Finite Group Databasebeta

← #34445092

Subgroups — #34445092

Order 20224.

654 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23775748C211yesmaximal, characteristic, complemented
2#23765623C211yesmaximal, characteristic, complemented
3#23765578C212yesmaximal, complemented
4#23765537C211yesmaximal, characteristic, complemented
5#23765484C212yesmaximal, complemented
6#23765432C211yesmaximal, characteristic, complemented
7#23765312C212yesmaximal, complemented
8C4×#14579102C211yesmaximal, characteristic, complemented
9D4×C79⋊C16C211yesmaximal, characteristic, complemented
10C2×#19098364C212yesmaximal, complemented
11D4×D8×C79C211yesmaximal, characteristic, complemented
12#19099217—21no—
13C2×#14579102C2211yescharacteristic
14C2×#14579102C2212yescomplemented
15C22×C79⋊C16C2212yescomplemented
16#19098968C2211yescharacteristic, complemented
17#19098877—22no—
18#19098844—21no—
19#19098799C2212yescomplemented
20C4×D632C2211yescharacteristic, complemented
21C2×#14579095C2212yescomplemented
22#19098388C2212yescomplemented
23#19098388C2211yescharacteristic, complemented
24C4⋊(C79⋊C16)C2211yescharacteristic, complemented
25#19098364C2212yes—
26#19098364C2211yescharacteristic
27#19098364C2212yescomplemented
28#19098364C2212yescomplemented
29#19098364C2212yescomplemented
30#19098363C2211yescharacteristic, complemented
31C22⋊(C79⋊C16)C2212yescomplemented
32C4×C79⋊C16C2211yescharacteristic, complemented
33C22×D8×C79C2212yes—
34D42×C79—21no—
35#493×C79C2211yescharacteristic
36#466×C79C2212yes—
37#459×C79—21no—
38#447×C79—22no—
39C4×D8×C79C2211yescharacteristic
40C8×D4×C79C2211yescharacteristic
41C2×#11929623—41no—
42C2×D632C2311yescharacteristic
43#14579161—21no—
44#14579145—42no—
45#14579141—41no—
46#14579135—22no—
47C4×D316—21no—
48C2×C4⋊Dic79—22no—
49#14579102—22no—
50#14579102—42no—
51C2×C79⋊C16—42no—
52C2×C79⋊C16C2312yes—
53C2×C79⋊C16C2311yescharacteristic
54C2×C79⋊C16C2311yescharacteristic
55#14579095—21no—
56#14579095—22no—
57#14579095C2311yescharacteristic
58#14579095C2312yes—
59#14579083—21no—
60#14579083—22no—
61C22×D4×C79—22no—
62C22×D4×C79—41no—
63C2×D8×C79—24no—
64C2×D8×C79—22no—
65C2×D8×C79—21no—
66C2×D8×C79—42no—
67C2×D8×C79C2312yes—
68C2×D8×C79C2311yescharacteristic
69C22×C8×C79C2312yes—
70#127×C79—21no—
71#121×C79—22no—
72#121×C79—41no—
73#120×C79—42no—
74C4×D4×C79—21no—
75C4×D4×C79D411yescharacteristic
76#107×C79C2311yescharacteristic
77C4⋊C8×C79D411yescharacteristic
78#102×C79—21no—
79#102×C79—22no—
80C22⋊C8×C79D412yes—
81C4×C8×C79C2311yescharacteristic
82#11929623—84no—
83C2×Dic158—41no—
84C2×D316—21no—
85C2×C22⋊D79—42no—
86C22×Dic79—42no—
87C2×C4×D79—41no—
88D632—21no—
89D632—21no—
90C4⋊Dic79—21no—
91C4⋊Dic79—22no—
92C4⋊Dic79—42no—
93C4⋊Dic79—41no—
94(C2×D79)⋊C4—41no—
95(C2×D79)⋊C4—42no—
96C79⋊C16—21no—
97C79⋊C16—21no—
98C79⋊C16—22no—
99C24×C79—42no—
100C2×D4×C79—21no—
101C2×D4×C79—22no—
102C2×D4×C79—44no—
103C2×D4×C79—42no—
104C2×D4×C79—42no—
105C2×D4×C79—41no—
106C2×D4×C79—41no—
107C2×D4×C79—82no—
108C2×D4×C79—84no—
109C2×D4×C79D811yescharacteristic
110C22×C4×C79—41no—
111C22×C4×C79C2×D412yes—
112D8×C79—21no—
113D8×C79—22no—
114D8×C79—22no—
115D8×C79—44no—
116D8×C79—81no—
117D8×C79#4112yescomplemented
118C2×C8×C79—22no—
119C2×C8×C79—22no—
120C2×C8×C79C2×D411yescharacteristic
