Finite Group Databasebeta

← #32005442

Subgroups — #32005442

Order 18176.

858 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986353C211yesmaximal, characteristic
2#22590895C211yesmaximal, characteristic, complemented
3#124×Dic71C211yesmaximal, characteristic
4#22589305C211yesmaximal, characteristic, complemented
5#22589020C212yesmaximal
6#22589013C212yesmaximal, complemented
7#22588470C211yesmaximal, characteristic, complemented
8#103×Dic71C212yesmaximal, complemented
9#102×Dic71C212yesmaximal
10C4×C8×Dic71C211yesmaximal, characteristic, complemented
11C71×#1958C211yesmaximal, characteristic
12C2×#14556120C2211yescharacteristic
13C2×Q8×Dic71C2211yescharacteristic
14C2×D4×Dic71C2211yescharacteristic
15#17986353C418yescomplemented
16C2×#14556052C2212yes—
17C2×#14556049C2212yes—
18C2×C4×C71⋊C8C2211yescharacteristic
19C2×C8×Dic71C2212yes—
20C2×C8×Dic71C2212yes—
21#17985906C2211yescharacteristic
22#17985902C2211yescharacteristic
23#17985895C2212yes—
24#17985892—22no—
25#17985881—22no—
26#17985865C2211yescharacteristic
27C4⋊C4×Dic71C2212yes—
28#17985850—22no—
29C22⋊C4×Dic71—22no—
30C42×Dic71C2211yescharacteristic
31#17985762C2212yes—
32#17985762C2212yes—
33#17985761C2212yes—
34#17985761C2212yes—
35C2×#124×C71C2211yescharacteristic
36C2×#103×C71C2212yes—
37C2×#102×C71C2212yes—
38C2×C4×C8×C71C2211yescharacteristic
39#382×C71C2211yescharacteristic
40C23×Dic71—21no—
41C22×C71⋊C8C2312yes—
42#14556120—22no—
43#14556120C2×C414yescomplemented
44Q8×Dic71—24no—
45D4×Dic71—24no—
46C2×C4⋊Dic71—21no—
47C2×C4⋊Dic71C2311yescharacteristic
48C2×C4⋊Dic71C2311yescharacteristic
49C2×C4⋊Dic71C2312yes—
50C2×Dic71⋊C4—22no—
51C2×Dic71⋊C4—22no—
52C2×C22⋊Dic71—22no—
53C2×C22⋊Dic71—21no—
54C2×C4×Dic71—21no—
55C2×C4×Dic71—22no—
56C2×C4×Dic71D412yes—
57C2×C4×Dic71D412yes—
58C2×C4×Dic71C2311yescharacteristic
59C2×C4×Dic71C2311yescharacteristic
60#14556052C2×C418yescomplemented
61#14556049C2×C418yes—
62C4×C71⋊C8C2×C414yescomplemented
63C8×Dic71—24no—
64C8×Dic71—24no—
65C71⋊#95—22no—
66C71⋊#95—22no—
67C71⋊#95—22no—
68C71⋊#95—22no—
69C22×Q8×C71C2311yescharacteristic
70C22×D4×C71C2311yescharacteristic
71C22×C8×C71C2312yes—
72#124×C71—22no—
73#124×C71C2×C414yes—
74C2×C4⋊C4×C71C2312yes—
75C2×C22⋊C4×C71—22no—
76C2×C42×C71C2311yescharacteristic
77#103×C71—24no—
78#102×C71—24no—
79C4×C8×C71—22no—
80#95×C71—22no—
81C22×Dic71—22no—
82C22×Dic71—21no—
83C22×Dic71—21no—
84C22×Dic71—42no—
85C22×Dic71—44no—
86C22×Dic71C2×D411yescharacteristic
87C22×Dic71C2×D412yes—
88C22×Dic71C2×D411yescharacteristic
89C2×C71⋊C8—22no—
90C2×C71⋊C8—22no—
91C2×C71⋊C8C22×C414yes—
92C4⋊Dic71—22no—
93C4⋊Dic71—22no—
94C4⋊Dic71—22no—
95C4⋊Dic71—42no—
96C4⋊Dic71C22×C414yes—
97Dic71⋊C4—28no—
98Dic71⋊C4—44no—
99C22⋊Dic71—42no—
100C22⋊Dic71—44no—
101C4×Dic71—22no—
102C4×Dic71—24no—
103C4×Dic71—24no—
104C4×Dic71—42no—
105C4×Dic71—44no—
106C4×Dic71#4114yes—
107C24×C71—21no—
108C2×Q8×C71—21no—
109C2×Q8×C71—22no—
110C2×Q8×C71—22no—
111C2×Q8×C71C22×C412yes—
112C2×D4×C71—22no—
113C2×D4×C71—22no—
114C2×D4×C71—21no—
115C2×D4×C71C22×C412yes—
