Finite Group Databasebeta

← #34199178

Subgroups — #34199178

Order 20224.

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

normal subgroups only

This group has 376 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#23763483C211yesmaximal, characteristic
3#23761431C212yesmaximal, complemented
4#23761407C212yesmaximal, complemented
5C79×#1458C211yesmaximal, characteristic
6C2×C4×Dic158C411yescharacteristic
7#19098527—21no—
8#19098493C2211yescharacteristic
9#19098488C411yescharacteristic
10#19098483C2211yescharacteristic
11#19098475C411yescharacteristic
12#19098451—21no—
13#19098447—21no—
14#19098444—21no—
15#19098436—21no—
16#19098431C411yescharacteristic
17#19098332—21no—
18#19098332C2212yes—
19C2×#117×C79C2211yescharacteristic
20#431×C79—21no—
21#324×C79C2212yes—
22#323×C79—21no—
23C22×Dic158C2×C411yescharacteristic
24C2×#11929608—22no—
25C4×Dic158—41no—
26C4×Dic158—41no—
27C2×C4⋊Dic79C2×C411yescharacteristic
28C2×Dic79⋊C4—21no—
29C2×Dic79⋊C4—21no—
30C2×Dic79⋊C4—21no—
31C2×C22⋊Dic79—21no—
32C2×C22⋊Dic79—21no—
33C2×C4×Dic79—21no—
34C22⋊(C79⋊C8)—41no—
35C22⋊(C79⋊C8)—41no—
36C79⋊#95—21no—
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—41no—
46#117×C79—41no—
47C2×C4⋊C4×C79D411yescharacteristic
48C2×C4⋊C4×C79C2311yescharacteristic
49C2×C22⋊C4×C79C2×C411yescharacteristic
50C2×C22⋊C4×C79D411yescharacteristic
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—
62#11929608—42no—
63#11929608—42no—
64C2×C79⋊C8—42no—
65C4⋊Dic79—41no—
66Dic79⋊C4—42no—
67Dic79⋊C4—41no—
68Dic79⋊C4—41no—
69Dic79⋊C4—42no—
70C22⋊Dic79—42no—
71C22⋊Dic79—42no—
72C4×Dic79—42no—
73C24×C79C22⋊C411yescharacteristic
74C22×C4×C79—21no—
75C22×C4×C79—21no—
76C22×C4×C79—22no—
77C22×C4×C79—41no—
78C22×C4×C79C22⋊C411yescharacteristic
79C22×C4×C79C22⋊C411yescharacteristic
80C22×C4×C79C22⋊C411yescharacteristic
81C22×C4×C79C22×C411yescharacteristic
82C22×C4×C79C2×D411yescharacteristic
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—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—
128Dic158—42no—
129Dic158—42no—
130C8×C79—82no—
131C79⋊C8—82no—
132C22×C79—21no—
133C22×C79—21no—
134C22×C79—42no—
135C22×C79—41no—
136C22×C79—41no—
137C22×C79—41no—
138C22×C79—81no—
139C22×C79C2×#9911yescharacteristic
140C22×C79#41611yescharacteristic
141C22×C79#42111yescharacteristic
142C4×C79—21no—
143C4×C79—21no—
144C4×C79—22no—
145C4×C79—41no—
146C4×C79—41no—
147C4×C79—41no—
148C4×C79—41no—
149Dic79—42no—
150Dic79—81no—
151Dic79—81no—
152Dic79—81no—
153#11849—791nomaximal, Sylow
154C2×C79—21no—
155C2×C79—21no—
156C2×C79—81no—
157C2×C79#151611yescharacteristic
158C2×C79#212011yescharacteristic
159C2×C79#213511yescharacteristic
160#2599—791no—
161#1822—791no—
162#1458D7911yescharacteristic
163#1354—792no—
164#1346—792no—
165C79#1184911yescharacteristic, complemented, Sylow
166C2×C4×Q8—791no—
167C2×#117C2×D7911yescharacteristic
168#431—21no—
169#407—1581no—
170#398—791no—
171#397—791no—