121C2×C8×C79C2×D412yes—
122C2×C8×C79C2411yescharacteristic
123C4⋊C4×C79—21no—
124C4⋊C4×C79D811yescharacteristic
125C22⋊C4×C79—41no—
126C22⋊C4×C79—42no—
127C22⋊C4×C79D812yes—
128C42×C79C2×D411yescharacteristic
129C23×C79—41no—
130C23×C79—42no—
131C23×C79—84no—
132C23×C79—84no—
133C23×C79—162no—
134C23×C79C2×D812yes—
135C22×D79—41no—
136D4×C79—22no—
137D4×C79—42no—
138D4×C79—41no—
139D4×C79—42no—
140D4×C79—84no—
141D4×C79—84no—
142D4×C79—82no—
143D4×C79—161no—
144D4×C79D1614yescomplemented
145C2×C4×C79—22no—
146C2×C4×C79—22no—
147C2×C4×C79—41no—
148C2×C4×C79—82no—
149C2×C4×C79C2×D811yescharacteristic
150C2×C4×C79C2×D812yes—
151C2×C4×C79C2×D811yescharacteristic
152C2×C4×C79C22×D411yescharacteristic
153C22⋊D79—84no—
154C2×Dic79—41no—
155C2×Dic79—42no—
156C2×Dic79—84no—
157D316—41no—
158D316—41no—
159C4×D79—82no—
160Dic158—82no—
161C8×C79—22no—
162C8×C79—21no—
163C8×C79C2×#4111yescharacteristic
164C8×C79H(5,𝔽2)11yescharacteristic
165C22×C79—22no—
166C22×C79—22no—
167C22×C79—42no—
168C22×C79—81no—
169C22×C79—81no—
170C22×C79—82no—
171C22×C79—82no—
172C22×C79—82no—
173C22×C79—168no—
174C22×C79C2×D1614yes—
175C22×C79C22×D811yescharacteristic
176C2×D79—81no—
177C2×D79—81no—
178C2×D79—81no—
179C4×C79—21no—
180C4×C79—22no—
181C4×C79—81no—
182C4×C79C2×D1612yes—
183C4×C79#54611yescharacteristic
184C4×C79#57711yescharacteristic
185Dic79—82no—
186Dic79—81no—
187#33861—791nomaximal, Sylow
188C2×C79—24no—
189C2×C79—82no—
190C2×C79—162no—
191C2×C79#329411yescharacteristic
192C2×C79C22×D1611yescharacteristic
193C2×C79#341611yescharacteristic
194D79—161no—
195#3334—791no—
196D4×D8D7911yescharacteristic, complemented
197#2240—791no—
198#2232—792no—
199#2224—791no—
200#2215—792no—
201#2206—791no—
202#2184—792no—
203C4×D16—791no—
204D4×C16—791no—
205C2×#357—792no—
206C79#3386111yescharacteristic, complemented, Sylow
207C22×D8C2×D7912yes—
208#547—1581no—
209D42—21no—
210C2×D16—792no—
211C2×D16—791no—
212C22×C16—792no—
213#500—791no—
214#493C2×D7911yescharacteristic
215#480—1582no—
216#474—1581no—
217#466—792no—
218#466C2×D7912yes—
219#459—21no—
220#447—22no—
221C4×D8—791no—
222C4×D8C2×D7911yescharacteristic
223C8×D4C2×D7911yescharacteristic
224C2×#107—792no—
225#366—791no—
226#366—792no—
227C4⋊C16—791no—
228#357—791no—
229#357—792no—
230#357—792no—
231#357—792no—
232#357—792no—
233#357—791no—
234C22⋊C16—792no—
235C4×C16—791no—
236C2×#41—3161no—
237C22×D4—22no—
238C22×D4—41no—
239C2×D8—24no—
240C2×D8—22no—
241C2×D8—21no—
242C2×D8—42no—
243C2×D8—791no—
244C2×D8C22×D7912yes—
245C2×D8C22×D7911yescharacteristic
246C22×C8C22×D7912yes—
247C4⋊Q8—1581no—
248#127—21no—
249#123—3162no—
250C22⋊Q8—3161no—
251#121—22no—
252#121—41no—
253#121—1582no—
254#120—42no—
255C4×D4—21no—
256C4×D4—1581no—
257C4×D4C22⋊D7911yescharacteristic
258C2×C4⋊C4—1582no—
259D16—3162no—
260D16—1582no—
261C2×C16—3162no—
262C2×C16—791no—
263C2×C16—791no—
264C2×C16—792no—
265#107—791no—
266#107—792no—
267#107—1582no—
268#107—1581no—
269#107C22×D7911yescharacteristic
270C4⋊C8C22⋊D7911yescharacteristic
271#102—21no—
272#102—22no—
273#102—1581no—
274#102—1582no—
275C22⋊C8C22⋊D7912yes—
276C4×C8C22×D7911yescharacteristic
277C24—42no—
278#41—6324no—
279C2×Q8—3161no—
280C2×D4—21no—
281C2×D4—22no—
282C2×D4—44no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—82no—
