Finite Group Databasebeta

← #32152251

Subgroups — #32152251

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
1#22597921C211yesmaximal, characteristic
2#22590548C212yesmaximal, complemented
3#22590542C211yesmaximal, characteristic, complemented
4#22590537C211yesmaximal, characteristic, complemented
5#22590531C212yesmaximal, complemented
6#22590525C211yesmaximal, characteristic, complemented
7#22590520C211yesmaximal, characteristic, complemented
8#22590122C212yesmaximal, complemented
9#22590045C212yesmaximal, complemented
10C2×C22⋊(C71⋊C16)C211yesmaximal, characteristic, complemented
11C2×#427×C71C211yesmaximal, characteristic, complemented
12C2×#14556150C2211yescharacteristic
13#17986510—21no—
14C22×C71⋊C16C2212yes—
15#17986240C2211yescharacteristic
16#17986240C2211yescharacteristic
17#17986235C2211yescharacteristic
18#17986235C2211yescharacteristic
19#17986228—21no—
20#17986228—21no—
21#17986037C2211yescharacteristic
22#17985996—21no—
23#17985826C2212yescomplemented
24#17985826C2212yescomplemented
25#17985824C2212yescomplemented
26#17985824C2212yescomplemented
27#17985802C2212yescomplemented
28#17985802C2212yescomplemented
29#17985799C2212yescomplemented
30#17985799C2212yescomplemented
31C22⋊(C71⋊C16)C2211yescharacteristic
32C22⋊(C71⋊C16)C2211yescharacteristic
33C22⋊(C71⋊C16)C2212yes—
34C23×C8×C71C2211yescharacteristic
35C2×#117×C71—21no—
36#427×C71—21no—
37#427×C71—21no—
38#427×C71C2212yes—
39#427×C71C2212yes—
40C2×#107×C71C2211yescharacteristic
41C2×#107×C71C2211yescharacteristic
42C2×#106×C71—21no—
43C2×#11765865—21no—
44#14556150—24no—
45C2×Dic284C2311yescharacteristic
46C2×#11765854—21no—
47C2×D568C2311yescharacteristic
48#14556111—41no—
49#14556107—21no—
50#14556107—21no—
51#14556102—21no—
52#14556102—21no—
53#14556097—41no—
54C4×C22⋊D71—41no—
55C4×C22⋊D71—41no—
56#14556087—21no—
57C2×C71⋊C16—24no—
58C2×C71⋊C16C2312yes—
59C2×C71⋊C16C2312yes—
60#14556062C2311yescharacteristic
61#14556062C2311yescharacteristic
62#14556060—21no—
63#14556053—21no—
64#14556053—21no—
65#14556050—21no—
66#14556050—21no—
67C23×C4×C71D411yescharacteristic
68C22×C8×C71—24no—
69C22×C8×C71D411yescharacteristic
70C22×C8×C71D411yescharacteristic
71C22×C8×C71D411yescharacteristic
72C22×C8×C71C2311yescharacteristic
73C22×C8×C71C2311yescharacteristic
74C22×C8×C71C2311yescharacteristic
75#117×C71—22no—
76#117×C71—22no—
77#117×C71—41no—
78#117×C71—41no—
79C2×C4⋊C4×C71—21no—
80C2×C4⋊C4×C71—21no—
81C2×C22⋊C4×C71—41no—
82C2×C42×C71—41no—
83#107×C71—21no—
84#107×C71—21no—
85#107×C71C2312yes—
86#107×C71C2312yes—
87#106×C71—22no—
88#106×C71—22no—
89#11765865—42no—
90#11765865—42no—
91C2×Dic142—21no—
92C2×D284—21no—
93C2×C22⋊D71—41no—
94C2×C22⋊D71—41no—
95C2×C22⋊D71—41no—
96C2×C4×D71—41no—
97Dic284—22no—
98#11765854—42no—
99D568—22no—
100C4⋊Dic71—21no—
101C4⋊Dic71—21no—
102Dic71⋊C4—41no—
103Dic71⋊C4—41no—
104C22⋊Dic71—41no—
105C22⋊Dic71—41no—
106C22⋊Dic71—41no—
107(C2×D71)⋊C4—41no—
108(C2×D71)⋊C4—41no—
109C4×Dic71—41no—
110C71⋊C16—22no—
111C71⋊C16—22no—
112C24×C71D811yescharacteristic
113C22×C4×C71—22no—
114C22×C4×C71—22no—
115C22×C4×C71—41no—
116C22×C4×C71—41no—
117C22×C4×C71D811yescharacteristic
