Finite Group Databasebeta

← #34185619

Subgroups — #34185619

Order 20224.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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#23762001C212yesmaximal, complemented
3#23761734C211yesmaximal, characteristic
4#23761294C212yesmaximal, complemented
5C79×#2921C211yesmaximal, characteristic
6C2×#14579153C2211yescharacteristic
7#19098620C2212yes—
8#19098502C411yescharacteristic
9#19098499C411yescharacteristic
10#19098493C411yescharacteristic
11#19098488C411yescharacteristic
12#19098458—21no—
13#19098457—21no—
14#19098447—21no—
15#19098436—21no—
16#19098409C2211yescharacteristic
17#19098304—24no—
18#19098289—21no—
19#19098287—21no—
20C2×#117×C79C2211yescharacteristic
21#433×C79—24no—
22#423×C79—21no—
23#407×C79—21no—
24C4×M16×C79C2212yes—
25#14579153—22no—
26#14579153—22no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×C4⋊Dic79C2×C411yescharacteristic
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×Dic79⋊C4—21no—
32C2×Dic79⋊C4—21no—
33C2×C22⋊Dic79—21no—
34C2×C22⋊Dic79—21no—
35C2×C4×Dic79—21no—
36C4⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38#14579072—22no—
39C79⋊#95—21no—
40C79⋊#95—21no—
41C79⋊#95—21no—
42C79⋊#95—21no—
43C79⋊#95—21no—
44C79⋊#95C2×C411yescharacteristic
45C79⋊#95C2×C411yescharacteristic
46C23×C4×C79D411yescharacteristic
47C2×M16×C79—22no—
48#117×C79—22no—
49#117×C79—22no—
50C2×C4⋊C4×C79C2×C411yescharacteristic
51C2×C4⋊C4×C79D411yescharacteristic
52C2×C22⋊C4×C79C2×C411yescharacteristic
53C2×C22⋊C4×C79D411yescharacteristic
54C2×C42×C79D411yescharacteristic
55C2×C42×C79C2311yescharacteristic
56C4⋊C8×C79—24no—
57C22⋊C8×C79—24no—
58#97×C79D414yes—
59C4×C8×C79—22no—
60C22×Dic79—21no—
61C22×Dic79—21no—
62C22×Dic79—21no—
63C22×Dic79—21no—
64C2×C79⋊C8—42no—
65C4⋊Dic79—22no—
66C4⋊Dic79—22no—
67Dic79⋊C4—42no—
68Dic79⋊C4—42no—
69Dic79⋊C4—41no—
70Dic79⋊C4—42no—
71Dic79⋊C4—41no—
72C22⋊Dic79—42no—
73C22⋊Dic79—42no—
74C4×Dic79—42no—
75C24×C79C22⋊C411yescharacteristic
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—41no—
79C22×C4×C79—41no—
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86M16×C79—42no—
87M16×C79—42no—
88C2×C8×C79—24no—
89C4⋊C4×C79—22no—
90C4⋊C4×C79—22no—
91C22⋊C4×C79—22no—
92C22⋊C4×C79—22no—
93C42×C79—21no—
94C42×C79—21no—
95C42×C79C2×D412yes—
96C42×C79#4112yes—
97C23×C79—21no—
98C23×C79—21no—
99C23×C79—21no—
100C23×C79—41no—
101C23×C79—41no—
102C23×C79C2×C22⋊C411yescharacteristic
103C2×C4×C79—21no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—42no—
113C2×C4×C79—42no—
114C2×C4×C79—41no—
115C2×C4×C79—41no—
116C2×C4×C79C4×D412yes—
117C2×C4×C79#12011yescharacteristic
118C2×C4×C79#12112yes—
119C2×C4×C79#12311yescharacteristic
120C2×Dic79—22no—
121C2×Dic79—41no—
122C2×Dic79—41no—
123C2×Dic79—41no—
124C2×Dic79—41no—
125C2×Dic79—41no—
126C2×Dic79—41no—
127C2×Dic79—41no—
128C2×Dic79—41no—
129C2×Dic79—41no—
130C2×Dic79—41no—
131C2×Dic79—41no—
132C8×C79—44no—
133C79⋊C8—82no—
134C22×C79—21no—
135C22×C79—21no—
136C22×C79—42no—
137C22×C79—41no—
138C22×C79—41no—
139C22×C79—41no—
140C22×C79—81no—
141C22×C79#38611yescharacteristic
142C22×C79#42111yescharacteristic
143C22×C79#42111yescharacteristic
144C4×C79—22no—
145C4×C79—21no—
146C4×C79—21no—
147C4×C79—22no—
148C4×C79—41no—
149C4×C79—41no—
150C4×C79—41no—
151Dic79—42no—
152Dic79—81no—
153Dic79—81no—
