Finite Group Databasebeta

← #32155607

Subgroups — #32155607

Order 18176.

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#22601489C211yesmaximal, characteristic, complemented
2#22601478C211yesmaximal, characteristic, complemented
3#22590659C212yesmaximal, complemented
4#22590655C211yesmaximal, characteristic, complemented
5#22590654C212yesmaximal, complemented
6#22590650C212yesmaximal, complemented
7#22590647C211yesmaximal, characteristic, complemented
8C2×#17985836C211yesmaximal, characteristic, complemented
9SD64×D71C212yesmaximal, complemented
10#22590635C211yesmaximal, characteristic, complemented
11C71×#2264C211yesmaximal, characteristic, complemented
12#17986812—21no—
13#17986801—21no—
14#17986441—22no—
15#17986440—21no—
16#17986436—22no—
17#17986434—21no—
18#17986432C2212yes—
19#17986432C2212yescomplemented
20#17986431C2211yescharacteristic, complemented
21Q32×D71C2212yescomplemented
22C2×D1136C2211yescharacteristic
23D16×D71C2212yes—
24#17986415C2211yescharacteristic
25#17985839C2211yescharacteristic, complemented
26#17985838C2212yescomplemented
27#17985837C2211yescharacteristic, complemented
28#17985836C2212yescomplemented
29#17985836C2212yescomplemented
30#17985836C2211yescharacteristic, complemented
31D2272C2212yescomplemented
32#17985834C2211yescharacteristic, complemented
33#17985832C2212yescomplemented
34C2×C71⋊C32C2211yescharacteristic, complemented
35C32×D71C2212yescomplemented
36#508×C71C2211yescharacteristic
37C2×Q32×C71C2211yescharacteristic, complemented
38Q64×C71C2212yescomplemented
39SD64×C71C2212yescomplemented
40M64×C71C2211yescharacteristic, complemented
41#14556176—41no—
42#14556174—41no—
43#14556160—42no—
44#14556159—41no—
45#14556155—42no—
46#14556153—41no—
47#14556151—22no—
48#14556151—22no—
49#14556150—21no—
50Q16×D71—22no—
51C2×D568—21no—
52D8×D71—22no—
53#14556134D411yescharacteristic
54Dic568C2311yescharacteristic
55#14556074—22no—
56#14556073—21no—
57#14556071—22no—
58#14556071—22no—
59D1136—21no—
60D1136C2311yescharacteristic
61D1136C2312yes—
62#14556069—22no—
63#14556068D411yescharacteristic
64#14556067D412yes—
65C16×D71C2312yes—
66C71⋊C32C2311yescharacteristic
67C71⋊C32C2311yescharacteristic
68#135×C71—21no—
69C2×Q16×C71—21no—
70Q32×C71—21no—
71Q32×C71C2312yes—
72Q32×C71C2311yescharacteristic
73SD32×C71—21no—
74D16×C71C2311yescharacteristic
75C2×C16×C71C2311yescharacteristic
76C32×C71C2312yes—
77#11765866—42no—
78#11765866—42no—
79#11765866—81no—
80#11765865—41no—
81#11765865—82no—
82#11765865—81no—
83#11765865D811yescharacteristic
84Q8×D71—42no—
85Q8×D71—81no—
86C2×D284—41no—
87C2×D284—81no—
88D4×D71—42no—
89D4×D71—82no—
90Dic284—21no—
91#11765855—42no—
92#11765854—41no—
93#11765852—42no—
94#11765852—42no—
95D568—22no—
96D568—21no—
97D568—41no—
98#11765850—42no—
99#11765849D811yescharacteristic
100#11765848D812yes—
101C8×D71C2×D412yes—
102C71⋊C16C2×D412yes—
103#41×C71—41no—
104C2×Q8×C71—41no—
105Q16×C71—21no—
106Q16×C71—22no—
107Q16×C71—41no—
108SD16×C71—41no—
109D8×C71—21no—
110C2×C8×C71C2×D411yescharacteristic
111C16×C71C2×D411yescharacteristic
112C16×C71C2411yescharacteristic
113C22×D71—81no—
114C22×D71—82no—
115Q8×C71—41no—
116Q8×C71—42no—
117Q8×C71—81no—
118D4×C71—41no—
119D4×C71—81no—
120C2×C4×C71—81no—
121C2×C4×C71—81no—
122C2×C4×C71C2×D811yescharacteristic
123C22⋊D71—81no—
124C22⋊D71—82no—
125C22⋊D71D1612yes—
126D284—42no—
127D284—41no—
128D284—82no—
129D284—82no—
130D284—81no—
131D284D1611yescharacteristic
132C4×D71—82no—
133C4×D71—81no—
134C4×D71—82no—
135C4×D71—82no—
136C4×D71C2×D812yes—
137Dic142—41no—
138Dic142—82no—
139Dic142D1611yescharacteristic
140C8×C71C2×D811yescharacteristic
