Finite Group Databasebeta

← #32152247

Subgroups — #32152247

Order 18176.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17986037C211yesmaximal, characteristic, complemented
2#22590545C212yesmaximal, complemented
3#22590540C211yesmaximal, characteristic, complemented
4#22590535C211yesmaximal, characteristic, complemented
5#22590528C212yesmaximal, complemented
6#22590523C211yesmaximal, characteristic, complemented
7#22590518C211yesmaximal, characteristic, complemented
8#22590122C212yesmaximal, complemented
9#22590047C212yesmaximal, complemented
10C2×C22⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
11C71×#3050C211yesmaximal, characteristic
12C2×#14556087—21no—
13C22×C71⋊C16C2212yes—
14#17986037—21no—
15#17986037—21no—
16#17986037C2212yes—
17#17986037C2212yes—
18C2×#14556062C2211yescharacteristic
19C2×#14556062C2211yescharacteristic
20C2×#14556060—21no—
21#17985826C2212yescomplemented
22#17985826C2212yescomplemented
23#17985824C2212yescomplemented
24#17985824C2212yescomplemented
25#17985803C2212yescomplemented
26#17985803C2212yescomplemented
27#17985800C2212yescomplemented
28#17985800C2212yescomplemented
29C22⋊(C71⋊C16)C2211yescharacteristic
30C22⋊(C71⋊C16)C2211yescharacteristic
31C22⋊(C71⋊C16)C2212yes—
32C2×#135×C71C2211yescharacteristic
33C23×C8×C71C2211yescharacteristic
34#525×C71—21no—
35#467×C71C2211yescharacteristic
36#467×C71C2211yescharacteristic
37#466×C71C2211yescharacteristic
38#466×C71C2211yescharacteristic
39#465×C71—21no—
40#465×C71—21no—
41#427×C71C2211yescharacteristic
42#416×C71—21no—
43#14556087—22no—
44#14556087—22no—
45#14556087—41no—
46#14556087—41no—
47C2×C4⋊Dic71—21no—
48C2×C4⋊Dic71—21no—
49C2×C22⋊Dic71—41no—
50C2×C4×Dic71—41no—
51C2×C71⋊C16—24no—
52C2×C71⋊C16C2312yes—
53C2×C71⋊C16C2312yes—
54#14556062—21no—
55#14556062—21no—
56#14556062C2312yes—
57#14556062C2312yes—
58#14556060—22no—
59#14556060—22no—
60C2×#41×C71—21no—
61C23×C4×C71D411yescharacteristic
62#135×C71—24no—
63C2×Q16×C71C2311yescharacteristic
64C2×SD16×C71—21no—
65C2×D8×C71C2311yescharacteristic
66C22×C8×C71—24no—
67C22×C8×C71D411yescharacteristic
68C22×C8×C71D411yescharacteristic
69C22×C8×C71D411yescharacteristic
70C22×C8×C71C2311yescharacteristic
71C22×C8×C71C2311yescharacteristic
72C22×C8×C71C2311yescharacteristic
73#123×C71—41no—
74C22⋊Q8×C71—21no—
75C22⋊Q8×C71—21no—
76#121×C71—21no—
77#121×C71—21no—
78#120×C71—41no—
79C4×D4×C71—41no—
80C4×D4×C71—41no—
81#117×C71—21no—
82#107×C71C2311yescharacteristic
83#107×C71C2311yescharacteristic
84#106×C71—21no—
85#103×C71—21no—
86#103×C71—21no—
87#102×C71—21no—
88#102×C71—21no—
89C22×Dic71—41no—
90C22×Dic71—41no—
91C4⋊Dic71—22no—
92C4⋊Dic71—22no—
93C4⋊Dic71—41no—
94C4⋊Dic71—41no—
95C22⋊Dic71—42no—
96C22⋊Dic71—42no—
97C4×Dic71—42no—
98C4×Dic71—42no—
99C71⋊C16—22no—
100C71⋊C16—22no—
101C24×C71D811yescharacteristic
102#41×C71—42no—
103#41×C71—42no—
104C2×Q8×C71—21no—
105C2×D4×C71—21no—
106C2×D4×C71—41no—
107C2×D4×C71—41no—
108C2×D4×C71—41no—
109C22×C4×C71—22no—
110C22×C4×C71—22no—
111C22×C4×C71—41no—
112C22×C4×C71D811yescharacteristic
113C22×C4×C71D811yescharacteristic
114C22×C4×C71D811yescharacteristic
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71C2×D411yescharacteristic
117C22×C4×C71C2×D411yescharacteristic
118Q16×C71—22no—
119SD16×C71—42no—
120D8×C71—22no—
121C2×C8×C71—22no—
122C2×C8×C71—22no—
123C2×C8×C71—22no—
124C2×C8×C71—22no—
