Finite Group Databasebeta

← #34446422

Subgroups — #34446422

Order 20224.

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

normal subgroups only

This group has 462 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#23776550C211yesmaximal, characteristic, complemented
2#23776538C211yesmaximal, characteristic, complemented
3#23765719C212yesmaximal, complemented
4#23765717C211yesmaximal, characteristic, complemented
5#23765712C211yesmaximal, characteristic, complemented
6#23765711C212yesmaximal, complemented
7#23765708C212yesmaximal, complemented
8C2×#19098402C211yesmaximal, characteristic, complemented
9SD64×D79C212yesmaximal, complemented
10#23765693C211yesmaximal, characteristic, complemented
11C79×#2262C211yesmaximal, characteristic, complemented
12#19099377—21no—
13#19099365—21no—
14#19099003—21no—
15#19099002—22no—
16#19099000—22no—
17#19098998—21no—
18#19098996C2212yes—
19#19098992C2212yescomplemented
20C2×Dic632C2211yescharacteristic, complemented
21Q32×D79C2212yescomplemented
22C2×D1264C2211yescharacteristic
23D16×D79C2212yes—
24#19098979C2211yescharacteristic
25#19098403C2212yescomplemented
26#19098402C2212yescomplemented
27#19098402C2212yescomplemented
28#19098401C2212yescomplemented
29#19098400C2212yescomplemented
30#19098398C2212yescomplemented
31#19098397C2211yescharacteristic, complemented
32#19098396C2212yescomplemented
33C32×D79C2212yescomplemented
34#508×C79C2211yescharacteristic
35#508×C79C2211yescharacteristic, complemented
36Q64×C79C2211yescharacteristic, complemented
37SD64×C79C2211yescharacteristic, complemented
38SD64×C79C2211yescharacteristic, complemented
39D32×C79C2211yescharacteristic, complemented
40C2×C32×C79C2211yescharacteristic, complemented
41#14579208—41no—
42#14579207—41no—
43#14579191—41no—
44#14579190—42no—
45#14579188—42no—
46#14579186—41no—
47#14579184—22no—
48#14579180—22no—
49C2×Dic316—21no—
50Q16×D79—22no—
51C2×D632—21no—
52D8×D79—22no—
53#14579167D411yescharacteristic
54Dic632—21no—
55Dic632C2312yes—
56#14579107—22no—
57#14579105—22no—
58#14579104—22no—
59D1264—21no—
60D1264C2312yes—
61#14579102—22no—
62#14579101D411yescharacteristic
63#14579100D412yes—
64C16×D79C2312yes—
65C79⋊C32C2312yes—
66#135×C79—21no—
67#135×C79—21no—
68Q32×C79C2311yescharacteristic
69Q32×C79C2311yescharacteristic
70SD32×C79—21no—
71SD32×C79—21no—
72D16×C79C2311yescharacteristic
73D16×C79C2311yescharacteristic
74C2×C16×C79C2311yescharacteristic
75C32×C79C2311yescharacteristic
76C32×C79C2311yescharacteristic
77#11929625—42no—
78#11929624—81no—
79#11929624—81no—
80#11929624D811yescharacteristic
81#11929623—42no—
82#11929623—82no—
83C2×Dic158—41no—
84C2×Dic158—81no—
85Q8×D79—42no—
86C2×D316—41no—
87C2×D316—81no—
88D4×D79—42no—
89D4×D79—82no—
90Dic316—22no—
91Dic316—41no—
92#11929614—42no—
93#11929612—42no—
94#11929611—42no—
95D632—22no—
96D632—41no—
97#11929609—42no—
98#11929608D811yescharacteristic
