Finite Group Databasebeta

← #31912427

Subgroups — #31912427

Order 18176.

826 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985973C211yesmaximal, characteristic
2#22590932C211yesmaximal, characteristic
3#22586464C214yesmaximal, complemented
4C2×#410×C71C211yesmaximal, characteristic
5C2×#14556087—21no—
6C2×#14556087C411yescharacteristic, complemented
7C22×C4⋊Dic71C411yescharacteristic, complemented
8#17985973—22no—
9#17985973C414yescomplemented
10C2×#14556043—21no—
11C2×#14556043C2212yes—
12#17985906C412yescomplemented
13#17985902C2212yes—
14#17985879—21no—
15#17985879—21no—
16#17985859—21no—
17#17985859—21no—
18C22×#41×C71C2211yescharacteristic
19C2×#117×C71—21no—
20#410×C71—22no—
21#410×C71C414yescomplemented
22C2×#99×C71—21no—
23C2×#99×C71C2212yes—
24C23×Dic71—21no—
25#14556087—24no—
26#14556087—42no—
27#14556087—42no—
28C2×C4⋊Dic71—21no—
29C2×C4⋊Dic71—21no—
30C2×C4⋊Dic71—41no—
31C2×C4⋊Dic71—41no—
32C2×C4⋊Dic71—41no—
33C2×C4⋊Dic71C2×C412yes—
34C2×C22⋊Dic71—21no—
35C2×C22⋊Dic71—22no—
36C2×C22⋊Dic71—22no—
37C2×C22⋊Dic71—22no—
38C2×C4×Dic71—41no—
39C2×C4×Dic71C2×C412yes—
40#14556043—24no—
41#14556043—22no—
42#14556043C2×C414yescomplemented
43C71⋊#95—22no—
44C71⋊#95—22no—
45C2×#41×C71—41no—
46C2×#41×C71—41no—
47C2×#41×C71—42no—
48C2×#41×C71C2×C412yes—
49C2×#41×C71C2×C412yes—
50C2×#41×C71D412yes—
51C2×#41×C71Q812yes—
52C22×Q8×C71C2×C411yescharacteristic
53C22×D4×C71D411yescharacteristic
54C22×D4×C71D411yescharacteristic
55C22×D4×C71C2311yescharacteristic
56C23×C4×C71C2×C411yescharacteristic
57C23×C4×C71D411yescharacteristic
58C23×C4×C71D411yescharacteristic
59#117×C71—24no—
60#117×C71—42no—
61C2×C4⋊C4×C71—41no—
62C2×C22⋊C4×C71—22no—
63C2×C42×C71—41no—
64#99×C71—24no—
65#99×C71—22no—
66#99×C71C2×C414yescomplemented
67C22×Dic71—22no—
68C22×Dic71—22no—
69C22×Dic71—42no—
70C22×Dic71—42no—
71C22×Dic71—82no—
72C4⋊Dic71—44no—
73C4⋊Dic71—41no—
74C4⋊Dic71—41no—
75C4⋊Dic71—42no—
76C4⋊Dic71—42no—
77C22⋊Dic71—24no—
78C22⋊Dic71—42no—
79C22⋊Dic71—42no—
80C22⋊Dic71—82no—
81C22⋊Dic71—82no—
82C4×Dic71—44no—
83C4×Dic71—42no—
84C24×C71C22⋊C411yescharacteristic
85C24×C71C2×D411yescharacteristic
86C24×C71C2×D411yescharacteristic
87#41×C71—42no—
88#41×C71—42no—
89#41×C71—42no—
90#41×C71—42no—
91#41×C71—42no—
92#41×C71—42no—
93#41×C71—42no—
94#41×C71—42no—
95C2×Q8×C71—41no—
96C2×Q8×C71—41no—
97C2×Q8×C71C4212yes—
98C2×Q8×C71C4⋊C412yes—
99C2×D4×C71—41no—
100C2×D4×C71—41no—
101C2×D4×C71—41no—
102C2×D4×C71—41no—
103C2×D4×C71—41no—
104C2×D4×C71—41no—
105C2×D4×C71C22⋊C412yes—
106C2×D4×C71C22⋊C412yes—
107C2×D4×C71C4⋊C412yes—
108C2×D4×C71C4⋊C412yes—
109C2×D4×C71C22×C411yescharacteristic
