Finite Group Databasebeta

← #34446419

Subgroups — #34446419

Order 20224.

472 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 127 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#23776538C211yesmaximal, characteristic, complemented
2#23776537C211yesmaximal, characteristic, complemented
3#23765715C211yesmaximal, characteristic, complemented
4#23765714C212yesmaximal, complemented
5#23765711C212yesmaximal, complemented
6#23765708C212yesmaximal, complemented
7#23765707C211yesmaximal, characteristic, complemented
8C2×#19098398C211yesmaximal, characteristic, complemented
9D32×D79C212yesmaximal, complemented
10#23765695C211yesmaximal, characteristic, complemented
11C79×#2263C211yesmaximal, characteristic, complemented
12#19099365—21no—
13#19099364—21no—
14#19099000—22no—
15#19098999—21no—
16#19098998—21no—
17#19098997—22no—
18#19098996C2212yescomplemented
19#19098995C2211yescharacteristic, complemented
20#19098992C2212yescomplemented
21C2×D1264C2211yescharacteristic, complemented
22D16×D79C2212yescomplemented
23D16×D79C2212yescomplemented
24#19098979C2211yescharacteristic
25#19098403C2211yescharacteristic, complemented
26#19098402C2212yescomplemented
27#19098401C2211yescharacteristic, complemented
28#19098400C2211yescharacteristic, complemented
29D2528C2212yescomplemented
30#19098398C2211yescharacteristic, complemented
31#19098398C2212yescomplemented
32#19098398C2212yescomplemented
33#19098396C2212yescomplemented
34C2×C79⋊C32C2211yescharacteristic, complemented
35C32×D79C2212yescomplemented
36#508×C79C2211yescharacteristic, complemented
37C2×D16×C79C2211yescharacteristic, complemented
38SD64×C79C2212yescomplemented
39D32×C79C2212yescomplemented
40M64×C79C2211yescharacteristic, complemented
41#14579207—41no—
42#14579206—41no—
43#14579188—42no—
44#14579187—41no—
45#14579186—41no—
46#14579185—42no—
47#14579184—22no—
48#14579183—21no—
49#14579180—22no—
50C2×D632—21no—
51D8×D79—22no—
52D8×D79—22no—
53#14579167D411yescharacteristic
54Dic632C2311yescharacteristic
55#14579106—21no—
56#14579105—22no—
57#14579104—22no—
58D1264—21no—
59D1264C2312yes—
60D1264C2311yescharacteristic
61#14579102—22no—
62#14579102—22no—
63#14579101D411yescharacteristic
64#14579100D412yes—
65C16×D79C2312yes—
66C79⋊C32C2311yescharacteristic
67C79⋊C32C2311yescharacteristic
68#135×C79—21no—
69C2×D8×C79—21no—
70Q32×C79C2311yescharacteristic
71SD32×C79—21no—
72D16×C79—21no—
73D16×C79C2312yes—
74D16×C79C2311yescharacteristic
75C2×C16×C79C2311yescharacteristic
76C32×C79C2312yes—
77#11929625—42no—
78#11929624—41no—
79#11929624—81no—
80#11929624D811yescharacteristic
81#11929623—42no—
82#11929623—81no—
83C2×D316—41no—
84C2×D316—81no—
85C2×C22⋊D79—82no—
86D4×D79—42no—
87D4×D79—42no—
88D4×D79—81no—
89D4×D79—82no—
90Dic316—21no—
91#11929613—41no—
92#11929612—42no—
93#11929611—42no—
94D632—21no—
95D632—22no—
96D632—41no—
97#11929609—42no—
98#11929609—42no—
