Finite Group Databasebeta

← #31888359

Subgroups — #31888359

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
2C2×#17985734C212yesmaximal
3#22586567C211yesmaximal, characteristic, complemented
4#22586220C212yesmaximal, complemented
5C4×C22⋊C8×C71C211yesmaximal, characteristic, complemented
6C2×#14556102C2211yescharacteristic
7C2×C4×D284C411yescharacteristic, complemented
8C2×C4×C22⋊D71—21no—
9#17985901C411yescharacteristic, complemented
10#17985897C411yescharacteristic, complemented
11#17985890—21no—
12#17985878—21no—
13#17985859C2211yescharacteristic
14C4×C4⋊Dic71C411yescharacteristic, complemented
15C4×(C2×D71)⋊C4—21no—
16C2×C71⋊#95—21no—
17C2×C71⋊#95—21no—
18C2×C71⋊#95C2212yes—
19#17985734C418yescomplemented
20C22×C42×C71C2211yescharacteristic
21C2×C22⋊C8×C71C2212yes—
22C2×C4×C8×C71—21no—
23#336×C71—21no—
24C22×D284C2×C411yescharacteristic
25C22×C22⋊D71—21no—
26C23×Dic71—22no—
27C22×C4×D71—21no—
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—
43C71⋊#95—21no—
44C71⋊#95—21no—
45C71⋊#95—21no—
46C71⋊#95—22no—
47C71⋊#95—42no—
48C71⋊#95—42no—
49C71⋊#95C2×C414yescomplemented
50C23×C4×C71D411yescharacteristic
51C23×C4×C71D411yescharacteristic
52C23×C4×C71C2311yescharacteristic
53C22×C8×C71—22no—
54C2×C42×C71—42no—
55C2×C42×C71C2×C411yescharacteristic
56C2×C42×C71C2×C411yescharacteristic
57C2×C42×C71D411yescharacteristic
58C2×C42×C71D411yescharacteristic
59C22⋊C8×C71—22no—
60C22⋊C8×C71C2×C414yescomplemented
61C4×C8×C71—42no—
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—
74C22×Dic71—24no—
75C22×Dic71—42no—
76C22×Dic71—42no—
77C22×Dic71—42no—
78C22×Dic71—42no—
79C22×Dic71—42no—
80C22×Dic71—42no—
81C2×C4×D71—42no—
82C2×C4×D71—42no—
83C2×C4×D71—42no—
84C4⋊Dic71—21no—
85C4⋊Dic71—41no—
86C4⋊Dic71—41no—
87C4⋊Dic71C4212yes—
88Dic71⋊C4—42no—
89C22⋊Dic71—41no—
90C22⋊Dic71—41no—
91(C2×D71)⋊C4—24no—
92(C2×D71)⋊C4—42no—
93(C2×D71)⋊C4—42no—
94(C2×D71)⋊C4—42no—
95C4×Dic71—42no—
96C24×C71C2×D411yescharacteristic
97C22×C4×C71—22no—
98C22×C4×C71—22no—
99C22×C4×C71—41no—
100C22×C4×C71—41no—
101C22×C4×C71—41no—
102C22×C4×C71—41no—
103C22×C4×C71—41no—
104C22×C4×C71C22⋊C411yescharacteristic
105C22×C4×C71C22⋊C411yescharacteristic
106C22×C4×C71C22⋊C411yescharacteristic
107C22×C4×C71C22⋊C411yescharacteristic
108C22×C4×C71D812yes—
109C22×C4×C71SD1612yes—
110C22×C4×C71C22×C411yescharacteristic
111C22×C4×C71C22×C412yes—
112C22×C4×C71C2×D411yescharacteristic
113C22×C4×C71#4112yes—
114C2×C8×C71—24no—
115C2×C8×C71—42no—
116C2×C8×C71—42no—
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#9912yes—
156C2×C4×C71#10212yes—
157C2×C4×C71#10212yes—
158C2×C4×C71C2×C4211yescharacteristic
159C2×C4×C71#11711yescharacteristic
160C2×C4×C71C4×D412yes—
161C22⋊D71—44no—
162C22⋊D71—44no—
163C2×Dic71—22no—
164C2×Dic71—41no—
165C2×Dic71—41no—
166C2×Dic71—44no—
167C2×Dic71—44no—
168C2×Dic71—41no—
169C2×Dic71—41no—
170C2×Dic71—41no—
171C2×Dic71—41no—
172C2×Dic71—41no—
173C2×Dic71—41no—
174C2×Dic71—44no—
175C2×Dic71—82no—
176C2×Dic71—82no—
177C2×Dic71—82no—
178C2×Dic71—82no—
179D284—42no—
180D284—42no—
181C4×D71—84no—
182C8×C71—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—
214Dic71—42no—
215Dic71—84no—
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⋊C8D7111yescharacteristic, complemented
231C2×#327—712no—
232#1434—711no—
233#1294—712no—
234C71#822011yescharacteristic, complemented, Sylow
235C2×#121—711no—
236C2×C4×D4—711no—
237C2×C4×D4—1421no—
238C22×C42C2×D7111yescharacteristic
239C2×C22⋊C8C2×D7112yes—
240C2×C4×C8—21no—
