Finite Group Databasebeta

← #32150796

Subgroups — #32150796

Order 18176.

696 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986437C211yesmaximal, characteristic
2#22597217C211yesmaximal, characteristic, complemented
3#22597193C211yesmaximal, characteristic, complemented
4#22590217C212yesmaximal, complemented
5#22590206C212yesmaximal, complemented
6C4×#14556073C212yesmaximal, complemented
7C4×D1136C212yesmaximal, complemented
8#22590128C211yesmaximal, characteristic, complemented
9#22590048C211yesmaximal, characteristic, complemented
10C2×#17985801C211yesmaximal, characteristic, complemented
11C71×#2108C211yesmaximal, characteristic, complemented
12C2×#14556150C2211yescharacteristic
13C22×D568C2211yescharacteristic
14#17986479—21no—
15#17986473—21no—
16#17986437C418yescomplemented
17C2×#14556073C2212yes—
18C2×D1136C2212yes—
19#17986112—22no—
20#17986101—22no—
21C4×Dic284C2212yescomplemented
22C4×D568C2212yes—
23C4×D568C2212yes—
24C4×D568C2212yescomplemented
25#17986037C2211yescharacteristic
26C2×#14556062C2211yescharacteristic
27#17986005—21no—
28#17986001—21no—
29#17985827C2212yescomplemented
30#17985825C2212yescomplemented
31#17985804C2211yescharacteristic, complemented
32#17985804C2211yescharacteristic, complemented
33#17985801C2212yes—
34#17985801C2212yes—
35#17985801C2211yescharacteristic, complemented
36#17985801C2211yescharacteristic, complemented
37C2×M32×C71C2211yescharacteristic
38C22×C16×C71C2211yescharacteristic
39#405×C71C2211yescharacteristic
40#346×C71C2212yescomplemented
41C4×C16×C71C2212yescomplemented
42C2×#11765865—21no—
43C22×D284—21no—
44#14556150—22no—
45#14556150C2×C414yescomplemented
46C2×Dic284C2311yescharacteristic
47C2×#11765854—21no—
48C2×D568—21no—
49C2×D568—22no—
50C2×D568C2×C414yescomplemented
51C2×D568C2312yes—
52C2×D568C2311yescharacteristic
53C4×Dic142—22no—
54C4×D284—22no—
55C4×D284—22no—
56C4×D284—22no—
57#14556090—41no—
58#14556090—41no—
59(C22⋊D71)⋊C4—42no—
60#14556087—21no—
61#14556086—41no—
62C2×C4⋊Dic71—21no—
63C4⋊C4×D71—41no—
64C22⋊C4×D71—42no—
65#14556073C2×C418yescomplemented
66D1136C2×C418yescomplemented
67#14556062—21no—
68#14556062C2312yes—
69#14556062C2311yescharacteristic
70#14556062C2311yescharacteristic
71#14556060—21no—
72#14556053—22no—
73#14556050—22no—
74#14556050—22no—
75#14556050—22no—
76C2×M16×C71D411yescharacteristic
77C22×C8×C71C2311yescharacteristic
78#117×C71D411yescharacteristic
79M32×C71C2×C414yescomplemented
80C2×C16×C71—22no—
81C2×C16×C71C2311yescharacteristic
82C2×C16×C71C2311yescharacteristic
83C2×C16×C71C2312yes—
84#97×C71D412yes—
85C4×C8×C71C2312yes—
86C23×D71—41no—
87#11765865—24no—
88#11765865—42no—
89C2×Dic142—21no—
90C2×D284—24no—
91C2×D284—22no—
92C2×D284—21no—
93C2×D284—41no—
94C2×D284—42no—
95C2×C22⋊D71—41no—
96C22×Dic71—41no—
97C2×C4×D71—41no—
98C2×C4×D71—42no—
99C2×C4×D71—42no—
100C2×C4×D71—42no—
101Dic284—21no—
102Dic284C22×C412yes—
103#11765854—24no—
104D568—24no—
105D568—22no—
106D568—21no—
107D568C22×C412yes—
108D568C22×C414yes—
109C4⋊Dic71—21no—
110C4⋊Dic71—21no—
111C4⋊Dic71—22no—
112C4⋊Dic71—41no—
113Dic71⋊C4—42no—
114C22⋊Dic71—41no—
115(C2×D71)⋊C4—42no—
116(C2×D71)⋊C4—42no—
117(C2×D71)⋊C4—42no—
118C4×Dic71—42no—
119C4×Dic71—41no—
120C22×C4×C71C2×D411yescharacteristic
121M16×C71—22no—
122C2×C8×C71C22×C412yes—
123C2×C8×C71C2×D412yes—
124C2×C8×C71C2×D411yescharacteristic
125C2×C8×C71#4112yes—
126C2×C8×C71C2411yescharacteristic
127C4⋊C4×C71D811yescharacteristic
128C4⋊C4×C71D811yescharacteristic
129C22⋊C4×C71D812yes—
130C42×C71C2×D412yes—
131C16×C71—21no—
132C16×C71—21no—
133C16×C71C22×C414yes—
134C23×C71C2×D811yescharacteristic
135C22×D71—41no—
136C22×D71—42no—
137C22×D71—44no—
138C22×D71—161no—
139C22×D71—161no—
140C2×C4×C71C4×D412yes—
141C2×C4×C71C2×D812yes—
142C2×C4×C71C2×D812yes—
143C2×C4×C71C2×D811yescharacteristic
144C2×C4×C71#13512yes—
145C2×C4×C71C22×D411yescharacteristic
146C22⋊D71—44no—
147C2×Dic71—41no—
148C2×Dic71—41no—
149C2×Dic71—41no—
150C2×Dic71—42no—
151C2×Dic71—161no—
152D284—24no—
153D284—22no—
154D284—42no—
155D284—41no—
156D284—44no—
157C4×D71—44no—
158C4×D71—162no—
159C4×D71—162no—
160C4×D71—162no—
161Dic142—22no—
162Dic142—41no—
163C8×C71—22no—
164C8×C71C4×D412yes—
165C8×C71C23×C411yescharacteristic
166C8×C71C2×#4111yescharacteristic
167C22×C71—21no—
168C22×C71—21no—
169C22×C71C4×D812yes—
170C22×C71C22×D811yescharacteristic
171C2×D71—42no—
172C2×D71—44no—
173C2×D71—81no—
174C2×D71—82no—
