Finite Group Databasebeta

← #34199195

Subgroups — #34199195

Order 20224.

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

normal subgroups only

This group has 368 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#23769098C211yesmaximal, characteristic
2#23761714C211yesmaximal, characteristic
3#23761426C212yesmaximal, complemented
4#23761405C212yesmaximal, complemented
5C79×#1822C211yesmaximal, characteristic
6C2×#14579153C411yescharacteristic
7#19098616—21no—
8#19098502C2211yescharacteristic
9#19098499C411yescharacteristic
10#19098493C2211yescharacteristic
11#19098488C411yescharacteristic
12#19098458—21no—
13#19098457—21no—
14#19098447—21no—
15#19098436—21no—
16#19098409C411yescharacteristic
17#19098289C2212yes—
18#19098287—21no—
19C2×#117×C79C2211yescharacteristic
20#407×C79—21no—
21#343×C79—21no—
22#343×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—
36C4×C79⋊C8—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
45C2×M16×C79—22no—
46#117×C79—41no—
47#117×C79—41no—
48C2×C4⋊C4×C79D411yescharacteristic
49C2×C4⋊C4×C79C2311yescharacteristic
50C2×C22⋊C4×C79C2×C411yescharacteristic
51C2×C22⋊C4×C79D411yescharacteristic
52C2×C42×C79C2×C411yescharacteristic
53C2×C42×C79D411yescharacteristic
54C22⋊C8×C79—41no—
55C22⋊C8×C79—41no—
56C22×Dic79—21no—
57C22×Dic79—21no—
58C22×Dic79—21no—
59C22×Dic79—21no—
60C2×C79⋊C8—42no—
61C4⋊Dic79—41no—
62C4⋊Dic79—41no—
63Dic79⋊C4—42no—
64Dic79⋊C4—42no—
65Dic79⋊C4—41no—
66Dic79⋊C4—41no—
67Dic79⋊C4—42no—
68C22⋊Dic79—42no—
69C22⋊Dic79—42no—
70C4×Dic79—42no—
71C24×C79C22⋊C411yescharacteristic
72C22×C4×C79—21no—
73C22×C4×C79—21no—
74C22×C4×C79—22no—
75C22×C4×C79—41no—
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22×C411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81C22×C4×C79C2×D411yescharacteristic
82M16×C79—42no—
83M16×C79—42no—
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—21no—
96C23×C79—41no—
97C23×C79—41no—
98C23×C79C2×C22⋊C411yescharacteristic
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—42no—
108C2×C4×C79—42no—
109C2×C4×C79—41no—
110C2×C4×C79—41no—
111C2×C4×C79—41no—
112C2×C4×C79—41no—
113C2×C4×C79#9911yescharacteristic
114C2×C4×C79#9911yescharacteristic
115C2×C4×C79#13512yes—
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—
128C8×C79—82no—
129C79⋊C8—82no—
130C22×C79—21no—
131C22×C79—21no—
132C22×C79—42no—
133C22×C79—41no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—81no—
137C22×C79C2×#9911yescharacteristic
138C22×C79#41611yescharacteristic
139C22×C79#42111yescharacteristic
140C4×C79—21no—
141C4×C79—21no—
142C4×C79—22no—
143C4×C79—41no—
144C4×C79—41no—
145C4×C79—41no—
146C4×C79—41no—
147Dic79—42no—
148Dic79—81no—
149Dic79—81no—
150Dic79—81no—
151#11853—791nomaximal, Sylow
152C2×C79—21no—
153C2×C79—21no—
154C2×C79—81no—
155C2×C79#151611yescharacteristic
156C2×C79#212311yescharacteristic
157C2×C79#213211yescharacteristic
158#2611—791no—
159#1822D7911yescharacteristic
160#1458—791no—
161#1353—792no—
162#1346—792no—
163C79#1185311yescharacteristic, complemented, Sylow
164C2×#125—791no—
165C2×#117C2×D7911yescharacteristic
166#431—1581no—
167#407—21no—
168#400—791no—
169#399—791no—
170#398—791no—
171#397—791no—
172#389—1581no—
173#388—1581no—
174#387—1581no—
175#385—1581no—
176#376—791no—
177#343—21no—
178#343C2×D7912yes—
179#324—792no—
180#323—1581no—
181C23×C4C22⋊D7911yescharacteristic
182C2×M16—22no—
183#125—3161no—
184#125—3161no—
185#117—41no—
186#117—41no—
187C2×C4⋊C4—791no—
188C2×C4⋊C4—791no—
189C2×C4⋊C4—1581no—
190C2×C4⋊C4—1581no—
191C2×C4⋊C4—1581no—
192C2×C4⋊C4—1581no—
193C2×C4⋊C4C22⋊D7911yescharacteristic
194C2×C4⋊C4C22×D7911yescharacteristic
195C2×C22⋊C4—1581no—
196C2×C22⋊C4—1581no—
197C2×C22⋊C4C4×D7911yescharacteristic
198C2×C22⋊C4D31611yescharacteristic
199C2×C42—1581no—
200C2×C42C4×D7911yescharacteristic
201C2×C42D31611yescharacteristic
202C4⋊C8—3161no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—1582no—
206C4×C8—3161no—
207#95—791no—
208#95—791no—
209#95—1581no—
210#95—1581no—
211#95—1581no—
212#95—1581no—
213#95—1581no—
214C24(C2×D79)⋊C411yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—1581no—
220C22×C4—1581no—
221C22×C4—1581no—
222C22×C4—1581no—
223C22×C4(C2×D79)⋊C411yescharacteristic
224C22×C4(C2×D79)⋊C411yescharacteristic
225C22×C4(C2×D79)⋊C411yescharacteristic
226C22×C4C2×C4×D7911yescharacteristic
227C22×C4C2×C22⋊D7911yescharacteristic
228C22×C4C2×D31611yescharacteristic
229M16—42no—
230M16—42no—
231C2×C8—42no—
232C2×C8—3162no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—3161no—
236C4⋊C4—3161no—
237C4⋊C4—3162no—
238C4⋊C4—3162no—
239C4⋊C4—3161no—
240C4⋊C4—3161no—
241C4⋊C4—3162no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—3162no—
245C22⋊C4—3162no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—3162no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×(C2×D79)⋊C411yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—3161no—
272C2×C4—3161no—
273C2×C4—3161no—
274C2×C4—3161no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—3161no—
278C2×C4—3161no—
279C2×C4—3161no—
280C2×C4—3161no—
281C2×C4—3161no—
282C2×C4—1582no—
283C2×C4#1457907511yescharacteristic
284C2×C4#1457907511yescharacteristic
285C2×C4#1457918112yes—
286C8—82no—
287C8—6322no—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22C2×#1457907511yescharacteristic
296C22#1909855711yescharacteristic
297C22#1909858111yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—3162no—
306C4—6321no—
307C4—6321no—
308C4—6321no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2#2376195411yescharacteristic
313C2#2376505511yescharacteristic
314C2#2376508311yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1822C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1822D79
hypocenterC79#11853
LCSG ▷ C22×C4×C79 ▷ C23×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #23761426C2semidirect
4. #23761405C2semidirect
163. C79#11853semidirect

Maximal subgroups

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

subgroupindex
#237690982
#237617142
#237614262
#237614052
C79×#18222
#1185379