116C22×C4×C71—21no—
117C22×C4×C71—22no—
118C22×C4×C71C2×D412yes—
119C22×C4×C71C2411yescharacteristic
120C2×C8×C71—22no—
121C2×C8×C71—22no—
122C2×C8×C71#4114yes—
123C4⋊C4×C71—22no—
124C4⋊C4×C71—22no—
125C22⋊C4×C71—28no—
126C42×C71C22×C412yes—
127C42×C71#4112yes—
128C23×C71—22no—
129C23×C71—41no—
130C23×C71—41no—
131C23×C71—42no—
132C23×C71—42no—
133C23×C71C22×D411yescharacteristic
134Q8×C71—24no—
135Q8×C71—42no—
136D4×C71—24no—
137D4×C71—42no—
138C2×C4×C71—22no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—42no—
146C2×C4×C71C4×D414yes—
147C2×C4×C71C23×C411yescharacteristic
148C2×C4×C71C2×#4112yes—
149C2×C4×C71C2×#4112yes—
150C2×C4×C71C2×#4111yescharacteristic
151C2×Dic71—22no—
152C2×Dic71—22no—
153C2×Dic71—22no—
154C2×Dic71—24no—
155C2×Dic71—42no—
156C2×Dic71—42no—
157C2×Dic71—41no—
158C2×Dic71—41no—
159C2×Dic71—42no—
160C2×Dic71—42no—
161C2×Dic71—84no—
162C2×Dic71#12412yes—
163C2×Dic71#12412yes—
164C2×Dic71#13514yes—
165C2×Dic71#13514yes—
166C8×C71—24no—
167C71⋊C8—24no—
168C22×C71—44no—
169C22×C71—42no—
170C22×C71—81no—
171C22×C71—81no—
172C22×C71—82no—
173C22×C71C2×C4×D411yescharacteristic
174C22×C71C2×#12411yescharacteristic
175C22×C71#53211yescharacteristic
176C22×C71C2×#13512yes—
177C22×C71C2×#13512yes—
178C4×C71—24no—
179C4×C71—42no—
180C4×C71—82no—
181C4×C71C4×#4112yes—
182C4×C71#52912yes—
183Dic71—22no—
184Dic71—24no—
185Dic71—44no—
186Dic71—81no—
187Dic71—81no—
188Dic71#48814yescomplemented
189C4×#488—711nomaximal, Sylow
190C2×C71—82no—
191C2×C71C4×#12411yescharacteristic
192C2×C71C4×#13512yes—
193C2×C71C2×#48811yescharacteristic
194C2×C71#313111yescharacteristic
195C2×C71#316812yes—
196C2×#488—711no—
197C4×#125—711no—
198C4×#124—711no—
199#1991—711no—
200#1958D7111yescharacteristic
201#1934—712no—
202#1933—712no—
203#1835—711no—
204C4×#103—712no—
205C4×#102—712no—
206C42×C8—711no—
207C71C4×#48811yescharacteristic, complemented, Sylow
208C2×#125—711no—
209C2×#124C2×D7111yescharacteristic
210C2×C4×Q8—711no—
211C2×C4×D4—711no—
212#488—718no—
213C2×#103—712no—
214C2×#103C2×D7112yes—
215C2×#102—712no—
216C2×#102C2×D7112yes—
217C2×C4×C8—711no—
218C2×C4×C8—712no—
219C2×C4×C8—712no—
220C2×C4×C8C2×D7111yescharacteristic
221#391—711no—
222#390—711no—
223#389—712no—
224#388—1422no—
225#387—1422no—
226#382—711no—
227#382C2×D7111yescharacteristic
228C4×C4⋊C4—712no—
229C4×C4⋊C4—1422no—
230C4×C22⋊C4—1422no—
231C43—711no—
232#340—712no—
233#340—712no—
234#340—712no—
235#340—712no—
236C22×Q8C22×D7111yescharacteristic
237C22×D4C22×D7111yescharacteristic
238C23×C4—1421no—
239C22×C8—712no—
240C22×C8C22×D7112yes—
241#125—714no—
242#125—1422no—
243#124—22no—
244#124C4×D7114yes—
245C4×Q8—1424no—
246C4×D4—1424no—
247C2×C4⋊C4—712no—
248C2×C4⋊C4—711no—
249C2×C4⋊C4—711no—
250C2×C4⋊C4—1421no—
251C2×C4⋊C4—1422no—
252C2×C4⋊C4—1422no—
253C2×C4⋊C4C22×D7112yes—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—1421no—
256C2×C22⋊C4—1422no—
257C2×C42—711no—
258C2×C42—712no—
259C2×C42—712no—
260C2×C42—711no—
261C2×C42—1421no—
262C2×C42—1422no—
263C2×C42C22×D7111yescharacteristic
264#103—24no—
