Finite Group Databasebeta

← #32861401

Subgroups — #32861401

Order 18176.

2706 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986765C212yesmaximal, complemented
2#22601673C211yesmaximal, characteristic
3#22597064C2116yesmaximal, complemented
4#22597057C214yesmaximal, complemented
5#22597051C214yesmaximal, complemented
6C2×#17985967C212yesmaximal, complemented
7C2×#17985955C211yesmaximal, characteristic, complemented
8C71×#2905C211yesmaximal, characteristic, complemented
9C22×#11765865C2212yes—
10#17986765—216no—
11C2×#14556136C2218yescomplemented
12C2×#14556134C2218yes—
13C22×#11765849C2211yescharacteristic
14C2×M16×D71C2218yescomplemented
15C23×C71⋊C8C2211yescharacteristic
16#17986580C414yes—
17#17986575C414yes—
18#17986568C414yes—
19#17986561C414yes—
20#17986549C2214yes—
21#17986511C2218yes—
22#17986076C22116yescomplemented
23#17986069C22132yescomplemented
24#17986064C22116yescomplemented
25#17985967C2218yes—
26#17985967C2214yescomplemented
27#17985967C2214yescomplemented
28C2×(C2×D71)⋊C8C2218yescomplemented
29#17985955C2218yescomplemented
30C2×#14556039C2212yescomplemented
31C2×C4×C71⋊C8C2212yescomplemented
32C22×M16×C71C2212yes—
33C2×#117×C71C2211yescharacteristic
34#424×C71C2218yescomplemented
35#423×C71C2214yescomplemented
36C2×#11765865—28no—
37C2×#11765865—28no—
38C2×#11765865C2×C414yes—
39C2×#11765865C2×C414yes—
40C2×#11765865C2314yes—
41C2×#11765865C2314yes—
42C22×Dic142C2×C412yes—
43C22×D284C2×C412yes—
44C22×C22⋊D71C2×C414yes—
45C22×C4×D71C2314yes—
46#14556136—232no—
47#14556134—232no—
48C2×#11765849D418yescomplemented
49C2×#11765849C2314yes—
50C2×#11765848C23116yes—
51M16×D71—232no—
52C22×C71⋊C8—24no—
53C22×C71⋊C8C2311yescharacteristic
54C22×C71⋊C8C2311yescharacteristic
55C22×C71⋊C8C2314yes—
56C22×C71⋊C8C2312yes—
57C2×C8×D71C23116yes—
58#14556128C2×C414yes—
59#14556125C2×C414yes—
60#14556116C2×C418yes—
61#14556112C2×C4116yes—
62#14556108C2×C4116yes—
63#14556103C2×C4116yes—
64#14556102C2×C4116yes—
65#14556098C2×C4116yes—
66C4×D284C23116yes—
67(C2×D71)⋊C8C23116yes—
68(C2×D71)⋊C8C23116yes—
69#14556039C2318yes—
70C4×C71⋊C8C2318yes—
71C23×C4×C71C2311yescharacteristic
72C2×M16×C71—28no—
73C2×M16×C71C2314yes—
74C2×M16×C71C2314yes—
75C22×C8×C71C2314yes—
76#117×C71C2×C414yes—
77#117×C71C2314yes—
78C2×C4⋊C4×C71C2×C412yes—
79C2×C22⋊C4×C71C2×C412yes—
80C2×C42×C71C2312yes—
81C4⋊C8×C71C23116yes—
82C23×D71C22×C414yes—
83#11765865—216no—
84#11765865—216no—
85#11765865—28no—
86#11765865—28no—
87#11765865—28no—
88#11765865—28no—
89C2×Dic142—28no—
90C2×Dic142C22×C414yes—
91C2×Dic142C22×C414yes—
92C2×D284—28no—
93C2×D284C22×C414yes—
94C2×D284C22×C414yes—
95C2×D284C22×C4116yes—
96C2×C22⋊D71—216no—
97C2×C22⋊D71C22×C418yes—
98C2×C22⋊D71C22×C418yes—
99C22×Dic71C22×C414yes—
100C2×C4×D71—28no—
101C2×C4×D71—28no—
102C2×C4×D71C22×C418yes—
103C2×C4×D71C2418yes—
104#11765849C2×D4116yes—
105#11765848—232no—
106C2×C71⋊C8—28no—
107C2×C71⋊C8—22no—
108C2×C71⋊C8—22no—
109C2×C71⋊C8C2×D418yes—
110C2×C71⋊C8C2412yes—
111C2×C71⋊C8C2412yes—
112C2×C71⋊C8C2414yes—
113C8×D71—232no—
114C4⋊Dic71C22×C4116yes—
115(C2×D71)⋊C4C22×C4132yes—
116C24×C71C22×C411yescharacteristic
