41no—
159C4×C79—41no—
160C4×C79—41no—
161C4×C79—81no—
162C4×C79—81no—
163C4×C79—81no—
164Dic79—84no—
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#2611D7911yescharacteristic
174#1740—791no—
175#1464—791no—
176#1352—792no—
177#1344—792no—
178C79#1176011yescharacteristic, complemented, Sylow
179C2×#125Dic7911yescharacteristic, complemented
180C2×#117—1581no—
181C2×#117C2×D7911yescharacteristic
182#432—1581no—
183#400Dic7911yescharacteristic, complemented
184#399C2×D7911yescharacteristic
185#398Dic7911yescharacteristic, complemented
186#397C2×D7911yescharacteristic
187#389—21no—
188#388—21no—
189#387—21no—
190#385—21no—
191#376Dic7911yescharacteristic, complemented
192#342—792no—
193#342—1581no—
194#324—1581no—
195#323—792no—
196C23×C4C22⋊D7911yescharacteristic
197#125—41no—
198#125—41no—
199#117—41no—
200#117—41no—
201#117—3162no—
202#117—1584no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—3161no—
208C2×C4⋊C4C2×Dic7911yescharacteristic
209C2×C4⋊C4C2×Dic7911yescharacteristic
210C2×C4⋊C4C2×Dic7911yescharacteristic
211C2×C4⋊C4C22⋊D7911yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—1582no—
215C2×C22⋊C4C22⋊D7911yescharacteristic
216C2×C22⋊C4C22×D7911yescharacteristic
217C2×C42—21no—
218C2×C42—3161no—
219C2×C42C2×Dic7911yescharacteristic
220C2×C42C22⋊D7911yescharacteristic
221C4⋊C8—3161no—
222C22⋊C8—1582no—
223#97—3161no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×Dic7911yescharacteristic
230#95C2×Dic7911yescharacteristic
231C24C2×C22⋊D7911yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—3162no—
242C22×C4C22⋊Dic7911yescharacteristic
243C22×C4C22⋊Dic7911yescharacteristic
244C22×C4C22⋊Dic7911yescharacteristic
245C22×C4C22⋊Dic7911yescharacteristic
246C22×C4C22×Dic7911yescharacteristic
247C22×C4C2×C22⋊D7911yescharacteristic
248C2×C8—3162no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—3164no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—42no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—3162no—
265C22⋊C4—1584no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—42no—
271C42—3164no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×C22⋊Dic7911yescharacteristic
277C23#1457918611yescharacteristic
278C23#1457919111yescharacteristic
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—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—3164no—
308C2×C4—6322no—
309C2×C4—6322no—
310C2×C4#1457907611yescharacteristic
311C2×C4#1457907611yescharacteristic
312C2×C4#1457908812yes—
313C8—6322no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—81no—
323C22C2×#1457907611yescharacteristic
324C22#1909856411yescharacteristic
325C22C2×#1457908811yescharacteristic
326C4—21no—
327C4—21no—
328C4—22no—
329C4—42no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—6324no—
338C2—21no—
339C2—21no—
340C2—41no—
341C2—41no—
342C2#2376190411yescharacteristic
343C2#2376505411yescharacteristic
344C2#2376506511yescharacteristic
subgroupG/H
centerC22C2×#14579076
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#2611C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#2611D79
hypocenterC79#11760
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C79C4semidirect
14. #400×C79C4semidirect
16. #398×C79C4semidirect
22. #376×C79C4semidirect
178. C79#11760semidirect
179. C2×#125Dic79semidirect
183. #400Dic79semidirect
185. #398Dic79semidirect
191. #376Dic79semidirect

Maximal subgroups

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

subgroupindex
#237631332
#237617342
#237614242
#237614002
C79×#26112
#1176079