Finite Group Databasebeta

← #32150913

Subgroups — #32150913

Order 18176.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22601476C211yesmaximal, characteristic
2#22597258C211yesmaximal, characteristic, complemented
3#22597225C211yesmaximal, characteristic, complemented
4#22590240C214yesmaximal, complemented
5#22590233C214yesmaximal, complemented
6C2×#17985822C211yesmaximal, characteristic, complemented
7#22590090C211yesmaximal, characteristic, complemented
8C2×#17985810C211yesmaximal, characteristic, complemented
9C71×#2110C211yesmaximal, characteristic, complemented
10C2×#14556150C2211yescharacteristic
11C2×#14556150C2211yescharacteristic
12#17986765—21no—
13#17986479—21no—
14#17986442C412yescomplemented
15#17986442C2212yes—
16#17986437C412yescomplemented
17#17986437C2212yes—
18#17986431C414yescomplemented
19#17986122C2214yes—
20#17986117—22no—
21#17986112C2212yescomplemented
22#17986107C2212yescomplemented
23C4×#11765854—22no—
24C2×#14556064C2211yescharacteristic
25C2×#14556060C2211yescharacteristic
26#17986017—21no—
27#17986001—21no—
28#17985822C2214yescomplemented
29#17985816C2212yescomplemented
30#17985813C2212yescomplemented
31#17985810C2214yes—
32C2×M32×C71C2211yescharacteristic
33C2×M32×C71C2211yescharacteristic
34#405×C71C2211yescharacteristic
35#347×C71C2214yescomplemented
36C2×#11765865—21no—
37C2×#11765865—21no—
38#14556150—21no—
39#14556150—21no—
40#14556150—21no—
41#14556150—21no—
42#14556150C2×C412yes—
43#14556150C2×C412yes—
44#14556150C2×C412yescomplemented
45#14556150C2312yes—
46C2×Dic284C2×C411yescharacteristic, complemented
47C2×Dic284C2311yescharacteristic
48C2×#11765854—21no—
49C2×#11765854—21no—
50C2×D568C2×C411yescharacteristic, complemented
51C2×D568C2311yescharacteristic
52#14556136—42no—
53#14556134—24no—
54C2×#11765849—21no—
55M16×D71—42no—
56C4×Dic142—22no—
57C4×D284—22no—
58(C22⋊D71)⋊C4—42no—
59#14556086—41no—
60C2×C4⋊Dic71—21no—
61C4⋊C4×D71—41no—
62Dic568C2×C414yescomplemented
63#14556073C2×C414yescomplemented
64#14556073C2×C414yescomplemented
65D1136C2×C414yescomplemented
66#14556064—21no—
67#14556064C2312yes—
68#14556060—21no—
69#14556060C2312yes—
70#14556056—24no—
71#14556053—22no—
72#14556050—22no—
73C2×M16×C71D411yescharacteristic
74C22×C8×C71C2311yescharacteristic
75#117×C71D411yescharacteristic
76M32×C71—21no—
77M32×C71C2×C412yescomplemented
78M32×C71C2312yes—
79M32×C71C2312yes—
80C2×C16×C71—21no—
81C2×C16×C71C2×C412yescomplemented
82#97×C71C2312yes—
83C4×C8×C71D412yes—
84#11765865—22no—
85#11765865—22no—
86#11765865—22no—
87#11765865—22no—
88#11765865—41no—
89#11765865—41no—
90#11765865—41no—
91#11765865—41no—
92C2×Dic142—21no—
93C2×Dic142—21no—
94C2×D284—21no—
95C2×D284—21no—
96C2×C22⋊D71—41no—
97C2×C22⋊D71—41no—
98C22×Dic71—41no—
99C2×C4×D71—41no—
100C2×C4×D71—41no—
101C2×C4×D71—42no—
102Dic284—21no—
103Dic284—21no—
104Dic284C22×C412yes—
105Dic284C22×C412yes—
106#11765854—22no—
107#11765854—22no—
108#11765854—22no—
109#11765854—22no—
110D568—21no—
111D568—21no—
112D568C22×C412yes—
113D568C22×C412yes—
114#11765849—22no—
115#11765849—41no—
116#11765848—44no—
117C2×C71⋊C8—41no—
118C8×D71—44no—
119C4⋊Dic71—22no—
120C4⋊Dic71—41no—
121Dic71⋊C4—42no—
122(C2×D71)⋊C4—42no—
123C4×Dic71—42no—
124C22×C4×C71C2×D411yescharacteristic
125M16×C71—21no—
126M16×C71—21no—
127C2×C8×C71C22×C411yescharacteristic
128C2×C8×C71C22×C411yescharacteristic
129C2×C8×C71C2×D412yes—
130C2×C8×C71C2×D411yescharacteristic