99#11929608D811yescharacteristic
100#11929607D812yes—
101C8×D79C2×D412yes—
102C79⋊C16C2×D412yes—
103#41×C79—41no—
104C2×D4×C79—41no—
105Q16×C79—21no—
106SD16×C79—41no—
107D8×C79—22no—
108D8×C79—21no—
109D8×C79—41no—
110C2×C8×C79C2×D411yescharacteristic
111C16×C79C2×D411yescharacteristic
112C16×C79C2411yescharacteristic
113C23×C79—81no—
114C22×D79—81no—
115C22×D79—82no—
116C22×D79—82no—
117Q8×C79—41no—
118D4×C79—41no—
119D4×C79—42no—
120D4×C79—81no—
121D4×C79—81no—
122C2×C4×C79—81no—
123C2×C4×C79C2×D811yescharacteristic
124C22⋊D79—81no—
125C22⋊D79—82no—
126C22⋊D79—82no—
127C22⋊D79—82no—
128C22⋊D79D1612yes—
129C2×Dic79—82no—
130D316—41no—
131D316—42no—
132D316—82no—
133D316—81no—
134D316D1611yescharacteristic
135C4×D79—81no—
136C4×D79—82no—
137C4×D79C2×D812yes—
138Dic158—41no—
139Dic158D1611yescharacteristic
140C8×C79C2×D811yescharacteristic
141C8×C79C22×D411yescharacteristic
142C79⋊C8C2×D812yes—
143C22×C79—81no—
144C22×C79—82no—
145C22×C79—321no—
146C22×C79C2×D1611yescharacteristic
147C2×D79—81no—
148C2×D79—82no—
149C2×D79—322no—
150C2×D79—322no—
151C2×D79—321no—
152C2×D79C2×D1612yes—
153C4×C79—81no—
154C4×C79C2×D1611yescharacteristic
155C4×C79C22×D811yescharacteristic
156Dic79—81no—
157Dic79C2×D1612yes—
158#33982—791nomaximal, Sylow
159C2×C79—21no—
160C2×C79—161no—
161C2×C79—162no—
162C2×C79C22×D1611yescharacteristic
163D79—22no—
164D79—161no—
165D79—162no—
166#3415—791no—
167#3415—791no—
168#2263—792no—
169#2263—792no—
170#2263—791no—
171#2263D7911yescharacteristic, complemented
172#2262—791no—
173#2262—792no—
174C2×D32—791no—
175C2×D32—792no—
176#2258—791no—
177C79#3398211yescharacteristic, complemented, Sylow
178#576—1581no—
179#576—1581no—
180#509—1582no—
181#509—1582no—
182#509—1581no—
183#509—1581no—
184#508—792no—
185#508—792no—
186#508—791no—
187#508C2×D7911yescharacteristic, complemented
188C2×D16—792no—
189C2×D16—792no—
190C2×D16—791no—
191C2×D16C2×D7911yescharacteristic, complemented
192#504—791no—
193Q64—791no—
194SD64—791no—
195SD64—791no—
196SD64—792no—
197SD64C2×D7912yescomplemented
198D32—792no—
199D32—791no—
200D32—792no—
201D32—792no—
202D32C2×D7912yescomplemented
203M64—792no—
204M64C2×D7911yescharacteristic, complemented
205C2×C32—791no—
206C2×C32—792no—
207H(5,𝔽2)—3161no—
208H(5,𝔽2)—3161no—
209AGL(1,ℤ/8)—3162no—
210AGL(1,ℤ/8)—3162no—
211AGL(1,ℤ/8)—3161no—
212AGL(1,ℤ/8)—3161no—
213#135—21no—
214#135—1582no—
215#135—1582no—
216#135—1581no—
217C2×D8—21no—
218C2×D8—1582no—
219C2×D8—1582no—
220C2×D8—1581no—
221#131—791no—
222Q32—791no—
223Q32C22×D7911yescharacteristic
224SD32—21no—
225SD32—1582no—
226SD32—1582no—
227SD32—1581no—
228D16—21no—
229D16—792no—
230D16—791no—
231D16—1582no—
232D16—1582no—
233D16—1581no—
234D16C22×D7912yes—
235D16C22×D7911yescharacteristic
236M32—791no—
237M32—792no—
238C2×C16—792no—
239C2×C16C22×D7911yescharacteristic
240C32—791no—
241C32—791no—
242C32C22×D7912yes—