110C2×D4×C71C22×C411yescharacteristic
111C2×D4×C71C2×D411yescharacteristic
112C2×D4×C71C2×Q811yescharacteristic
113C22×C4×C71—21no—
114C22×C4×C71—21no—
115C22×C4×C71—22no—
116C22×C4×C71—21no—
117C22×C4×C71—21no—
118C22×C4×C71—22no—
119C22×C4×C71—21no—
120C22×C4×C71—21no—
121C22×C4×C71—22no—
122C22×C4×C71—42no—
123C22×C4×C71C4212yes—
124C22×C4×C71C22⋊C411yescharacteristic
125C22×C4×C71C22⋊C411yescharacteristic
126C22×C4×C71C22⋊C412yes—
127C22×C4×C71C22⋊C411yescharacteristic
128C22×C4×C71C22⋊C412yes—
129C22×C4×C71C4⋊C412yes—
130C22×C4×C71C4⋊C412yes—
131C22×C4×C71C4⋊C412yes—
132C22×C4×C71C22×C411yescharacteristic
133C4⋊C4×C71—44no—
134C22⋊C4×C71—24no—
135C22⋊C4×C71—42no—
136C42×C71—44no—
137C23×C71—21no—
138C23×C71—21no—
139C23×C71—21no—
140C23×C71—21no—
141C23×C71—21no—
142C23×C71—21no—
143C23×C71—41no—
144C23×C71—41no—
145C23×C71—41no—
146C23×C71—41no—
147C23×C71—41no—
148C23×C71—41no—
149C23×C71#9512yes—
150C23×C71C2×C22⋊C411yescharacteristic
151C23×C71C2×C22⋊C411yescharacteristic
152C23×C71C2×C22⋊C411yescharacteristic
153C23×C71C2×C4⋊C411yescharacteristic
154C23×C71C2×C4⋊C411yescharacteristic
155Q8×C71—42no—
156Q8×C71—42no—
157D4×C71—42no—
158D4×C71—42no—
159D4×C71—42no—
160D4×C71—42no—
161D4×C71—41no—
162D4×C71—41no—
163D4×C71—41no—
164D4×C71—41no—
165C2×C4×C71—21no—
166C2×C4×C71—21no—
167C2×C4×C71—21no—
168C2×C4×C71—21no—
169C2×C4×C71—21no—
170C2×C4×C71—21no—
171C2×C4×C71—22no—
172C2×C4×C71—22no—
173C2×C4×C71—22no—
174C2×C4×C71—22no—
175C2×C4×C71—22no—
176C2×C4×C71—22no—
177C2×C4×C71—22no—
178C2×C4×C71—22no—
179C2×C4×C71—22no—
180C2×C4×C71—22no—
181C2×C4×C71—22no—
182C2×C4×C71—22no—
183C2×C4×C71—44no—
184C2×C4×C71—82no—
185C2×C4×C71—82no—
186C2×C4×C71#9512yes—
187C2×C4×C71#9512yes—
188C2×C4×C71#9512yes—
189C2×C4×C71#9912yes—
190C2×C4×C71#9911yescharacteristic
191C2×C4×C71#9911yescharacteristic
192C2×C4×C71C2×C4211yescharacteristic
193C2×C4×C71C2×C4⋊C411yescharacteristic
194C2×Dic71—44no—
195C2×Dic71—42no—
196C2×Dic71—42no—
197C2×Dic71—42no—
198C2×Dic71—42no—
199C2×Dic71—42no—
200C2×Dic71—42no—
201C2×Dic71—82no—
202C2×Dic71—82no—
203C2×Dic71—84no—
204C22×C71—21no—
205C22×C71—21no—
206C22×C71—21no—
207C22×C71—21no—
208C22×C71—22no—
209C22×C71—21no—
210C22×C71—21no—
211C22×C71—42no—
212C22×C71—41no—
213C22×C71—41no—
214C22×C71—41no—
215C22×C71—41no—
216C22×C71—42no—
217C22×C71—41no—
218C22×C71—41no—
219C22×C71—41no—
220C22×C71—41no—
221C22×C71—41no—
222C22×C71—41no—
223C22×C71—81no—
224C22×C71—81no—
225C22×C71C2×#9511yescharacteristic
226C22×C71C2×#9911yescharacteristic
227C22×C71C2×#9911yescharacteristic
228C4×C71—22no—
229C4×C71—22no—
