Finite Group Databasebeta

← #32005427

Subgroups — #32005427

Order 18176.

878 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
1C2×#17986351C211yesmaximal, characteristic
2C4⋊Q8×Dic71C211yesmaximal, characteristic
3#22590907C211yesmaximal, characteristic
4#22589142C211yesmaximal, characteristic
5#22589017C214yesmaximal
6#22588467C211yesmaximal, characteristic
7#22588159C214yesmaximal
8C4×C8×Dic71C211yesmaximal, characteristic
9C71×#1990C211yesmaximal, characteristic
10C2×#14556128C2211yescharacteristic
11C2×Q8×Dic71—21no—
12C2×#14556093C2212yes—
13#17986351C418yescomplemented
14C2×#14556053C2214yes—
15C2×#14556051C2214yes—
16C2×C4×C71⋊C8C2211yescharacteristic
17C2×C8×Dic71C2212yes—
18C2×C8×Dic71C2212yes—
19#17985906—21no—
20#17985904C2212yes—
21#17985898—21no—
22#17985898C2212yes—
23#17985895—21no—
24#17985895—21no—
25#17985869C2211yescharacteristic
26C4⋊C4×Dic71—21no—
27C4⋊C4×Dic71—21no—
28C4⋊C4×Dic71—21no—
29C4⋊C4×Dic71C2212yes—
30C42×Dic71C2211yescharacteristic
31#17985761C2214yes—
32C2×C4⋊Q8×C71C2211yescharacteristic
33C2×C4×C8×C71C2211yescharacteristic
34#384×C71C2211yescharacteristic
35#340×C71C2214yes—
36C22×Dic142C2312yes—
37C22×C71⋊C8C2312yes—
38#14556128—22no—
39#14556128C2×C414yes—
40Q8×Dic71—44no—
41#14556093—28no—
42C2×C4⋊Dic71—22no—
43C2×C4⋊Dic71—21no—
44C2×C4⋊Dic71—21no—
45C2×C4⋊Dic71—21no—
46C2×C4⋊Dic71—21no—
47C2×C4⋊Dic71C2312yes—
48C2×Dic71⋊C4—22no—
49C2×C4×Dic71—22no—
50C2×C4×Dic71—22no—
51C2×C4×Dic71D411yescharacteristic
52C2×C4×Dic71D411yescharacteristic
53C2×C4×Dic71D411yescharacteristic
54C2×C4×Dic71D411yescharacteristic
55C2×C4×Dic71C2311yescharacteristic
56C2×C4×Dic71C2311yescharacteristic
57#14556053—28no—
58#14556051C2×C4116yes—
59C4×C71⋊C8C2×C414yes—
60C8×Dic71—24no—
61C8×Dic71—24no—
62C71⋊#95—22no—
63C71⋊#95—22no—
64C71⋊#95—22no—
65C71⋊#95—22no—
66C22×Q8×C71—21no—
67C22×C8×C71C2312yes—
68C4⋊Q8×C71—22no—
69C4⋊Q8×C71C2×C414yes—
70C2×C4⋊C4×C71—21no—
71C2×C4⋊C4×C71—21no—
72C2×C4⋊C4×C71—21no—
73C2×C4⋊C4×C71C2312yes—
74C2×C4⋊C4×C71C2312yes—
75C2×C42×C71C2311yescharacteristic
76C4×C8×C71—22no—
77C2×Dic142—24no—
78C2×Dic142—24no—
79C2×Dic142—22no—
80C2×Dic142C22×C414yes—
81C22×Dic71—22no—
82C22×Dic71—22no—
83C22×Dic71C2×D411yescharacteristic
84C22×Dic71C2×D411yescharacteristic
85C22×Dic71C2×D411yescharacteristic
86C22×Dic71C2×D411yescharacteristic
87C2×C71⋊C8—22no—
88C2×C71⋊C8—22no—
89C2×C71⋊C8C22×C414yes—
90C4⋊Dic71—24no—
91C4⋊Dic71—24no—
92C4⋊Dic71—24no—
93C4⋊Dic71—42no—
94C4⋊Dic71—42no—
95C4⋊Dic71—44no—
96Dic71⋊C4—44no—
97C4×Dic71—22no—
98C4×Dic71—22no—
99C4×Dic71—22no—
100C4×Dic71—44no—
101C4×Dic71—44no—
102C4×Dic71SD1614yes—
103C4×Dic71Q1614yes—
104C4×Dic71#4114yes—
105C2×Q8×C71—24no—
106C2×Q8×C71—42no—
107C2×Q8×C71—42no—
108C22×C4×C71—22no—
109C22×C4×C71—22no—
110C22×C4×C71C2×D411yescharacteristic
111C22×C4×C71C2×D411yescharacteristic
112C22×C4×C71C2411yescharacteristic
113C2×C8×C71—22no—
114C2×C8×C71—22no—
115C2×C8×C71#4114yes—
116C4⋊C4×C71—24no—
117C4⋊C4×C71—22no—
118C4⋊C4×C71—24no—
119C4⋊C4×C71—42no—
