Finite Group Databasebeta

← #32017151

Subgroups — #32017151

Order 18176.

722 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556150C211yesmaximal, characteristic
2#22597008C212yesmaximal, complemented
3#22589226C212yesmaximal
4#22588987C214yesmaximal, complemented
5#22588357C212yesmaximal, complemented
6#22588222C212yesmaximal, complemented
7C8×C4⋊Dic71C211yesmaximal, characteristic, complemented
8C71×#1848C211yesmaximal, characteristic, complemented
9C2×#14556150C2211yescharacteristic
10C4×#11765865C2212yes—
11C4×Dic284C411yescharacteristic, complemented
12C4×Dic284C411yescharacteristic, complemented
13C4×#11765854C412yescomplemented
14C4×#11765854C412yescomplemented
15C4×D568C411yescharacteristic, complemented
16C4×D568C411yescharacteristic, complemented
17#17986059—24no—
18#17986055—22no—
19#17986051—22no—
20#17986037C2211yescharacteristic
21C2×C4⋊(C71⋊C8)C2211yescharacteristic
22C2×#14556054C2214yes—
23#17985996C2212yes—
24#17985967C2214yes—
25#17985957C2214yes—
26C2×C8×Dic71C2212yes—
27C4×C4⋊Dic71C2211yescharacteristic
28#17985760C2214yes—
29#17985753C2212yes—
30C2×#108×C71C2212yes—
31C2×#108×C71C2212yes—
32#423×C71C2212yes—
33C2×C4×C8×C71C2211yescharacteristic
34C2×#11765865C2312yes—
35#14556150—22no—
36#14556150—22no—
37C2×Dic284C2×C411yescharacteristic
38C2×#11765854C2×C412yes—
39C2×D568C2×C411yescharacteristic
40C22×C71⋊C8C2312yes—
41C4×Dic142C2×C412yes—
42C4×Dic142C2×C412yes—
43C4×D284C2×C412yes—
44C4×D284C2×C412yes—
45C4×C22⋊D71—22no—
46C4×C22⋊D71—22no—
47#14556087C2312yes—
48#14556085—22no—
49C2×C4⋊Dic71—21no—
50C2×C4×Dic71C2312yes—
51C42×D71—22no—
52#14556062C2×C411yescharacteristic
53#14556062C2×C411yescharacteristic
54#14556060C2×C412yes—
55C4⋊(C71⋊C8)D411yescharacteristic
56C4⋊(C71⋊C8)D411yescharacteristic
57C4⋊(C71⋊C8)D412yes—
58#14556054—44no—
59#14556053C2×C412yes—
60#14556053C2×C412yes—
61#14556050C2×C412yes—
62#14556050C2×C412yes—
63(C2×D71)⋊C8—24no—
64(C2×D71)⋊C8—24no—
65#14556038—24no—
66C8×Dic71—24no—
67C8×Dic71—24no—
68C71⋊#95—21no—
69C2×M16×C71C2314yes—
70C22×C8×C71C2311yescharacteristic
71C22×C8×C71C2311yescharacteristic
72C2×C42×C71C2311yescharacteristic
73#108×C71—24no—
74#108×C71—42no—
75C4⋊C8×C71—22no—
76C4⋊C8×C71—22no—
77C4×C8×C71C2×C411yescharacteristic
78C4×C8×C71C2×C411yescharacteristic
79C4×C8×C71C2×C411yescharacteristic
80C4×C8×C71C2×C411yescharacteristic
81#11765865—22no—
82#11765865—22no—
83#11765865—22no—
84#11765865—22no—
85C2×Dic142C22×C412yes—
86C2×D284C22×C412yes—
87C2×C22⋊D71—22no—
88C22×Dic71—22no—
89C2×C4×D71—22no—
90C2×C4×D71—22no—
91C2×C4×D71—22no—
92Dic284—22no—
93#11765854—24no—
94D568—22no—
95C2×C71⋊C8—22no—
96C2×C71⋊C8—22no—
97C2×C71⋊C8C2×D412yes—
98C2×C71⋊C8C2×D412yes—
99C4⋊Dic71—24no—
100C4⋊Dic71C22×C412yes—
101C4⋊Dic71C22×C412yes—
102Dic71⋊C4—22no—
103Dic71⋊C4—22no—
104C22⋊Dic71—22no—
105(C2×D71)⋊C4—22no—
106(C2×D71)⋊C4—22no—
107C4×Dic71—22no—
108C4×Dic71—22no—
109C4×Dic71—22no—
110C4×Dic71—22no—
111C4×Dic71—22no—
112C22×C4×C71#4111yescharacteristic
113C22×C4×C71#4111yescharacteristic
114C22×C4×C71C2411yescharacteristic
115M16×C71—24no—
116M16×C71—24no—
117C2×C8×C71—21no—
118C2×C8×C71—21no—
119C2×C8×C71—24no—
120C2×C8×C71C22×C411yescharacteristic
121C2×C8×C71C22×C411yescharacteristic
122C2×C8×C71C22×C411yescharacteristic
123C2×C8×C71C22×C411yescharacteristic
