Finite Group Databasebeta

← #32155602

Subgroups — #32155602

Order 18176.

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#22601478C211yesmaximal, characteristic, complemented
2#22601477C211yesmaximal, characteristic, complemented
3#22590655C211yesmaximal, characteristic, complemented
4#22590654C212yesmaximal, complemented
5#22590651C212yesmaximal, complemented
6#22590648C212yesmaximal, complemented
7#22590647C211yesmaximal, characteristic, complemented
8C2×#17985834C211yesmaximal, characteristic, complemented
9D32×D71C212yesmaximal, complemented
10#22590635C211yesmaximal, characteristic, complemented
11C71×#2263C211yesmaximal, characteristic, complemented
12#17986801—21no—
13#17986800—21no—
14#17986436—22no—
15#17986435—21no—
16#17986434—21no—
17#17986433—22no—
18#17986432C2212yescomplemented
19#17986431C2211yescharacteristic, complemented
20#17986428C2212yescomplemented
21C2×D1136C2211yescharacteristic, complemented
22D16×D71C2212yescomplemented
23D16×D71C2212yescomplemented
24#17986415C2211yescharacteristic
25#17985839C2211yescharacteristic, complemented
26#17985838C2212yescomplemented
27#17985837C2211yescharacteristic, complemented
28#17985836C2211yescharacteristic, complemented
29D2272C2212yescomplemented
30#17985834C2211yescharacteristic, complemented
31#17985834C2212yescomplemented
32#17985834C2212yescomplemented
33#17985832C2212yescomplemented
34C2×C71⋊C32C2211yescharacteristic, complemented
35C32×D71C2212yescomplemented
36#508×C71C2211yescharacteristic, complemented
37C2×D16×C71C2211yescharacteristic, complemented
38SD64×C71C2212yescomplemented
39D32×C71C2212yescomplemented
40M64×C71C2211yescharacteristic, complemented
41#14556174—41no—
42#14556173—41no—
43#14556155—42no—
44#14556154—41no—
45#14556153—41no—
46#14556152—42no—
47#14556151—22no—
48#14556150—21no—
49#14556147—22no—
50C2×D568—21no—
51D8×D71—22no—
52D8×D71—22no—
53#14556134D411yescharacteristic
54Dic568C2311yescharacteristic
55#14556073—21no—
56#14556072—22no—
57#14556071—22no—
58D1136—21no—
59D1136C2312yes—
60D1136C2311yescharacteristic
61#14556069—22no—
62#14556069—22no—
63#14556068D411yescharacteristic
64#14556067D412yes—
65C16×D71C2312yes—
66C71⋊C32C2311yescharacteristic
67C71⋊C32C2311yescharacteristic
68#135×C71—21no—
69C2×D8×C71—21no—
70Q32×C71C2311yescharacteristic
71SD32×C71—21no—
72D16×C71—21no—
73D16×C71C2312yes—
74D16×C71C2311yescharacteristic
75C2×C16×C71C2311yescharacteristic
76C32×C71C2312yes—
77#11765866—42no—
78#11765865—41no—
79#11765865—81no—
80#11765865D811yescharacteristic
81#11765864—42no—
82#11765864—81no—
83C2×D284—41no—
84C2×D284—81no—
85C2×C22⋊D71—82no—
86D4×D71—42no—
87D4×D71—42no—
88D4×D71—82no—
89D4×D71—81no—
90Dic284—21no—
91#11765854—41no—
92#11765853—42no—
93#11765852—42no—
94D568—21no—
95D568—22no—
96D568—41no—
97#11765850—42no—
98#11765850—42no—
99#11765849D811yescharacteristic
100#11765848D812yes—
101C8×D71C2×D412yes—
102C71⋊C16C2×D412yes—
103#41×C71—41no—
104C2×D4×C71—41no—
105Q16×C71—21no—
106SD16×C71—41no—
107D8×C71—22no—
108D8×C71—21no—
109D8×C71—41no—
110C2×C8×C71C2×D411yescharacteristic
111C16×C71C2×D411yescharacteristic
112C16×C71C2411yescharacteristic
113C23×C71—81no—
114C22×D71—81no—
115C22×D71—82no—
116C22×D71—82no—
117Q8×C71—41no—
118D4×C71—42no—
119D4×C71—41no—
120D4×C71—81no—
121D4×C71—81no—
122C2×C4×C71—81no—
123C2×C4×C71C2×D811yescharacteristic
124C22⋊D71—81no—
125C22⋊D71—82no—
126C22⋊D71—82no—
127C22⋊D71—82no—
128C22⋊D71D1612yes—
129C2×Dic71—82no—
130D284—41no—
131D284—42no—
132D284—82no—
133D284—81no—
134D284D1611yescharacteristic
135C4×D71—81no—
136C4×D71—82no—
137C4×D71C2×D812yes—
138Dic142—41no—
139Dic142D1611yescharacteristic
140C8×C71C2×D811yescharacteristic
