Finite Group Databasebeta

← #34185946

Subgroups — #34185946

Order 20224.

518 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 103 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#23772429C211yesmaximal, characteristic
2#23768392C211yesmaximal, characteristic
3#23762125C212yesmaximal, complemented
4#23761255C212yesmaximal, complemented
5C79×#1466C211yesmaximal, characteristic
6C2×#14579161C2211yescharacteristic
7C2×#14579148C412yes—
8#19098635—24no—
9#19098620C2212yes—
10#19098527—21no—
11C2×C22⋊(C79⋊C8)—21no—
12#19098488—21no—
13#19098488—21no—
14#19098468C412yes—
15#19098423—21no—
16#19098423—21no—
17C4×C4⋊Dic79C2211yescharacteristic
18#19098311—24no—
19C2×C4×D4×C79C2211yescharacteristic
20#432×C79C2212yes—
21#323×C79—21no—
22#323×C79—21no—
23C22×Dic158C2×C412yes—
24C2×#11929608—22no—
25C22×C79⋊C8—22no—
26#14579161—22no—
27#14579161—22no—
28#14579148—44no—
29C2×C4⋊Dic79—21no—
30C2×C4⋊Dic79—21no—
31C2×C4⋊Dic79—21no—
32C2×C22⋊Dic79—22no—
33C2×C22⋊Dic79—22no—
34C2×C4×Dic79C2×C412yes—
35C4⋊(C79⋊C8)D414yes—
36C22⋊(C79⋊C8)—22no—
37C22⋊(C79⋊C8)—22no—
38C22⋊(C79⋊C8)—22no—
39#14579072—22no—
40C79⋊#95—22no—
41C79⋊#95—22no—
42C79⋊#95—21no—
43C22×D4×C79C2×C411yescharacteristic
44C23×C4×C79D411yescharacteristic
45C23×C4×C79D411yescharacteristic
46C4×D4×C79—22no—
47C4×D4×C79—22no—
48C2×C4⋊C4×C79C2×C411yescharacteristic
49C2×C22⋊C4×C79D411yescharacteristic
50C2×C22⋊C4×C79D411yescharacteristic
51C2×C42×C79C2311yescharacteristic
52C4⋊C8×C79—22no—
53C22⋊C8×C79—22no—
54#97×C79—22no—
55C2×Dic158—24no—
56C2×Dic158—42no—
57C2×Dic158—42no—
58C22×Dic79—22no—
59C22×Dic79—22no—
60#11929608—82no—
61C2×C79⋊C8—24no—
62C2×C79⋊C8—82no—
63C4⋊Dic79—42no—
64C4⋊Dic79—41no—
65C4⋊Dic79—42no—
66C4⋊Dic79—41no—
67C22⋊Dic79—44no—
68C22⋊Dic79—44no—
69C4×Dic79—42no—
70C24×C79C22⋊C411yescharacteristic
71C24×C79C22⋊C411yescharacteristic
72C2×D4×C79—22no—
73C2×D4×C79—22no—
74C2×D4×C79—41no—
75C22×C4×C79—21no—
76C22×C4×C79—21no—
77C22×C4×C79—21no—
78C22×C4×C79—21no—
79C22×C4×C79—42no—
80C22×C4×C79—42no—
81C22×C4×C79C22⋊C411yescharacteristic
82C22×C4×C79C22⋊C411yescharacteristic
83C22×C4×C79C22×C411yescharacteristic
84C22×C4×C79C2×D411yescharacteristic
85C22×C4×C79C2×D411yescharacteristic
86C2×C8×C79—42no—
87C4⋊C4×C79—22no—
88C22⋊C4×C79—22no—
89C22⋊C4×C79—22no—
90C42×C79C2×D412yes—
91C42×C79#4112yes—
92C23×C79—21no—
93C23×C79—21no—
94C23×C79—21no—
95C23×C79—21no—
96C23×C79—21no—
97C23×C79—41no—
98C23×C79—41no—
99C23×C79—41no—
100C23×C79—41no—
101C23×C79#10012yes—
102C23×C79C2×C22⋊C411yescharacteristic
103D4×C79—44no—
104C2×C4×C79—21no—
105C2×C4×C79—21no—
106C2×C4×C79—22no—
107C2×C4×C79—21no—
108C2×C4×C79—21no—
109C2×C4×C79—41no—
110C2×C4×C79—42no—
111C2×C4×C79—42no—
112C2×C4×C79—41no—
113C2×C4×C79—41no—
114C2×C4×C79—81no—
115C2×C4×C79—81no—
116C2×C4×C79C4×D412yes—
117C2×C4×C79#12011yescharacteristic
118C2×C4×C79#12112yes—
119C2×C4×C79#12311yescharacteristic
120C2×Dic79—42no—
121C2×Dic79—42no—
122C2×Dic79—42no—
123C2×Dic79—42no—
124C2×Dic79—42no—
125C2×Dic79—42no—
126Dic158—84no—
127C8×C79—82no—
128C79⋊C8—44no—
129C22×C79—21no—
130C22×C79—21no—
131C22×C79—22no—
132C22×C79—42no—
133C22×C79—42no—
134C22×C79—41no—
135C22×C79—41no—
136C22×C79—41no—
137C22×C79—41no—
138C22×C79—41no—
139C22×C79—81no—
140C22×C79—81no—
141C22×C79#38611yescharacteristic
142C22×C79C2×#10011yescharacteristic
143C22×C79#41311yescharacteristic
144C4×C79—21no—
145C4×C79—22no—
146C4×C79—22no—
