Finite Group Databasebeta

← #31888337

Subgroups — #31888337

Order 18176.

712 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556102C211yesmaximal, characteristic
2#22588102C211yesmaximal, characteristic, complemented
3C2×#17985732C212yesmaximal
4#22586219C212yesmaximal, complemented
5C71×#1434C211yesmaximal, characteristic, complemented
6C2×#14556102C2211yescharacteristic
7C2×C4×D284C411yescharacteristic, complemented
8C2×C4×C22⋊D71—21no—
9C2×C22⋊(C71⋊C8)C2212yes—
10C2×#14556039—21no—
11#17985901C411yescharacteristic, complemented
12#17985897C411yescharacteristic, complemented
13#17985890—21no—
14#17985878—21no—
15#17985859C2211yescharacteristic
16C4×C4⋊Dic71C411yescharacteristic, complemented
17C4×(C2×D71)⋊C4—21no—
18#17985754—21no—
19#17985732C418yescomplemented
20C22×C42×C71C2211yescharacteristic
21C2×#95×C71—21no—
22C2×#95×C71—21no—
23C2×#95×C71C2212yes—
24C22×D284C2×C411yescharacteristic
25C22×C22⋊D71—21no—
26C22×C4×D71—21no—
27C22×C71⋊C8—22no—
28#14556102—22no—
29#14556102C2×C414yes—
30C4×D284—42no—
31C4×C22⋊D71—44no—
32C2×C4⋊Dic71C2×C411yescharacteristic
33C2×C4⋊Dic71C2×C411yescharacteristic
34C2×C4⋊Dic71C2×C411yescharacteristic
35C2×Dic71⋊C4—21no—
36C2×C22⋊Dic71—21no—
37C2×(C2×D71)⋊C4—21no—
38C2×(C2×D71)⋊C4—21no—
39C2×(C2×D71)⋊C4—21no—
40C2×(C2×D71)⋊C4—21no—
41C2×C4×Dic71—21no—
42C22⋊(C71⋊C8)—22no—
43C22⋊(C71⋊C8)C2×C414yescomplemented
44#14556039—42no—
45C71⋊#95—21no—
46C71⋊#95—21no—
47C71⋊#95—21no—
48C23×C4×C71—22no—
49C23×C4×C71D411yescharacteristic
50C23×C4×C71D411yescharacteristic
51C23×C4×C71C2311yescharacteristic
52C2×C42×C71—42no—
53C2×C42×C71C2×C411yescharacteristic
54C2×C42×C71C2×C411yescharacteristic
55C2×C42×C71D411yescharacteristic
56C2×C42×C71D411yescharacteristic
57#95×C71—22no—
58#95×C71—42no—
59#95×C71—42no—
60#95×C71C2×C414yescomplemented
61C23×D71—21no—
62C2×D284—21no—
63C2×D284—41no—
64C2×D284—41no—
65C2×D284C4212yes—
66C2×C22⋊D71—24no—
67C2×C22⋊D71—41no—
68C2×C22⋊D71—42no—
69C2×C22⋊D71—41no—
70C22×Dic71—21no—
71C22×Dic71—21no—
72C22×Dic71—21no—
73C2×C4×D71—42no—
74C2×C4×D71—42no—
75C2×C4×D71—42no—
76C2×C71⋊C8—24no—
77C2×C71⋊C8—42no—
78C2×C71⋊C8—42no—
79C4⋊Dic71—21no—
80C4⋊Dic71—41no—
81C4⋊Dic71—41no—
82C4⋊Dic71C4212yes—
83Dic71⋊C4—42no—
84C22⋊Dic71—42no—
85(C2×D71)⋊C4—24no—
86(C2×D71)⋊C4—42no—
87(C2×D71)⋊C4—42no—
88(C2×D71)⋊C4—42no—
89C4×Dic71—42no—
90C24×C71C2×D411yescharacteristic
91C22×C4×C71—21no—
92C22×C4×C71—22no—
93C22×C4×C71—21no—
94C22×C4×C71—22no—
95C22×C4×C71—24no—
96C22×C4×C71—41no—
97C22×C4×C71—41no—
98C22×C4×C71—41no—
99C22×C4×C71—41no—
100C22×C4×C71—41no—
101C22×C4×C71—42no—
102C22×C4×C71—42no—
103C22×C4×C71—42no—
104C22×C4×C71—42no—
105C22×C4×C71—42no—
106C22×C4×C71—42no—
107C22×C4×C71C22⋊C411yescharacteristic
108C22×C4×C71C22⋊C411yescharacteristic
109C22×C4×C71C22⋊C411yescharacteristic
110C22×C4×C71C22⋊C411yescharacteristic
111C22×C4×C71C22×C411yescharacteristic
112C22×C4×C71C22×C412yes—
113C22×C4×C71C2×D411yescharacteristic
114C22×C4×C71#4112yes—
115C42×C71—22no—
116C42×C71—22no—
117C42×C71—41no—
118C42×C71—41no—
119C23×C71—21no—
120C23×C71—21no—
121C23×C71—21no—
122C23×C71—21no—
123C23×C71C2×C22⋊C411yescharacteristic
124C23×C71C4×D412yes—
125C23×C71AGL(1,ℤ/8)12yes—
126C23×C71AGL(1,ℤ/8)11yescharacteristic
127C23×C71#13711yescharacteristic
128C22×D71—22no—
