Finite Group Databasebeta

← #34199143

Subgroups — #34199143

Order 20224.

390 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 79 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#23768977C211yesmaximal, characteristic
2#23763477C211yesmaximal, characteristic
3#23761422C212yesmaximal
4#23761401C212yesmaximal
5C79×#1458C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic, complemented
7#19098493C411yescharacteristic, complemented
8#19098488C2211yescharacteristic
9#19098483C411yescharacteristic, complemented
10#19098475C2211yescharacteristic
11#19098451—21no—
12#19098447—21no—
13#19098444—21no—
14#19098436—21no—
15#19098431C411yescharacteristic, complemented
16#19098426—21no—
17#19098330—21no—
18#19098330C2212yes—
19C2×#117×C79C2211yescharacteristic
20#431×C79—21no—
21#324×C79—21no—
22#323×C79C2212yes—
23C22×Dic158C2×C411yescharacteristic
24C4×Dic158—41no—
25C4×Dic158—41no—
26C2×C4⋊Dic79C2×C411yescharacteristic
27C2×Dic79⋊C4—21no—
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×C22⋊Dic79—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—22no—
33C2×C4×Dic79—21no—
34C79⋊#95—21no—
35C79⋊#95—21no—
36C79⋊#95—21no—
37C79⋊#95—21no—
38C79⋊#95—21no—
39C79⋊#95—21no—
40C79⋊#95—41no—
41C79⋊#95—41no—
42C79⋊#95C2×C411yescharacteristic
43C79⋊#95C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45#117×C79—41no—
46#117×C79—41no—
47C2×C4⋊C4×C79C2×C411yescharacteristic
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C22⋊C4×C79C2311yescharacteristic
51C2×C42×C79C2×C411yescharacteristic
52C2×C42×C79D411yescharacteristic
53C4⋊C8×C79—41no—
54C22⋊C8×C79—22no—
55C4×C8×C79—41no—
56C2×Dic158—41no—
57C2×Dic158—42no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—42no—
63C4⋊Dic79—41no—
64Dic79⋊C4—42no—
65Dic79⋊C4—42no—
66Dic79⋊C4—41no—
67Dic79⋊C4—41no—
68C22⋊Dic79—42no—
69C22⋊Dic79—42no—
70C22⋊Dic79—42no—
71C22⋊Dic79—42no—
72C4×Dic79—42no—
73C24×C79C2×D411yescharacteristic
74C22×C4×C79—22no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—41no—
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22×C411yescharacteristic
83C22×C4×C79C2×D411yescharacteristic
84C2×C8×C79—42no—
85C4⋊C4×C79—41no—
86C4⋊C4×C79—41no—
87C22⋊C4×C79—41no—
88C22⋊C4×C79—41no—
89C42×C79—21no—
90C42×C79—21no—
91C42×C79—21no—
92C42×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—41no—
96C23×C79—41no—
97C23×C79C2×C22⋊C411yescharacteristic
98C23×C79#13512yes—
99C2×C4×C79—21no—
100C2×C4×C79—21no—
101C2×C4×C79—21no—
102C2×C4×C79—21no—
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—42no—
109C2×C4×C79—42no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79#9911yescharacteristic
115C2×C4×C79#9911yescharacteristic
116C2×Dic79—22no—
117C2×Dic79—41no—
118C2×Dic79—41no—
119C2×Dic79—41no—
120C2×Dic79—41no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—82no—
129C2×Dic79—82no—
130C2×Dic79—82no—
131Dic158—42no—
132Dic158—42no—
133C8×C79—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—22no—
137C22×C79—42no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—81no—
141C22×C79C2×#9911yescharacteristic
142C22×C79#41611yescharacteristic
143C22×C79#42111yescharacteristic
144C4×C79—21no—
145C4×C79—21no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151Dic79—42no—
152Dic79—81no—
153Dic79—81no—
154Dic79—81no—
155Dic79—162no—
156#11842—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—21no—
