Finite Group Databasebeta

← #32150794

Subgroups — #32150794

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×#17986435C211yesmaximal, characteristic
2#22597217C211yesmaximal, characteristic, complemented
3#22590214C212yesmaximal, complemented
4#22590204C212yesmaximal, complemented
5C4×#14556072C212yesmaximal, complemented
6C4×#14556069C212yesmaximal, complemented
7#22590129C211yesmaximal, characteristic, complemented
8#22590045C211yesmaximal, characteristic, complemented
9C2×#17985800C211yesmaximal, characteristic, complemented
10#22589937C211yesmaximal, characteristic, complemented
11C71×#2948C211yesmaximal, characteristic, complemented
12C2×#14556150C2211yescharacteristic
13#17986479—21no—
14#17986435C418yescomplemented
15C2×#14556072C2212yes—
16C2×#14556069C2212yes—
17C2×#14556068C2211yescharacteristic
18C22×C71⋊C16C2211yescharacteristic
19#17986101—22no—
20C4×Dic284C2212yescomplemented
21C4×D568C2212yescomplemented
22C2×#14556062C2211yescharacteristic
23#17986001—21no—
24#17985826C2212yescomplemented
25#17985824C2212yescomplemented
26#17985802C2211yescharacteristic, complemented
27#17985802C2211yescharacteristic, complemented
28#17985800C2212yes—
29#17985800C2212yes—
30#17985799C2211yescharacteristic, complemented
31#17985799C2211yescharacteristic, complemented
32#17985775C2212yescomplemented
33C4×C71⋊C16C2212yescomplemented
34C22×D8×C71C2211yescharacteristic
35#518×C71—21no—
36#442×C71—22no—
37C4×D8×C71C2212yes—
38C4×D8×C71C2212yes—
39#427×C71C2211yescharacteristic
40#418×C71—21no—
41#405×C71C2211yescharacteristic
42C2×#11765865—21no—
43#14556150—22no—
44#14556150C2×C414yescomplemented
45C2×Dic284C2311yescharacteristic
46C2×#11765854—21no—
47C2×D568C2311yescharacteristic
48C4×Dic142—22no—
49C4×D284—22no—
50(C22⋊D71)⋊C4—42no—
51#14556086—41no—
52C2×C4⋊Dic71—21no—
53C4⋊C4×D71—41no—
54#14556072C2×C418yescomplemented
55#14556069C2×C418yescomplemented
56#14556068C2×C414yescomplemented
57C2×C71⋊C16—22no—
58C2×C71⋊C16C2311yescharacteristic
59C2×C71⋊C16C2311yescharacteristic
60C2×C71⋊C16C2312yes—
61#14556062—21no—
62#14556062C2312yes—
63#14556053—22no—
64#14556050—22no—
65C22×D4×C71—21no—
66C2×D8×C71—21no—
67C2×D8×C71—22no—
68C2×D8×C71C2×C414yescomplemented
69C2×D8×C71C2312yes—
70C2×M16×C71D411yescharacteristic
71C22×C8×C71C2311yescharacteristic
72C4×D4×C71—22no—
73C4×D4×C71—22no—
74C4×D4×C71—41no—
75C4×D4×C71—41no—
76#117×C71—21no—
77#117×C71D411yescharacteristic
78C2×C22⋊C4×C71—42no—
79#107×C71C2311yescharacteristic
80#107×C71C2311yescharacteristic
81#106×C71—21no—
82#102×C71—22no—
83#102×C71—22no—
84#97×C71D412yes—
85C4×C8×C71C2312yes—
86#11765865—24no—
87#11765865—42no—
88C2×Dic142—21no—
89C2×D284—21no—
90C2×C22⋊D71—41no—
91C22×Dic71—41no—
92C2×C4×D71—41no—
93C2×C4×D71—42no—
94Dic284—21no—
95Dic284C22×C412yes—
96#11765854—24no—
97D568—21no—
98D568C22×C412yes—
99C4⋊Dic71—22no—
100C4⋊Dic71—41no—
101Dic71⋊C4—42no—
102(C2×D71)⋊C4—42no—
103C4×Dic71—42no—
104C71⋊C16—21no—
105C71⋊C16—21no—
106C71⋊C16C22×C414yes—
107C24×C71—41no—
108C2×D4×C71—24no—
109C2×D4×C71—22no—
110C2×D4×C71—41no—
111C2×D4×C71—42no—
112C22×C4×C71—42no—
113C22×C4×C71—42no—
114C22×C4×C71C2×D411yescharacteristic
115D8×C71—24no—
116D8×C71—22no—
117D8×C71C22×C414yes—
