Finite Group Databasebeta

← #31888357

Subgroups — #31888357

Order 18176.

724 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14556102C211yesmaximal, characteristic
2C4×C22⋊(C71⋊C8)C211yesmaximal, 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×C4×C71⋊C8—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—42no—
32C4×C22⋊D71—42no—
33C2×C4⋊Dic71C2×C411yescharacteristic
34C2×C4⋊Dic71C2×C411yescharacteristic
35C2×C4⋊Dic71C2×C411yescharacteristic
36C2×Dic71⋊C4—21no—
37C2×C22⋊Dic71—21no—
38C2×(C2×D71)⋊C4—21no—
39C2×(C2×D71)⋊C4—21no—
40C2×(C2×D71)⋊C4—21no—
41C2×(C2×D71)⋊C4—21no—
42C2×C4×Dic71—21no—
43C22⋊(C71⋊C8)—22no—
44C22⋊(C71⋊C8)C2×C414yescomplemented
45C4×C71⋊C8—42no—
46C71⋊#95—21no—
47C71⋊#95—21no—
48C71⋊#95—21no—
49C23×C4×C71—22no—
50C23×C4×C71D411yescharacteristic
51C23×C4×C71D411yescharacteristic
52C23×C4×C71C2311yescharacteristic
53C2×C42×C71—42no—
54C2×C42×C71C2×C411yescharacteristic
55C2×C42×C71C2×C411yescharacteristic
56C2×C42×C71D411yescharacteristic
57C2×C42×C71D411yescharacteristic
58#95×C71—22no—
59#95×C71—42no—
60#95×C71—42no—
61#95×C71C2×C414yescomplemented
62C23×D71—21no—
63C2×D284—21no—
64C2×D284—41no—
65C2×D284—41no—
66C2×D284C4212yes—
67C2×C22⋊D71—24no—
68C2×C22⋊D71—41no—
69C2×C22⋊D71—41no—
70C2×C22⋊D71—42no—
71C22×Dic71—21no—
72C22×Dic71—21no—
73C22×Dic71—21no—
74C2×C4×D71—42no—
75C2×C4×D71—42no—
76C2×C4×D71—42no—
77C2×C71⋊C8—24no—
78C2×C71⋊C8—42no—
79C2×C71⋊C8—42no—
80C4⋊Dic71—21no—
81C4⋊Dic71—41no—
82C4⋊Dic71—41no—
83C4⋊Dic71C4212yes—
84Dic71⋊C4—42no—
85C22⋊Dic71—41no—
86C22⋊Dic71—41no—
87(C2×D71)⋊C4—24no—
88(C2×D71)⋊C4—42no—
89(C2×D71)⋊C4—42no—
90(C2×D71)⋊C4—42no—
91C4×Dic71—42no—
92C24×C71C2×D411yescharacteristic
93C22×C4×C71—22no—
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×C71D812yes—
112C22×C4×C71SD1612yes—
113C22×C4×C71C22×C411yescharacteristic
114C22×C4×C71C22×C412yes—
115C22×C4×C71C2×D411yescharacteristic
116C22×C4×C71#4112yes—
117C42×C71—22no—
118C42×C71—22no—
119C42×C71—41no—
120C42×C71—41no—
121C23×C71—21no—
122C23×C71—21no—
123C23×C71—21no—
124C23×C71—21no—
125C23×C71C2×C22⋊C411yescharacteristic
126C23×C71C4×D412yes—
127C23×C71C2×D811yescharacteristic
128C23×C71C2×SD1611yescharacteristic
129C23×C71#13512yes—
130C22×D71—22no—
131C22×D71—41no—
132C22×D71—41no—
133C22×D71—41no—
134C22×D71—41no—
135C22×D71—41no—
136C22×D71—41no—
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—42no—
153C2×C4×C71—42no—
154C2×C4×C71—44no—
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×C71#10212yes—
164C2×C4×C71#10212yes—
165C2×C4×C71C2×C4211yescharacteristic
166C2×C4×C71#11711yescharacteristic
167C2×C4×C71C4×D412yes—
168C22⋊D71—44no—
169C22⋊D71—44no—
170C2×Dic71—22no—
171C2×Dic71—41no—
172C2×Dic71—41no—
173C2×Dic71—41no—
174C2×Dic71—41no—
175C2×Dic71—41no—
176C2×Dic71—41no—
177C2×Dic71—41no—
178C2×Dic71—41no—
179D284—42no—
