Finite Group Databasebeta

← #32155605

Subgroups — #32155605

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#22601490C211yesmaximal, characteristic, complemented
2#22601478C211yesmaximal, characteristic, complemented
3#22590659C212yesmaximal, complemented
4#22590657C211yesmaximal, characteristic, complemented
5#22590652C211yesmaximal, characteristic, complemented
6#22590651C212yesmaximal, complemented
7#22590648C212yesmaximal, complemented
8C2×#17985838C211yesmaximal, characteristic, complemented
9SD64×D71C212yesmaximal, complemented
10#22590633C211yesmaximal, characteristic, complemented
11C71×#2262C211yesmaximal, characteristic, complemented
12#17986813—21no—
13#17986801—21no—
14#17986439—21no—
15#17986438—22no—
16#17986436—22no—
17#17986434—21no—
18#17986432C2212yes—
19#17986428C2212yescomplemented
20C2×Dic568C2211yescharacteristic, complemented
21Q32×D71C2212yescomplemented
22C2×D1136C2211yescharacteristic
23D16×D71C2212yes—
24#17986415C2211yescharacteristic
25#17985839C2212yescomplemented
26#17985838C2212yescomplemented
27#17985838C2212yescomplemented
28#17985837C2212yescomplemented
29#17985836C2212yescomplemented
30#17985834C2212yescomplemented
31#17985833C2211yescharacteristic, complemented
32#17985832C2212yescomplemented
33C32×D71C2212yescomplemented
34#508×C71C2211yescharacteristic
35#508×C71C2211yescharacteristic, complemented
36Q64×C71C2211yescharacteristic, complemented
37SD64×C71C2211yescharacteristic, complemented
38SD64×C71C2211yescharacteristic, complemented
39D32×C71C2211yescharacteristic, complemented
40C2×C32×C71C2211yescharacteristic, complemented
41#14556175—41no—
42#14556174—41no—
43#14556158—41no—
44#14556157—42no—
45#14556155—42no—
46#14556153—41no—
47#14556151—22no—
48#14556147—22no—
49C2×Dic284—21no—
50Q16×D71—22no—
51C2×D568—21no—
52D8×D71—22no—
53#14556134D411yescharacteristic
54Dic568—21no—
55Dic568C2312yes—
56#14556074—22no—
57#14556072—22no—
58#14556071—22no—
59D1136—21no—
60D1136C2312yes—
61#14556069—22no—
62#14556068D411yescharacteristic
63#14556067D412yes—
64C16×D71C2312yes—
65C71⋊C32C2312yes—
66#135×C71—21no—
67#135×C71—21no—
68Q32×C71C2311yescharacteristic
69Q32×C71C2311yescharacteristic
70SD32×C71—21no—
71SD32×C71—21no—
72D16×C71C2311yescharacteristic
73D16×C71C2311yescharacteristic
74C2×C16×C71C2311yescharacteristic
75C32×C71C2311yescharacteristic
76C32×C71C2311yescharacteristic
77#11765866—42no—
78#11765865—81no—
79#11765865—81no—
80#11765865D811yescharacteristic
81#11765864—42no—
82#11765864—82no—
83C2×Dic142—41no—
84C2×Dic142—81no—
85Q8×D71—42no—
86C2×D284—41no—
87C2×D284—81no—
88D4×D71—42no—
89D4×D71—82no—
90Dic284—22no—
91Dic284—41no—
92#11765855—42no—
93#11765853—42no—
94#11765852—42no—
95D568—22no—
96D568—41no—
97#11765850—42no—
98#11765849D811yescharacteristic
99#11765848D812yes—
100C8×D71C2×D412yes—
101C71⋊C16C2×D412yes—
102#41×C71—41no—
103#41×C71—41no—
104Q16×C71—21no—
105Q16×C71—21no—
106SD16×C71—41no—
107SD16×C71—41no—
108D8×C71—21no—
109D8×C71—21no—
110C2×C8×C71C2×D411yescharacteristic
111C16×C71C2×D411yescharacteristic
112C16×C71C2411yescharacteristic
113C22×D71—81no—
114C22×D71—82no—
115Q8×C71—41no—
116Q8×C71—41no—
117D4×C71—41no—
118D4×C71—41no—
119D4×C71—81no—
120D4×C71—81no—
121C2×C4×C71—81no—
122C2×C4×C71—81no—
123C2×C4×C71C2×D811yescharacteristic
124C22⋊D71—82no—
125C22⋊D71—82no—
126C22⋊D71D1612yes—
127C2×Dic71—81no—
128C2×Dic71—82no—
129D284—42no—
130D284—82no—
131D284—81no—
132D284D1611yescharacteristic
133C4×D71—82no—
134C4×D71—82no—
135C4×D71C2×D812yes—
136Dic142—42no—
137Dic142—81no—
138Dic142—82no—
139Dic142D1611yescharacteristic
140C8×C71C2×D811yescharacteristic
141C8×C71C22×D411yescharacteristic
142C71⋊C8C2×D812yes—
143C22×C71—81no—
144C22×C71—81no—
145C22×C71C2×D1611yescharacteristic
146C2×D71—82no—
147C2×D71—322no—
148C2×D71—321no—
149C2×D71C2×D1612yes—
150C4×C71—81no—
151C4×C71—81no—
152C4×C71C2×D1611yescharacteristic
153C4×C71C22×D811yescharacteristic
154Dic71—82no—
155Dic71C2×D1612yes—
156#33983—711nomaximal, Sylow
157C2×C71—21no—
158C2×C71—161no—
159C2×C71—161no—
160C2×C71C22×D1611yescharacteristic
161D71—22no—
162D71—162no—
163#3417—711no—
164#3415—711no—
165#2264—711no—
166#2264—712no—
167#2263—711no—
168#2263—712no—
169#2262—712no—
170#2262D7111yescharacteristic, complemented
171C2×SD64—712no—
172C2×SD64—711no—
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#508C2×D7111yescharacteristic
184#508C2×D7111yescharacteristic, complemented
185C2×Q32—712no—
186C2×Q32—711no—
187C2×D16—712no—
188C2×D16—711no—
189#504—711no—
190Q64—712no—
191Q64C2×D7111yescharacteristic, complemented
192SD64—712no—
193SD64—712no—
194SD64—712no—
195SD64—712no—
196SD64C2×D7111yescharacteristic, complemented
197SD64C2×D7111yescharacteristic, complemented
198D32—712no—
199D32C2×D7111yescharacteristic, complemented
200M64—711no—
201M64—712no—
202C2×C32—712no—
203C2×C32C2×D7111yescharacteristic, complemented
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—21no—
212#135—1422no—
213#135—1422no—
214C2×Q16—1422no—
215C2×Q16—1421no—
216C2×D8—1422no—
217C2×D8—1421no—
218#131—711no—
219Q32—712no—
220Q32—1422no—
221Q32—1421no—
222Q32C22×D7111yescharacteristic
223Q32C22×D7111yescharacteristic
224SD32—21no—
225SD32—21no—
226SD32—1422no—
227SD32—1422no—
228D16—712no—
229D16—1422no—
230D16—1421no—
231D16C22×D7111yescharacteristic
232D16C22×D7111yescharacteristic
233M32—711no—
234M32—712no—
235C2×C16—712no—
236C2×C16C22×D7111yescharacteristic
237C32—712no—
238C32C22×D7111yescharacteristic
239C32C22×D7111yescharacteristic
240#41—41no—
241#41—41no—
242#41—2842no—
243#41—2842no—
244#41—5681no—
245#41—5681no—
246#41—5682no—
247#41—711no—
248C2×Q8—2842no—
249C2×Q8—2841no—
250C2×Q8—5681no—
251C2×D4—2842no—
252C2×D4—2841no—
253C2×D4—5681no—
254C2×D4—5682no—
255Q16—21no—
256Q16—21no—
257Q16—2842no—
258Q16—2841no—
259Q16—1422no—
260SD16—41no—
261SD16—41no—
262SD16—2842no—
263SD16—2842no—
264D8—21no—
265D8—21no—
266D8—2842no—
267D8—2841no—
268D8—1422no—
269M16—711no—
270M16—712no—
271C2×C8—712no—
272C2×C8D4×D7111yescharacteristic
273C16—712no—
274C16D4×D7111yescharacteristic
275C16C23×D7111yescharacteristic
276C23—5682no—
277C23—5681no—
278Q8—41no—
279Q8—41no—
280Q8—2842no—
281Q8—5682no—
282Q8—5681no—
283Q8—711no—
284D4—41no—
285D4—41no—
286D4—81no—
287D4—81no—
288D4—2842no—
289D4—5682no—
290D4—5682no—
291D4—5682no—
292D4—5681no—
293D4—712no—
294D4—711no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—5682no—
298C2×C4—5682no—
299C2×C4—5681no—
300C2×C4—5682no—
301C2×C4—712no—
302C2×C4D8×D7111yescharacteristic
303C8—712no—
304C8D8×D7111yescharacteristic
305C8C2×D4×D7111yescharacteristic
306C22—81no—
307C22—81no—
308C22—5682no—
309C22—712no—
310C22—22721no—
311C22—22722no—
312C22D16×D7111yescharacteristic
313C4—81no—
314C4—81no—
315C4—5682no—
316C4—712no—
317C4D16×D7111yescharacteristic
318C4C2×D8×D7111yescharacteristic
319C2—21no—
320C2—161no—
321C2—161no—
322C2—11362no—
323C2—1422no—
324C2C2×D16×D7111yescharacteristic
subgroupG/H
centerC2C2×D16×D71
commutator, cosocleC16×C71C24
FrattiniC16C23×D71
FittingC71×#2262C2
socleC2×C71C22×D16
hypercenter#2262D71
hypocenterC71#33983
LCSG ▷ C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2262