172#395—791no—
173#393—791no—
174#388—1581no—
175#387—1581no—
176#387—1581no—
177#385—1581no—
178#383—791no—
179#343—792no—
180#343—1581no—
181#324C2×D7912yes—
182#323—21no—
183C22×Q8—791no—
184C23×C4D31611yescharacteristic
185C2×M16—1582no—
186C4×Q8—3161no—
187C4×Q8—3161no—
188#117—41no—
189#117—41no—
190C2×C4⋊C4—791no—
191C2×C4⋊C4—1581no—
192C2×C4⋊C4—1581no—
193C2×C4⋊C4—1581no—
194C2×C4⋊C4D31611yescharacteristic
195C2×C4⋊C4C22×D7911yescharacteristic
196C2×C22⋊C4—1581no—
197C2×C22⋊C4—1581no—
198C2×C22⋊C4C4×D7911yescharacteristic
199C2×C22⋊C4C22⋊D7911yescharacteristic
200C2×C42—1581no—
201C2×C42C4×D7911yescharacteristic
202C2×C42C22⋊D7911yescharacteristic
203C4⋊C8—41no—
204C22⋊C8—22no—
205C22⋊C8—3161no—
206C22⋊C8—3161no—
207C4×C8—41no—
208#95—791no—
209#95—791no—
210#95—1581no—
211#95—1581no—
212#95—1581no—
213#95—1581no—
214#95—1581no—
215#95—1581no—
216C24(C2×D79)⋊C411yescharacteristic
217C2×Q8—3161no—
218C2×Q8—3162no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—1581no—
224C22×C4—1581no—
225C22×C4—1581no—
226C22×C4—1581no—
227C22×C4(C2×D79)⋊C411yescharacteristic
228C22×C4(C2×D79)⋊C411yescharacteristic
229C22×C4(C2×D79)⋊C411yescharacteristic
230C22×C4C2×C4×D7911yescharacteristic
231C22×C4C2×C22⋊D7911yescharacteristic
232C22×C4C2×D31611yescharacteristic
233M16—3162no—
234M16—3162no—
235C2×C8—42no—
236C2×C8—3162no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—3161no—
240C4⋊C4—3162no—
241C4⋊C4—3161no—
242C4⋊C4—3161no—
243C4⋊C4—3162no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—3162no—
247C22⋊C4—3162no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—3162no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×(C2×D79)⋊C411yescharacteristic
259Q8—3162no—
260Q8—3162no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—3161no—
276C2×C4—3161no—
277C2×C4—3161no—
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—1582no—
287C2×C4#1457907511yescharacteristic
288C2×C4#1457907511yescharacteristic
289C2×C4#1457918312yes—
290C8—82no—
291C8—6322no—
292C22—21no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22C2×#1457907511yescharacteristic
300C22#1909856011yescharacteristic
301C22#1909857911yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—3162no—
310C4—6321no—
311C4—6321no—
312C4—6321no—
313C2—21no—
314C2—21no—
315C2—81no—
316C2#2376195711yescharacteristic
317C2#2376503911yescharacteristic
318C2#2376509411yescharacteristic
subgroupG/H
centerC22C2×#14579075
commutatorC22×C4×C79C22×C4
FrattiniC2×C4⋊C4C22×D79
FittingC79×#1458C2
socleC22×C79C2×#99
prime radicalC24×C79C22⋊C4
cosocleC2×C4⋊C4×C79C23
hypercenter#1458D79
hypocenterC79#11849
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
3. #23761431C2semidirect
4. #23761407C2semidirect
165. C79#11849semidirect

Maximal subgroups

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

subgroupindex
#237689772
#237634832
#237614312
#237614072
C79×#14582
#1184979