141C8×C71C22×D411yescharacteristic
142C71⋊C8C2×D812yes—
143C22×C71—81no—
144C22×C71C2×D1611yescharacteristic
145C2×D71—81no—
146C2×D71—82no—
147C2×D71—322no—
148C2×D71—321no—
149C2×D71C2×D1612yes—
150C4×C71—82no—
151C4×C71—81no—
152C4×C71C2×D1611yescharacteristic
153C4×C71C22×D811yescharacteristic
154Dic71—81no—
155Dic71C2×D1612yes—
156#33983—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—161no—
159C2×C71C22×D1611yescharacteristic
160D71—22no—
161D71—161no—
162D71—162no—
163#3417—711no—
164#3415—711no—
165#2264—712no—
166#2264D7111yescharacteristic, complemented
167#2263—712no—
168#2263—711no—
169#2262—711no—
170#2262—712no—
171C2×SD64—711no—
172C2×SD64—712no—
173#2258—711no—
174C71#3398311yescharacteristic, complemented, Sylow
175#578—1421no—
176#576—1421no—
177#510—1422no—
178#510—1421no—
179#509—1422no—
180#509—1421no—
181#508—712no—
182#508—712no—
183#508—711no—
184#508C2×D7111yescharacteristic
185C2×Q32—712no—
186C2×Q32C2×D7111yescharacteristic, complemented
187C2×D16—712no—
188C2×D16—711no—
189#504—711no—
190Q64—711no—
191Q64C2×D7112yescomplemented
192SD64—712no—
193SD64—712no—
194SD64—711no—
195SD64—711no—
196SD64—712no—
197SD64C2×D7112yescomplemented
198D32—711no—
199D32—712no—
200M64—712no—
201M64C2×D7111yescharacteristic, complemented
202C2×C32—711no—
203C2×C32—712no—
2042−1+4—2841no—
205H(5,𝔽2)—2841no—
206#137—2842no—
207#137—2841no—
208AGL(1,ℤ/8)—2842no—
209AGL(1,ℤ/8)—2841no—
210#135—21no—
211#135—1422no—
212#135—1422no—
213#135—1421no—
214C2×Q16—21no—
215C2×Q16—1422no—
216C2×D8—1422no—
217C2×D8—1421no—
218#131—711no—
219Q32—21no—
220Q32—711no—
221Q32—1422no—
222Q32C22×D7112yes—
223Q32C22×D7111yescharacteristic
224SD32—21no—
225SD32—1422no—
226SD32—1422no—
227SD32—1421no—
228D16—711no—
229D16—712no—
230D16—1422no—
231D16—1421no—
232D16C22×D7111yescharacteristic
233M32—711no—
234M32—712no—
235C2×C16—712no—
236C2×C16C22×D7111yescharacteristic
237C32—711no—
238C32—711no—
239C32C22×D7112yes—
240#41—41no—
241#41—2842no—
242#41—2842no—
243#41—2841no—
244#41—5681no—
245#41—5682no—
246#41—5681no—
247#41—711no—
248C2×Q8—41no—
249C2×Q8—2842no—
250C2×Q8—5681no—
251C2×D4—2842no—
252C2×D4—2841no—
253C2×D4—5681no—
254C2×D4—5682no—
255Q16—21no—
256Q16—22no—
257Q16—41no—
258Q16—2842no—
259Q16—1421no—
260SD16—41no—
261SD16—2842no—
262SD16—2842no—
263SD16—2841no—
264D8—21no—
265D8—2842no—
266D8—2841no—
267D8—1422no—
268D8—1421no—
269M16—711no—
270M16—712no—
271C2×C8—712no—
272C2×C8D4×D7111yescharacteristic
273C16—712no—
274C16D4×D7111yescharacteristic
275C16C23×D7111yescharacteristic
276C23—5682no—
277C23—5681no—
278Q8—41no—
279Q8—42no—
280Q8—81no—
281Q8—2841no—
282Q8—5682no—
283Q8—711no—
284D4—41no—
285D4—81no—
286D4—2842no—
287D4—2841no—
288D4—5682no—
289D4—5682no—
290D4—5681no—
291D4—5682no—
292D4—5681no—
293D4—712no—
294D4—711no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—5682no—
298C2×C4—5682no—
299C2×C4—5682no—
300C2×C4—5681no—
301C2×C4—712no—
302C2×C4D8×D7111yescharacteristic
303C8—712no—
304C8D8×D7111yescharacteristic
305C8C2×D4×D7111yescharacteristic
306C22—81no—
307C22—5681no—
308C22—5682no—
309C22—712no—
310C22—22721no—
311C22—22722no—
312C22D16×D7111yescharacteristic
313C4—81no—
314C4—82no—
315C4—5681no—
316C4—712no—
317C4D16×D7111yescharacteristic
318C4C2×D8×D7111yescharacteristic
319C2—21no—
320C2—161no—
321C2—11362no—
322C2—11361no—
323C2—1422no—
324C2C2×D16×D7111yescharacteristic
subgroupG/H
centerC2C2×D16×D71
commutator, cosocleC16×C71C24
FrattiniC16C23×D71
FittingC71×#2264C2
socleC2×C71C22×D16
hypercenter#2264D71
hypocenterC71#33983
LCSG ▷ C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2264