180D284—42no—
181C4×D71—84no—
182C71⋊C8—44no—
183C22×C71—22no—
184C22×C71—22no—
185C22×C71—22no—
186C22×C71—21no—
187C22×C71—21no—
188C22×C71—21no—
189C22×C71—21no—
190C22×C71—22no—
191C22×C71C4×C22⋊C411yescharacteristic
192C22×C71C2×#9911yescharacteristic
193C22×C71#41011yescharacteristic
194C22×C71C2×#10211yescharacteristic
195C22×C71#41611yescharacteristic
196C22×C71#42111yescharacteristic
197C22×C71#42111yescharacteristic
198C22×C71C4×D812yes—
199C22×C71C4×SD1612yes—
200C2×D71—42no—
201C2×D71—42no—
202C2×D71—42no—
203C2×D71—81no—
204C2×D71—81no—
205C2×D71—81no—
206C2×D71—81no—
207C4×C71—22no—
208C4×C71—21no—
209C4×C71—21no—
210C4×C71—22no—
211C4×C71—22no—
212C4×C71—22no—
213C4×C71—41no—
214C4×C71—84no—
215Dic71—42no—
216Dic71—81no—
217Dic71—81no—
218#8220—711nomaximal, Sylow
219C2×C71—22no—
220C2×C71—22no—
221C2×C71#151611yescharacteristic
222C2×C71#151711yescharacteristic
223C2×C71#151911yescharacteristic
224C2×C71#152111yescharacteristic
225C2×C71C4×#9911yescharacteristic
226C2×C71#175911yescharacteristic
227C2×C71C4×#10211yescharacteristic
228D71—82no—
229C4×#121—711no—
230C4×C22⋊C8—711no—
231C2×#327—712no—
232#1434D7111yescharacteristic, complemented
233#1294—712no—
234C71#822011yescharacteristic, complemented, Sylow
235C2×#121—711no—
236C2×C4×D4—711no—
237C2×C4×D4—1421no—
238C22×C42C2×D7111yescharacteristic
239C2×C22⋊C8—712no—
240C2×C4×C8—1421no—
241#390—711no—
242#389—711no—
243#388—1421no—
244#386—1421no—
245#380—711no—
246C4×C4⋊C4—711no—
247C4×C22⋊C4—1421no—
248C2×#95—21no—
249C2×#95—21no—
250C2×#95C2×D7112yes—
251#336—1421no—
252#327—718no—
253C22×D4—711no—
254C22×D4—1421no—
255C23×C4—22no—
256C23×C4—1421no—
257C23×C4C22⋊D7111yescharacteristic
258C23×C4C22⋊D7111yescharacteristic
259C23×C4C22×D7111yescharacteristic
260C22×C8—1422no—
261#121—714no—
262#121—1422no—
263C4×D4—2842no—
264C4×D4—2842no—
265C4×D4—2842no—
266C2×C4⋊C4—711no—
267C2×C4⋊C4—711no—
268C2×C4⋊C4—711no—
269C2×C4⋊C4—1421no—
270C2×C22⋊C4—1421no—
271C2×C22⋊C4—1421no—
272C2×C22⋊C4—1421no—
273C2×C22⋊C4—1421no—
274C2×C22⋊C4—1421no—
275C2×C42—42no—
276C2×C42—1421no—
277C2×C42C4×D7111yescharacteristic
278C2×C42C4×D7111yescharacteristic
279C2×C42D28411yescharacteristic
280C2×C42D28411yescharacteristic
281C22⋊C8—714no—
282C22⋊C8—1422no—
283C4×C8—2842no—
284#95—22no—
285#95—42no—
286#95—42no—
287#95—1421no—
288#95—1421no—
289#95—1421no—
290#95C4×D7114yescomplemented
291C24—1421no—
292C24C2×C22⋊D7111yescharacteristic
293C2×D4—2841no—
294C2×D4—2841no—
295C2×D4—2841no—
296C2×D4—2841no—
297C2×D4—2842no—
298C2×D4—712no—
299C2×D4—1424no—
300C2×D4—1421no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—2842no—
316C22×C4—2842no—
317C22×C4—2842no—
318C22×C4—1421no—
319C22×C4—1421no—
320C22×C4—1421no—
321C22×C4(C2×D71)⋊C411yescharacteristic
322C22×C4(C2×D71)⋊C411yescharacteristic
323C22×C4(C2×D71)⋊C411yescharacteristic
