Finite Group Databasebeta

← #31967290

Subgroups — #31967290

Order 18176.

662 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22597423C211yesmaximal, characteristic, complemented
2#22597380C211yesmaximal, characteristic
3C2×C4×#11765854C211yesmaximal, characteristic
4#22587244C214yesmaximal, complemented
5#22587232C214yesmaximal, complemented
6#22587088C211yesmaximal, characteristic, complemented
7#22586736C211yesmaximal, characteristic, complemented
8#22586732C211yesmaximal, characteristic
9C71×#1450C211yesmaximal, characteristic, complemented
10C22×#11765854C411yescharacteristic
11C2×C4×Dic142C2211yescharacteristic
12C2×C4×D284C2211yescharacteristic
13#17986129C2214yescomplemented
14#17986126—22no—
15C4×#11765854C412yes—
16C4×#11765854C412yes—
17C4×#11765854C2212yes—
18C4×#11765854C2212yes—
19#17986078C2214yes—
20#17986076—22no—
21#17986053—22no—
22#17986052C2211yescharacteristic
23#17986052C2211yescharacteristic
24C2×#14556060C411yescharacteristic
25C2×#14556053C411yescharacteristic
26C2×#14556050C411yescharacteristic
27#17985961—22no—
28#17985752C2212yescomplemented
29#17985750C2212yescomplemented
30#17985731C2214yescomplemented
31#17985728C2214yes—
32#431×C71C2211yescharacteristic
33#431×C71C2211yescharacteristic
34C2×C4×C8×C71C2211yescharacteristic
35#322×C71C2212yescomplemented
36C82×C71C2212yescomplemented
37C22×Dic142C2×C411yescharacteristic
38C22×D284C2×C411yescharacteristic
39C22×C4×D71—21no—
40C2×#11765854—24no—
41C2×#11765854C2×C412yes—
42C2×#11765854C2×C412yes—
43C2×#11765848—24no—
44C4×Dic142—21no—
45C4×Dic142—21no—
46C4×Dic142C2×C412yes—
47C4×Dic142C2312yes—
48C4×D284—21no—
49C4×D284—21no—
50C4×D284C2×C412yes—
51C4×D284C2312yes—
52C2×C4⋊Dic71C2×C411yescharacteristic
53C2×C4⋊Dic71C2×C411yescharacteristic
54C2×Dic71⋊C4—21no—
55C2×(C2×D71)⋊C4—21no—
56C2×C4×Dic71—21no—
57#14556060C2×C412yes—
58#14556060C2×C412yes—
59C4⋊(C71⋊C8)C2312yes—
60C4⋊(C71⋊C8)C2312yes—
61Dic71⋊C8—24no—
62#14556053C2×C412yes—
63#14556053C2×C412yes—
64#14556050C2×C412yes—
65#14556050C2×C412yes—
66C22⋊(C71⋊C8)—21no—
67C22⋊(C71⋊C8)—21no—
68(C2×D71)⋊C8—24no—
69#14556038—24no—
70C4×C71⋊C8—21no—
71C4×C71⋊C8—21no—
72C22×C8×C71C2×C411yescharacteristic
73C22×C8×C71C2×C411yescharacteristic
74C2×C42×C71C2311yescharacteristic
75C4⋊C8×C71—21no—
76C4⋊C8×C71D412yes—
77C22⋊C8×C71—21no—
78C22⋊C8×C71D412yes—
79C4×C8×C71C2×C411yescharacteristic
80C4×C8×C71C2×C411yescharacteristic
81C4×C8×C71C2312yes—
82C4×C8×C71C2311yescharacteristic
83C4×C8×C71C2311yescharacteristic
84C4×C8×C71C2312yes—
85C23×D71—21no—
86C2×Dic142—22no—
87C2×Dic142—22no—
88C2×Dic142—21no—
89C2×Dic142C22×C412yes—
90C2×D284—22no—
91C2×D284—22no—
92C2×D284—21no—
93C2×D284C22×C412yes—
94C22×Dic71—21no—
95C22×Dic71—21no—
96C22×Dic71—21no—
97C2×C4×D71—22no—
98C2×C4×D71—22no—
99C2×C4×D71—82no—
100#11765854—24no—
101#11765854—24no—
102#11765848—84no—
103C2×C71⋊C8—22no—
104C2×C71⋊C8—22no—
105C4⋊Dic71—21no—
106C4⋊Dic71—21no—
107C4⋊Dic71C22×C412yes—
108C4⋊Dic71C22×C412yes—
109Dic71⋊C4—22no—
110Dic71⋊C4—22no—
111(C2×D71)⋊C4—22no—
112(C2×D71)⋊C4—22no—
113C4×Dic71—22no—
114C4×Dic71—22no—
115C22×C4×C71C22×C411yescharacteristic
116C22×C4×C71C2×D411yescharacteristic
117C22×C4×C71#4111yescharacteristic
118C2×C8×C71—22no—
119C2×C8×C71—22no—
120C2×C8×C71M1614yescomplemented
121C2×C8×C71C22×C411yescharacteristic