131C2×C8×C71#4111yescharacteristic
132C2×C8×C71#4111yescharacteristic
133C2×C8×C71C2411yescharacteristic
134C4⋊C4×C71—21no—
135C22⋊C4×C71—21no—
136C42×C71C2×D412yes—
137C16×C71—22no—
138C16×C71C22×C414yes—
139C23×C71AGL(1,ℤ/8)11yescharacteristic
140C22×D71—41no—
141C22×D71—41no—
142C2×C4×C71—22no—
143C2×C4×C71C4×D411yescharacteristic
144C2×C4×C71C4×D411yescharacteristic
145C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C71AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C71C22×D411yescharacteristic
149C22⋊D71—42no—
150C22⋊D71—42no—
151C22⋊D71—42no—
152C22⋊D71—42no—
153C2×Dic71—41no—
154C2×Dic71—41no—
155C2×Dic71—42no—
156C2×Dic71—161no—
157D284—22no—
158D284—22no—
159D284—41no—
160D284—41no—
161C4×D71—42no—
162C4×D71—42no—
163C4×D71—42no—
164C4×D71—42no—
165C4×D71—162no—
166Dic142—22no—
167Dic142—22no—
168Dic142—41no—
169Dic142—41no—
170C8×C71—22no—
171C8×C71C4×D411yescharacteristic
172C8×C71C4×D411yescharacteristic
173C8×C71C23×C411yescharacteristic
174C8×C71C2×#4111yescharacteristic
175C71⋊C8—42no—
176C22×C71—41no—
177C22×C71#44211yescharacteristic
178C22×C71#44211yescharacteristic
179C22×C71C2×AGL(1,ℤ/8)11yescharacteristic
180C2×D71—42no—
181C2×D71—42no—
182C2×D71—81no—
183C2×D71—81no—
184C2×D71—161no—
185C4×C71—42no—
186C4×C71#44211yescharacteristic
187C4×C71#44211yescharacteristic
188C4×C71C2×C4×D411yescharacteristic
189C4×C71C2×AGL(1,ℤ/8)11yescharacteristic
190Dic71—42no—
191Dic71—42no—
192Dic71—82no—
193#33469—711nomaximal, Sylow
194C2×C71—21no—
195C2×C71—21no—
196C2×C71—21no—
197C2×C71C2×#44211yescharacteristic
198D71—82no—
199D71—82no—
200#3414—711no—
201#2957—711no—
202#2952—711no—
203AGL(1,ℤ/16)—714no—
204#2180—714no—
205C2×#365—711no—
206#2148—711no—
207C2×#360—711no—
208#2110D7111yescharacteristic, complemented
209C71#3346911yescharacteristic, complemented, Sylow
210C2×#135—711no—
211C2×#135—711no—
212#568—1421no—
213#520—1421no—
214#510—712no—
215#510—712no—
216#509—712no—
217#509—712no—
218#508—714no—
219C2×M32C2×D7111yescharacteristic
220C2×M32C2×D7111yescharacteristic
221#444—714no—
222#443—1422no—
223#442—712no—
224#441—712no—
225C4×SD16—1422no—
226C2×#108—711no—
227C2×#106—711no—
228#421—1421no—
229#417—1421no—
230#405C2×D7111yescharacteristic
231#365—714no—
232#362—712no—
233#361—712no—
234#360—714no—
235#347C2×D7114yescomplemented
236C2×#41—1421no—
237C2×#41—1421no—
238#135—712no—
239#135—712no—
240#135—712no—
241#135—712no—
242#135—1421no—
243#135—1421no—
244#135—1421no—
245#135—1421no—
246C2×Q16—711no—
247C2×Q16—711no—
248C2×SD16—1421no—
249C2×SD16—1421no—
250C2×D8—711no—
251C2×D8—711no—
252#131—2842no—
253#131—1424no—
254C2×M16—2842no—
255C2×M16—1421no—
256C2×M16D28411yescharacteristic
257C22×C8C22×D7111yescharacteristic
258C4×Q8—1422no—
259C4×D4—2842no—
260C4×D4—1422no—
261#117—2841no—
262#117D28411yescharacteristic
263C2×C4⋊C4—2841no—
264C2×C4⋊C4—1421no—
265Q32—714no—
266SD32—714no—
267SD32—714no—
268D16—714no—
269M32—21no—
270M32C4×D7112yescomplemented
271M32C22×D7112yes—
272M32C22×D7112yes—
273C2×C16—21no—
274C2×C16C4×D7112yescomplemented
275#108—712no—
276#108—1421no—
277#106—712no—
278#106—1421no—
279#104—1424no—
280#103—1422no—
281#102—1422no—
282#97C22×D7112yes—
283C4×C8D28412yes—
284#41—2841no—
285#41—2841no—
286#41—2841no—
287#41—2841no—
288#41—1422no—
289#41—1422no—
290#41—1422no—
291#41—1422no—
292C2×Q8—1421no—
293C2×Q8—1421no—
294C2×D4—2841no—
295C2×D4—2841no—