324C22×C4(C2×D71)⋊C411yescharacteristic
325C22×C4#1176585012yes—
326C22×C4#1176585212yes—
327C22×C4C2×C4×D7111yescharacteristic
328C22×C4C2×C4×D7112yes—
329C22×C4C2×D28411yescharacteristic
330C22×C4#1176586512yes—
331C2×C8—2842no—
332C2×C8—2842no—
333C2×C8—1424no—
334C4⋊C4—2841no—
335C4⋊C4—2841no—
336C4⋊C4—2842no—
337C4⋊C4—712no—
338C4⋊C4—1421no—
339C22⋊C4—2842no—
340C22⋊C4—2841no—
341C22⋊C4—2841no—
342C22⋊C4—2842no—
343C22⋊C4—2842no—
344C22⋊C4—1424no—
345C42—22no—
346C42—22no—
347C42—41no—
348C42—41no—
349C42—2842no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—2841no—
355C23—2841no—
356C23—2841no—
357C23—2841no—
358C23—2841no—
359C23—2841no—
360C23—1422no—
361C23C2×(C2×D71)⋊C411yescharacteristic
362C23C4×C22⋊D7112yes—
363C23C2×#1176585011yescharacteristic
364C23C2×#1176585211yescharacteristic
365C23#1455614812yes—
366D4—2842no—
367D4—2842no—
368D4—2844no—
369D4—2844no—
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—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—2841no—
396C2×C4—2841no—
397C2×C4—2841no—
398C2×C4—2841no—
399C2×C4—2841no—
400C2×C4—2841no—
401C2×C4—2841no—
402C2×C4—2841no—
403C2×C4—5684no—
404C2×C4—1422no—
405C2×C4#1455604212yes—
406C2×C4#1455604812yes—
407C2×C4#1455604812yes—
408C2×C4C42×D7111yescharacteristic
409C2×C4#1455608511yescharacteristic
410C2×C4C4×D28412yes—
411C8—2844no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—2842no—
421C22—2842no—
422C22—2842no—
423C22—5681no—
424C22—5681no—
425C22—5681no—
426C22—5681no—
427C22C4×(C2×D71)⋊C411yescharacteristic
428C22C2×#1455604211yescharacteristic
429C22#1798597211yescharacteristic
430C22C2×#1455604811yescharacteristic
431C22#1798599311yescharacteristic
432C22#1798601711yescharacteristic
433C22#1798601711yescharacteristic
434C22C4×#1176585012yes—
435C22C4×#1176585212yes—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—84no—
444C4—2842no—
445C4—5681no—
446C4—5681no—
447C2—22no—
448C2—22no—
449C2—5682no—
450C2#2258689411yescharacteristic
451C2#2258689911yescharacteristic
452C2#2258690711yescharacteristic
453C2#2258691511yescharacteristic
454C2C4×#1455604211yescharacteristic
455C2#2258815111yescharacteristic
456C2C4×#1455604811yescharacteristic
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#8220
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. C4×C22⋊(C71⋊C8)C2semidirect
4. #22586219C2semidirect
5. C71×#1434C2semidirect
7. C2×C4×D284C4semidirect
11. #17985901C4semidirect
12. #17985897C4semidirect
16. C4×C4⋊Dic71C4semidirect
19. #17985732C4semidirect
44. C22⋊(C71⋊C8)C2×C4semidirect
61. #95×C71C2×C4semidirect
232. #1434D71semidirect
234. C71#8220semidirect
290. #95C4×D71semidirect

Maximal subgroups

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

subgroupindex
C4×#145561022
C4×C22⋊(C71⋊C8)2
C2×#179857322
#225862192
C71×#14342
#822071