122C2×C8×C71C22×C411yescharacteristic
123C2×C8×C71C22×C411yescharacteristic
124C2×C8×C71C22×C411yescharacteristic
125C2×C8×C71C2×D414yes—
126C42×C71C22×C411yescharacteristic
127C42×C71C2×D411yescharacteristic
128C42×C71#4111yescharacteristic
129C42×C71C2411yescharacteristic
130C23×C71C4×D411yescharacteristic
131C22×D71—22no—
132C22×D71—42no—
133C22×D71—81no—
134C22×D71—81no—
135C2×C4×C71—22no—
136C2×C4×C71—41no—
137C2×C4×C71C4×D411yescharacteristic
138C2×C4×C71C4×D411yescharacteristic
139C2×C4×C71C4×D411yescharacteristic
140C2×C4×C71C2×M1612yes—
141C2×C4×C71#13112yes—
142C2×C4×C71C23×C411yescharacteristic
143C2×C4×C71C22×D411yescharacteristic
144C2×C4×C71C2×#4111yescharacteristic
145C2×Dic71—22no—
146C2×Dic71—22no—
147C2×Dic71—22no—
148C2×Dic71—42no—
149C2×Dic71—81no—
150C2×Dic71—81no—
151D284—22no—
152D284—22no—
153D284—42no—
154C4×D71—44no—
155C4×D71—82no—
156Dic142—22no—
157Dic142—22no—
158Dic142—42no—
159C8×C71—21no—
160C8×C71—21no—
161C8×C71—24no—
162C8×C71—42no—
163C8×C71C2×M1612yes—
164C8×C71C2×M1612yes—
165C71⋊C8—42no—
166C71⋊C8—42no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71#43612yes—
170C22×C71C2×C4×D411yescharacteristic
171C2×D71—42no—
172C2×D71—42no—
173C2×D71—42no—
174C2×D71—84no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71#43612yes—
179C4×C71C22×M1611yescharacteristic
180C4×C71C2×#13111yescharacteristic
181C4×C71#57712yes—
182Dic71—42no—
183Dic71—42no—
184Dic71—42no—
185#17896—711nomaximal, Sylow
186C2×C71—22no—
187C2×C71C2×#43611yescharacteristic
188C2×C71#294911yescharacteristic
189C2×C71#295711yescharacteristic
190D71—82no—
191#2983—711no—
192#2975—711no—
193C2×C4×SD16—711no—
194#1592—714no—
195#1590—714no—
196#1564—711no—
197#1479—711no—
198#1478—711no—
199#1450D7111yescharacteristic, complemented
200C71#1789611yescharacteristic, complemented, Sylow
201C22×SD16—711no—
202C2×C4×Q8—711no—
203C2×C4×D4—711no—
204#446—714no—
205#446—1422no—
206C4×SD16—712no—
207C4×SD16—712no—
208C4×SD16—712no—
209C4×SD16—712no—
210#436—714no—
211#436—1422no—
212#432—1422no—
213#431—711no—
214#431—711no—
215#431C2×D7111yescharacteristic
216#431C2×D7111yescharacteristic
217C2×#106—711no—
218C2×#103—711no—
219C2×#102—711no—
220#407—1422no—
221C2×C4×C8C2×D7111yescharacteristic
222#335—712no—
223#334—712no—
224Q8⋊C8—714no—
225D4⋊C8—714no—
226#322C2×D7112yescomplemented
227C82C2×D7112yescomplemented
228C22×Q8—711no—
229C22×D4—711no—
230C23×C4—1421no—
231C2×SD16—712no—
232C2×SD16—712no—
233C2×SD16—1424no—
234C2×M16—1424no—
235C22×C8C4×D7111yescharacteristic
236C22×C8C4×D7111yescharacteristic
237C4×Q8—712no—
238C4×Q8—712no—
239C4×Q8—1421no—
240C4×Q8—1421no—
241C4×D4—712no—
242C4×D4—712no—
243C4×D4—1421no—
244C4×D4—1421no—
245C2×C4⋊C4—711no—
246C2×C4⋊C4—711no—
247C2×C4⋊C4—1421no—
248C2×C22⋊C4—1421no—
249C2×C42—1421no—
250C2×C42C22×D7111yescharacteristic
251#106—712no—
252#106—712no—
253C4⋊C8—21no—
254C4⋊C8—712no—
255C4⋊C8—712no—
256C4⋊C8—1424no—
257C4⋊C8D28412yes—
258#103—712no—
259#103—712no—
260#102—712no—
261#102—712no—
262C22⋊C8—21no—
263C22⋊C8—1421no—
264C22⋊C8—1421no—
265C22⋊C8—1424no—
266C22⋊C8D28412yes—
267#97—1424no—
268C4×C8—1421no—
269C4×C8—1421no—
270C4×C8C4×D7111yescharacteristic
271C4×C8C4×D7111yescharacteristic
272C4×C8C22×D7112yes—
273C4×C8C22×D7111yescharacteristic