124C2×C8×C71#4112yes—
125C2×C8×C71#4112yes—
126C42×C71C22×C411yescharacteristic
127C42×C71C22×C411yescharacteristic
128C42×C71C2×D411yescharacteristic
129C42×C71C2×D411yescharacteristic
130C23×C71C2×#4111yescharacteristic
131C22×D71—22no—
132C2×C4×C71—22no—
133C2×C4×C71—41no—
134C2×C4×C71C4×D411yescharacteristic
135C2×C4×C71C4×D411yescharacteristic
136C2×C4×C71C4×D411yescharacteristic
137C2×C4×C71C4×D411yescharacteristic
138C2×C4×C71C23×C411yescharacteristic
139C2×C4×C71C22×D411yescharacteristic
140C2×C4×C71C2×#4111yescharacteristic
141C2×C4×C71C2×#4111yescharacteristic
142C2×C4×C71H(5,𝔽2)11yescharacteristic
143C2×C4×C712−1+411yescharacteristic
144C22⋊D71—42no—
145C22⋊D71—42no—
146C2×Dic71—24no—
147C2×Dic71—22no—
148C2×Dic71—22no—
149C2×Dic71—22no—
150C2×Dic71—42no—
151C2×Dic71—42no—
152D284—22no—
153D284—22no—
154C4×D71—42no—
155C4×D71—42no—
156C4×D71—44no—
157C4×D71—82no—
158Dic142—22no—
159Dic142—22no—
160C8×C71—21no—
161C8×C71—21no—
162C8×C71—44no—
163C8×C71#11712yes—
164C8×C71#11712yes—
165C71⋊C8—22no—
166C71⋊C8—22no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71C2×C4×D411yescharacteristic
170C22×C71#55511yescharacteristic
171C22×C71#55611yescharacteristic
172C2×D71—42no—
173C2×D71—42no—
174C2×D71—42no—
175C4×C71—21no—
176C4×C71—21no—
177C4×C71—21no—
178C4×C71#44312yes—
179C4×C71C2×#11711yescharacteristic
180C4×C71#52011yescharacteristic
181C4×C71#54711yescharacteristic
182C4×C71#55011yescharacteristic
183Dic71—44no—
184Dic71—42no—
185Dic71—42no—
186Dic71—42no—
187#22802—711nomaximal, Sylow
188C2×C71—21no—
189C2×C71—21no—
190C2×C71#235411yescharacteristic
191C2×C71C2×#44311yescharacteristic
192C2×C71#295711yescharacteristic
193D71—82no—
194C4×#135—711no—
195#2925—712no—
196#1974—712no—
197#1929—714no—
198#1848D7111yescharacteristic, complemented
199#1806—712no—
200#1772—712no—
201C8×C4⋊C4—711no—
202C71#2280211yescharacteristic, complemented, Sylow
203C2×#135—711no—
204C4×#41—712no—
205C4×Q16—711no—
206C4×Q16—711no—
207C4×SD16—712no—
208C4×SD16—712no—
209C4×D8—711no—
210C4×D8—711no—
211#433—1424no—
212#432—1422no—
213#431—1422no—
214C2×#108C2×D7112yes—
215C2×#108C2×D7112yes—
216#427—711no—
217#423C2×D7112yes—
218C2×C4⋊C8—711no—
219C2×#104—714no—
220#416—712no—
221#408—714no—
222#405—714no—
223C2×C4×C8—712no—
224C2×C4×C8C2×D7111yescharacteristic
225C4×C4⋊C4—711no—
226#339—714no—
227#336—712no—
228C2×#41—712no—
229#135—1422no—
230#135—1422no—
231C2×Q16—711no—
232C2×SD16—712no—
233C2×D8—711no—
234C2×M16C22×D7114yes—
235C22×C8—712no—
236C22×C8C22×D7111yescharacteristic
237C22×C8C22×D7111yescharacteristic
238C4×Q8—712no—
239C4×Q8—712no—
240C4×D4—712no—
241C4×D4—712no—
242C4×D4—1422no—
243C4×D4—1422no—
244#117—712no—
245#117—1422no—
246C2×C4⋊C4—1421no—
247C2×C42—712no—
248C2×C42—1422no—
249C2×C42C22×D7111yescharacteristic
250#108—24no—
251#108—42no—
252#107—711no—
253#107—711no—
254#106—712no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8—711no—
258C4⋊C8—711no—
259C4⋊C8—712no—
260#104—2844no—
261#103—712no—
262#103—712no—
263#102—712no—
264#102—712no—
265C22⋊C8—1424no—
266C22⋊C8—1424no—
267#97—1424no—
268C4×C8—1424no—
269C4×C8—1424no—
270C4×C8C4×D7111yescharacteristic
271C4×C8C4×D7111yescharacteristic
272C4×C8C4×D7111yescharacteristic
273C4×C8C4×D7111yescharacteristic
274#95—1421no—
275#41—1422no—
276#41—1422no—
277#41—1422no—
