Finite Group Databasebeta

← #34269556

Subgroups — #34269556

Order 20224.

960 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 171 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#23774387C212yesmaximal, complemented
2#23773803C211yesmaximal, characteristic, complemented
3#23762723C214yesmaximal, complemented
4#23762693C214yesmaximal, complemented
5#23762027C211yesmaximal, characteristic, complemented
6#23761998C212yesmaximal, complemented
7C2×#562×C79C211yesmaximal, characteristic, complemented
8#19099113C2211yescharacteristic
9#19098953C2214yescomplemented
10#19098933C2212yescomplemented
11#19098929C2212yescomplemented
12#19098922C2214yescomplemented
13#19098920—22no—
14#19098775—24no—
15#19098702—22no—
16#19098697—22no—
17#19098620C2212yes—
18#19098562—22no—
19C4×#11929608C2211yescharacteristic
20#19098305C2214yescomplemented
21#19098303C2218yescomplemented
22#562×C79—22no—
23#562×C79—24no—
24#562×C79C2214yes—
25C2×#125×C79C2211yescharacteristic
26C2×#124×C79C2212yes—
27C2×#123×C79—22no—
28C2×C22⋊Q8×C79—22no—
29C2×#121×C79—22no—
30C2×#11929624—21no—
31C2×#11929611—24no—
32C2×#11929609—24no—
33C2×#11929608—21no—
34#14579161C2311yescharacteristic
35#14579158C2311yescharacteristic
36#14579149—21no—
37#14579135—22no—
38C4×D316—22no—
39C4⋊(C79⋊C8)—22no—
40#14579085—24no—
41#14579084—24no—
42#14579082—24no—
43#14579081—24no—
44C22⋊(C79⋊C8)—22no—
45#14579072C2314yes—
46#14579072C2312yes—
47C4×C79⋊C8—21no—
48C22×Q8×C79D412yes—
49C22×D4×C79—22no—
50C22×D4×C79D412yes—
51C23×C4×C79—22no—
52#125×C79—22no—
53#125×C79D412yes—
54#125×C79C2312yes—
55#124×C79—22no—
56#124×C79—22no—
57#124×C79D414yes—
58#124×C79C2314yes—
59#123×C79—28no—
60#123×C79—28no—
61C22⋊Q8×C79—28no—
62C22⋊Q8×C79—28no—
63#121×C79—28no—
64#121×C79—28no—
65C2×C4⋊C4×C79—22no—
66C2×C4⋊C4×C79—22no—
67C2×C4⋊C4×C79D412yes—
68C2×C22⋊C4×C79—22no—
69C2×C22⋊C4×C79—22no—
70C2×C22⋊C4×C79—22no—
71C2×C42×C79C2311yescharacteristic
72#11929624—44no—
73C2×Dic158—21no—
74C2×D316—21no—
75C2×D316—22no—
76C2×C22⋊D79—41no—
77C2×C4×D79—41no—
78#11929611—84no—
79#11929609—84no—
80#11929608—42no—
81C2×C79⋊C8—22no—
82C2×C79⋊C8—24no—
83C4⋊Dic79—22no—
84(C2×D79)⋊C4—42no—
85C24×C79—22no—
86C2×Q8×C79—22no—
87C2×Q8×C79—82no—
88C2×Q8×C79C2×D414yes—
89C2×D4×C79—28no—
90C2×D4×C79—22no—
91C2×D4×C79—82no—
92C2×D4×C79—84no—
93C2×D4×C79C2×D414yes—
94C22×C4×C79—22no—
95C22×C4×C79—22no—
96C22×C4×C79—28no—
97C22×C4×C79—84no—
98C22×C4×C79C2×D411yescharacteristic
99C22×C4×C79C2×D412yes—
100C4⋊C4×C79—28no—
101C4⋊C4×C79—24no—
102C4⋊C4×C79—22no—
103C4⋊C4×C79—24no—
104C4⋊C4×C79C2×D414yes—
105C22⋊C4×C79—28no—
106C22⋊C4×C79—24no—
107C22⋊C4×C79—24no—
108C22⋊C4×C79—24no—
109C22⋊C4×C79—24no—
110C42×C79C2×D411yescharacteristic
111C42×C79C2×D412yes—
112C42×C79C2411yescharacteristic
113C23×C79—24no—
114C23×C79—44no—
115C23×C79—44no—
116C23×C79—44no—
117C23×C79#12011yescharacteristic
118C22×D79—41no—
119Q8×C79—44no—
120Q8×C79—82no—
121D4×C79—44no—
122D4×C79—82no—
123D4×C79—88no—
124C2×C4×C79—22no—
125C2×C4×C79—24no—
126C2×C4×C79—24no—
127C2×C4×C79—42no—
128C2×C4×C79—44no—
129C2×C4×C79—44no—
130C2×C4×C79—88no—
131C2×C4×C79#12012yes—
132C2×C4×C79#12011yescharacteristic
133C2×C4×C79C22×D412yes—
134C2×C4×C79C22×D411yescharacteristic
135C22⋊D79—82no—
136C2×Dic79—41no—
137D316—44no—
138D316—42no—
139C4×D79—82no—
140Dic158—42no—
141C79⋊C8—42no—
142C79⋊C8—44no—
143C22×C79—22no—
144C22×C79—44no—
145C22×C79—44no—
146C22×C79—44no—
147C22×C79—88no—
148C22×C79#45412yes—
149C22×C79C2×#12011yescharacteristic
150C2×D79—81no—
151C2×D79—82no—
152C4×C79—21no—
153C4×C79—21no—
154C4×C79—22no—
155C4×C79—22no—
156C4×C79—44no—
157C4×C79—44no—
158Dic79—81no—
159#19001—791nomaximal, Sylow
160C2×C79—22no—
161C2×C79—84no—
162C2×C79C2×#45411yescharacteristic
163C2×C79#301912yes—
