Finite Group Databasebeta

← #34186159

Subgroups — #34186159

Order 20224.

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

normal subgroups only

This group has 458 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#23772038C211yesmaximal, characteristic
2#23761847C212yesmaximal, complemented
3#23761734C211yesmaximal, characteristic
4#23761264C212yesmaximal, complemented
5C79×#2599C211yesmaximal, characteristic
6#19098624—24no—
7#19098620C2212yes—
8#19098587—21no—
9#19098527—21no—
10C4×#11929608C2212yes—
11Q8⋊(C79⋊C8)—24no—
12#19098289—21no—
13#19098287—21no—
14C2×C4×Q8×C79C2211yescharacteristic
15C2×#117×C79C2211yescharacteristic
16#398×C79C411yescharacteristic
17#397×C79C411yescharacteristic
18#395×C79C411yescharacteristic
19#393×C79C411yescharacteristic
20#388×C79—21no—
21#387×C79—21no—
22#387×C79—21no—
23#385×C79—21no—
24#383×C79C2211yescharacteristic
25C2×#11929608—22no—
26C4⋊(C79⋊C8)—24no—
27C4⋊(C79⋊C8)—22no—
28C22⋊(C79⋊C8)—24no—
29C22⋊(C79⋊C8)—22no—
30#14579072—22no—
31#14579072—22no—
32C4×C79⋊C8D414yes—
33C22×Q8×C79C2×C411yescharacteristic
34C23×C4×C79D411yescharacteristic
35C4×Q8×C79—22no—
36C4×Q8×C79—22no—
37#117×C79—22no—
38#117×C79—22no—
39C2×C4⋊C4×C79—21no—
40C2×C4⋊C4×C79—21no—
41C2×C4⋊C4×C79—21no—
42C2×C4⋊C4×C79C2×C411yescharacteristic
43C2×C4⋊C4×C79C2×C411yescharacteristic
44C2×C4⋊C4×C79D411yescharacteristic
45C2×C22⋊C4×C79—21no—
46C2×C22⋊C4×C79—21no—
47C2×C22⋊C4×C79C2×C411yescharacteristic
48C2×C22⋊C4×C79D411yescharacteristic
49C2×C42×C79—21no—
50C2×C42×C79D411yescharacteristic
51C2×C42×C79C2311yescharacteristic
52#95×C79—21no—
53#95×C79—21no—
54#95×C79—21no—
55#95×C79—21no—
56#95×C79—21no—
57#95×C79—21no—
58#95×C79C2×C411yescharacteristic
59#95×C79C2×C411yescharacteristic
60#11929608—42no—
61#11929608—42no—
62C2×C79⋊C8—24no—
63C2×C79⋊C8—42no—
64C24×C79C22⋊C411yescharacteristic
65C2×Q8×C79—22no—
66C2×Q8×C79—42no—
67C22×C4×C79—21no—
68C22×C4×C79—21no—
69C22×C4×C79—21no—
70C22×C4×C79—21no—
71C22×C4×C79—21no—
72C22×C4×C79—21no—
73C22×C4×C79—41no—
74C22×C4×C79—41no—
75C22×C4×C79C22⋊C411yescharacteristic
76C22×C4×C79C22⋊C411yescharacteristic
77C22×C4×C79C22⋊C411yescharacteristic
78C22×C4×C79C22×C411yescharacteristic
79C22×C4×C79C2×D411yescharacteristic
80C22×C4×C79C2×D411yescharacteristic
81C4⋊C4×C79—22no—
82C4⋊C4×C79—22no—
83C4⋊C4×C79—22no—
84C4⋊C4×C79—42no—
85C4⋊C4×C79—42no—
86C4⋊C4×C79—41no—
87C4⋊C4×C79—41no—
88C22⋊C4×C79—22no—
89C22⋊C4×C79—22no—
90C22⋊C4×C79—42no—
91C22⋊C4×C79—42no—
92C42×C79—21no—
93C42×C79—21no—
94C42×C79—42no—
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
103Q8×C79—24no—
104Q8×C79—42no—
105C2×C4×C79—21no—
106C2×C4×C79—21no—
107C2×C4×C79—22no—
108C2×C4×C79—21no—
109C2×C4×C79—21no—
110C2×C4×C79—21no—
111C2×C4×C79—21no—
112C2×C4×C79—21no—
113C2×C4×C79—41no—
114C2×C4×C79—41no—
115C2×C4×C79—42no—
116C2×C4×C79—42no—
117C2×C4×C79—41no—
118C2×C4×C79—41no—
119C2×C4×C79—41no—
120C2×C4×C79—41no—
121C2×C4×C79—41no—
122C2×C4×C79—41no—
123C2×C4×C79—41no—
124C2×C4×C79—41no—
125C2×C4×C79—41no—
126C2×C4×C79—41no—
127C2×C4×C79—41no—
128C2×C4×C79—41no—
129C2×C4×C79—41no—
130C2×C4×C79C4×D412yes—
131C2×C4×C79#12011yescharacteristic
132C2×C4×C79#12112yes—
133C2×C4×C79#12311yescharacteristic
134C79⋊C8—44no—
135C79⋊C8—82no—
136C22×C79—21no—
137C22×C79—21no—
138C22×C79—42no—
139C22×C79—41no—
140C22×C79—41no—
141C22×C79—41no—
142C22×C79—81no—
143C22×C79#38611yescharacteristic
144C22×C79#42111yescharacteristic
145C22×C79#42111yescharacteristic
146C4×C79—22no—