125C2×C8×C71—22no—
126C2×C8×C71C2×D411yescharacteristic
127C2×C8×C71C2×D411yescharacteristic
128C2×C8×C71C2×D411yescharacteristic
129C2×C8×C71#4114yes—
130C2×C8×C71C2411yescharacteristic
131C4⋊C4×C71—21no—
132C4⋊C4×C71—21no—
133C4⋊C4×C71—41no—
134C4⋊C4×C71—41no—
135C22⋊C4×C71—41no—
136C22⋊C4×C71—41no—
137C22⋊C4×C71—41no—
138C22⋊C4×C71—41no—
139C22⋊C4×C71—41no—
140C42×C71—41no—
141C23×C71—22no—
142C23×C71—22no—
143C23×C71—41no—
144C23×C71—41no—
145C23×C71C2×D811yescharacteristic
146C23×C71C2×D811yescharacteristic
147C23×C71C2×D811yescharacteristic
148Q8×C71—41no—
149Q8×C71—41no—
150D4×C71—41no—
151D4×C71—41no—
152D4×C71—82no—
153D4×C71—82no—
154D4×C71—82no—
155C2×C4×C71—22no—
156C2×C4×C71—21no—
157C2×C4×C71—21no—
158C2×C4×C71—21no—
159C2×C4×C71—21no—
160C2×C4×C71—22no—
161C2×C4×C71—22no—
162C2×C4×C71—41no—
163C2×C4×C71—41no—
164C2×C4×C71—41no—
165C2×C4×C71—82no—
166C2×C4×C71#12312yes—
167C2×C4×C71C2×D811yescharacteristic
168C2×C4×C71C2×D811yescharacteristic
169C2×C4×C71C2×D811yescharacteristic
170C2×C4×C71#13712yes—
171C2×C4×C71C22×D411yescharacteristic
172C2×Dic71—42no—
173C2×Dic71—42no—
174C2×Dic71—82no—
175C2×Dic71—82no—
176C8×C71—21no—
177C8×C71—21no—
178C8×C71#12312yes—
179C8×C71C2×#4112yes—
180C22×C71—21no—
181C22×C71—21no—
182C22×C71—21no—
183C22×C71—21no—
184C22×C71—21no—
185C22×C71—21no—
186C22×C71—21no—
187C22×C71—44no—
188C22×C71—81no—
189C22×C71—81no—
190C22×C71—81no—
191C22×C71#48012yes—
192C22×C71#50812yes—
193C22×C71C22×D811yescharacteristic
194C4×C71—21no—
195C4×C71—21no—
196C4×C71—81no—
197C4×C71—81no—
198C4×C71—81no—
199C4×C71#48012yes—
200C4×C71C2×#12311yescharacteristic
201C4×C71C2×#13711yescharacteristic
202Dic71—82no—
203Dic71—82no—
204#33645—711nomaximal, Sylow
205C2×C71—22no—
206C2×C71—22no—
207C2×C71—41no—
208C2×C71—161no—
209C2×C71C2×#48011yescharacteristic
210C2×C71C2×#50811yescharacteristic
211C2×C71#341411yescharacteristic
212#3050D7111yescharacteristic
213C2×#427—711no—
214#2237—712no—
215#2236—711no—
216#2235—711no—
217#2234—712no—
218#2233—711no—
219#2232—711no—
220#2158—712no—
221#2138—712no—
222C2×C22⋊C16—711no—
223C71#3364511yescharacteristic, complemented, Sylow
224C2×#135C2×D7111yescharacteristic
225C23×C8C2×D7111yescharacteristic
226#525—21no—
227C2×#117—1421no—
228C22×C16—712no—
229#467C2×D7111yescharacteristic
230#467C2×D7111yescharacteristic
231#466C2×D7111yescharacteristic
232#466C2×D7111yescharacteristic
233#465—21no—
234#465—21no—
235#427—712no—
236#427—712no—
237#427—1421no—
238#427—1421no—
239#427C2×D7111yescharacteristic
240C2×#107—711no—
241C2×#107—711no—
242C2×#106—1421no—
243#416—21no—
244#367—712no—
245#367—712no—
246#366—712no—
247#366—712no—
248#358—712no—
249#358—712no—
250#357—712no—
251#357—712no—
252C22⋊C16—711no—
253C22⋊C16—711no—
254C22⋊C16—712no—
255C2×#41—21no—
256C23×C4C22⋊D7111yescharacteristic
257#135—24no—
258C2×Q16C22×D7111yescharacteristic
259C2×SD16—21no—
260C2×D8C22×D7111yescharacteristic
261C22×C8—24no—
262C22×C8C22⋊D7111yescharacteristic
263C22×C8C22⋊D7111yescharacteristic
264C22×C8C22⋊D7111yescharacteristic
265C22×C8C22×D7111yescharacteristic
266C22×C8C22×D7111yescharacteristic
267C22×C8C22×D7111yescharacteristic
268#123—41no—
269C22⋊Q8—21no—
270C22⋊Q8—21no—
271#121—21no—
272#121—21no—
273#120—41no—
274C4×D4—41no—
275C4×D4—41no—
276#117—21no—
277#117—2841no—
278#117—2841no—
279#117—1422no—
280#117—1422no—