164D79—161no—
165C2×#562D7911yescharacteristic, complemented
166#3213—792no—
167#3142—791no—
168#1662—794no—
169#1656—794no—
170#1534—791no—
171#1527—792no—
172C79#1900111yescharacteristic, complemented, Sylow
173#562—22no—
174#562—24no—
175#562C2×D7914yes—
176#532—791no—
177C2×#125C2×D7911yescharacteristic
178C2×#124C2×D7912yes—
179C2×#123—22no—
180C2×C22⋊Q8—22no—
181C2×#121—22no—
182#496—794no—
183#491—792no—
184#490—792no—
185#489—794no—
186#488—1582no—
187#463—1584no—
188#448—1582no—
189#447—1582no—
190#432—792no—
191#417—1582no—
192C4×M16—791no—
193#330—794no—
194#329—798no—
195C2×#41—1581no—
196C22×Q8C22⋊D7912yes—
197C22×D4—22no—
198C22×D4C22⋊D7912yes—
199C23×C4—22no—
200C2×SD16—1584no—
201C2×D8—1584no—
202C2×M16—1581no—
203C4⋊Q8—791no—
204#127—791no—
205#125—22no—
206#125C22⋊D7912yes—
207#125C22×D7912yes—
208#124—22no—
209#124—22no—
210#124—1581no—
211#124C22⋊D7914yes—
212#124C22×D7914yes—
213#123—28no—
214#123—28no—
215C22⋊Q8—28no—
216C22⋊Q8—28no—
217#121—28no—
218#121—28no—
219#121—1582no—
220C4×D4—1582no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C22⋊D7912yes—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42C22×D7911yescharacteristic
228C4⋊C8—1582no—
229#103—1584no—
230#103—1584no—
231#102—1584no—
232#102—1584no—
233C22⋊C8—1582no—
234#97—792no—
235#97—794no—
236C4×C8—1581no—
237C24—22no—
238#41—3164no—
239C2×Q8—22no—
240C2×Q8—82no—
241C2×Q8—1581no—
242C2×Q8C2×C22⋊D7914yes—
243C2×D4—28no—
244C2×D4—22no—
245C2×D4—82no—
246C2×D4—84no—
247C2×D4—3161no—
248C2×D4—1581no—
249C2×D4—1582no—
250C2×D4C2×C22⋊D7914yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—84no—
255C22×C4—3161no—
256C22×C4C2×C22⋊D7911yescharacteristic
257C22×C4C2×C22⋊D7912yes—
258SD16—6324no—
259D8—6324no—
260M16—3162no—
261C2×C8—1582no—
262C2×C8—1584no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—24no—
267C4⋊C4—1582no—
268C4⋊C4C2×C22⋊D7914yes—
269C22⋊C4—28no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—3162no—
275C42C2×C22⋊D7911yescharacteristic
276C42C2×C22⋊D7912yes—
277C42C23×D7911yescharacteristic
278C23—24no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—3161no—
283C23#1457913011yescharacteristic
284Q8—44no—
285Q8—82no—
286Q8—3162no—
287D4—44no—
288D4—82no—
289D4—88no—
290D4—3164no—
291D4—3162no—
292D4—6322no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—88no—
300C2×C4—3161no—
301C2×C4—6322no—
302C2×C4#1457913011yescharacteristic
303C2×C4#1457913012yes—
304C2×C4C22×C22⋊D7911yescharacteristic
305C2×C4C22×C22⋊D7912yes—
306C8—3162no—
307C8—3164no—
308C22—22no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—88no—
313C22—6321no—
314C22—6322no—
315C22#1909873612yes—
316C22C2×#1457913011yescharacteristic
317C4—21no—
318C4—21no—
319C4—22no—
320C4—22no—
321C4—44no—
322C4—44no—
323C4—6321no—
324C2—22no—
325C2—84no—
326C2—12641no—
327C2C2×#1909873611yescharacteristic
328C2#2377275712yes—
subgroupG/H
centerC22C2×#14579130
commutator, cosocleC42×C79C24
FrattiniC42C23×D79
FittingC2×#562×C79C2
socleC22×C79C2×#120
hypercenterC2×#562D79
hypocenterC79#19001
LCSG ▷ C42×C79 ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#562

Complements

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

NH ≅ G/Ntype
1. #23774387C2semidirect
2. #23773803C2semidirect
3. #23762723C2semidirect
4. #23762693C2semidirect
5. #23762027C2semidirect
6. #23761998C2semidirect
7. C2×#562×C79C2semidirect
9. #19098953C22semidirect
10. #19098933C22semidirect
11. #19098929C22semidirect
12. #19098922C22semidirect
20. #19098305C22semidirect
21. #19098303C22semidirect
165. C2×#562D79semidirect
172. C79#19001semidirect

Maximal subgroups

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

subgroupindex
#237743872
#237738032
#237627232
#237626932
#237620272
#237619982
C2×#562×C792
#1900179