117C22×C4×C71C22×C414yes—
118C22×C4×C71C22×C412yes—
119C22×C4×C71C22×C411yescharacteristic
120C22×C4×C71C2×D414yes—
121C22×C4×C71#4114yes—
122C22×C4×C71C2412yes—
123C22×C4×C71C2411yescharacteristic
124M16×C71—28no—
125M16×C71—28no—
126C2×C8×C71—28no—
127C2×C8×C71C2418yes—
128C4⋊C4×C71C22×C418yes—
129C22⋊C4×C71C22×C418yes—
130C42×C71C22×C414yes—
131C42×C71C2414yes—
132C23×C71—42no—
133C23×C71C4×D414yes—
134C23×C71C23×C412yes—
135C23×C71C23×C411yescharacteristic
136C22×D71—28no—
137C22×D71—48no—
138C22×D71C23×C418yes—
139C2×C4×C71—44no—
140C2×C4×C71C4×D418yes—
141C2×C4×C71C4×D414yes—
142C2×C4×C71C23×C418yes—
143C2×C4×C71C23×C411yescharacteristic
144C2×C4×C71C23×C412yes—
145C2×C4×C71C22×D414yes—
146C2×C4×C71C22×D412yes—
147C2×C4×C71C2×#4114yes—
148C2×C4×C71C2×#4112yes—
149C2×C4×C71C2511yescharacteristic
150C22⋊D71—216no—
151C22⋊D71—216no—
152C2×Dic71—28no—
153C2×Dic71C23×C418yes—
154D284—28no—
155D284—28no—
156D284—416no—
157C4×D71—216no—
158C4×D71—28no—
159C4×D71—28no—
160Dic142—28no—
161Dic142—28no—
162C8×C71—28no—
163C71⋊C8—22no—
164C71⋊C8—22no—
165C71⋊C8C22×D418yes—
166C22×C71—22no—
167C22×C71—24no—
168C22×C71—44no—
169C22×C71C2×C4×D414yes—
170C22×C71C2×C4×D412yes—
171C22×C71C24×C411yescharacteristic
172C2×D71—28no—
173C2×D71—216no—
174C2×D71—88no—
175C4×C71—28no—
176C4×C71C2×C4×D412yes—
177C4×C71C2×C4×D414yes—
178C4×C71C23×D411yescharacteristic
179C4×C71C22×#4111yescharacteristic
180Dic71—28no—
181#60143—711nomaximal, Sylow
182C2×C71—24no—
183C2×C71—22no—
184C2×C71C22×C4×D411yescharacteristic
185C2×C71#357312yes—
186D71—48no—
187C2×#568—712no—
188#3449—711no—
189#2933—7116no—
190#2932—714no—
191#2931—714no—
192#2905D7111yescharacteristic, complemented
193C2×#408—712no—
194C2×#405—711no—
195C71#6014311yescharacteristic, complemented, Sylow
196C22×#41—712no—
197#568—14216no—
198C2×#131—718no—
199C2×#131—718no—
200C22×M16—711no—
201C22×M16—718no—
202C22×M16C2×D7112yes—
203C23×C8—711no—
204#537—714no—
205#536—714no—
206#535—714no—
207#534—714no—
208#532—714no—
209#525—718no—
210C2×#117C2×D7111yescharacteristic
211#436—7116no—
212#435—7132no—
213C8×D4—7116no—
214#424C2×D7118yescomplemented
215#423C2×D7114yescomplemented
216#408—714no—
217#408—714no—
218#408—718no—
219C2×C22⋊C8—718no—
220#405—718no—
221C2×#97—712no—
222C2×C4×C8—712no—
223C2×#41—714no—
224C2×#41—714no—
225C2×#41—714no—
226C2×#41—714no—
227C2×#41—1428no—
228C2×#41—1428no—
229C22×Q8—712no—
230C22×D4—714no—
231C22×D4—712no—
232C23×C4—714no—
233C23×C4C22×D7111yescharacteristic
234#131—14232no—
235#131—14232no—
236C2×M16—28no—
237C2×M16—718no—
238C2×M16—714no—
239C2×M16—7116no—
240C2×M16—14232no—
241C2×M16C22×D7114yes—
242C2×M16C22×D7114yes—
243C22×C8—712no—
244C22×C8—711no—
245C22×C8—711no—
246C22×C8—714no—
247C22×C8—7116no—
248C22×C8—1424no—
249C22×C8C22×D7114yes—
250C4⋊Q8—714no—
251#127—714no—
252#124—718no—
253#123—7116no—
254C22⋊Q8—7116no—
255#121—7116no—
256#121—7116no—
257#120—7116no—
258C4×D4—7116no—
259#117C4×D7114yes—
260#117C22×D7114yes—
261C2×C4⋊C4C4×D7112yes—
262C2×C22⋊C4C4×D7112yes—
263C2×C42C22×D7112yes—
264C4⋊C8C22×D71116yes—
265C22⋊C8—7116no—
266C22⋊C8—7116no—
267#97—718no—
268C4×C8—718no—