265#103—718no—
266#102—24no—
267#102—718no—
268C4×C8—22no—
269C4×C8—714no—
270C4×C8—1424no—
271C4×C8—1424no—
272#95—22no—
273#95—1422no—
274#95—1422no—
275#95—1422no—
276#95—1422no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8C2×C4×D7112yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4C2×C4×D7112yes—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—2842no—
289C22×C4—2844no—
290C22×C4—711no—
291C22×C4—711no—
292C22×C4—712no—
293C22×C4—1421no—
294C22×C4—1421no—
295C22×C4—1422no—
296C22×C4D4×D7112yes—
297C22×C4C23×D7111yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—714no—
301C2×C8—1422no—
302C2×C8—1422no—
303C2×C8#1176586414yes—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—2844no—
307C4⋊C4—2842no—
308C4⋊C4—714no—
309C4⋊C4—1422no—
310C4⋊C4—1422no—
311C4⋊C4—1422no—
312C4⋊C4—1428no—
313C22⋊C4—28no—
314C22⋊C4—2844no—
315C22⋊C4—2842no—
316C42—2844no—
317C42—2842no—
318C42—714no—
319C42—1424no—
320C42—1424no—
321C42—1422no—
322C42C2×C4×D7112yes—
323C42#1176586412yes—
324C23—22no—
325C23—41no—
326C23—42no—
327C23—42no—
328C23—41no—
329C23C2×D4×D7111yescharacteristic
330Q8—24no—
331Q8—42no—
332D4—24no—
333D4—42no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—2842no—
343C2×C4—2842no—
344C2×C4—2841no—
345C2×C4—2841no—
346C2×C4—2842no—
347C2×C4—2842no—
348C2×C4—5684no—
349C2×C4—712no—
350C2×C4—714no—
351C2×C4—712no—
352C2×C4—714no—
353C2×C4—1424no—
354C2×C4—1422no—
355C2×C4—1422no—
356C2×C4—1422no—
357C2×C4(C22⋊D71)⋊C414yes—
358C2×C4C22×C4×D7111yescharacteristic
359C2×C4#41×D7112yes—
360C2×C4C2×#1176586412yes—
361C2×C4C2×#1176586411yescharacteristic
362C8—24no—
363C8—1424no—
364C22—42no—
365C22—44no—
366C22—81no—
367C22—82no—
368C22—81no—
369C22C2×(C22⋊D71)⋊C411yescharacteristic
370C22#124×D7111yescharacteristic
371C22#1798654711yescharacteristic
372C22#135×D7112yes—
373C22#135×D7112yes—
374C4—24no—
375C4—42no—
376C4—82no—
377C4—2844no—
378C4—5681no—
379C4—5681no—
380C4—714no—
381C4—1422no—
382C4—1424no—
383C4C4×#1176586412yes—
384C4#1798653012yes—
385C2—82no—
386C2#2259088711yescharacteristic
387C2#2259710812yes—
388C2#488×D7111yescharacteristic
389C2#2259866111yescharacteristic
390C2#2259890712yes—
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1958C2
socleC23×C71C22×D4
prime radicalC22×D4×C71C23
cosocleC22×C4×C71C24
hypercenter#1958D71
hypocenterC71C4×#488
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1958

Complements

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

NH ≅ G/Ntype
2. #22590895C2semidirect
4. #22589305C2semidirect
6. #22589013C2semidirect
7. #22588470C2semidirect
8. #103×Dic71C2semidirect
10. C4×C8×Dic71C2semidirect
15. #17986353C4semidirect
43. #14556120C2×C4semidirect
60. #14556052C2×C4semidirect
62. C4×C71⋊C8C2×C4semidirect
188. Dic71#488semidirect
207. C71C4×#488semidirect

Maximal subgroups

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

subgroupindex
C2×#179863532
#225908952
#124×Dic712
#225893052
#225890202
#225890132
#225884702
#103×Dic712
#102×Dic712
C4×C8×Dic712
C71×#19582
C4×#48871