99#11929607D812yes—
100C8×D79C2×D412yes—
101C79⋊C16C2×D412yes—
102#41×C79—41no—
103#41×C79—41no—
104Q16×C79—21no—
105Q16×C79—21no—
106SD16×C79—41no—
107SD16×C79—41no—
108D8×C79—21no—
109D8×C79—21no—
110C2×C8×C79C2×D411yescharacteristic
111C16×C79C2×D411yescharacteristic
112C16×C79C2411yescharacteristic
113C22×D79—81no—
114C22×D79—82no—
115Q8×C79—41no—
116Q8×C79—41no—
117D4×C79—41no—
118D4×C79—41no—
119D4×C79—81no—
120D4×C79—81no—
121C2×C4×C79—81no—
122C2×C4×C79—81no—
123C2×C4×C79C2×D811yescharacteristic
124C22⋊D79—82no—
125C22⋊D79—82no—
126C22⋊D79D1612yes—
127C2×Dic79—81no—
128C2×Dic79—82no—
129D316—42no—
130D316—82no—
131D316—81no—
132D316D1611yescharacteristic
133C4×D79—82no—
134C4×D79—82no—
135C4×D79C2×D812yes—
136Dic158—42no—
137Dic158—81no—
138Dic158—82no—
139Dic158D1611yescharacteristic
140C8×C79C2×D811yescharacteristic
141C8×C79C22×D411yescharacteristic
142C79⋊C8C2×D812yes—
143C22×C79—81no—
144C22×C79—81no—
145C22×C79C2×D1611yescharacteristic
146C2×D79—82no—
147C2×D79—322no—
148C2×D79—321no—
149C2×D79C2×D1612yes—
150C4×C79—81no—
151C4×C79—81no—
152C4×C79C2×D1611yescharacteristic
153C4×C79C22×D811yescharacteristic
154Dic79—82no—
155Dic79C2×D1612yes—
156#33983—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—161no—
159C2×C79—161no—
160C2×C79C22×D1611yescharacteristic
161D79—22no—
162D79—162no—
163#3417—791no—
164#3415—791no—
165#2264—791no—
166#2264—792no—
167#2263—791no—
168#2263—792no—
169#2262—792no—
170#2262D7911yescharacteristic, complemented
171C2×SD64—792no—
172C2×SD64—791no—
173#2258—791no—
174C79#3398311yescharacteristic, complemented, Sylow
175#578—1581no—
176#576—1581no—
177#510—1582no—
178#510—1581no—
179#509—1582no—
180#509—1581no—
181#508—792no—
182#508—792no—
183#508C2×D7911yescharacteristic
184#508C2×D7911yescharacteristic, complemented
185C2×Q32—792no—
186C2×Q32—791no—
187C2×D16—792no—
188C2×D16—791no—
189#504—791no—
190Q64—792no—
191Q64C2×D7911yescharacteristic, complemented
192SD64—792no—
193SD64—792no—
194SD64—792no—
195SD64—792no—
196SD64C2×D7911yescharacteristic, complemented
197SD64C2×D7911yescharacteristic, complemented
198D32—792no—
199D32C2×D7911yescharacteristic, complemented
200M64—791no—
201M64—792no—
202C2×C32—792no—
203C2×C32C2×D7911yescharacteristic, complemented
2042−1+4—3161no—
205H(5,𝔽2)—3161no—
206#137—3162no—
207#137—3161no—
208AGL(1,ℤ/8)—3162no—
209AGL(1,ℤ/8)—3161no—
210#135—21no—
211#135—21no—
212#135—1582no—
213#135—1582no—
214C2×Q16—1582no—
215C2×Q16—1581no—
216C2×D8—1582no—
217C2×D8—1581no—
218#131—791no—
219Q32—792no—
220Q32—1582no—
221Q32—1581no—
222Q32C22×D7911yescharacteristic
223Q32C22×D7911yescharacteristic
224SD32—21no—
225SD32—21no—
226SD32—1582no—
227SD32—1582no—
228D16—792no—
229D16—1582no—
230D16—1581no—
231D16C22×D7911yescharacteristic
232D16C22×D7911yescharacteristic
233M32—791no—
234M32—792no—
235C2×C16—792no—