243#41—41no—
244#41—3162no—
245#41—3162no—
246#41—3161no—
247#41—791no—
248#41—6321no—
249#41—6321no—
250C2×D4—41no—
251C2×D4—3162no—
252C2×D4—3162no—
253C2×D4—3161no—
254C2×D4—6321no—
255C2×D4—6322no—
256C2×D4—6321no—
257C2×D4—6322no—
258Q16—21no—
259Q16—1581no—
260SD16—41no—
261SD16—3162no—
262SD16—3162no—
263SD16—3161no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—3162no—
268D8—3162no—
269D8—3161no—
270D8—1581no—
271D8—1582no—
272M16—791no—
273M16—792no—
274C2×C8—792no—
275C2×C8D4×D7911yescharacteristic
276C16—792no—
277C16D4×D7911yescharacteristic
278C16C23×D7911yescharacteristic
279C23—81no—
280C23—6322no—
281C23—6322no—
282C23—6321no—
283Q8—41no—
284Q8—3161no—
285Q8—791no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—3161no—
291D4—3162no—
292D4—792no—
293D4—791no—
294D4—6322no—
295D4—6322no—
296D4—6322no—
297D4—6322no—
298D4—6321no—
299D4—6321no—
300C2×C4—81no—
301C2×C4—792no—
302C2×C4—6322no—
303C2×C4—6322no—
304C2×C4—6321no—
305C2×C4D8×D7911yescharacteristic
306C8—792no—
307C8D8×D7911yescharacteristic
308C8C2×D4×D7911yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—792no—
313C22—6322no—
314C22—6321no—
315C22—25281no—
316C22—25282no—
317C22—25282no—
318C22D16×D7911yescharacteristic
319C4—81no—
320C4—792no—
321C4—6321no—
322C4D16×D7911yescharacteristic
323C4C2×D8×D7911yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—1582no—
328C2—12642no—
329C2—12641no—
330C2C2×D16×D7911yescharacteristic
subgroupG/H
centerC2C2×D16×D79
commutator, cosocleC16×C79C24
FrattiniC16C23×D79
FittingC79×#2263C2
socleC2×C79C22×D16
hypercenter#2263D79
hypocenterC79#33982
LCSG ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. #23776538C2semidirect
2. #23776537C2semidirect
3. #23765715C2semidirect
4. #23765714C2semidirect
5. #23765711C2semidirect
6. #23765708C2semidirect
7. #23765707C2semidirect
8. C2×#19098398C2semidirect
9. D32×D79C2semidirect
10. #23765695C2semidirect
11. C79×#2263C2semidirect
18. #19098996C22semidirect
19. #19098995C22semidirect
20. #19098992C22semidirect
21. C2×D1264C22semidirect
22. D16×D79C22semidirect
23. D16×D79C22semidirect
25. #19098403C22semidirect
26. #19098402C22semidirect
27. #19098401C22semidirect
28. #19098400C22semidirect
29. D2528C22semidirect
30. #19098398C22semidirect
31. #19098398C22semidirect
32. #19098398C22semidirect
33. #19098396C22semidirect
34. C2×C79⋊C32C22semidirect
35. C32×D79C22semidirect
36. #508×C79C22semidirect
37. C2×D16×C79C22semidirect
38. SD64×C79C22semidirect
39. D32×C79C22semidirect
40. M64×C79C22semidirect
171. #2263D79semidirect
177. C79#33982semidirect
187. #508C2×D79semidirect
191. C2×D16C2×D79semidirect
197. SD64C2×D79semidirect
202. D32C2×D79semidirect
204. M64C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237765382
#237765372
#237657152
#237657142
#237657112
#237657082
#237657072
C2×#190983982
D32×D792
#237656952
C79×#22632
#3398279