118M16×C71—22no—
119C2×C8×C71C22×C412yes—
120C2×C8×C71C2×D412yes—
121C2×C8×C71C2×D411yescharacteristic
122C2×C8×C71#4112yes—
123C2×C8×C71C2411yescharacteristic
124C4⋊C4×C71—21no—
125C4⋊C4×C71—21no—
126C4⋊C4×C71D811yescharacteristic
127C4⋊C4×C71D811yescharacteristic
128C22⋊C4×C71—41no—
129C22⋊C4×C71—42no—
130C22⋊C4×C71—42no—
131C22⋊C4×C71D812yes—
132C42×C71—41no—
133C42×C71C2×D412yes—
134C23×C71—42no—
135C23×C71—44no—
136C23×C71—161no—
137C23×C71—161no—
138C23×C71C2×D811yescharacteristic
139C22×D71—41no—
140D4×C71—24no—
141D4×C71—42no—
142D4×C71—44no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—162no—
146C2×C4×C71—162no—
147C2×C4×C71C4×D412yes—
148C2×C4×C71C2×D812yes—
149C2×C4×C71C2×D812yes—
150C2×C4×C71C2×D811yescharacteristic
151C2×C4×C71#13512yes—
152C2×C4×C71C22×D411yescharacteristic
153C22⋊D71—44no—
154C2×Dic71—41no—
155C2×Dic71—42no—
156C2×Dic71—161no—
157D284—22no—
158D284—41no—
159C4×D71—44no—
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×C71—44no—
170C22×C71—82no—
171C22×C71—82no—
172C22×C71—164no—
173C22×C71C4×D812yes—
174C22×C71C22×D811yescharacteristic
175C2×D71—42no—
176C2×D71—81no—
177C2×D71—81no—
178C4×C71—22no—
179C4×C71—22no—
180C4×C71—81no—
181C4×C71—81no—
182C4×C71C4×D812yes—
183C4×C71C2×C4×D411yescharacteristic
184C4×C71C2×#13511yescharacteristic
185Dic71—42no—
186Dic71—82no—
187#33454—711nomaximal, Sylow
188C2×C71—22no—
189C2×C71—84no—
190C2×C71C2×C4×D811yescharacteristic
191C2×C71#341511yescharacteristic
192C2×C71#341611yescharacteristic
193D71—82no—
194C2×#509—711no—
195#2951—711no—
196#2948D7111yescharacteristic, complemented
197#2177—712no—
198#2175—712no—
199C4×SD32—712no—
200C4×D16—712no—
201#2159—711no—
202#2138—711no—
203C2×#357—711no—
204#2108—711no—
205C71#3345411yescharacteristic, complemented, Sylow
206C2×#135—711no—
207C22×D8C2×D7111yescharacteristic
208#520—1421no—
209#518—21no—
210#509—718no—
211C2×SD32—712no—
212C2×D16—712no—
213C2×M32—711no—
214C22×C16—711no—
215#442—22no—
216#440—1422no—
217C4×Q16—712no—
218C4×D8—712no—
219C4×D8C2×D7112yes—
220C4×D8C2×D7112yes—
221#427C2×D7111yescharacteristic
222C2×#107—711no—
223#418—21no—
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#346—712no—
235C4×C16—712no—
236C2×#41—1421no—
237C22×D4—21no—
238#135—714no—
239#135—1422no—
240C2×Q16—711no—
241C2×SD16—1421no—
242C2×D8—21no—
243C2×D8—22no—
244C2×D8—711no—
245C2×D8C4×D7114yescomplemented
246C2×D8C22×D7112yes—
247C2×M16C22⋊D7111yescharacteristic
248C22×C8C22×D7111yescharacteristic
249C4×Q8—1422no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—2842no—
255C4×D4—1422no—
256#117—21no—
257#117—2841no—
258#117C22⋊D7111yescharacteristic
259C2×C4⋊C4—2841no—
260C2×C4⋊C4—1421no—
261C2×C22⋊C4—42no—
262SD32—718no—
263D16—718no—
264M32—714no—
265C2×C16—712no—
266C2×C16—711no—
267C2×C16—711no—
268C2×C16—1422no—
269#107—712no—
270#107—1421no—
271#107C22×D7111yescharacteristic
272#107C22×D7111yescharacteristic
273#106—21no—
274#103—1422no—
275#102—22no—
276#102—22no—
277#102—1422no—
278#97C22⋊D7112yes—
279C4×C8C22×D7112yes—
280C24—41no—
281#41—2842no—
282#41—1424no—
283C2×Q8—1421no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—2841no—
289C2×D4—1421no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—2841no—
