Finite Group Databasebeta

← #31890839

Subgroups — #31890839

Order 18176.

868 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588250C211yesmaximal, characteristic
2#22586563C211yesmaximal, characteristic
3#22586218C212yesmaximal, complemented
4#22586218C212yesmaximal, complemented
5C71×#2280C211yesmaximal, characteristic
6C22×C22⋊Dic71—21no—
7C2×C22⋊(C71⋊C8)—21no—
8C2×C22⋊(C71⋊C8)—21no—
9C2×C22⋊(C71⋊C8)C2212yes—
10C2×C71⋊#95—21no—
11C2×C71⋊#95C2212yes—
12C24×C4×C71C2211yescharacteristic
13C2×#117×C71C411yescharacteristic, complemented
14C2×#117×C71C411yescharacteristic, complemented
15#381×C71—21no—
16#380×C71C411yescharacteristic, complemented
17#380×C71C411yescharacteristic, complemented
18#380×C71C2211yescharacteristic
19#380×C71C2211yescharacteristic
20#379×C71—21no—
21C4×C22⋊C4×C71—21no—
22C4×C22⋊C4×C71—21no—
23C23×Dic71—22no—
24C22×C71⋊C8—22no—
25C2×C22⋊Dic71—24no—
26C2×C22⋊Dic71—42no—
27C2×C22⋊Dic71—42no—
28C22⋊(C71⋊C8)—24no—
29C22⋊(C71⋊C8)—42no—
30C22⋊(C71⋊C8)—42no—
31C71⋊#95—24no—
32C71⋊#95—42no—
33C25×C71D411yescharacteristic
34C23×C4×C71—24no—
35C23×C4×C71—42no—
36C23×C4×C71—42no—
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71C2×C411yescharacteristic
39C23×C4×C71D411yescharacteristic
40C23×C4×C71D411yescharacteristic
41C23×C4×C71D411yescharacteristic
42C23×C4×C71C2311yescharacteristic
43#117×C71—42no—
44#117×C71—42no—
45C2×C4⋊C4×C71C2×C411yescharacteristic
46C2×C4⋊C4×C71C2×C411yescharacteristic
47C2×C4⋊C4×C71C2×C411yescharacteristic
48C2×C4⋊C4×C71C2×C411yescharacteristic
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71—21no—
51C2×C22⋊C4×C71—21no—
52C2×C22⋊C4×C71—21no—
53C2×C22⋊C4×C71—21no—
54C2×C22⋊C4×C71—21no—
55C2×C42×C71—21no—
56C2×C42×C71—21no—
57#95×C71—21no—
58#95×C71—21no—
59#95×C71—21no—
60#95×C71—21no—
61C22×Dic71—24no—
62C22×Dic71—42no—
63C22×Dic71—42no—
64C22×Dic71—42no—
65C22×Dic71—42no—
66C22×Dic71—42no—
67C22×Dic71—42no—
68C2×C71⋊C8—42no—
69C2×C71⋊C8—42no—
70C2×C71⋊C8—42no—
71C22⋊Dic71—48no—
72C24×C71—22no—
73C24×C71—22no—
74C24×C71—21no—
75C24×C71—21no—
76C24×C71—22no—
77C24×C71—22no—
78C24×C71—41no—
79C24×C71—41no—
80C24×C71C22⋊C411yescharacteristic
81C24×C71C22⋊C411yescharacteristic
82C24×C71C2×D411yescharacteristic
83C22×C4×C71—22no—
84C22×C4×C71—22no—
85C22×C4×C71—22no—
86C22×C4×C71—22no—
87C22×C4×C71—22no—
88C22×C4×C71—22no—
89C22×C4×C71—22no—
90C22×C4×C71—22no—
91C22×C4×C71—22no—
92C22×C4×C71—22no—
93C22×C4×C71—21no—
94C22×C4×C71—21no—
95C22×C4×C71—21no—
96C22×C4×C71—21no—
97C22×C4×C71—41no—
98C22×C4×C71—41no—
99C22×C4×C71—41no—
100C22×C4×C71—41no—
101C22×C4×C71—41no—
102C22×C4×C71—41no—
103C22×C4×C71—41no—
104C22×C4×C71—416no—