269C24—714no—
270C24C2×C4×D7111yescharacteristic
271#41—1428no—
272#41—1428no—
273#41—1428no—
274#41—1428no—
275#41—14216no—
276#41—14216no—
277C2×Q8—714no—
278C2×Q8—714no—
279C2×Q8—1428no—
280C2×D4—718no—
281C2×D4—718no—
282C2×D4—7116no—
283C2×D4—714no—
284C2×D4—714no—
285C2×D4—14216no—
286C2×D4—1428no—
287C22×C4—714no—
288C22×C4—718no—
289C22×C4—718no—
290C22×C4—1428no—
291C22×C4—1428no—
292C22×C4C2×C4×D7114yes—
293C22×C4C2×C4×D7111yescharacteristic
294C22×C4C2×C4×D7112yes—
295C22×C4D4×D7114yes—
296C22×C4#1176586614yes—
297C22×C4C23×D7111yescharacteristic
298C22×C4C23×D7112yes—
299M16—28no—
300M16—28no—
301M16—7116no—
302M16—14232no—
303C2×C8—28no—
304C2×C8—718no—
305C2×C8—714no—
306C2×C8—712no—
307C2×C8—712no—
308C2×C8—14232no—
309C2×C8—1428no—
310C2×C8—1422no—
311C2×C8—1422no—
312C2×C8C23×D7118yes—
313C4⋊C4—7116no—
314C4⋊C4C2×C4×D7118yes—
315C22⋊C4—7132no—
316C22⋊C4C2×C4×D7118yes—
317C42C2×C4×D7114yes—
318C42C23×D7114yes—
319C23—42no—
320C23—2848no—
321C23—718no—
322C23—1428no—
323C23#1455609014yes—
324C23C22×C4×D7112yes—
325C23C22×C4×D7111yescharacteristic
326Q8—1428no—
327Q8—1428no—
328D4—28416no—
329D4—14216no—
330D4—14216no—
331D4—1428no—
332D4—1428no—
333C2×C4—44no—
334C2×C4—718no—
335C2×C4—1428no—
336C2×C4—1428no—
337C2×C4—1428no—
338C2×C4—14216no—
339C2×C4#1455609018yes—
340C2×C4#1455609014yes—
341C2×C4C22×C4×D7118yes—
342C2×C4C22×C4×D7111yescharacteristic
343C2×C4C22×C4×D7112yes—
344C2×C4C2×D4×D7114yes—
345C2×C4C2×D4×D7112yes—
346C2×C4C2×#1176586614yes—
347C2×C4C2×#1176586612yes—
348C2×C4C24×D7111yescharacteristic
349C8—28no—
350C8—718no—
351C8—1422no—
352C8—1422no—
353C22—22no—
354C22—24no—
355C22—44no—
356C22—5688no—
357C22—1428no—
358C22—14216no—
359C22C2×#1455609014yes—
360C22C2×#1455609012yes—
361C22C23×C4×D7111yescharacteristic
362C4—28no—
363C4—1428no—
364C4C2×#1455609014yes—
365C4C2×#1455609012yes—
366C4C22×D4×D7111yescharacteristic
367C4C22×#1176586611yescharacteristic
368C2—24no—
369C2—22no—
370C2—2848no—
371C2C22×#1455609011yescharacteristic
372C2#2260305312yes—
subgroupG/H
center, FrattiniC2×C4C24×D71
commutator, socleC22×C71C24×C4
FittingC71×#2905C2
prime radical#22601673C2
cosocleC2×C4×C71C25
hypercenter#2905D71
hypocenterC71#60143
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C2×#17986765C2semidirect
3. #22597064C2semidirect
4. #22597057C2semidirect
5. #22597051C2semidirect
6. C2×#17985967C2semidirect
7. C2×#17985955C2semidirect
8. C71×#2905C2semidirect
11. C2×#14556136C22semidirect
14. C2×M16×D71C22semidirect
22. #17986076C22semidirect
23. #17986069C22semidirect
24. #17986064C22semidirect
26. #17985967C22semidirect
27. #17985967C22semidirect
28. C2×(C2×D71)⋊C8C22semidirect
29. #17985955C22semidirect
30. C2×#14556039C22semidirect
31. C2×C4×C71⋊C8C22semidirect
34. #424×C71C22semidirect
35. #423×C71C22semidirect
48. C2×#11765849D4semidirect
192. #2905D71semidirect
195. C71#60143semidirect
214. #424C2×D71semidirect
215. #423C2×D71semidirect

Maximal subgroups

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

subgroupindex
C2×#179867652
#226016732
#225970642
#225970572
#225970512
C2×#179859672
C2×#179859552
C71×#29052
#6014371