129C22×D71—41no—
130C22×D71—41no—
131C22×D71—41no—
132C22×D71—41no—
133C22×D71—41no—
134C22×D71—41no—
135C2×C4×C71—21no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—21no—
139C2×C4×C71—21no—
140C2×C4×C71—21no—
141C2×C4×C71—21no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—21no—
146C2×C4×C71—21no—
147C2×C4×C71—21no—
148C2×C4×C71—21no—
149C2×C4×C71—21no—
150C2×C4×C71—42no—
151C2×C4×C71—42no—
152C2×C4×C71—44no—
153C2×C4×C71—42no—
154C2×C4×C71—42no—
155C2×C4×C71—44no—
156C2×C4×C71—44no—
157C2×C4×C71—44no—
158C2×C4×C71—82no—
159C2×C4×C71—82no—
160C2×C4×C71—82no—
161C2×C4×C71—82no—
162C2×C4×C71#9912yes—
163C2×C4×C71C2×C4211yescharacteristic
164C2×C4×C71#11711yescharacteristic
165C2×C4×C71C4×D412yes—
166C22⋊D71—44no—
167C22⋊D71—44no—
168C2×Dic71—22no—
169C2×Dic71—41no—
170C2×Dic71—41no—
171C2×Dic71—41no—
172C2×Dic71—41no—
173C2×Dic71—41no—
174C2×Dic71—41no—
175C2×Dic71—41no—
176C2×Dic71—41no—
177D284—42no—
178D284—42no—
179C4×D71—84no—
180C71⋊C8—44no—
181C22×C71—22no—
182C22×C71—22no—
183C22×C71—22no—
184C22×C71—21no—
185C22×C71—21no—
186C22×C71—21no—
187C22×C71—21no—
188C22×C71—22no—
189C22×C71C4×C22⋊C411yescharacteristic
190C22×C71C2×#9911yescharacteristic
191C22×C71#41011yescharacteristic
192C22×C71#41711yescharacteristic
193C22×C71#41811yescharacteristic
194C22×C71#42111yescharacteristic
195C22×C71#42111yescharacteristic
196C22×C71#44012yes—
197C22×C71#44212yes—
198C2×D71—42no—
199C2×D71—42no—
200C2×D71—42no—
201C2×D71—81no—
202C2×D71—81no—
203C2×D71—81no—
204C2×D71—81no—
205C4×C71—22no—
206C4×C71—21no—
207C4×C71—21no—
208C4×C71—22no—
209C4×C71—41no—
210C4×C71—41no—
211C4×C71—41no—
212C4×C71—84no—
213Dic71—42no—
214Dic71—81no—
215Dic71—81no—
216#8216—711nomaximal, Sylow
217C2×C71—22no—
218C2×C71—22no—
219C2×C71#151011yescharacteristic
220C2×C71#151211yescharacteristic
221C2×C71#151311yescharacteristic
222C2×C71#151411yescharacteristic
223C2×C71C4×#9911yescharacteristic
224C2×C71#175911yescharacteristic
225C2×C71#176211yescharacteristic
226D71—82no—
227C4×#121—711no—
228#1749—711no—
229C2×#327—712no—
230#1434D7111yescharacteristic, complemented
231#1294—712no—
232C71#821611yescharacteristic, complemented, Sylow
233C2×#121—711no—
234C2×C4×D4—711no—
235C2×C4×D4—1421no—
236C22×C42C2×D7111yescharacteristic
237C2×C22⋊C8—712no—
238C2×#97—1421no—
239#390—711no—
240#389—711no—
241#388—1421no—
242#386—1421no—
243#380—711no—
244C4×C4⋊C4—711no—
245C4×C22⋊C4—1421no—
246C2×#95—21no—
247C2×#95—21no—
248C2×#95C2×D7112yes—
249#336—1421no—
250#327—718no—
251C22×D4—711no—
252C22×D4—1421no—
253C23×C4—22no—
254C23×C4—1421no—
255C23×C4C22⋊D7111yescharacteristic
256C23×C4C22⋊D7111yescharacteristic
257C23×C4C22×D7111yescharacteristic
258C22×C8—1422no—
259#121—714no—
260#121—1422no—
261C4×D4—2842no—
262C4×D4—2844no—
263C2×C4⋊C4—711no—
264C2×C4⋊C4—711no—
265C2×C4⋊C4—711no—
266C2×C4⋊C4—1421no—
267C2×C22⋊C4—1421no—
268C2×C22⋊C4—1421no—
269C2×C22⋊C4—1421no—
270C2×C22⋊C4—1421no—
271C2×C22⋊C4—1421no—
272C2×C42—42no—
273C2×C42—1421no—
274C2×C42C4×D7111yescharacteristic
275C2×C42C4×D7111yescharacteristic
276C2×C42D28411yescharacteristic
277C2×C42D28411yescharacteristic
278C22⋊C8—714no—
279C22⋊C8—1422no—
280#97—2842no—
281#95—22no—
282#95—42no—
283#95—42no—
284#95—1421no—
285#95—1421no—
286#95—1421no—
287#95C4×D7114yescomplemented
