Finite Group Databasebeta

← #32008053

Subgroups — #32008053

Order 18176.

762 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986060C211yesmaximal, characteristic
2#22590949C211yesmaximal, characteristic
3#22589106C212yesmaximal, complemented
4#22589096C212yesmaximal, complemented
5#22588539C212yesmaximal
6#22588528C212yesmaximal
7#22588509C211yesmaximal, characteristic, complemented
8#22588499C211yesmaximal, characteristic, complemented
9#22588475C211yesmaximal, characteristic, complemented
10#22588470C211yesmaximal, characteristic, complemented
11C71×#2317C211yesmaximal, characteristic
12#17986060C418yescomplemented
13C2×#14556061C2212yes—
14C2×#14556061C2212yes—
15C2×#14556061C2212yes—
16C2×#14556059C2212yes—
17C2×C4⋊(C71⋊C8)C2212yes—
18C2×C22⋊(C71⋊C8)C2212yes—
19C2×#14556039C2211yescharacteristic
20C2×C4×C71⋊C8C2211yescharacteristic
21#17985892—22no—
22#17985885—22no—
23#17985869C2211yescharacteristic
24#17985865C2211yescharacteristic
25#17985859C2212yes—
26C4×C4⋊Dic71C2212yes—
27#17985761C2212yes—
28#17985761C2212yes—
29#17985761C2212yes—
30#17985761C2212yes—
31C2×#117×C71C2211yescharacteristic
32C2×#117×C71C2211yescharacteristic
33C22×C4⋊C4×C71C2211yescharacteristic
34#389×C71—22no—
35#385×C71—22no—
36#384×C71C2211yescharacteristic
37#384×C71C2211yescharacteristic
38#382×C71C2211yescharacteristic
39#382×C71C2211yescharacteristic
40C22×C71⋊C8C2312yes—
41C22×C71⋊C8C2312yes—
42C2×C4⋊Dic71—22no—
43C2×C4⋊Dic71C2312yes—
44C2×C4⋊Dic71C2312yes—
45C2×C22⋊Dic71—22no—
46C2×C4×Dic71—22no—
47#14556061—42no—
48#14556061—42no—
49#14556061—42no—
50#14556059—42no—
51C4⋊(C71⋊C8)C2×C418yescomplemented
52C22⋊(C71⋊C8)C2×C418yes—
53#14556039C2×C414yescomplemented
54C4×C71⋊C8C2×C414yescomplemented
55C71⋊#95—22no—
56C71⋊#95—22no—
57C71⋊#95—22no—
58C23×C4×C71—21no—
59C23×C4×C71C2311yescharacteristic
60#117×C71—42no—
61#117×C71C2×C414yes—
62#117×C71Q814yes—
63C2×C4⋊C4×C71—21no—
64C2×C4⋊C4×C71—22no—
65C2×C4⋊C4×C71—22no—
66C2×C4⋊C4×C71—42no—
67C2×C4⋊C4×C71D412yes—
68C2×C4⋊C4×C71C2311yescharacteristic
69C2×C4⋊C4×C71C2311yescharacteristic
70C2×C4⋊C4×C71C2311yescharacteristic
71C2×C4⋊C4×C71C2311yescharacteristic
72C2×C22⋊C4×C71—21no—
73C2×C22⋊C4×C71D412yes—
74C2×C42×C71—21no—
75C2×C42×C71C2311yescharacteristic
76C2×C42×C71C2311yescharacteristic
77#95×C71—22no—
78#95×C71—22no—
79C22×Dic71—22no—
80C22×Dic71—22no—
81C2×C71⋊C8—22no—
82C2×C71⋊C8—22no—
83C2×C71⋊C8—22no—
84C2×C71⋊C8—22no—
85C2×C71⋊C8C22×C414yes—
86C2×C71⋊C8C22×C414yes—
87C4⋊Dic71—22no—
88C4⋊Dic71—22no—
89C4⋊Dic71—24no—
90C4⋊Dic71—82no—
91C22⋊Dic71—82no—
92C4×Dic71—82no—
93C24×C71C2×D411yescharacteristic
94C22×C4×C71—22no—
95C22×C4×C71—22no—
96C22×C4×C71—21no—
97C22×C4×C71—21no—
98C22×C4×C71—21no—
99C22×C4×C71—21no—
100C22×C4×C71—42no—
101C22×C4×C71—82no—
102C22×C4×C71C22×C412yes—
103C22×C4×C71C2×D412yes—
104C22×C4×C71C2×D412yes—
105C22×C4×C71C2×D411yescharacteristic
106C22×C4×C71C2×Q812yes—
107C22×C4×C71C2411yescharacteristic