118C22×C4×C71D811yescharacteristic
119C22×C4×C71D811yescharacteristic
120C22×C4×C71C2×D411yescharacteristic
121C22×C4×C71C2×D411yescharacteristic
122C22×C4×C71C2×D411yescharacteristic
123C2×C8×C71—22no—
124C2×C8×C71—22no—
125C2×C8×C71—22no—
126C2×C8×C71—22no—
127C2×C8×C71—22no—
128C2×C8×C71C2×D411yescharacteristic
129C2×C8×C71C2×D411yescharacteristic
130C2×C8×C71C2×D411yescharacteristic
131C2×C8×C71#4114yes—
132C2×C8×C71C2411yescharacteristic
133C4⋊C4×C71—22no—
134C4⋊C4×C71—22no—
135C4⋊C4×C71—41no—
136C4⋊C4×C71—41no—
137C22⋊C4×C71—42no—
138C22⋊C4×C71—42no—
139C42×C71—42no—
140C42×C71—42no—
141C23×C71—22no—
142C23×C71—22no—
143C23×C71—41no—
144C23×C71C2×D811yescharacteristic
145C23×C71C2×D811yescharacteristic
146C23×C71C2×D811yescharacteristic
147C22×D71—41no—
148C2×C4×C71—22no—
149C2×C4×C71—21no—
150C2×C4×C71—21no—
151C2×C4×C71—21no—
152C2×C4×C71—21no—
153C2×C4×C71—22no—
154C2×C4×C71—22no—
155C2×C4×C71—42no—
156C2×C4×C71—42no—
157C2×C4×C71—82no—
158C2×C4×C71—82no—
159C2×C4×C71#12312yes—
160C2×C4×C71C2×D811yescharacteristic
161C2×C4×C71C2×D811yescharacteristic
162C2×C4×C71C2×D811yescharacteristic
163C2×C4×C71#13712yes—
164C2×C4×C71C22×D411yescharacteristic
165C22⋊D71—82no—
166C22⋊D71—82no—
167C22⋊D71—82no—
168C2×Dic71—41no—
169C2×Dic71—41no—
170C2×Dic71—41no—
171D284—41no—
172D284—41no—
173C4×D71—82no—
174Dic142—41no—
175Dic142—41no—
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#48012yes—
189C22×C71#50812yes—
190C22×C71C22×D811yescharacteristic
191C2×D71—81no—
192C2×D71—81no—
193C2×D71—81no—
194C4×C71—21no—
195C4×C71—21no—
196C4×C71—82no—
197C4×C71—82no—
198C4×C71#48012yes—
199C4×C71C2×#12311yescharacteristic
200C4×C71C2×#13711yescharacteristic
201Dic71—81no—
202Dic71—81no—
203Dic71—81no—
204#33645—711nomaximal, Sylow
205C2×C71—22no—
206C2×C71—22no—
207C2×C71—41no—
208C2×C71C2×#48011yescharacteristic
209C2×C71C2×#50811yescharacteristic
210C2×C71#341411yescharacteristic
211D71—161no—
212#3050—711no—
213C2×#427D7111yescharacteristic, complemented
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×#135—711no—
225C23×C8C2×D7111yescharacteristic
226#525—1421no—
227C2×#117—21no—
228C22×C16—712no—
229#467—711no—
230#467—711no—
231#466—711no—
232#466—711no—
233#465—1421no—
234#465—1421no—
235#427—21no—
236#427—21no—
237#427—711no—
238#427C2×D7112yes—
239#427C2×D7112yes—
240C2×#107C2×D7111yescharacteristic
241C2×#107C2×D7111yescharacteristic
242C2×#106—21no—
243#416—1421no—
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—1421no—
256C23×C4C22⋊D7111yescharacteristic
257#135—1424no—
258C2×Q16—711no—
259C2×SD16—1421no—
260C2×D8—711no—
261C22×C8—24no—
262C22×C8C22⋊D7111yescharacteristic
263C22×C8C22⋊D7111yescharacteristic
264C22×C8C22⋊D7111yescharacteristic
265C22×C8C22×D7111yescharacteristic
266C22×C8C22×D7111yescharacteristic
267C22×C8C22×D7111yescharacteristic
268#123—2841no—
269C22⋊Q8—1421no—
270C22⋊Q8—1421no—
271#121—1421no—
272#121—1421no—
273#120—2841no—
274C4×D4—2841no—
275C4×D4—2841no—
276#117—22no—
277#117—22no—
278#117—41no—
279#117—41no—
280#117—1421no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C22⋊C4—41no—
284C2×C42—41no—
285C2×C16—712no—
286C2×C16—712no—