236C2×C16C22×D7911yescharacteristic
237C32—792no—
238C32C22×D7911yescharacteristic
239C32C22×D7911yescharacteristic
240#41—41no—
241#41—41no—
242#41—3162no—
243#41—3162no—
244#41—791no—
245#41—6321no—
246#41—6321no—
247#41—6322no—
248C2×Q8—3162no—
249C2×Q8—3161no—
250C2×Q8—6321no—
251C2×D4—3162no—
252C2×D4—3161no—
253C2×D4—6321no—
254C2×D4—6322no—
255Q16—21no—
256Q16—21no—
257Q16—3162no—
258Q16—3161no—
259Q16—1582no—
260SD16—41no—
261SD16—41no—
262SD16—3162no—
263SD16—3162no—
264D8—21no—
265D8—21no—
266D8—3162no—
267D8—3161no—
268D8—1582no—
269M16—791no—
270M16—792no—
271C2×C8—792no—
272C2×C8D4×D7911yescharacteristic
273C16—792no—
274C16D4×D7911yescharacteristic
275C16C23×D7911yescharacteristic
276C23—6322no—
277C23—6321no—
278Q8—41no—
279Q8—41no—
280Q8—3162no—
281Q8—791no—
282Q8—6322no—
283Q8—6321no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—3162no—
289D4—792no—
290D4—791no—
291D4—6322no—
292D4—6322no—
293D4—6322no—
294D4—6321no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—792no—
298C2×C4—6322no—
299C2×C4—6322no—
300C2×C4—6322no—
301C2×C4—6321no—
302C2×C4D8×D7911yescharacteristic
303C8—792no—
304C8D8×D7911yescharacteristic
305C8C2×D4×D7911yescharacteristic
306C22—81no—
307C22—81no—
308C22—792no—
309C22—6322no—
310C22—25281no—
311C22—25282no—
312C22D16×D7911yescharacteristic
313C4—81no—
314C4—81no—
315C4—792no—
316C4—6322no—
317C4D16×D7911yescharacteristic
318C4C2×D8×D7911yescharacteristic
319C2—21no—
320C2—161no—
321C2—161no—
322C2—1582no—
323C2—12642no—
324C2C2×D16×D7911yescharacteristic
subgroupG/H
centerC2C2×D16×D79
commutator, cosocleC16×C79C24
FrattiniC16C23×D79
FittingC79×#2262C2
socleC2×C79C22×D16
hypercenter#2262D79
hypocenterC79#33983
LCSG ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #23776550C2semidirect
2. #23776538C2semidirect
3. #23765719C2semidirect
4. #23765717C2semidirect
5. #23765712C2semidirect
6. #23765711C2semidirect
7. #23765708C2semidirect
8. C2×#19098402C2semidirect
9. SD64×D79C2semidirect
10. #23765693C2semidirect
11. C79×#2262C2semidirect
19. #19098992C22semidirect
20. C2×Dic632C22semidirect
21. Q32×D79C22semidirect
25. #19098403C22semidirect
26. #19098402C22semidirect
27. #19098402C22semidirect
28. #19098401C22semidirect
29. #19098400C22semidirect
30. #19098398C22semidirect
31. #19098397C22semidirect
32. #19098396C22semidirect
33. C32×D79C22semidirect
35. #508×C79C22semidirect
36. Q64×C79C22semidirect
37. SD64×C79C22semidirect
38. SD64×C79C22semidirect
39. D32×C79C22semidirect
40. C2×C32×C79C22semidirect
170. #2262D79semidirect
174. C79#33983semidirect
184. #508C2×D79semidirect
191. Q64C2×D79semidirect
196. SD64C2×D79semidirect
197. SD64C2×D79semidirect
199. D32C2×D79semidirect
203. C2×C32C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237765502
#237765382
#237657192
#237657172
#237657122
#237657112
#237657082
C2×#190984022
SD64×D792
#237656932
C79×#22622
#3398379