141C8×C71C22×D411yescharacteristic
142C71⋊C8C2×D812yes—
143C22×C71—82no—
144C22×C71—81no—
145C22×C71—321no—
146C22×C71C2×D1611yescharacteristic
147C2×D71—82no—
148C2×D71—81no—
149C2×D71—322no—
150C2×D71—322no—
151C2×D71—321no—
152C2×D71C2×D1612yes—
153C4×C71—81no—
154C4×C71C2×D1611yescharacteristic
155C4×C71C22×D811yescharacteristic
156Dic71—81no—
157Dic71C2×D1612yes—
158#33982—711nomaximal, Sylow
159C2×C71—21no—
160C2×C71—161no—
161C2×C71—162no—
162C2×C71C22×D1611yescharacteristic
163D71—22no—
164D71—162no—
165D71—161no—
166#3415—711no—
167#3415—711no—
168#2263—712no—
169#2263—712no—
170#2263—711no—
171#2263D7111yescharacteristic, complemented
172#2262—711no—
173#2262—712no—
174C2×D32—711no—
175C2×D32—712no—
176#2258—711no—
177C71#3398211yescharacteristic, complemented, Sylow
178#576—1421no—
179#576—1421no—
180#509—1422no—
181#509—1422no—
182#509—1421no—
183#509—1421no—
184#508—712no—
185#508—712no—
186#508—711no—
187#508C2×D7111yescharacteristic, complemented
188C2×D16—712no—
189C2×D16—712no—
190C2×D16—711no—
191C2×D16C2×D7111yescharacteristic, complemented
192#504—711no—
193Q64—711no—
194SD64—711no—
195SD64—711no—
196SD64—712no—
197SD64C2×D7112yescomplemented
198D32—712no—
199D32—711no—
200D32—712no—
201D32—712no—
202D32C2×D7112yescomplemented
203M64—712no—
204M64C2×D7111yescharacteristic, complemented
205C2×C32—711no—
206C2×C32—712no—
207H(5,𝔽2)—2841no—
208H(5,𝔽2)—2841no—
209AGL(1,ℤ/8)—2842no—
210AGL(1,ℤ/8)—2842no—
211AGL(1,ℤ/8)—2841no—
212AGL(1,ℤ/8)—2841no—
213#135—21no—
214#135—1422no—
215#135—1422no—
216#135—1421no—
217C2×D8—21no—
218C2×D8—1422no—
219C2×D8—1422no—
220C2×D8—1421no—
221#131—711no—
222Q32—711no—
223Q32C22×D7111yescharacteristic
224SD32—21no—
225SD32—1422no—
226SD32—1422no—
227SD32—1421no—
228D16—21no—
229D16—712no—
230D16—711no—
231D16—1422no—
232D16—1422no—
233D16—1421no—
234D16C22×D7112yes—
235D16C22×D7111yescharacteristic
236M32—711no—
237M32—712no—
238C2×C16—712no—
239C2×C16C22×D7111yescharacteristic
240C32—711no—
241C32—711no—
242C32C22×D7112yes—
243#41—41no—
244#41—2842no—
245#41—2842no—
246#41—2841no—
247#41—5681no—
248#41—5681no—
249#41—711no—
250C2×D4—41no—
251C2×D4—2842no—
252C2×D4—2842no—
253C2×D4—2841no—
254C2×D4—5681no—
255C2×D4—5682no—
256C2×D4—5681no—
257C2×D4—5682no—
258Q16—21no—
259Q16—1421no—
260SD16—41no—
261SD16—2842no—
262SD16—2842no—
263SD16—2841no—
264D8—22no—
265D8—21no—
266D8—41no—
267D8—2842no—
268D8—2842no—
269D8—2841no—
270D8—1421no—
271D8—1422no—
272M16—711no—
273M16—712no—
274C2×C8—712no—
275C2×C8D4×D7111yescharacteristic
276C16—712no—
277C16D4×D7111yescharacteristic
278C16C23×D7111yescharacteristic
279C23—81no—
280C23—5682no—
281C23—5682no—
282C23—5681no—
283Q8—41no—
284Q8—2841no—
285Q8—711no—
286D4—42no—
287D4—41no—
288D4—81no—
289D4—81no—
290D4—2841no—
291D4—2842no—
292D4—5682no—
293D4—5682no—
294D4—5681no—
295D4—5682no—
296D4—5682no—
297D4—5681no—
298D4—712no—
299D4—711no—
300C2×C4—81no—
301C2×C4—5682no—
302C2×C4—5682no—
303C2×C4—5681no—
304C2×C4—712no—
305C2×C4D8×D7111yescharacteristic
306C8—712no—
307C8D8×D7111yescharacteristic
308C8C2×D4×D7111yescharacteristic
309C22—81no—
310C22—82no—
311C22—321no—
312C22—5682no—
313C22—5681no—
314C22—712no—
315C22—22721no—
316C22—22722no—
317C22—22722no—
318C22D16×D7111yescharacteristic
319C4—81no—
320C4—5681no—
321C4—712no—
322C4D16×D7111yescharacteristic
323C4C2×D8×D7111yescharacteristic
324C2—21no—
325C2—162no—
326C2—161no—
327C2—11362no—
328C2—11361no—
329C2—1422no—
330C2C2×D16×D7111yescharacteristic
subgroupG/H
centerC2C2×D16×D71
commutator, cosocleC16×C71C24
FrattiniC16C23×D71
FittingC71×#2263C2
socleC2×C71C22×D16
hypercenter#2263D71
hypocenterC71#33982
LCSG ▷ C16×C71 ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #2263