175C2×D71—82no—
176C2×D71—81no—
177C2×D71—164no—
178C4×C71—22no—
179C4×C71—22no—
180C4×C71C4×D812yes—
181C4×C71C2×C4×D411yescharacteristic
182C4×C71C2×#13511yescharacteristic
183Dic71—42no—
184Dic71—81no—
185Dic71—82no—
186Dic71—81no—
187#33454—711nomaximal, Sylow
188C2×C71—22no—
189C2×C71C2×C4×D811yescharacteristic
190C2×C71#341511yescharacteristic
191C2×C71#341611yescharacteristic
192D71—82no—
193D71—84no—
194C2×#509—711no—
195#2951—711no—
196#2948—711no—
197#2177—712no—
198#2175—712no—
199C4×SD32—712no—
200C4×D16—712no—
201#2159—711no—
202#2138—711no—
203C2×#357—711no—
204#2108D7111yescharacteristic, complemented
205C71#3345411yescharacteristic, complemented, Sylow
206C2×#135—711no—
207C22×D8—711no—
208#520—1421no—
209#518—1421no—
210#509—718no—
211C2×SD32—712no—
212C2×D16—712no—
213C2×M32C2×D7111yescharacteristic
214C22×C16C2×D7111yescharacteristic
215#442—1422no—
216#440—1422no—
217C4×Q16—712no—
218C4×D8—712no—
219C4×D8—712no—
220C4×D8—712no—
221#427—711no—
222C2×#107—711no—
223#418—1421no—
224#417—1421no—
225#405C2×D7111yescharacteristic
226#367—712no—
227#366—712no—
228#358—711no—
229#358—711no—
230#357—712no—
231#357—712no—
232#357—711no—
233#357—711no—
234#346C2×D7112yescomplemented
235C4×C16C2×D7112yescomplemented
236C2×#41—1421no—
237C22×D4—1421no—
238#135—714no—
239#135—1422no—
240C2×Q16—711no—
241C2×SD16—1421no—
242C2×D8—714no—
243C2×D8—712no—
244C2×D8—711no—
245C2×D8—1421no—
246C2×D8—1422no—
247C2×M16D28411yescharacteristic
248C22×C8C22×D7111yescharacteristic
249C4×Q8—1422no—
250C4×D4—2841no—
251C4×D4—2841no—
252C4×D4—2842no—
253C4×D4—1422no—
254C4×D4—1422no—
255C4×D4—1422no—
256#117—2841no—
257#117—1421no—
258#117D28411yescharacteristic
259C2×C4⋊C4—2841no—
260C2×C4⋊C4—1421no—
261C2×C22⋊C4—2842no—
262SD32—718no—
263D16—718no—
264M32C4×D7114yescomplemented
265C2×C16—22no—
266C2×C16C22×D7111yescharacteristic
267C2×C16C22×D7111yescharacteristic
268C2×C16C22×D7112yes—
269#107—712no—
270#107—711no—
271#107—711no—
272#107—1421no—
273#106—1421no—
274#103—1422no—
275#102—1422no—
276#102—1422no—
277#102—1422no—
278#97D28412yes—
279C4×C8C22×D7112yes—
280C24—2841no—
281#41—2842no—
282#41—1424no—
283C2×Q8—1421no—
284C2×D4—2841no—
285C2×D4—2841no—
286C2×D4—2842no—
287C2×D4—1422no—
288C2×D4—1421no—
289C2×D4—1424no—
290C22×C4—2841no—
291C22×C4—2841no—
292C22×C4—2842no—
293C22×C4—2842no—
294C22×C4—2842no—
295C22×C4C2×D28411yescharacteristic
296Q16—712no—
297Q16—1421no—
298SD16—1424no—
299D8—712no—
300D8—714no—
301D8—1422no—
302D8—1421no—
303D8—1424no—
304M16—22no—
305C2×C8C2×C4×D7112yes—
306C2×C8C2×D28412yes—
307C2×C8C2×D28411yescharacteristic
308C2×C8#1176586512yes—
309C2×C8C23×D7111yescharacteristic
310C4⋊C4—2842no—
311C4⋊C4—2841no—
312C4⋊C4—1421no—
313C4⋊C4—1421no—
314C4⋊C4—1422no—
315C4⋊C4D56811yescharacteristic
316C4⋊C4D56811yescharacteristic
317C22⋊C4—2841no—
318C22⋊C4—2842no—
319C22⋊C4—2842no—
320C22⋊C4—2842no—
321C22⋊C4D56812yes—
322C42—2842no—
323C42—2841no—
324C42C2×D28412yes—
325C16—21no—
326C16—21no—
327C16C2×C4×D7114yes—
328C23—2841no—
329C23—2842no—
330C23—2844no—
331C23—11361no—
332C23—11361no—
333C23C2×D56811yescharacteristic
334Q8—2841no—
335Q8—1422no—
336D4—2844no—
337D4—2844no—
338D4—2842no—
339D4—2841no—
340D4—1424no—
341D4—1422no—
342C2×C4—2841no—
343C2×C4—2844no—
344C2×C4—2841no—
345C2×C4—2841no—
346C2×C4—2842no—
347C2×C4—11362no—
348C2×C4—11362no—
349C2×C4—11362no—
350C2×C4—11361no—
351C2×C4C4×D28412yes—
352C2×C4C2×D56812yes—
353C2×C4C2×D56812yes—
354C2×C4C2×D56811yescharacteristic
355C2×C4#1455615012yes—
356C2×C4C22×D28411yescharacteristic
357C8—22no—
358C8C4×D28412yes—
359C8C22×C4×D7111yescharacteristic
360C8C2×#1176586511yescharacteristic
361C22—21no—
362C22—21no—
363C22—2842no—
364C22—2844no—
365C22—5681no—
366C22—5682no—
367C22—5682no—
368C22—5681no—
369C22—11364no—
370C22C4×D56812yes—
371C22C22×D56811yescharacteristic
372C4—22no—
373C4—22no—
374C4—2842no—
375C4—5681no—
376C4—5682no—
377C4—5681no—
378C4C4×D56812yes—
379C4C2×C4×D28411yescharacteristic
380C4C2×#1455615011yescharacteristic
381C2—22no—
382C2—5684no—
383C2—5682no—
384C2C2×C4×D56811yescharacteristic
385C2#2260148111yescharacteristic
386C2#2260148811yescharacteristic
subgroupG/H
centerC22C22×D568
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×#2108C2
socleC22×C71C22×D8
prime radicalC2×#17986437C2
cosocleC2×C8×C71C24
hypercenter#2108D71
hypocenterC71#33454
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108