281C2×C4⋊C4—1421no—
282C2×C4⋊C4—1421no—
283C2×C22⋊C4—2841no—
284C2×C42—2841no—
285C2×C16—712no—
286C2×C16—712no—
287C2×C16—1424no—
288#107—712no—
289#107—712no—
290#107—1421no—
291#107—1421no—
292#107C22×D7111yescharacteristic
293#107C22×D7111yescharacteristic
294#106—21no—
295#106—1422no—
296#106—1422no—
297#103—21no—
298#103—21no—
299#102—21no—
300#102—21no—
301C24#1176585011yescharacteristic
302#41—42no—
303#41—42no—
304C2×Q8—21no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—2841no—
313C22×C4—2841no—
314C22×C4#1176585011yescharacteristic
315C22×C4#1176585011yescharacteristic
316C22×C4#1176585011yescharacteristic
317C22×C4C2×C22⋊D7111yescharacteristic
318C22×C4C2×C22⋊D7111yescharacteristic
319C22×C4C2×C22⋊D7111yescharacteristic
320Q16—22no—
321SD16—42no—
322D8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×C22⋊D7111yescharacteristic
329C2×C8C2×C22⋊D7111yescharacteristic
330C2×C8C2×C22⋊D7111yescharacteristic
331C2×C8#1176586414yes—
332C2×C8C23×D7111yescharacteristic
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—2841no—
338C4⋊C4—2841no—
339C4⋊C4—1422no—
340C4⋊C4—1422no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—2842no—
347C22⋊C4—2842no—
348C42—41no—
349C42—2842no—
350C42—2842no—
351C16—1422no—
352C16—1422no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—41no—
357C23C2×#1176585011yescharacteristic
358C23C2×#1176585011yescharacteristic
359C23C2×#1176585011yescharacteristic
360Q8—41no—
361Q8—41no—
362D4—41no—
363D4—41no—
364D4—82no—
365D4—82no—
366D4—82no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—82no—
378C2×C4—2842no—
379C2×C4—2842no—
380C2×C4—5682no—
381C2×C4—5682no—
382C2×C4#1455610912yes—
383C2×C4C2×#1176585011yescharacteristic
384C2×C4C2×#1176585011yescharacteristic
385C2×C4C2×#1176585011yescharacteristic
386C2×C4#1455615812yes—
387C2×C4C22×C22⋊D7111yescharacteristic
388C8—21no—
389C8—21no—
390C8#1455610912yes—
391C8C2×#1176586412yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22#1798631012yes—
404C22#1798642912yes—
405C22C22×#1176585011yescharacteristic
406C4—21no—
407C4—21no—
408C4—81no—
409C4—81no—
410C4—81no—
411C4—5682no—
412C4—5682no—
413C4#1798631012yes—
414C4C2×#1455610911yescharacteristic
415C4C2×#1455615811yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—161no—
420C2C2×#1798631011yescharacteristic
421C2C2×#1798642911yescharacteristic
422C2#2260147411yescharacteristic
subgroupG/H
centerC2×C4C2×#11765850
commutator, cosocleC2×C8×C71C24
FrattiniC2×C8C23×D71
Fitting, prime radicalC71×#3050C2
socleC22×C71C22×D8
hypercenter#3050D71
hypocenterC71#33645
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C2×#17986037C2semidirect
2. #22590545C2semidirect
3. #22590540C2semidirect
4. #22590535C2semidirect
5. #22590528C2semidirect
6. #22590523C2semidirect
7. #22590518C2semidirect
8. #22590122C2semidirect
9. #22590047C2semidirect
10. C2×C22⋊(C71⋊C16)C2semidirect
21. #17985826C22semidirect
22. #17985826C22semidirect
23. #17985824C22semidirect
24. #17985824C22semidirect
25. #17985803C22semidirect
26. #17985803C22semidirect
27. #17985800C22semidirect
28. #17985800C22semidirect
223. C71#33645semidirect

Maximal subgroups

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

subgroupindex
C2×#179860372
#225905452
#225905402
#225905352
#225905282
#225905232
#225905182
#225901222
#225900472
C2×C22⋊(C71⋊C16)2
C71×#30502
#3364571