120C4⋊C4×C71—42no—
121C4⋊C4×C71C22×C414yes—
122C42×C71C22×C412yes—
123C42×C71#4112yes—
124C23×C71C22×D411yescharacteristic
125Q8×C71—48no—
126C2×C4×C71—24no—
127C2×C4×C71—42no—
128C2×C4×C71—42no—
129C2×C4×C71—42no—
130C2×C4×C71—42no—
131C2×C4×C71—42no—
132C2×C4×C71C4×D412yes—
133C2×C4×C71C4×D412yes—
134C2×C4×C71C2×SD1612yes—
135C2×C4×C71C2×Q1612yes—
136C2×C4×C71#13512yes—
137C2×C4×C71#13512yes—
138C2×C4×C71C23×C411yescharacteristic
139C2×C4×C71C2×#4112yes—
140C2×C4×C71C2×#4111yescharacteristic
141C2×C4×C71C2×#4112yes—
142C2×Dic71—21no—
143C2×Dic71—21no—
144C2×Dic71—21no—
145C2×Dic71—21no—
146C2×Dic71—21no—
147C2×Dic71—21no—
148C2×Dic71—24no—
149C2×Dic71—42no—
150C2×Dic71—42no—
151C2×Dic71—42no—
152C2×Dic71—42no—
153C2×Dic71—42no—
154C2×Dic71#12412yes—
155C2×Dic71#12412yes—
156C2×Dic71C2×SD1612yes—
157C2×Dic71C2×SD1612yes—
158C2×Dic71C2×Q1612yes—
159C2×Dic71C2×Q1612yes—
160Dic142—28no—
161Dic142—44no—
162C8×C71—24no—
163C71⋊C8—24no—
164C22×C71C2×C4×D411yescharacteristic
165C22×C71C2×#12411yescharacteristic
166C22×C71#53211yescharacteristic
167C22×C71C22×SD1611yescharacteristic
168C22×C71C22×Q1611yescharacteristic
169C22×C71C2×#13511yescharacteristic
170C22×C71C2×#13511yescharacteristic
171C4×C71—44no—
172C4×C71—82no—
173C4×C71C4×SD1614yes—
174C4×C71C4×Q1614yes—
175C4×C71C4×#4112yes—
176C4×C71#52912yes—
177Dic71—22no—
178Dic71—22no—
179Dic71—22no—
180Dic71—44no—
181Dic71—82no—
182Dic71#48714yescomplemented
183C4×#487—711nomaximal, Sylow
184C2×C71C4×#12411yescharacteristic
185C2×C71C2×C4×SD1611yescharacteristic
186C2×C71C2×C4×Q1611yescharacteristic
187C2×C71C2×#48711yescharacteristic
188C2×C71#313111yescharacteristic
189C2×C71#316711yescharacteristic
190C2×C71#317011yescharacteristic
191C2×#487—711no—
192C4×C4⋊Q8—711no—
193C4×C4⋊Q8—711no—
194#1990D7111yescharacteristic
195#1956—711no—
196#1933—714no—
197#1834—711no—
198C4×#103—714no—
199C42×C8—711no—
200C71C4×#48711yescharacteristic, complemented, Sylow
201C2×C4⋊Q8—711no—
202C2×C4⋊Q8C2×D7111yescharacteristic
203C2×C4×Q8—712no—
204C2×C4×Q8—1421no—
205#487—718no—
206C2×#103—714no—
207C2×#103—714no—
208C2×C4×C8—711no—
209C2×C4×C8—712no—
210C2×C4×C8—712no—
211C2×C4×C8C2×D7111yescharacteristic
212#391—712no—
213#391—1421no—
214#389—712no—
215#389—1421no—
216#389—1421no—
217#389—1421no—
218#384—711no—
219#384C2×D7111yescharacteristic
220C4×C4⋊C4—712no—
221C4×C4⋊C4—1421no—
222C4×C4⋊C4—1421no—
223C4×C4⋊C4—1421no—
224C43—711no—
225#340—714no—
226#340C2×D7114yes—
227C22×Q8—21no—
228C22×Q8—712no—
229C22×C8—712no—
230C22×C8C22×D7112yes—
231C4⋊Q8—22no—
232C4⋊Q8—714no—
233C4⋊Q8—1422no—
234C4⋊Q8C4×D7114yes—
235C4×Q8—2844no—
236C4×Q8—1428no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—712no—
241C2×C4⋊C4—1422no—
242C2×C4⋊C4—1422no—
243C2×C4⋊C4—1421no—
244C2×C4⋊C4—1421no—
245C2×C4⋊C4—1421no—
246C2×C4⋊C4—1421no—
247C2×C4⋊C4C22×D7112yes—
248C2×C4⋊C4C22×D7112yes—
249C2×C42—711no—
250C2×C42—711no—
251C2×C42—711no—
252C2×C42—711no—
253C2×C42—711no—
254C2×C42—711no—
255C2×C42—1422no—
256C2×C42—1422no—