230C4×C71—21no—
231C4×C71—21no—
232C4×C71—84no—
233C4×C71#34212yes—
234C4×C71#34212yes—
235Dic71—82no—
236Dic71—84no—
237Dic71—82no—
238#12583—711nomaximal, Sylow
239C2×C71—21no—
240C2×C71—21no—
241C2×C71—41no—
242C2×C71—41no—
243C2×C71—42no—
244C2×C71—41no—
245C2×C71—41no—
246C2×C71C2×#34211yescharacteristic
247C2×C71#212811yescharacteristic
248C2×C71#212911yescharacteristic
249C2×#410—711no—
250C2×#410D7111yescharacteristic
251#2311—711no—
252#1393—714no—
253C71#1258311yescharacteristic, complemented, Sylow
254C22×#41C2×D7111yescharacteristic
255C2×#117—21no—
256C2×#117—711no—
257C2×#117—1421no—
258C22×C4⋊C4—711no—
259#410—22no—
260#410—714no—
261#410—1422no—
262#410Dic7114yescomplemented
263C2×#99—21no—
264C2×#99—712no—
265C2×#99—1421no—
266C2×#99C2×D7112yes—
267#391—712no—
268#390—712no—
269#386—1421no—
270#386—1421no—
271#380—1421no—
272#380—1421no—
273C2×#41—41no—
274C2×#41—41no—
275C2×#41—42no—
276C2×#41Dic14212yes—
277C2×#41C4×D7112yes—
278C2×#41C2×Dic7112yes—
279C2×#41C22⋊D7112yes—
280C22×Q8C4×D7111yescharacteristic
281C22×D4D28411yescharacteristic
282C22×D4C22⋊D7111yescharacteristic
283C22×D4C22×D7111yescharacteristic
284C23×C4—1421no—
285C23×C4C4×D7111yescharacteristic
286C23×C4D28411yescharacteristic
287C23×C4C22⋊D7111yescharacteristic
288#117—24no—
289#117—42no—
290#117—2842no—
291#117—2842no—
292#117—1424no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—2841no—
295C2×C4⋊C4—2841no—
296C2×C4⋊C4—2841no—
297C2×C4⋊C4—712no—
298C2×C4⋊C4—1421no—
299C2×C4⋊C4—1421no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—1421no—
302C2×C22⋊C4—1422no—
303C2×C22⋊C4—1422no—
304C2×C22⋊C4—1422no—
305C2×C42—41no—
306C2×C42—2841no—
307C2×C42—712no—
308#99—24no—
309#99—22no—
310#99—714no—
311#99—1424no—
312#99—1422no—
313#99C2×Dic7114yescomplemented
314#95—1422no—
315#95—1422no—
316C24(C2×D71)⋊C411yescharacteristic
317C24C2×C22⋊D7111yescharacteristic
318C24C2×D28411yescharacteristic
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—42no—
324#41—42no—
325#41—42no—
326#41—42no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8C4×Dic7112yes—
330C2×Q8Dic71⋊C412yes—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4(C2×D71)⋊C412yes—
338C2×D4C22⋊Dic7112yes—
339C2×D4Dic71⋊C412yes—
340C2×D4C4⋊Dic7112yes—
341C2×D4C2×C4×D7111yescharacteristic
342C2×D4C22×Dic7111yescharacteristic
343C2×D4C2×C22⋊D7111yescharacteristic
344C2×D4C2×Dic14211yescharacteristic
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—2842no—
356C22×C4—2842no—
357C22×C4—5682no—
358C22×C4—1422no—
359C22×C4—1422no—
360C22×C4C4×Dic7112yes—
361C22×C4(C2×D71)⋊C411yescharacteristic
