Finite Group Databasebeta

← #31912402

Subgroups — #31912402

Order 18176.

810 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985983C211yesmaximal, characteristic
2#22590932C211yesmaximal, characteristic
3#22586457C214yesmaximal, complemented
4C2×#410×C71C211yesmaximal, characteristic
5C22×#11765849—21no—
6C2×#14556087C411yescharacteristic, complemented
7C22×C4⋊Dic71C411yescharacteristic, complemented
8#17985983—22no—
9#17985983C414yescomplemented
10C2×#14556047—21no—
11C2×#14556045C2212yes—
12#17985906—21no—
13#17985902—21no—
14#17985879C412yescomplemented
15#17985879C2212yes—
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×Dic71C2×C412yes—
25C2×#11765849—24no—
26C2×#11765849—22no—
27C2×#11765849—41no—
28C2×#11765849—42no—
29C22×C71⋊C8—41no—
30#14556087—42no—
31C2×C4⋊Dic71—21no—
32C2×C4⋊Dic71—21no—
33C2×C4⋊Dic71—21no—
34C2×C4⋊Dic71—41no—
35C2×C4⋊Dic71—41no—
36C2×C22⋊Dic71—22no—
37C2×C22⋊Dic71—22no—
38C2×C22⋊Dic71C2×C412yes—
39C2×C4×Dic71—21no—
40#14556047—24no—
41#14556045—22no—
42#14556045C2×C414yes—
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×C71D411yescharacteristic
53C22×D4×C71C2×C411yescharacteristic
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×C414yes—
67C22×Dic71—22no—
68C22×Dic71—22no—
69C22×Dic71—42no—
70C22×Dic71—82no—
71#11765849—24no—
72#11765849—42no—
73#11765849—44no—
74C2×C71⋊C8—42no—
75C2×C71⋊C8—44no—
76C4⋊Dic71—42no—
77C4⋊Dic71—41no—
78C4⋊Dic71—41no—
79C4⋊Dic71—42no—
80C22⋊Dic71—42no—
81C22⋊Dic71—82no—
82C22⋊Dic71—82no—
83C4×Dic71—42no—
84C24×C71C22⋊C411yescharacteristic
85C24×C71C22×C411yescharacteristic
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×C71C22⋊C412yes—
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×C71C4212yes—
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×C71C2×D411yescharacteristic
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×C4211yescharacteristic
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×C22⋊C411yescharacteristic
193C2×C4×C71C2×C4⋊C411yescharacteristic
194C2×Dic71—42no—
195C2×Dic71—42no—
196C2×Dic71—42no—
197C2×Dic71—42no—
198C2×Dic71—42no—
199C2×Dic71—42no—
200C2×Dic71—84no—
201C71⋊C8—44no—
202C22×C71—21no—
203C22×C71—21no—
204C22×C71—21no—
205C22×C71—22no—
206C22×C71—21no—
207C22×C71—21no—
208C22×C71—21no—
209C22×C71—42no—
210C22×C71—41no—
211C22×C71—41no—
212C22×C71—41no—
213C22×C71—41no—
214C22×C71—42no—
215C22×C71—41no—
216C22×C71—41no—
217C22×C71—41no—
218C22×C71—41no—
219C22×C71—41no—
220C22×C71—41no—
221C22×C71—81no—
222C22×C71—81no—
223C22×C71C2×#9511yescharacteristic
224C22×C71C2×#9911yescharacteristic
225C22×C71C2×#9911yescharacteristic
226C4×C71—21no—
227C4×C71—22no—
228C4×C71—22no—
229C4×C71—21no—
230C4×C71—84no—
231C4×C71#34212yes—
232C4×C71#34212yes—
233Dic71—82no—
234Dic71—82no—
235#12575—711nomaximal, Sylow
236C2×C71—21no—
237C2×C71—21no—
238C2×C71—41no—
239C2×C71—41no—
240C2×C71—41no—
241C2×C71—42no—
242C2×C71—41no—
243C2×C71C2×#34211yescharacteristic
244C2×C71#212211yescharacteristic
245C2×C71#212311yescharacteristic
246C2×#413—711no—
247C2×#410D7111yescharacteristic
248#2311—711no—
249#1390—714no—
250C71#1257511yescharacteristic, complemented, Sylow
251C22×#41C2×D7111yescharacteristic
252C22×M16—1421no—
253C2×#117—21no—
254C2×#117—711no—
255C22×C4⋊C4—711no—
256#413—714no—
257#413—1422no—
258C2×#101—1421no—
259C2×#100—712no—
260#410—22no—
261#410Dic7114yescomplemented
262C2×#99—21no—
263C2×#99C2×D7112yes—
264#391—1421no—
265#390—1421no—
266#386—712no—
267#386—712no—
268#380—1421no—
269#380—1421no—
270C2×#41—41no—
271C2×#41—41no—
272C2×#41—42no—
273C2×#41Dic14212yes—
274C2×#41C4×D7112yes—
275C2×#41C2×Dic7112yes—
276C2×#41C22⋊D7112yes—
277C22×Q8D28411yescharacteristic
278C22×D4C4×D7111yescharacteristic
279C22×D4C22⋊D7111yescharacteristic
280C22×D4C22×D7111yescharacteristic
281C23×C4—712no—
282C23×C4C4×D7111yescharacteristic
283C23×C4D28411yescharacteristic
284C23×C4C22⋊D7111yescharacteristic