108C4⋊C4×C71—22no—
109C4⋊C4×C71—22no—
110C4⋊C4×C71—22no—
111C4⋊C4×C71—22no—
112C4⋊C4×C71—42no—
113C4⋊C4×C71—82no—
114C4⋊C4×C71—82no—
115C4⋊C4×C71C4⋊C418yes—
116C22⋊C4×C71—42no—
117C22⋊C4×C71C4⋊C418yes—
118C42×C71—42no—
119C42×C71C22×C412yes—
120C42×C71C22×C412yes—
121C42×C71C2×Q812yes—
122C42×C71C2×Q812yes—
123C23×C71—21no—
124C23×C71—21no—
125C23×C71—22no—
126C23×C71—22no—
127C23×C71C2×C4⋊C412yes—
128C23×C71C22×D411yescharacteristic
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—22no—
132C2×C4×C71—22no—
133C2×C4×C71—22no—
134C2×C4×C71—22no—
135C2×C4×C71—24no—
136C2×C4×C71—24no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—42no—
142C2×C4×C71—42no—
143C2×C4×C71—42no—
144C2×C4×C71—42no—
145C2×C4×C71—84no—
146C2×C4×C71C2×C4⋊C414yes—
147C2×C4×C71C2×C4⋊C414yes—
148C2×C4×C71C2×C4⋊C412yes—
149C2×C4×C71C23×C411yescharacteristic
150C2×C4×C71C22×Q811yescharacteristic
151C2×C4×C71H(5,𝔽2)12yes—
152C2×C4×C712−1+412yes—
153C2×Dic71—42no—
154C2×Dic71—42no—
155C2×Dic71—42no—
156C2×Dic71—42no—
157C2×Dic71—42no—
158C2×Dic71—42no—
159C71⋊C8—24no—
160C71⋊C8—24no—
161C22×C71—22no—
162C22×C71—24no—
163C22×C71—41no—
164C22×C71—41no—
165C22×C71—42no—
166C22×C71C22×C4⋊C411yescharacteristic
167C22×C71#53511yescharacteristic
168C22×C71#53611yescharacteristic
169C22×C71#57612yes—
170C22×C71#57812yes—
171C4×C71—22no—
172C4×C71—24no—
173C4×C71—24no—
174C4×C71—81no—
175C4×C71—81no—
176C4×C71—81no—
177C4×C71—81no—
178C4×C71#52012yes—
179C4×C71#54412yes—
180Dic71—82no—
181Dic71—82no—
182#21991—711nomaximal, Sylow
183C2×C71—42no—
184C2×C71#231711yescharacteristic
185C2×C71#291312yes—
186C2×C71#322011yescharacteristic
187C2×C71#322311yescharacteristic
188C2×C71#324312yes—
189C2×#433—711no—
190#2317D7111yescharacteristic
191#2314—711no—
192#1949—712no—
193#1947—712no—
194#1854—712no—
195#1852—712no—
196#1847—711no—
197#1845—711no—
198#1836—711no—
199#1835—711no—
200C71#2199111yescharacteristic, complemented, Sylow
201C2×#117C2×D7111yescharacteristic
202C2×#117C2×D7111yescharacteristic
203C22×C4⋊C4C2×D7111yescharacteristic
204#433—718no—
205C2×#107—712no—
206C2×#107—712no—
207C2×#107—712no—
208C2×#106—712no—
209C2×C4⋊C8—712no—
210C2×C22⋊C8—712no—
211C2×#97—711no—
212C2×C4×C8—711no—
213#389—22no—
214#388—1422no—
215#387—1422no—
216#385—22no—
217#384—711no—
218#384C2×D7111yescharacteristic
219#384C2×D7111yescharacteristic
220#382—711no—
221#382C2×D7111yescharacteristic
222#382C2×D7111yescharacteristic
223#380—712no—
224C4×C4⋊C4—712no—
225#340—712no—
226#340—712no—
227#340—712no—
228#340—712no—
229C23×C4—21no—
230C23×C4C22×D7111yescharacteristic
231C22×C8—712no—
232C22×C8—712no—
233#117—42no—
234#117Dic14214yes—
235#117C4×D7114yes—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—712no—
241C2×C4⋊C4—712no—
242C2×C4⋊C4—1422no—
243C2×C4⋊C4C22⋊D7112yes—
244C2×C4⋊C4C22×D7111yescharacteristic
245C2×C4⋊C4C22×D7111yescharacteristic
246C2×C4⋊C4C22×D7111yescharacteristic