147C4×C79—41no—
148C4×C79—41no—
149C4×C79#45512yes—
150Dic79—82no—
151Dic79—82no—
152#9451—791nomaximal, Sylow
153C2×C79—21no—
154C2×C79—21no—
155C2×C79—42no—
156C2×C79—81no—
157C2×C79#188511yescharacteristic
158C2×C79#190011yescharacteristic
159C2×C79#190911yescharacteristic
160#2973—791no—
161#2547—791no—
162#1558—792no—
163#1466D7911yescharacteristic
164#1285—792no—
165C79#945111yescharacteristic, complemented, Sylow
166C2×C4⋊Q8—791no—
167C2×#124—792no—
168C2×C4×D4C2×D7911yescharacteristic
169#435—1584no—
170#432—792no—
171#432C2×D7912yes—
172#407—1581no—
173C2×C22⋊C8—1581no—
174#397—1581no—
175#397—1581no—
176#391—792no—
177#380—1581no—
178#380—1581no—
179C4×C4⋊C4—791no—
180#333—1584no—
181#323—21no—
182#323—21no—
183C22×Q8—792no—
184C22×D4C4×D7911yescharacteristic
185C23×C4D31611yescharacteristic
186C23×C4D31611yescharacteristic
187C2×M16—1582no—
188C22×C8—1582no—
189C4⋊Q8—1582no—
190C4⋊Q8—1582no—
191#124—3164no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—1581no—
195C2×C4⋊C4—1581no—
196C2×C4⋊C4—1581no—
197C2×C4⋊C4C4×D7911yescharacteristic
198C2×C22⋊C4—1582no—
199C2×C22⋊C4—1582no—
200C2×C22⋊C4C22⋊D7911yescharacteristic
201C2×C22⋊C4C22⋊D7911yescharacteristic
202C2×C42—792no—
203C2×C42C22×D7911yescharacteristic
204C4⋊C8—22no—
205C4⋊C8—794no—
206C22⋊C8—22no—
207C22⋊C8—1582no—
208C22⋊C8—1582no—
209C22⋊C8—1582no—
210#97—22no—
211#97—1582no—
212#95—1582no—
213#95—1582no—
214#95—1581no—
215C24(C2×D79)⋊C411yescharacteristic
216C24(C2×D79)⋊C411yescharacteristic
217C2×Q8—3162no—
218C2×Q8—3162no—
219C2×Q8—1584no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—1582no—
230C22×C4—1582no—
231C22×C4(C2×D79)⋊C411yescharacteristic
232C22×C4(C2×D79)⋊C411yescharacteristic
233C22×C4C2×C4×D7911yescharacteristic
234C22×C4C2×C22⋊D7911yescharacteristic
235C22×C4C2×D31611yescharacteristic
236M16—6322no—
237C2×C8—42no—
238C2×C8—6322no—
239C2×C8—1584no—
240C4⋊C4—22no—
241C4⋊C4—3162no—
242C4⋊C4—3161no—
243C4⋊C4—3162no—
244C4⋊C4—3161no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—3164no—
248C22⋊C4—3164no—
249C42—3162no—
250C42D4×D7912yes—
251C42#1192962312yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1457907712yes—
262C23C2×(C2×D79)⋊C411yescharacteristic
263Q8—6324no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—3162no—
278C2×C4—3162no—
279C2×C4—3162no—
280C2×C4—3162no—
281C2×C4—3162no—
282C2×C4—3162no—
283C2×C4(C22⋊D79)⋊C412yes—
284C2×C4#1457913111yescharacteristic
285C2×C4#1457913312yes—
286C2×C4#1457914511yescharacteristic
287C8—82no—
288C8—3164no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1909844011yescharacteristic
302C22C2×#1457907711yescharacteristic
303C22#1909854511yescharacteristic
304C4—21no—
305C4—22no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—6322no—
310C4—6322no—
311C4#1909874312yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2376378611yescharacteristic
317C2#2376386111yescharacteristic
318C2#2376392011yescharacteristic
subgroupG/H
centerC22#19098440
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#1466C2
socleC22×C79#386
prime radicalC22×D4×C79C2×C4
cosocleC2×C42×C79C23
hypercenter#1466D79
hypocenterC79#9451
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #23762125C2semidirect
4. #23761255C2semidirect
165. C79#9451semidirect

Maximal subgroups

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

subgroupindex
#237724292
#237683922
#237621252
#237612552
C79×#14662
#945179