293C22×C4—2841no—
294C22×C4—2842no—
295C22×C4C2×C22⋊D7111yescharacteristic
296Q16—712no—
297Q16—1421no—
298SD16—1424no—
299D8—22no—
300D8—24no—
301D8—712no—
302D8—1421no—
303D8C2×C4×D7114yes—
304M16—22no—
305C2×C8C2×C4×D7112yes—
306C2×C8C2×C22⋊D7112yes—
307C2×C8C2×C22⋊D7111yescharacteristic
308C2×C8#1176586512yes—
309C2×C8C23×D7111yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—2842no—
313C4⋊C4—2841no—
314C4⋊C4—1422no—
315C4⋊C4#1176585011yescharacteristic
316C4⋊C4#1176585011yescharacteristic
317C22⋊C4—41no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—2842no—
321C22⋊C4#1176585012yes—
322C42—41no—
323C42—2842no—
324C42C2×C22⋊D7112yes—
325C16—714no—
326C16—1421no—
327C16—1421no—
328C23—42no—
329C23—44no—
330C23—161no—
331C23—161no—
332C23—2841no—
333C23C2×#1176585011yescharacteristic
334Q8—2841no—
335Q8—1422no—
336D4—24no—
337D4—44no—
338D4—42no—
339D4—2844no—
340D4—2841no—
341D4—1422no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—162no—
345C2×C4—162no—
346C2×C4—2841no—
347C2×C4—2844no—
348C2×C4—2842no—
349C2×C4—11362no—
350C2×C4—11361no—
351C2×C4C4×C22⋊D7112yes—
352C2×C4C2×#1176585012yes—
353C2×C4C2×#1176585012yes—
354C2×C4C2×#1176585011yescharacteristic
355C2×C4#1455614812yes—
356C2×C4C22×C22⋊D7111yescharacteristic
357C8—22no—
358C8C4×C22⋊D7112yes—
359C8C22×C4×D7111yescharacteristic
360C8C2×#1176586511yescharacteristic
361C22—21no—
362C22—21no—
363C22—44no—
364C22—82no—
365C22—82no—
366C22—164no—
367C22—2842no—
368C22—5681no—
369C22—5681no—
370C22C4×#1176585012yes—
371C22C22×#1176585011yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—81no—
376C4—2842no—
377C4—5682no—
378C4C4×#1176585012yes—
379C4C2×C4×C22⋊D7111yescharacteristic
380C4C2×#1455614811yescharacteristic
381C2—22no—
382C2—84no—
383C2—5682no—
384C2C2×C4×#1176585011yescharacteristic
385C2#2260147911yescharacteristic
386C2#2260148711yescharacteristic
subgroupG/H
centerC22C22×#11765850
commutatorC8×C71C23×C4
FrattiniC2×C8C23×D71
FittingC71×#2948C2
socleC22×C71C22×D8
prime radicalC2×#17986435C2
cosocleC2×C8×C71C24
hypercenter#2948D71
hypocenterC71#33454
LCSG ▷ C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2948

Complements

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

NH ≅ G/Ntype
2. #22597217C2semidirect
3. #22590214C2semidirect
4. #22590204C2semidirect
5. C4×#14556072C2semidirect
6. C4×#14556069C2semidirect
7. #22590129C2semidirect
8. #22590045C2semidirect
9. C2×#17985800C2semidirect
10. #22589937C2semidirect
11. C71×#2948C2semidirect
14. #17986435C4semidirect
20. C4×Dic284C22semidirect
21. C4×D568C22semidirect
24. #17985826C22semidirect
25. #17985824C22semidirect
26. #17985802C22semidirect
27. #17985802C22semidirect
30. #17985799C22semidirect
31. #17985799C22semidirect
32. #17985775C22semidirect
33. C4×C71⋊C16C22semidirect
44. #14556150C2×C4semidirect
54. #14556072C2×C4semidirect
55. #14556069C2×C4semidirect
56. #14556068C2×C4semidirect
68. C2×D8×C71C2×C4semidirect
196. #2948D71semidirect
205. C71#33454semidirect
245. C2×D8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179864352
#225972172
#225902142
#225902042
C4×#145560722
C4×#145560692
#225901292
#225900452
C2×#179858002
#225899372
C71×#29482
#3345471