296C2×D4—1421no—
297C2×D4—1421no—
298C22×C4—2842no—
299C22×C4—2841no—
300C22×C4—2841no—
301C22×C4—2841no—
302C22×C4C2×D28411yescharacteristic
303Q16—712no—
304Q16—712no—
305Q16—1421no—
306Q16—1421no—
307SD16—1422no—
308SD16—1422no—
309SD16—1422no—
310SD16—1422no—
311D8—712no—
312D8—712no—
313D8—1421no—
314D8—1421no—
315M16—21no—
316M16—21no—
317M16—2844no—
318M16—2841no—
319M16—1422no—
320C2×C8—2844no—
321C2×C8—2841no—
322C2×C8C2×C4×D7111yescharacteristic
323C2×C8C2×C4×D7111yescharacteristic
324C2×C8C2×D28412yes—
325C2×C8C2×D28411yescharacteristic
326C2×C8#1176586511yescharacteristic
327C2×C8#1176586511yescharacteristic
328C2×C8C23×D7111yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—2842no—
331C4⋊C4—2841no—
332C4⋊C4—1422no—
333C22⋊C4—21no—
334C22⋊C4—2842no—
335C42—2842no—
336C42C2×D28412yes—
337C16—22no—
338C16C2×C4×D7114yes—
339C23—2841no—
340C23—2841no—
341C23#1455615611yescharacteristic
342Q8—2841no—
343Q8—2841no—
344Q8—1422no—
345Q8—1422no—
346D4—2842no—
347D4—2842no—
348D4—2842no—
349D4—2842no—
350D4—2841no—
351D4—2841no—
352D4—1422no—
353D4—1422no—
354C2×C4—22no—
355C2×C4—2841no—
356C2×C4—2841no—
357C2×C4—2842no—
358C2×C4—2842no—
359C2×C4—2842no—
360C2×C4—2842no—
361C2×C4—2842no—
362C2×C4—11362no—
363C2×C4—11361no—
364C2×C4C4×D28411yescharacteristic
365C2×C4C4×D28411yescharacteristic
366C2×C4#1455615611yescharacteristic
367C2×C4#1455615611yescharacteristic
368C2×C4#1455615611yescharacteristic
369C2×C4C22×D28411yescharacteristic
370C8—22no—
371C8—2842no—
372C8C4×D28411yescharacteristic
373C8C4×D28411yescharacteristic
374C8C22×C4×D7111yescharacteristic
375C8C2×#1176586511yescharacteristic
376C22—41no—
377C22—2842no—
378C22—2842no—
379C22—5681no—
380C22—5681no—
381C22—11361no—
382C22#1798611211yescharacteristic
383C22#1798611211yescharacteristic
384C22C2×#1455615611yescharacteristic
385C4—42no—
386C4—2842no—
387C4—2842no—
388C4—5682no—
389C4#1798611211yescharacteristic
390C4#1798611211yescharacteristic
391C4C2×C4×D28411yescharacteristic
392C4C2×#1455615611yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—5682no—
397C2—5682no—
398C2C2×#1798611211yescharacteristic
subgroupG/H
centerC2C2×#17986112
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×#2110C2
socleC2×C71C2×#442
prime radical#22601476C2
cosocleC2×C8×C71C24
hypercenter#2110D71
hypocenterC71#33469
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #22597258C2semidirect
3. #22597225C2semidirect
4. #22590240C2semidirect
5. #22590233C2semidirect
6. C2×#17985822C2semidirect
7. #22590090C2semidirect
8. C2×#17985810C2semidirect
9. C71×#2110C2semidirect
14. #17986442C4semidirect
16. #17986437C4semidirect
18. #17986431C4semidirect
21. #17986112C22semidirect
22. #17986107C22semidirect
28. #17985822C22semidirect
29. #17985816C22semidirect
30. #17985813C22semidirect
35. #347×C71C22semidirect
44. #14556150C2×C4semidirect
46. C2×Dic284C2×C4semidirect
50. C2×D568C2×C4semidirect
62. Dic568C2×C4semidirect
63. #14556073C2×C4semidirect
64. #14556073C2×C4semidirect
65. D1136C2×C4semidirect
77. M32×C71C2×C4semidirect
81. C2×C16×C71C2×C4semidirect
208. #2110D71semidirect
209. C71#33469semidirect
235. #347C2×D71semidirect
270. M32C4×D71semidirect
274. C2×C16C4×D71semidirect

Maximal subgroups

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

subgroupindex
#226014762
#225972582
#225972252
#225902402
#225902332
C2×#179858222
#225900902
C2×#179858102
C71×#21102
#3346971