288C2×D4—84no—
289C2×D4—3162no—
290C2×D4—1581no—
291C2×D4#1192960911yescharacteristic
292C22×C4—41no—
293C22×C4—3162no—
294C22×C4—3161no—
295C22×C4C2×C22⋊D7912yes—
296D8—22no—
297D8—21no—
298D8—22no—
299D8—44no—
300D8—81no—
301D8—1581no—
302D8—1581no—
303D8#1192962312yescomplemented
304C2×C8—22no—
305C2×C8—22no—
306C2×C8C2×C22⋊D7911yescharacteristic
307C2×C8C2×C22⋊D7912yes—
308C2×C8C23×D7911yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—3162no—
311C4⋊C4—3161no—
312C4⋊C4—1581no—
313C4⋊C4—1582no—
314C4⋊C4#1192960911yescharacteristic
315C22⋊C4—41no—
316C22⋊C4—42no—
317C22⋊C4—3161no—
318C22⋊C4—3162no—
319C22⋊C4#1192960912yes—
320C42C2×C22⋊D7911yescharacteristic
321C16—1581no—
322C16—1581no—
323C16—1582no—
324C23—41no—
325C23—42no—
326C23—84no—
327C23—84no—
328C23—162no—
329C23—3161no—
330C23C2×#1192960912yes—
331Q8—6322no—
332D4—22no—
333D4—42no—
334D4—41no—
335D4—42no—
336D4—84no—
337D4—84no—
338D4—82no—
339D4—161no—
340D4—3161no—
341D4—3161no—
342D4—6324no—
343D4#1457910214yescomplemented
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—82no—
348C2×C4—3161no—
349C2×C4—3162no—
350C2×C4—6324no—
351C2×C4—6322no—
352C2×C4C2×#1192960911yescharacteristic
353C2×C4C2×#1192960912yes—
354C2×C4C2×#1192960911yescharacteristic
355C2×C4C22×C22⋊D7911yescharacteristic
356C8—22no—
357C8—21no—
358C8C2×#1192962311yescharacteristic
359C8#1457920611yescharacteristic
360C22—22no—
361C22—22no—
362C22—42no—
363C22—81no—
364C22—81no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—168no—
369C22—6321no—
370C22—6321no—
371C22—6321no—
372C22C2×#1457910214yes—
373C22C22×#1192960911yescharacteristic
374C4—21no—
375C4—22no—
376C4—81no—
377C4—6321no—
378C4—6322no—
379C4C2×#1457910212yes—
380C4#1909920711yescharacteristic
381C4#1909937411yescharacteristic
382C2—24no—
383C2—82no—
384C2—162no—
385C2—12641no—
386C2#2377521311yescharacteristic
387C2C22×#1457910211yescharacteristic
388C2#2377654711yescharacteristic
subgroupG/H
centerC22C22×#11929609
commutator, cosocleC2×C8×C79C24
FrattiniC2×C8C23×D79
FittingD4×D8×C79C2
socleC22×C79C22×D8
hypercenterD4×D8D79
hypocenterC79#33861
LCSG ▷ C2×C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. #23775748C2semidirect
2. #23765623C2semidirect
3. #23765578C2semidirect
4. #23765537C2semidirect
5. #23765484C2semidirect
6. #23765432C2semidirect
7. #23765312C2semidirect
8. C4×#14579102C2semidirect
9. D4×C79⋊C16C2semidirect
10. C2×#19098364C2semidirect
11. D4×D8×C79C2semidirect
14. C2×#14579102C22semidirect
15. C22×C79⋊C16C22semidirect
16. #19098968C22semidirect
19. #19098799C22semidirect
20. C4×D632C22semidirect
21. C2×#14579095C22semidirect
22. #19098388C22semidirect
23. #19098388C22semidirect
24. C4⋊(C79⋊C16)C22semidirect
27. #19098364C22semidirect
28. #19098364C22semidirect
29. #19098364C22semidirect
30. #19098363C22semidirect
31. C22⋊(C79⋊C16)C22semidirect
32. C4×C79⋊C16C22semidirect
117. D8×C79#41semidirect
144. D4×C79D16semidirect
196. D4×D8D79semidirect
206. C79#33861semidirect
303. D8#11929623semidirect
343. D4#14579102semidirect

Maximal subgroups

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

subgroupindex
#237757482
#237656232
#237655782
#237655372
#237654842
#237654322
#237653122
C4×#145791022
D4×C79⋊C162
C2×#190983642
D4×D8×C792
#3386179