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—712no—
249C2×#95—1421no—
250C2×#95—1421no—
251#336—21no—
252#327—718no—
253C22×D4—711no—
254C22×D4—1421no—
255C23×C4—1422no—
256C23×C4—1421no—
257C23×C4D28411yescharacteristic
258C23×C4D28411yescharacteristic
259C23×C4C22×D7111yescharacteristic
260C22×C8—22no—
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×C42C22⋊D7111yescharacteristic
280C2×C42C22⋊D7111yescharacteristic
281C22⋊C8—22no—
282C22⋊C8C4×D7114yescomplemented
283C4×C8—42no—
284#95—2842no—
285#95—2842no—
286#95—714no—
287#95—1421no—
288#95—1421no—
289#95—1421no—
290#95—1422no—
291C24—1421no—
292C24C2×D28411yescharacteristic
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—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—2842no—
309C22×C4—2842no—
310C22×C4—2842no—
311C22×C4—2842no—
312C22×C4—2842no—
313C22×C4—2842no—
314C22×C4—2842no—
315C22×C4—2842no—
316C22×C4—2842no—
317C22×C4—1421no—
318C22×C4—1421no—
319C22×C4—1424no—
320C22×C4—1421no—
321C22×C4(C2×D71)⋊C411yescharacteristic
322C22×C4(C2×D71)⋊C411yescharacteristic
323C22×C4(C2×D71)⋊C411yescharacteristic
324C22×C4(C2×D71)⋊C411yescharacteristic
325C22×C4D56812yes—
326C22×C4#1176585412yes—
327C22×C4C2×C4×D7111yescharacteristic
328C22×C4C2×C4×D7112yes—
329C22×C4C2×C22⋊D7111yescharacteristic
330C22×C4#1176586512yes—
331C2×C8—24no—
332C2×C8—42no—
333C2×C8—42no—
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×D28412yes—
363C23C2×D56811yescharacteristic
364C23C2×#1176585411yescharacteristic
365C23#1455615012yes—
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—2841no—
389C2×C4—2841no—
390C2×C4—2844no—
391C2×C4—2844no—
392C2×C4—2844no—
393C2×C4—2841no—
394C2×C4—2841no—
395C2×C4—2841no—
396C2×C4—2841no—
397C2×C4—2841no—
398C2×C4—2841no—
399C2×C4—5684no—
400C2×C4—5682no—
401C2×C4—5682no—
402C2×C4—5682no—
403C2×C4—5682no—
404C2×C4—1422no—
405C2×C4#1455604212yes—
406C2×C4#1455605012yes—
407C2×C4#1455605012yes—
408C2×C4C42×D7111yescharacteristic
409C2×C4#1455608511yescharacteristic
410C2×C4C4×C22⋊D7112yes—
411C8—44no—
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×#1455605011yescharacteristic
431C22#1798599611yescharacteristic
432C22#1798601511yescharacteristic
433C22#1798601511yescharacteristic
434C22C4×D56812yes—
435C22C4×#1176585412yes—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—22no—
440C4—21no—
441C4—21no—
442C4—41no—
443C4—2842no—
444C4—5684no—
445C4—5681no—
446C4—5681no—
447C2—22no—
448C2—22no—
449C2—5682no—
450C2#2258689711yescharacteristic
451C2#2258690111yescharacteristic
452C2#2258690911yescharacteristic
453C2#2258691811yescharacteristic
454C2C4×#1455604211yescharacteristic
455C2#2258815311yescharacteristic
456C2C4×#1455605011yescharacteristic
subgroupG/H
centerC23C2×(C2×D71)⋊C4
commutatorC2×C4×C71C2×C42
FrattiniC23×C4C22×D71
FittingC4×C22⋊C8×C71C2
socleC23×C71C2×C22⋊C4
prime radicalC2×#14556102C22
cosocleC23×C4×C71C23
hypercenterC4×C22⋊C8D71
hypocenterC71#8220
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. #22586567C2semidirect
4. #22586220C2semidirect
5. C4×C22⋊C8×C71C2semidirect
7. C2×C4×D284C4semidirect
9. #17985901C4semidirect
10. #17985897C4semidirect
14. C4×C4⋊Dic71C4semidirect
19. #17985734C4semidirect
49. C71⋊#95C2×C4semidirect
60. C22⋊C8×C71C2×C4semidirect
230. C4×C22⋊C8D71semidirect
234. C71#8220semidirect
282. C22⋊C8C4×D71semidirect

Maximal subgroups

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

subgroupindex
C4×#145561022
C2×#179857342
#225865672
#225862202
C4×C22⋊C8×C712
#822071