257C2×C42C22×D7111yescharacteristic
258#103—7116no—
259#103—1428no—
260C4×C8—22no—
261C4×C8—714no—
262C4×C8—1424no—
263C4×C8—1424no—
264#95—1422no—
265#95—1422no—
266#95—1422no—
267#95—1422no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—714no—
272C2×Q8—1424no—
273C2×Q8—1424no—
274C2×Q8—1422no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—711no—
278C22×C4—711no—
279C22×C4—711no—
280C22×C4—711no—
281C22×C4—1422no—
282C22×C4—1422no—
283C22×C4D4×D7111yescharacteristic
284C22×C4D4×D7111yescharacteristic
285C22×C4C23×D7111yescharacteristic
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—714no—
289C2×C8—1422no—
290C2×C8—1422no—
291C2×C8#1176586614yes—
292C4⋊C4—22no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—2844no—
298C4⋊C4—2844no—
299C4⋊C4—2842no—
300C4⋊C4—2842no—
301C4⋊C4—1424no—
302C4⋊C4—1424no—
303C4⋊C4—1424no—
304C4⋊C4C2×C4×D7114yes—
305C42—2844no—
306C42—2844no—
307C42—714no—
308C42—714no—
309C42—714no—
310C42—1422no—
311C42—1422no—
312C42—1422no—
313C42C2×C4×D7112yes—
314C42#1176586412yes—
315C23C2×D4×D7111yescharacteristic
316Q8—48no—
317Q8—2844no—
318Q8—1428no—
319C2×C4—24no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—2842no—
326C2×C4—2842no—
327C2×C4—2842no—
328C2×C4—2842no—
329C2×C4—2842no—
330C2×C4—712no—
331C2×C4—712no—
332C2×C4—712no—
333C2×C4—712no—
334C2×C4—712no—
335C2×C4—712no—
336C2×C4—1424no—
337C2×C4—1421no—
338C2×C4—1421no—
339C2×C4—1421no—
340C2×C4—1421no—
341C2×C4—1421no—
342C2×C4—1421no—
343C2×C4(C22⋊D71)⋊C412yes—
344C2×C4(C22⋊D71)⋊C412yes—
345C2×C4SD16×D7112yes—
346C2×C4Q16×D7112yes—
347C2×C4#1455614712yes—
348C2×C4#1455614912yes—
349C2×C4C22×C4×D7111yescharacteristic
350C2×C4#41×D7112yes—
351C2×C4C2×#1176586411yescharacteristic
352C2×C4C2×#1176586612yes—
353C8—24no—
354C8—1424no—
355C22C2×(C22⋊D71)⋊C411yescharacteristic
356C22#124×D7111yescharacteristic
357C22#1798655011yescharacteristic
358C22C2×SD16×D7111yescharacteristic
359C22C2×Q16×D7111yescharacteristic
360C22C2×#1455614711yescharacteristic
361C22C2×#1455614911yescharacteristic
362C4—44no—
363C4—82no—
364C4—2844no—
365C4—5682no—
366C4—714no—
367C4—1422no—
368C4—1422no—
369C4—1422no—
370C4#1798609014yes—
371C4#1798609114yes—
372C4C4×#1176586612yes—
373C4#1798653512yes—
374C2#2259088311yescharacteristic
375C2C2×#1798609011yescharacteristic
376C2C2×#1798609111yescharacteristic
377C2#487×D7111yescharacteristic
378C2#2259865911yescharacteristic
379C2#2259890111yescharacteristic
380C2#2259892711yescharacteristic
subgroupG/H
centerC23C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1990C2
socle, prime radicalC23×C71C22×D4
cosocleC22×C4×C71C24
hypercenter#1990D71
hypocenterC71C4×#487
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
13. #17986351C4semidirect
182. Dic71#487semidirect
200. C71C4×#487semidirect

Maximal subgroups

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

subgroupindex
C2×#179863512
C4⋊Q8×Dic712
#225909072
#225891422
#225890172
#225884672
#225881592
C4×C8×Dic712
C71×#19902
C4×#48771