285C2×M16—2841no—
286C2×M16—2842no—
287C2×M16—1424no—
288C2×M16—1422no—
289C22×C8—2841no—
290#117—24no—
291#117—42no—
292#117—2842no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—2841no—
295C2×C4⋊C4—2841no—
296C2×C4⋊C4—1421no—
297C2×C4⋊C4—1421no—
298C2×C4⋊C4—1421no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—712no—
301C2×C22⋊C4—1422no—
302C2×C22⋊C4—1422no—
303C2×C42—41no—
304C2×C42—1421no—
305#101—1424no—
306#100—714no—
307#100—1422no—
308#99—24no—
309#99—22no—
310#99C2×Dic7114yes—
311#95—1422no—
312#95—1422no—
313C24(C2×D71)⋊C411yescharacteristic
314C24C2×C4×D7111yescharacteristic
315C24C2×C22⋊D7111yescharacteristic
316#41—42no—
317#41—42no—
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—42no—
323#41—42no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8(C2×D71)⋊C412yes—
327C2×Q8C4⋊Dic7112yes—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4C4×Dic7112yes—
335C2×D4C22⋊Dic7112yes—
336C2×D4Dic71⋊C412yes—
337C2×D4Dic71⋊C412yes—
338C2×D4C2×C4×D7111yescharacteristic
339C2×D4C22×Dic7111yescharacteristic
340C2×D4C2×C22⋊D7111yescharacteristic
341C2×D4C2×Dic14211yescharacteristic
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—42no—
352C22×C4—2842no—
353C22×C4—5682no—
354C22×C4—1422no—
355C22×C4—1422no—
356C22×C4C4×Dic7112yes—
357C22×C4(C2×D71)⋊C411yescharacteristic
358C22×C4(C2×D71)⋊C411yescharacteristic
359C22×C4(C2×D71)⋊C412yes—
360C22×C4(C2×D71)⋊C411yescharacteristic
361C22×C4C22⋊Dic7112yes—
362C22×C4Dic71⋊C412yes—
363C22×C4Dic71⋊C412yes—
364C22×C4C4⋊Dic7112yes—
365C22×C4C2×D28411yescharacteristic
366M16—2842no—
367M16—2844no—
368M16—1424no—
369C2×C8—2842no—
370C2×C8—2844no—
371C4⋊C4—44no—
372C4⋊C4—2842no—
373C4⋊C4—2841no—
374C4⋊C4—2841no—
375C4⋊C4—2842no—
376C22⋊C4—24no—
377C22⋊C4—42no—
378C22⋊C4—2842no—
379C22⋊C4—5682no—
380C22⋊C4—5682no—
381C42—44no—
382C42—2842no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C71⋊#9512yes—
396C23C2×C4×Dic7111yescharacteristic
397C23C2×(C2×D71)⋊C411yescharacteristic
398C23C2×C22⋊Dic7111yescharacteristic
399C23C2×Dic71⋊C411yescharacteristic
400C23C2×Dic71⋊C411yescharacteristic
401Q8—42no—
402Q8—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—42no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
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—44no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—2842no—
433C2×C4—2842no—
434C2×C4—2842no—
435C2×C4—2842no—
436C2×C4—2842no—
437C2×C4—2842no—
438C2×C4—5684no—
439C2×C4C71⋊#9512yes—
440C2×C4C71⋊#9512yes—
441C2×C4C71⋊#9512yes—
442C2×C4#1455604212yes—
443C2×C4#1455604211yescharacteristic
444C2×C4#1455604211yescharacteristic
445C2×C4C2×(C2×D71)⋊C411yescharacteristic
446C2×C4C2×C4⋊Dic7111yescharacteristic
447C8—2844no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—42no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—81no—
468C22—81no—
469C22C2×C71⋊#9511yescharacteristic
470C22C2×#1455604211yescharacteristic
471C22C2×#1455604211yescharacteristic
472C4—21no—
473C4—22no—
474C4—22no—
475C4—21no—
476C4—84no—
477C4—5682no—
478C4—5682no—
479C4#1798576512yes—
480C4#1798576512yes—
481C2—21no—
482C2—21no—
483C2—41no—
484C2—41no—
485C2—41no—
486C2—42no—
487C2—41no—
488C2C2×#1798576511yescharacteristic
489C2#2258998911yescharacteristic
490C2#2258999311yescharacteristic
subgroupG/H
centerC22C2×#14556042
commutatorC23×C71C2×C42
FrattiniC22×D4C22×D71
FittingC2×#410×C71C2
socleC22×C71C2×#99
prime radicalC22×#41×C71C22
cosocleC22×D4×C71C23
hypercenterC2×#410D71
hypocenterC71#12575
LCSG ▷ C23×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. #22586457C2semidirect
6. C2×#14556087C4semidirect
7. C22×C4⋊Dic71C4semidirect
9. #17985983C4semidirect
14. #17985879C4semidirect
21. #410×C71C4semidirect
250. C71#12575semidirect
261. #410Dic71semidirect

Maximal subgroups

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

subgroupindex
C2×#179859832
#225909322
#225864572
C2×#410×C712
#1257571