Complements

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

NH ≅ G/Ntype
1. #22601489C2semidirect
2. #22601478C2semidirect
3. #22590659C2semidirect
4. #22590655C2semidirect
5. #22590654C2semidirect
6. #22590650C2semidirect
7. #22590647C2semidirect
8. C2×#17985836C2semidirect
9. SD64×D71C2semidirect
10. #22590635C2semidirect
11. C71×#2264C2semidirect
19. #17986432C22semidirect
20. #17986431C22semidirect
21. Q32×D71C22semidirect
25. #17985839C22semidirect
26. #17985838C22semidirect
27. #17985837C22semidirect
28. #17985836C22semidirect
29. #17985836C22semidirect
30. #17985836C22semidirect
31. D2272C22semidirect
32. #17985834C22semidirect
33. #17985832C22semidirect
34. C2×C71⋊C32C22semidirect
35. C32×D71C22semidirect
37. C2×Q32×C71C22semidirect
38. Q64×C71C22semidirect
39. SD64×C71C22semidirect
40. M64×C71C22semidirect
166. #2264D71semidirect
174. C71#33983semidirect
186. C2×Q32C2×D71semidirect
191. Q64C2×D71semidirect
197. SD64C2×D71semidirect
201. M64C2×D71semidirect

Maximal subgroups

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

subgroupindex
#226014892
#226014782
#225906592
#225906552
#225906542
#225906502
#225906472
C2×#179858362
SD64×D712
#225906352
C71×#22642
#3398371