247C2×C4⋊C4C22×D7111yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—1422no—
250C2×C22⋊C4C22⋊D7112yes—
251C2×C42—21no—
252C2×C42—1422no—
253C2×C42C22×D7111yescharacteristic
254C2×C42C22×D7111yescharacteristic
255#107—2842no—
256#107—2842no—
257#107—2842no—
258#106—2842no—
259C4⋊C8—718no—
260C22⋊C8—718no—
261#97—714no—
262C4×C8—714no—
263#95—22no—
264#95—22no—
265#95—1422no—
266#95—1422no—
267#95—1422no—
268C24C2×C22⋊D7111yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4—1422no—
278C22×C4—1422no—
279C22×C4C2×C4×D7112yes—
280C22×C4C2×C22⋊D7112yes—
281C22×C4C2×C22⋊D7112yes—
282C22×C4C2×C22⋊D7111yescharacteristic
283C22×C4C2×Dic14212yes—
284C22×C4C23×D7111yescharacteristic
285C2×C8—714no—
286C2×C8—714no—
287C2×C8—1422no—
288C2×C8—1422no—
289C2×C8—1422no—
290C2×C8—1422no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—42no—
296C4⋊C4—82no—
297C4⋊C4—82no—
298C4⋊C4—5682no—
299C4⋊C4—1422no—
300C4⋊C4—1422no—
301C4⋊C4—1424no—
302C4⋊C4Dic71⋊C418yes—
303C22⋊C4—42no—
304C22⋊C4—5682no—
305C22⋊C4Dic71⋊C418yes—
306C42—42no—
307C42—5682no—
308C42C2×C4×D7112yes—
309C42C2×C4×D7112yes—
310C42C2×Dic14212yes—
311C42C2×Dic14212yes—
312C23—21no—
313C23—21no—
314C23—22no—
315C23—22no—
316C23C2×Dic71⋊C412yes—
317C23C22×C22⋊D7111yescharacteristic
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—84no—
335C2×C4—2842no—
336C2×C4—2842no—
337C2×C4—2842no—
338C2×C4—2842no—
339C2×C4—2842no—
340C2×C4—2842no—
341C2×C4C2×Dic71⋊C414yes—
342C2×C4C2×Dic71⋊C414yes—
343C2×C4C2×Dic71⋊C412yes—
344C2×C4C22×C4×D7111yescharacteristic
345C2×C4C22×Dic14211yescharacteristic
346C2×C4#1455617312yes—
347C2×C4#1455617612yes—
348C8—1424no—
349C8—1424no—
350C22—24no—
351C22—22no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22C22×Dic71⋊C411yescharacteristic
356C22#1798656611yescharacteristic
357C22#1798657311yescharacteristic
358C22#1798680212yes—
359C22#1798681412yes—
360C4—24no—
361C4—24no—
362C4—22no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4—5682no—
368C4—5682no—
369C4#1798648012yes—
370C4#1798663012yes—
371C2—42no—
372C2#2259096411yescharacteristic
373C2#2259693812yes—
374C2#2259939011yescharacteristic
375C2#2259942311yescharacteristic
376C2#2259962312yes—
subgroupG/H
centerC23C22×C22⋊D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#2317C2
socleC23×C71C22×D4
prime radicalC24×C71C2×D4
cosocleC22×C4×C71C24
hypercenter#2317D71
hypocenterC71#21991
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #22589106C2semidirect
4. #22589096C2semidirect
7. #22588509C2semidirect
8. #22588499C2semidirect
9. #22588475C2semidirect
10. #22588470C2semidirect
12. #17986060C4semidirect
51. C4⋊(C71⋊C8)C2×C4semidirect
53. #14556039C2×C4semidirect
54. C4×C71⋊C8C2×C4semidirect
200. C71#21991semidirect

Maximal subgroups

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

subgroupindex
C2×#179860602
#225909492
#225891062
#225890962
#225885392
#225885282
#225885092
#225884992
#225884752
#225884702
C71×#23172
#2199171