159C2×C79—42no—
160C2×C79#151611yescharacteristic
161C2×C79#212011yescharacteristic
162C2×C79#212911yescharacteristic
163#2599—791no—
164#1820—791no—
165#1458D7911yescharacteristic
166#1351—792no—
167#1344—792no—
168C79#1184211yescharacteristic, complemented, Sylow
169C2×C4×Q8—791no—
170C2×#117C2×D7911yescharacteristic
171#431—21no—
172#398—791no—
173#397—791no—
174#395—791no—
175#393—791no—
176#388—1581no—
177#387—1581no—
178#387—1581no—
179#385—1581no—
180#383—791no—
181#381—1581no—
182#342—792no—
183#342—1581no—
184#324—21no—
185#323C2×D7912yes—
186C22×Q8—791no—
187C23×C4D31611yescharacteristic
188C4×Q8—3161no—
189C4×Q8—3161no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—791no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4C4×D7911yescharacteristic
197C2×C4⋊C4C22⋊D7911yescharacteristic
198C2×C22⋊C4—1582no—
199C2×C22⋊C4—1581no—
200C2×C22⋊C4—1581no—
201C2×C22⋊C4D31611yescharacteristic
202C2×C22⋊C4C22×D7911yescharacteristic
203C2×C42—1581no—
204C2×C42C4×D7911yescharacteristic
205C2×C42C22⋊D7911yescharacteristic
206C4⋊C8—41no—
207C22⋊C8—22no—
208C4×C8—41no—
209#95—3161no—
210#95—3161no—
211#95—791no—
212#95—791no—
213#95—1581no—
214#95—1581no—
215#95—1581no—
216#95—1581no—
217#95—1581no—
218#95—1581no—
219C24C2×D31611yescharacteristic
220C2×Q8—3161no—
221C2×Q8—3162no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—3162no—
227C22×C4—1581no—
228C22×C4—1581no—
229C22×C4—1581no—
230C22×C4—1581no—
231C22×C4(C2×D79)⋊C411yescharacteristic
232C22×C4(C2×D79)⋊C411yescharacteristic
233C22×C4(C2×D79)⋊C411yescharacteristic
234C22×C4(C2×D79)⋊C411yescharacteristic
235C22×C4C2×C4×D7911yescharacteristic
236C22×C4C2×C22⋊D7911yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—3161no—
241C4⋊C4—3162no—
242C4⋊C4—3162no—
243C4⋊C4—3161no—
244C4⋊C4—3161no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—3162no—
248C22⋊C4—3162no—
249C22⋊C4—3162no—
250C22⋊C4—3162no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—3162no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×(C2×D79)⋊C411yescharacteristic
261C23#1457918312yes—
262Q8—3162no—
263Q8—3162no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—3161no—
283C2×C4—3161no—
284C2×C4—3161no—
285C2×C4—3161no—
286C2×C4—3161no—
287C2×C4—3161no—
288C2×C4—3161no—
289C2×C4—3161no—
290C2×C4—6322no—
291C2×C4—6322no—
292C2×C4—6322no—
293C2×C4—1582no—
294C2×C4#1457907511yescharacteristic
295C2×C4#1457907511yescharacteristic
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1457907511yescharacteristic
305C22#1909856011yescharacteristic
306C22#1909857911yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—3162no—
315C4—6321no—
316C4—6321no—
317C4—6321no—
318C4—12642no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2376195711yescharacteristic
323C2#2376504211yescharacteristic
324C2#2376507411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C22⋊C4C22×D79
FittingC79×#1458C2
socleC22×C79C2×#99
prime radicalC24×C79C2×D4
cosocleC2×C22⋊C4×C79C23
hypercenter#1458D79
hypocenterC79#11842
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic158C4semidirect
7. #19098493C4semidirect
9. #19098483C4semidirect
15. #19098431C4semidirect
168. C79#11842semidirect

Maximal subgroups

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

subgroupindex
#237689772
#237634772
#237614222
#237614012
C79×#14582
#1184279