Complements

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

NH ≅ G/Ntype
1. #22601478C2semidirect
2. #22601477C2semidirect
3. #22590655C2semidirect
4. #22590654C2semidirect
5. #22590651C2semidirect
6. #22590648C2semidirect
7. #22590647C2semidirect
8. C2×#17985834C2semidirect
9. D32×D71C2semidirect
10. #22590635C2semidirect
11. C71×#2263C2semidirect
18. #17986432C22semidirect
19. #17986431C22semidirect
20. #17986428C22semidirect
21. C2×D1136C22semidirect
22. D16×D71C22semidirect
23. D16×D71C22semidirect
25. #17985839C22semidirect
26. #17985838C22semidirect
27. #17985837C22semidirect
28. #17985836C22semidirect
29. D2272C22semidirect
30. #17985834C22semidirect
31. #17985834C22semidirect
32. #17985834C22semidirect
33. #17985832C22semidirect
34. C2×C71⋊C32C22semidirect
35. C32×D71C22semidirect
36. #508×C71C22semidirect
37. C2×D16×C71C22semidirect
38. SD64×C71C22semidirect
39. D32×C71C22semidirect
40. M64×C71C22semidirect
171. #2263D71semidirect
177. C71#33982semidirect
187. #508C2×D71semidirect
191. C2×D16C2×D71semidirect
197. SD64C2×D71semidirect
202. D32C2×D71semidirect
204. M64C2×D71semidirect

Maximal subgroups

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

subgroupindex
#226014782
#226014772
#225906552
#225906542
#225906512
#225906482
#225906472
C2×#179858342
D32×D712
#225906352
C71×#22632
#3398271