Finite Group Databasebeta

← #34446425

Subgroups — #34446425

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#23776537C211yesmaximal, characteristic, complemented
3#23765720C211yesmaximal, characteristic, complemented
4#23765716C212yesmaximal, complemented
5#23765711C212yesmaximal, complemented
6#23765707C211yesmaximal, characteristic, complemented
7#23765706C212yesmaximal, complemented
8C2×#19098401C211yesmaximal, characteristic, complemented
9SD64×D79C212yesmaximal, complemented
10#23765695C211yesmaximal, characteristic, complemented
11C79×#2263C211yesmaximal, characteristic, complemented
12#19099377—21no—
13#19099364—21no—
14#19099003—21no—
15#19099002—22no—
16#19098999—21no—
17#19098997—22no—
18#19098995C2211yescharacteristic
19#19098992C2212yes—
20#19098992C2212yescomplemented
21C2×Dic632C2211yescharacteristic, complemented
22Q32×D79C2212yescomplemented
23D16×D79C2212yes—
24#19098979C2211yescharacteristic
25Dic1264C2212yescomplemented
26#19098403C2211yescharacteristic, complemented
27#19098402C2212yescomplemented
28#19098401C2212yescomplemented
29#19098401C2212yescomplemented
30#19098401C2211yescharacteristic, complemented
31#19098400C2211yescharacteristic, complemented
32#19098398C2211yescharacteristic, complemented
33#19098396C2212yescomplemented
34C2×C79⋊C32C2211yescharacteristic, complemented
35C32×D79C2212yescomplemented
36#508×C79C2211yescharacteristic, complemented
37C2×D16×C79C2211yescharacteristic
38SD64×C79C2212yescomplemented
39D32×C79C2212yescomplemented
40M64×C79C2211yescharacteristic, complemented
41#14579208—41no—
42#14579206—41no—
43#14579191—41no—
44#14579190—42no—
45#14579187—41no—
46#14579185—42no—
47#14579183—21no—
48#14579180—22no—
49#14579180—22no—
50C2×Dic316—21no—
51Q16×D79—22no—
52D8×D79—22no—
53#14579167D411yescharacteristic
54Dic632—21no—
55Dic632C2312yes—
56Dic632C2311yescharacteristic
57#14579107—22no—
58#14579106—21no—
59#14579105—22no—
60#14579105—22no—
61D1264C2311yescharacteristic
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×C79C2311yescharacteristic
74D16×C79C2312yes—
75C2×C16×C79C2311yescharacteristic
76C32×C79C2312yes—
77#11929624—41no—
78#11929624—81no—
79#11929624D811yescharacteristic
80#11929623—42no—
81#11929623—42no—
82#11929623—81no—
83#11929623—82no—
84C2×Dic158—41no—
85C2×Dic158—81no—
86Q8×D79—42no—
87C2×C22⋊D79—82no—
88D4×D79—42no—
89D4×D79—81no—
90Dic316—21no—
91Dic316—22no—
92Dic316—41no—
93#11929614—42no—
94#11929613—41no—
95#11929612—42no—
96#11929612—42no—
97D632—21no—
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—82no—
115Q8×C79—41no—
116D4×C79—42no—
117D4×C79—41no—
118D4×C79—81no—
119D4×C79—81no—
120C2×C4×C79—81no—
121C2×C4×C79C2×D811yescharacteristic
122C22⋊D79—81no—
123C22⋊D79—82no—
124C22⋊D79—82no—
125C22⋊D79—82no—
126C22⋊D79D1612yes—
127C2×Dic79—81no—
128C2×Dic79—82no—
129C2×Dic79—82no—
130D316—41no—
131D316D1611yescharacteristic
132C4×D79—82no—
133C4×D79—81no—
134C4×D79C2×D812yes—
135Dic158—41no—
136Dic158—42no—
137Dic158—81no—
138Dic158—82no—
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—322no—
149C2×D79C2×D1612yes—
150C4×C79—81no—
151C4×C79C2×D1611yescharacteristic
152C4×C79C22×D811yescharacteristic
153Dic79—81no—
154Dic79—82no—
155Dic79C2×D1612yes—
156#33983—791nomaximal, Sylow
157C2×C79—21no—
158C2×C79—162no—
159C2×C79—161no—
160C2×C79C22×D1611yescharacteristic
161D79—22no—
162D79—161no—
163#3417—791no—
164#3415—791no—
165#2264—792no—
166#2264—791no—
167#2263—792no—
168#2263D7911yescharacteristic, complemented
169#2262—791no—
170#2262—792no—
171C2×SD64—791no—
172C2×SD64—792no—
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#508—791no—
184#508C2×D7911yescharacteristic, complemented
185C2×Q32—792no—