274C4×C8C22×D7111yescharacteristic
275C4×C8C22×D7112yes—
276C24—1421no—
277C2×Q8—712no—
278C2×Q8—1422no—
279C2×Q8—1422no—
280C2×Q8—1421no—
281C2×D4—712no—
282C2×D4—1421no—
283C2×D4—1422no—
284C2×D4—1422no—
285C22×C4—5682no—
286C22×C4—1421no—
287C22×C4—1422no—
288C22×C4—1421no—
289C22×C4—1421no—
290C22×C4—1422no—
291C22×C4C2×C4×D7111yescharacteristic
292C22×C4C2×D28411yescharacteristic
293C22×C4#1176586511yescharacteristic
294SD16—1424no—
295SD16—1424no—
296M16—5684no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—1422no—
300C2×C8—1422no—
301C2×C8#1176584814yescomplemented
302C2×C8C2×C4×D7111yescharacteristic
303C2×C8C2×C4×D7111yescharacteristic
304C2×C8C2×C4×D7111yescharacteristic
305C2×C8C2×C4×D7111yescharacteristic
306C2×C8C2×D28414yes—
307C4⋊C4—712no—
308C4⋊C4—712no—
309C4⋊C4—1422no—
310C4⋊C4—1422no—
311C4⋊C4—1421no—
312C4⋊C4—1421no—
313C22⋊C4—1422no—
314C22⋊C4—1422no—
315C42—1422no—
316C42—1422no—
317C42C2×C4×D7111yescharacteristic
318C42C2×D28411yescharacteristic
319C42#1176586511yescharacteristic
320C42C23×D7111yescharacteristic
321C23—2842no—
322C23—5681no—
323C23—5681no—
324C23—1422no—
325C23C4×D28411yescharacteristic
326Q8—2842no—
327Q8—1422no—
328Q8—1422no—
329D4—2842no—
330D4—1422no—
331D4—1422no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—2842no—
335C2×C4—2844no—
336C2×C4—5682no—
337C2×C4—5681no—
338C2×C4—5681no—
339C2×C4—1422no—
340C2×C4—1422no—
341C2×C4—1422no—
342C2×C4C4×D28411yescharacteristic
343C2×C4C4×D28411yescharacteristic
344C2×C4C4×D28411yescharacteristic
345C2×C4C2×#1176584812yes—
346C2×C4#1455613412yes—
347C2×C4C22×C4×D7111yescharacteristic
348C2×C4C22×D28411yescharacteristic
349C2×C4C2×#1176586511yescharacteristic
350C8—21no—
351C8—21no—
352C8—24no—
353C8—42no—
354C8—2842no—
355C8—2842no—
356C8C2×#1176584812yes—
357C8C2×#1176584812yes—
358C22—21no—
359C22—21no—
360C22—2842no—
361C22—2842no—
362C22—2842no—
363C22—5684no—
364C22#1798607812yes—
365C22C2×C4×D28411yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—2842no—
370C4—2842no—
371C4—2842no—
372C4#1798607812yes—
373C4C22×#1176584811yescharacteristic
374C4C2×#1455613411yescharacteristic
375C4#1798681112yes—
376C2—22no—
377C2—5682no—
378C2C2×#1798607811yescharacteristic
379C2#2259720111yescharacteristic
380C2#2259725811yescharacteristic
subgroupG/H
centerC2×C4C22×D284
commutatorC2×C4×C71C23×C4
FrattiniC42C23×D71
FittingC71×#1450C2
socleC22×C71C2×C4×D4
prime radicalC22×D284C2×C4
cosocleC42×C71C24
hypercenter#1450D71
hypocenterC71#17896
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1450

Complements

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

NH ≅ G/Ntype
1. #22597423C2semidirect
4. #22587244C2semidirect
5. #22587232C2semidirect
6. #22587088C2semidirect
7. #22586736C2semidirect
9. C71×#1450C2semidirect
13. #17986129C22semidirect
28. #17985752C22semidirect
29. #17985750C22semidirect
30. #17985731C22semidirect
35. #322×C71C22semidirect
36. C82×C71C22semidirect
120. C2×C8×C71M16semidirect
199. #1450D71semidirect
200. C71#17896semidirect
226. #322C2×D71semidirect
227. C82C2×D71semidirect
301. C2×C8#11765848semidirect

Maximal subgroups

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

subgroupindex
#225974232
#225973802
C2×C4×#117658542
#225872442
#225872322
#225870882
#225867362
#225867322
C71×#14502
#1789671