278#41—1422no—
279C2×Q8—712no—
280C2×D4—712no—
281C2×D4—1422no—
282C22×C4—1422no—
283C22×C4—1422no—
284C22×C4—1422no—
285C22×C4—1422no—
286C22×C4#1176586611yescharacteristic
287C22×C4#1176586611yescharacteristic
288C22×C4C23×D7111yescharacteristic
289Q16—1422no—
290SD16—1424no—
291D8—1422no—
292M16—24no—
293M16—24no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—24no—
297C2×C8—712no—
298C2×C8—712no—
299C2×C8—1422no—
300C2×C8—1422no—
301C2×C8C2×C4×D7111yescharacteristic
302C2×C8C2×C4×D7111yescharacteristic
303C2×C8C2×C4×D7111yescharacteristic
304C2×C8C2×C4×D7111yescharacteristic
305C2×C8#1176586412yes—
306C2×C8#1176586612yes—
307C4⋊C4—712no—
308C4⋊C4—712no—
309C4⋊C4—1424no—
310C4⋊C4—1422no—
311C4⋊C4—1422no—
312C22⋊C4—1422no—
313C22⋊C4—1422no—
314C22⋊C4—1422no—
315C42—1422no—
316C42—1422no—
317C42—1422no—
318C42—1422no—
319C42—1422no—
320C42C2×C4×D7111yescharacteristic
321C42C2×C4×D7111yescharacteristic
322C42D4×D7111yescharacteristic
323C42D4×D7111yescharacteristic
324C23—1422no—
325C23C2×#1176586611yescharacteristic
326Q8—1422no—
327Q8—1422no—
328D4—2842no—
329D4—2842no—
330D4—1422no—
331D4—1422no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—2842no—
335C2×C4—2842no—
336C2×C4—2844no—
337C2×C4—2842no—
338C2×C4—2842no—
339C2×C4—5682no—
340C2×C4—1424no—
341C2×C4—1422no—
342C2×C4—1422no—
343C2×C4—1422no—
344C2×C4#1455609011yescharacteristic
345C2×C4#1455609011yescharacteristic
346C2×C4#1455609011yescharacteristic
347C2×C4#1455609011yescharacteristic
348C2×C4C22×C4×D7111yescharacteristic
349C2×C4C2×D4×D7111yescharacteristic
350C2×C4C2×#1176586411yescharacteristic
351C2×C4C2×#1176586611yescharacteristic
352C2×C4#1455617311yescharacteristic
353C2×C4#1455617611yescharacteristic
354C8—21no—
355C8—21no—
356C8—44no—
357C8—1422no—
358C8—1422no—
359C8#1455608612yes—
360C8#1455608612yes—
361C22—21no—
362C22—21no—
363C22—2842no—
364C22—2842no—
365C22—2842no—
366C22C2×#1455609011yescharacteristic
367C22#1798671511yescharacteristic
368C22#1798672311yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—2844no—
373C4—2842no—
374C4—2842no—
375C4—2842no—
376C4#1798611512yes—
377C4C2×#1455608611yescharacteristic
378C4#1798648011yescharacteristic
379C4#1798665211yescharacteristic
380C4#1798666911yescharacteristic
381C2—21no—
382C2—21no—
383C2—5682no—
384C2#2259129111yescharacteristic
385C2C2×#1798611511yescharacteristic
386C2#2259725611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D71
commutatorC2×C4×C71C23×C4
FrattiniC22×C4C23×D71
FittingC71×#1848C2
socleC22×C71C2×C4×D4
prime radicalC2×#14556150C22
cosocleC22×C4×C71C24
hypercenter#1848D71
hypocenterC71#22802
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1848

Complements

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

NH ≅ G/Ntype
2. #22597008C2semidirect
4. #22588987C2semidirect
5. #22588357C2semidirect
6. #22588222C2semidirect
7. C8×C4⋊Dic71C2semidirect
8. C71×#1848C2semidirect
11. C4×Dic284C4semidirect
12. C4×Dic284C4semidirect
13. C4×#11765854C4semidirect
14. C4×#11765854C4semidirect
15. C4×D568C4semidirect
16. C4×D568C4semidirect
198. #1848D71semidirect
202. C71#22802semidirect

Maximal subgroups

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

subgroupindex
C4×#145561502
#225970082
#225892262
#225889872
#225883572
#225882222
C8×C4⋊Dic712
C71×#18482
#2280271