186C2×Q32—791no—
187C2×D16—792no—
188C2×D16C2×D7911yescharacteristic
189#504—791no—
190Q64—791no—
191Q64—792no—
192SD64—791no—
193SD64—792no—
194SD64—792no—
195SD64—791no—
196SD64—792no—
197SD64C2×D7912yescomplemented
198D32—791no—
199D32C2×D7912yescomplemented
200M64—792no—
201M64C2×D7911yescharacteristic, complemented
202C2×C32—791no—
203C2×C32—792no—
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—1582no—
212#135—1582no—
213#135—1581no—
214C2×Q16—1582no—
215C2×Q16—1581no—
216C2×D8—21no—
217C2×D8—1582no—
218#131—791no—
219Q32—792no—
220Q32—791no—
221Q32—1582no—
222Q32—1581no—
223Q32C22×D7911yescharacteristic
224SD32—21no—
225SD32—1582no—
226SD32—1582no—
227SD32—1581no—
228D16—21no—
229D16—791no—
230D16—1582no—
231D16C22×D7911yescharacteristic
232D16C22×D7912yes—
233M32—791no—
234M32—792no—
235C2×C16—792no—
236C2×C16C22×D7911yescharacteristic
237C32—791no—
238C32—791no—
239C32C22×D7912yes—
240#41—41no—
241#41—3162no—
242#41—3162no—
243#41—3161no—
244#41—791no—
245#41—6321no—
246#41—6322no—
247#41—6321no—
248C2×Q8—3162no—
249C2×Q8—3161no—
250C2×Q8—6321no—
251C2×D4—41no—
252C2×D4—3162no—
253C2×D4—6321no—
254C2×D4—6322no—
255Q16—21no—
256Q16—3162no—
257Q16—3161no—
258Q16—1581no—
259Q16—1582no—
260SD16—41no—
261SD16—3162no—
262SD16—3162no—
263SD16—3161no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—3162no—
268D8—1581no—
269M16—791no—
270M16—792no—
271C2×C8—792no—
272C2×C8D4×D7911yescharacteristic
273C16—792no—
274C16D4×D7911yescharacteristic
275C16C23×D7911yescharacteristic
276C23—81no—
277C23—6322no—
278Q8—41no—
279Q8—3161no—
280Q8—3162no—
281Q8—791no—
282Q8—6322no—
283Q8—6321no—
284D4—42no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—3161no—
289D4—792no—
290D4—791no—
291D4—6322no—
292D4—6322no—
293D4—6322no—
294D4—6321no—
295C2×C4—81no—
296C2×C4—792no—
297C2×C4—6322no—
298C2×C4—6322no—
299C2×C4—6322no—
300C2×C4—6321no—
301C2×C4—6321no—
302C2×C4D8×D7911yescharacteristic
303C8—792no—
304C8D8×D7911yescharacteristic
305C8C2×D4×D7911yescharacteristic
306C22—81no—
307C22—82no—
308C22—321no—
309C22—792no—
310C22—6321no—
311C22—25282no—
312C22D16×D7911yescharacteristic
313C4—81no—
314C4—792no—
315C4—6321no—
316C4—6322no—
317C4D16×D7911yescharacteristic
318C4C2×D8×D7911yescharacteristic
319C2—21no—
320C2—162no—
321C2—161no—
322C2—1582no—
323C2—12641no—
324C2C2×D16×D7911yescharacteristic
subgroupG/H
centerC2C2×D16×D79
commutator, cosocleC16×C79C24
FrattiniC16C23×D79
FittingC79×#2263C2
socleC2×C79C22×D16
hypercenter#2263D79
hypocenterC79#33983
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. #23776550C2semidirect
2. #23776537C2semidirect
3. #23765720C2semidirect
4. #23765716C2semidirect
5. #23765711C2semidirect
6. #23765707C2semidirect
7. #23765706C2semidirect
8. C2×#19098401C2semidirect
9. SD64×D79C2semidirect
10. #23765695C2semidirect
11. C79×#2263C2semidirect
20. #19098992C22semidirect
21. C2×Dic632C22semidirect
22. Q32×D79C22semidirect
25. Dic1264C22semidirect
26. #19098403C22semidirect
27. #19098402C22semidirect
28. #19098401C22semidirect
29. #19098401C22semidirect
30. #19098401C22semidirect
31. #19098400C22semidirect
32. #19098398C22semidirect
33. #19098396C22semidirect
34. C2×C79⋊C32C22semidirect
35. C32×D79C22semidirect
36. #508×C79C22semidirect
38. SD64×C79C22semidirect
39. D32×C79C22semidirect
40. M64×C79C22semidirect
168. #2263D79semidirect
174. C79#33983semidirect
184. #508C2×D79semidirect
197. SD64C2×D79semidirect
199. D32C2×D79semidirect
201. M64C2×D79semidirect

Maximal subgroups

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

subgroupindex
#237765502
#237765372
#237657202
#237657162
#237657112
#237657072
#237657062
C2×#190984012
SD64×D792
#237656952
C79×#22632
#3398379