Complements

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

NH ≅ G/Ntype
1. #22601490C2semidirect
2. #22601478C2semidirect
3. #22590659C2semidirect
4. #22590657C2semidirect
5. #22590652C2semidirect
6. #22590651C2semidirect
7. #22590648C2semidirect
8. C2×#17985838C2semidirect
9. SD64×D71C2semidirect
10. #22590633C2semidirect
11. C71×#2262C2semidirect
19. #17986428C22semidirect
20. C2×Dic568C22semidirect
21. Q32×D71C22semidirect
25. #17985839C22semidirect
26. #17985838C22semidirect
27. #17985838C22semidirect
28. #17985837C22semidirect
29. #17985836C22semidirect
30. #17985834C22semidirect
31. #17985833C22semidirect
32. #17985832C22semidirect
33. C32×D71C22semidirect
35. #508×C71C22semidirect
36. Q64×C71C22semidirect
37. SD64×C71C22semidirect
38. SD64×C71C22semidirect
39. D32×C71C22semidirect
40. C2×C32×C71C22semidirect
170. #2262D71semidirect
174. C71#33983semidirect
184. #508C2×D71semidirect
191. Q64C2×D71semidirect
196. SD64C2×D71semidirect
197. SD64C2×D71semidirect
199. D32C2×D71semidirect
203. C2×C32C2×D71semidirect

Maximal subgroups

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

subgroupindex
#226014902
#226014782
#225906592
#225906572
#225906522
#225906512
#225906482
C2×#179858382
SD64×D712
#225906332
C71×#22622
#3398371