288C24—1421no—
289C24C2×C22⋊D7111yescharacteristic
290C2×D4—2841no—
291C2×D4—2841no—
292C2×D4—2842no—
293C2×D4—2841no—
294C2×D4—2841no—
295C2×D4—712no—
296C2×D4—1424no—
297C2×D4—1421no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—2842no—
315C22×C4—2842no—
316C22×C4—2842no—
317C22×C4—1421no—
318C22×C4—1421no—
319C22×C4—1421no—
320C22×C4(C2×D71)⋊C411yescharacteristic
321C22×C4(C2×D71)⋊C411yescharacteristic
322C22×C4(C2×D71)⋊C411yescharacteristic
323C22×C4(C2×D71)⋊C411yescharacteristic
324C22×C4C2×C4×D7111yescharacteristic
325C22×C4C2×C4×D7112yes—
326C22×C4C2×D28411yescharacteristic
327C22×C4#1176586512yes—
328C2×C8—2842no—
329C2×C8—2842no—
330C2×C8—1424no—
331C4⋊C4—2841no—
332C4⋊C4—2841no—
333C4⋊C4—2842no—
334C4⋊C4—712no—
335C4⋊C4—1421no—
336C22⋊C4—2842no—
337C22⋊C4—2842no—
338C22⋊C4—2842no—
339C22⋊C4—2842no—
340C22⋊C4—1424no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C42—2842no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—2841no—
351C23—2841no—
352C23—2841no—
353C23—2841no—
354C23—2841no—
355C23—2841no—
356C23—1422no—
357C23C2×(C2×D71)⋊C411yescharacteristic
358C23C4×C22⋊D7112yes—
359C23#1455615312yes—
360C23#1455615411yescharacteristic
361C23#1455615911yescharacteristic
362D4—2842no—
363D4—2842no—
364D4—2844no—
365D4—2844no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—2841no—
394C2×C4—2841no—
395C2×C4—2841no—
396C2×C4—2841no—
397C2×C4—2841no—
398C2×C4—2841no—
399C2×C4—2841no—
400C2×C4—2841no—
401C2×C4—5684no—
402C2×C4—1422no—
403C2×C4#1455604212yes—
404C2×C4C42×D7111yescharacteristic
405C2×C4#1455608511yescharacteristic
406C2×C4C4×D28412yes—
407C8—2844no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—21no—
415C22—21no—
416C22—2842no—
417C22—2842no—
418C22—2842no—
419C22—5681no—
420C22—5681no—
421C22—5681no—
422C22—5681no—
423C22C4×(C2×D71)⋊C411yescharacteristic
424C22C2×#1455604211yescharacteristic
425C22#1798597211yescharacteristic
426C22#1798599811yescharacteristic
427C22#1798600311yescharacteristic
428C22#1798601711yescharacteristic
429C22#1798601711yescharacteristic
430C22#1798609712yes—
431C22#1798610912yes—
432C4—22no—
433C4—21no—
434C4—21no—
435C4—22no—
436C4—41no—
437C4—41no—
438C4—41no—
439C4—84no—
440C4—2842no—
441C4—5681no—
442C4—5681no—
443C2—22no—
444C2—22no—
445C2—5682no—
446C2#2258686811yescharacteristic
447C2#2258687611yescharacteristic
448C2#2258688111yescharacteristic
449C2#2258688611yescharacteristic
450C2C4×#1455604211yescharacteristic
451C2#2258815111yescharacteristic
452C2#2258816811yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C2×C42
FrattiniC23×C4C22×D71
FittingC71×#1434C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#14556102C22
cosocleC23×C4×C71C23
hypercenter#1434D71
hypocenterC71#8216
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. #22588102C2semidirect
4. #22586219C2semidirect
5. C71×#1434C2semidirect
7. C2×C4×D284C4semidirect
11. #17985901C4semidirect
12. #17985897C4semidirect
16. C4×C4⋊Dic71C4semidirect
19. #17985732C4semidirect
43. C22⋊(C71⋊C8)C2×C4semidirect
60. #95×C71C2×C4semidirect
230. #1434D71semidirect
232. C71#8216semidirect
287. #95C4×D71semidirect

Maximal subgroups

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

subgroupindex
C4×#145561022
#225881022
C2×#179857322
#225862192
C71×#14342
#821671