Complements

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

NH ≅ G/Ntype
2. #22597217C2semidirect
3. #22597193C2semidirect
4. #22590217C2semidirect
5. #22590206C2semidirect
6. C4×#14556073C2semidirect
7. C4×D1136C2semidirect
8. #22590128C2semidirect
9. #22590048C2semidirect
10. C2×#17985801C2semidirect
11. C71×#2108C2semidirect
16. #17986437C4semidirect
21. C4×Dic284C22semidirect
24. C4×D568C22semidirect
29. #17985827C22semidirect
30. #17985825C22semidirect
31. #17985804C22semidirect
32. #17985804C22semidirect
35. #17985801C22semidirect
36. #17985801C22semidirect
40. #346×C71C22semidirect
41. C4×C16×C71C22semidirect
45. #14556150C2×C4semidirect
50. C2×D568C2×C4semidirect
65. #14556073C2×C4semidirect
66. D1136C2×C4semidirect
79. M32×C71C2×C4semidirect
204. #2108D71semidirect
205. C71#33454semidirect
234. #346C2×D71semidirect
235. C4×C16C2×D71semidirect
264. M32C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864372
#225972172
#225971932
#225902172
#225902062
C4×#145560732
C4×D11362
#225901282
#225900482
C2×#179858012
C71×#21082
#3345471