147C4×C79—21no—
148C4×C79—21no—
149C4×C79—22no—
150C4×C79—41no—
151C4×C79—42no—
152C4×C79—41no—
153C4×C79—41no—
154C4×C79—81no—
155C4×C79—81no—
156C4×C79—81no—
157#9493—791nomaximal, Sylow
158C2×C79—21no—
159C2×C79—21no—
160C2×C79—81no—
161C2×C79#189111yescharacteristic
162C2×C79#189711yescharacteristic
163C2×C79#190911yescharacteristic
164#2921—791no—
165#2599D7911yescharacteristic
166#1491—792no—
167#1464—791no—
168#1288—792no—
169C79#949311yescharacteristic, complemented, Sylow
170C2×C4×Q8C2×D7911yescharacteristic
171C2×#117C2×D7911yescharacteristic
172#433—1584no—
173#432—792no—
174#423—1581no—
175#407—1581no—
176C4×M16—792no—
177#398Dic7911yescharacteristic
178#397Dic7911yescharacteristic
179#395Dic7911yescharacteristic
180#393Dic7911yescharacteristic
181#388—21no—
182#387—21no—
183#387—21no—
184#385—21no—
185#383C2×D7911yescharacteristic
186Q8⋊C8—1584no—
187#324—1581no—
188#323—1581no—
189C22×Q8C2×Dic7911yescharacteristic
190C23×C4C22⋊D7911yescharacteristic
191C2×M16—1582no—
192C4×Q8—22no—
193C4×Q8—22no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×Dic7911yescharacteristic
200C2×C4⋊C4C2×Dic7911yescharacteristic
201C2×C4⋊C4C22⋊D7911yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C2×Dic7911yescharacteristic
205C2×C22⋊C4C22⋊D7911yescharacteristic
206C2×C42—21no—
207C2×C42C22⋊D7911yescharacteristic
208C2×C42C22×D7911yescharacteristic
209C4⋊C8—1584no—
210C4⋊C8—1582no—
211C22⋊C8—1582no—
212C22⋊C8—1584no—
213#97—1582no—
214#97—1582no—
215C4×C8—794no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95C2×Dic7911yescharacteristic
223#95C2×Dic7911yescharacteristic
224C24C22⋊Dic7911yescharacteristic
225C2×Q8—22no—
226C2×Q8—42no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4C22⋊Dic7911yescharacteristic
236C22×C4C22⋊Dic7911yescharacteristic
237C22×C4C22⋊Dic7911yescharacteristic
238C22×C4C22×Dic7911yescharacteristic
239C22×C4C2×C22⋊D7911yescharacteristic
240C22×C4C2×C22⋊D7911yescharacteristic
241M16—3162no—
242M16—3162no—
243C2×C8—3162no—
244C2×C8—1584no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C42—21no—
257C42—21no—
258C42—42no—
259C42D4×D7912yes—
260C42#1192962312yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊Dic7911yescharacteristic
267Q8—24no—
268Q8—42no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4D4×Dic7912yes—
295C2×C4#1457912911yescharacteristic
296C2×C4#1457913312yes—
297C2×C4#1457914211yescharacteristic
298C8—3164no—
299C8—6322no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1909844311yescharacteristic
308C22#1909858011yescharacteristic
309C22#1909858011yescharacteristic
310C4—22no—
311C4—21no—
312C4—21no—
313C4—22no—
314C4—42no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—81no—
319C4—81no—
320C4—81no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#2376381911yescharacteristic
325C2#2376384911yescharacteristic
326C2#2376392311yescharacteristic
subgroupG/H
centerC22#19098443
commutatorC22×C4×C79C22×C4
FrattiniC2×C42C22×D79
FittingC79×#2599C2
socleC22×C79#386
prime radicalC24×C79C22⋊C4
cosocleC2×C42×C79C23
hypercenter#2599D79
hypocenterC79#9493
LCSG ▷ C22×C4×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2599

Complements

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

NH ≅ G/Ntype
2. #23761847C2semidirect
4. #23761264C2semidirect
169. C79#9493semidirect

Maximal subgroups

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

subgroupindex
#237720382
#237618472
#237617342
#237612642
C79×#25992
#949379