362C22×C4(C2×D71)⋊C411yescharacteristic
363C22×C4(C2×D71)⋊C412yes—
364C22×C4(C2×D71)⋊C411yescharacteristic
365C22×C4C22⋊Dic7112yes—
366C22×C4Dic71⋊C412yes—
367C22×C4Dic71⋊C412yes—
368C22×C4C4⋊Dic7112yes—
369C22×C4C2×C4×D7111yescharacteristic
370C4⋊C4—44no—
371C4⋊C4—2842no—
372C4⋊C4—2844no—
373C4⋊C4—2841no—
374C4⋊C4—2841no—
375C4⋊C4—2842no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—2842no—
379C22⋊C4—2842no—
380C22⋊C4—5682no—
381C22⋊C4—5682no—
382C22⋊C4—1424no—
383C42—44no—
384C42—2842no—
385C42—2844no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23C71⋊#9512yes—
399C23C2×(C2×D71)⋊C411yescharacteristic
400C23C2×(C2×D71)⋊C411yescharacteristic
401C23C2×C22⋊Dic7111yescharacteristic
402C23C2×Dic71⋊C411yescharacteristic
403C23C2×C4⋊Dic7111yescharacteristic
404Q8—42no—
405Q8—42no—
406D4—42no—
407D4—42no—
408D4—42no—
409D4—42no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—44no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—2844no—
436C2×C4—2842no—
437C2×C4—2842no—
438C2×C4—2842no—
439C2×C4—2842no—
440C2×C4—2842no—
441C2×C4—2842no—
442C2×C4—5684no—
443C2×C4—5682no—
444C2×C4—5682no—
445C2×C4C71⋊#9512yes—
446C2×C4C71⋊#9512yes—
447C2×C4C71⋊#9512yes—
448C2×C4#1455604212yes—
449C2×C4#1455604211yescharacteristic
450C2×C4#1455604211yescharacteristic
451C2×C4C2×C4×Dic7111yescharacteristic
452C2×C4C2×Dic71⋊C411yescharacteristic
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—81no—
473C22—81no—
474C22C2×C71⋊#9511yescharacteristic
475C22C2×#1455604211yescharacteristic
476C22C2×#1455604211yescharacteristic
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—84no—
482C4—5682no—
483C4—5684no—
484C4—5682no—
485C4#1798576512yes—
486C4#1798576512yes—
487C2—21no—
488C2—21no—
489C2—41no—
490C2—41no—
491C2—42no—
492C2—41no—
493C2—41no—
494C2C2×#1798576511yescharacteristic
495C2#2259001111yescharacteristic
496C2#2259001411yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC2×C4×C71C2×C42
FrattiniC22×D4C22×D71
FittingC2×#410×C71C2
socleC22×C71C2×#99
prime radicalC22×#41×C71C22
cosocleC22×D4×C71C23
hypercenterC2×#410D71
hypocenterC71#12583
LCSG ▷ C2×C4×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#410

Complements

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

NH ≅ G/Ntype
3. #22586464C2semidirect
6. C2×#14556087C4semidirect
7. C22×C4⋊Dic71C4semidirect
9. #17985973C4semidirect
12. #17985906C4semidirect
21. #410×C71C4semidirect
42. #14556043C2×C4semidirect
66. #99×C71C2×C4semidirect
253. C71#12583semidirect
262. #410Dic71semidirect
313. #99C2×Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859732
#225909322
#225864642
C2×#410×C712
#1258371