154Dic79—81no—
155#9390—791nomaximal, Sylow
156C2×C79—21no—
157C2×C79—21no—
158C2×C79—81no—
159C2×C79#188411yescharacteristic
160C2×C79#189711yescharacteristic
161C2×C79#191511yescharacteristic
162#2921D7911yescharacteristic
163#2611—791no—
164#1528—792no—
165#1464—791no—
166#1300—792no—
167C79#939011yescharacteristic, complemented, Sylow
168C2×#125—791no—
169C2×#117C2×D7911yescharacteristic
170#433—24no—
171#432—792no—
172#423—21no—
173#407—21no—
174C4×M16C2×D7912yes—
175#400—791no—
176#399—791no—
177#398—791no—
178#397—791no—
179#389—1581no—
180#388—1581no—
181#387—1581no—
182#385—1581no—
183#376—791no—
184#330—1584no—
185#324—1581no—
186#323—1581no—
187C23×C4C22⋊D7911yescharacteristic
188C2×M16—22no—
189#125—1582no—
190#125—1582no—
191#117—22no—
192#117—22no—
193C2×C4⋊C4—791no—
194C2×C4⋊C4—791no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4—1581no—
197C2×C4⋊C4—1581no—
198C2×C4⋊C4—1581no—
199C2×C4⋊C4C4×D7911yescharacteristic
200C2×C4⋊C4D31611yescharacteristic
201C2×C22⋊C4—1581no—
202C2×C22⋊C4—1581no—
203C2×C22⋊C4C4×D7911yescharacteristic
204C2×C22⋊C4D31611yescharacteristic
205C2×C42—1581no—
206C2×C42C22⋊D7911yescharacteristic
207C2×C42C22×D7911yescharacteristic
208C4⋊C8—24no—
209C4⋊C8—1582no—
210C22⋊C8—24no—
211C22⋊C8—1582no—
212#97—1582no—
213#97C22⋊D7914yes—
214C4×C8—22no—
215#95—791no—
216#95—791no—
217#95—1581no—
218#95—1581no—
219#95—1581no—
220#95—1581no—
221#95—1581no—
222C24(C2×D79)⋊C411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—41no—
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×C4C2×C4×D7911yescharacteristic
235C22×C4C2×C22⋊D7911yescharacteristic
236C22×C4C2×D31611yescharacteristic
237M16—42no—
238M16—42no—
239C2×C8—24no—
240C2×C8—3162no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—3162no—
244C4⋊C4—3162no—
245C4⋊C4—3161no—
246C4⋊C4—3162no—
247C4⋊C4—3161no—
248C4⋊C4—1582no—
249C4⋊C4—1582no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—3162no—
253C22⋊C4—3162no—
254C42—21no—
255C42—21no—
256C42—3162no—
257C42C2×C22⋊D7912yes—
258C42#1192962412yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×(C2×D79)⋊C411yescharacteristic
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—3161no—
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—1582no—
290C2×C4C4×C22⋊D7912yes—
291C2×C4#1457913011yescharacteristic
292C2×C4#1457913512yes—
293C2×C4#1457914411yescharacteristic
294C8—44no—
295C8—6322no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22#1909844211yescharacteristic
304C22#1909858111yescharacteristic
305C22#1909858111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—22no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—3162no—
314C4—6321no—
315C4—6321no—
316C4—6321no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#2376378311yescharacteristic
321C2#2376384811yescharacteristic
322C2#2376395211yescharacteristic
subgroupG/H
centerC22#19098442
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2921C2
socleC22×C79#386
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#2921D79
hypocenterC79#9390
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #23762001C2semidirect
4. #23761294C2semidirect
167. C79#9390semidirect

Maximal subgroups

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

subgroupindex
#237690982
#237620012
#237617342
#237612942
C79×#29212
#939079