287C2×C16—1424no—
288#107—21no—
289#107—21no—
290#107—711no—
291#107—711no—
292#107C22×D7112yes—
293#107C22×D7112yes—
294#106—22no—
295#106—22no—
296#106—1421no—
297#103—1421no—
298#103—1421no—
299#102—1421no—
300#102—1421no—
301C24#1176585011yescharacteristic
302#41—2842no—
303#41—2842no—
304C2×Q8—1421no—
305C2×D4—2841no—
306C2×D4—2841no—
307C2×D4—2841no—
308C2×D4—1421no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—41no—
312C22×C4—41no—
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—1422no—
321SD16—2842no—
322D8—1422no—
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#1176586514yes—
332C2×C8C23×D7111yescharacteristic
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—2841no—
338C4⋊C4—2841no—
339C4⋊C4—1421no—
340C4⋊C4—1421no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—2841no—
344C22⋊C4—2841no—
345C22⋊C4—2841no—
346C22⋊C4—2841no—
347C22⋊C4—2841no—
348C42—42no—
349C42—42no—
350C42—2841no—
351C16—1422no—
352C16—1422no—
353C23—22no—
354C23—22no—
355C23—41no—
356C23—2841no—
357C23C2×#1176585011yescharacteristic
358C23C2×#1176585011yescharacteristic
359C23C2×#1176585011yescharacteristic
360Q8—2841no—
361Q8—2841no—
362D4—2841no—
363D4—2841no—
364D4—5682no—
365D4—5682no—
366D4—5682no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—2841no—
379C2×C4—2841no—
380C2×C4—2841no—
381C2×C4—5682no—
382C2×C4#1455611112yes—
383C2×C4C2×#1176585011yescharacteristic
384C2×C4C2×#1176585011yescharacteristic
385C2×C4C2×#1176585011yescharacteristic
386C2×C4#1455615912yes—
387C2×C4C22×C22⋊D7111yescharacteristic
388C8—21no—
389C8—21no—
390C8#1455611112yes—
391C8C2×#1176586512yes—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—5681no—
401C22—5681no—
402C22—5681no—
403C22#1798631212yes—
404C22#1798642912yes—
405C22C22×#1176585011yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—82no—
410C4—5681no—
411C4—5681no—
412C4—5681no—
413C4#1798631212yes—
414C4C2×#1455611111yescharacteristic
415C4C2×#1455615911yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—11361no—
420C2C2×#1798631211yescharacteristic
421C2C2×#1798642911yescharacteristic
422C2#2260147411yescharacteristic
subgroupG/H
centerC2×C4C2×#11765850
commutator, cosocleC2×C8×C71C24
FrattiniC2×C8C23×D71
FittingC2×#427×C71C2
socleC22×C71C22×D8
prime radical#22597921C2
hypercenterC2×#427D71
hypocenterC71#33645
LCSG ▷ C2×C8×C71 ▷ C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #22590548C2semidirect
3. #22590542C2semidirect
4. #22590537C2semidirect
5. #22590531C2semidirect
6. #22590525C2semidirect
7. #22590520C2semidirect
8. #22590122C2semidirect
9. #22590045C2semidirect
10. C2×C22⋊(C71⋊C16)C2semidirect
11. C2×#427×C71C2semidirect
23. #17985826C22semidirect
24. #17985826C22semidirect
25. #17985824C22semidirect
26. #17985824C22semidirect
27. #17985802C22semidirect
28. #17985802C22semidirect
29. #17985799C22semidirect
30. #17985799C22semidirect
213. C2×#427D71semidirect
223. C71#33645semidirect

Maximal subgroups

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

subgroupindex
#225979212
#225905482
#225905422
#225905372
#225905312
#225905252
#225905202
#225901222
#225900452
C2×C22⋊(C71⋊C16)2
C2×#427×C712
#3364571