105C22×C4×C71C22⋊C411yescharacteristic
106C22×C4×C71C22⋊C411yescharacteristic
107C22×C4×C71C22×C411yescharacteristic
108C22×C4×C71C2×D411yescharacteristic
109C22×C4×C71#4114yes—
110C4⋊C4×C71—41no—
111C4⋊C4×C71—41no—
112C4⋊C4×C71—41no—
113C4⋊C4×C71—41no—
114C22⋊C4×C71—42no—
115C22⋊C4×C71—42no—
116C22⋊C4×C71—42no—
117C22⋊C4×C71—42no—
118C22⋊C4×C71—42no—
119C22⋊C4×C71—41no—
120C22⋊C4×C71—41no—
121C42×C71—42no—
122C42×C71—42no—
123C23×C71—21no—
124C23×C71—21no—
125C23×C71—21no—
126C23×C71—21no—
127C23×C71—24no—
128C23×C71—24no—
129C23×C71—21no—
130C23×C71—21no—
131C23×C71—21no—
132C23×C71—21no—
133C23×C71—21no—
134C23×C71—21no—
135C23×C71—21no—
136C23×C71—21no—
137C23×C71—21no—
138C23×C71—21no—
139C23×C71—21no—
140C23×C71—21no—
141C23×C71—44no—
142C23×C71—44no—
143C23×C71—44no—
144C23×C71—44no—
145C23×C71—42no—
146C23×C71—42no—
147C23×C71—42no—
148C23×C71—42no—
149C23×C71#9912yes—
150C23×C71#10412yes—
151C23×C71C2×C22⋊C411yescharacteristic
152C23×C71#12312yes—
153C23×C71#13512yes—
154C23×C71AGL(1,ℤ/8)11yescharacteristic
155C23×C71#13711yescharacteristic
156C2×C4×C71—28no—
157C2×C4×C71—21no—
158C2×C4×C71—21no—
159C2×C4×C71—21no—
160C2×C4×C71—21no—
161C2×C4×C71—21no—
162C2×C4×C71—21no—
163C2×C4×C71—21no—
164C2×C4×C71—21no—
165C2×C4×C71—21no—
166C2×C4×C71—44no—
167C2×C4×C71—44no—
168C2×C4×C71—44no—
169C2×C4×C71—44no—
170C2×C4×C71—44no—
171C2×C4×C71—41no—
172C2×C4×C71—41no—
173C2×C4×C71—41no—
174C2×C4×C71—41no—
175C2×C4×C71—41no—
176C2×C4×C71—41no—
177C2×C4×C71—41no—
178C2×C4×C71—41no—
179C2×C4×C71—41no—
180C2×C4×C71—41no—
181C2×C4×C71—41no—
182C2×C4×C71—41no—
183C2×C4×C71#9912yes—
184C2×C4×C71#11712yes—
185C2×C4×C71#12312yes—
186C2×Dic71—44no—
187C2×Dic71—44no—
188C2×Dic71—44no—
189C2×Dic71—82no—
190C2×Dic71—82no—
191C2×Dic71—82no—
192C2×Dic71—82no—
193C71⋊C8—82no—
194C22×C71—22no—
195C22×C71—22no—
196C22×C71—22no—
197C22×C71—21no—
198C22×C71—21no—
199C22×C71—21no—
200C22×C71—21no—
201C22×C71—22no—
202C22×C71—22no—
203C22×C71—22no—
204C22×C71—22no—
205C22×C71—22no—
206C22×C71—22no—
207C22×C71—22no—
208C22×C71—48no—
209C22×C71—48no—
210C22×C71—41no—
211C22×C71—41no—
212C22×C71—41no—
213C22×C71—41no—
214C22×C71—41no—
215C22×C71—41no—
216C22×C71—41no—
217C22×C71—41no—
218C22×C71#38111yescharacteristic
219C22×C71C2×#9911yescharacteristic
220C22×C71C2×#9911yescharacteristic
221C22×C71#41611yescharacteristic
222C22×C71#41711yescharacteristic
223C22×C71C2×#10411yescharacteristic
224C22×C71#42111yescharacteristic
225C22×C71#48212yes—
226C22×C71#48512yes—
227C4×C71—22no—
228C4×C71—22no—
229C4×C71—22no—
230C4×C71—41no—
231C4×C71—81no—
232C4×C71—81no—
233C4×C71—81no—
234C4×C71—81no—
235Dic71—84no—
236#8715—711nomaximal, Sylow
237C2×C71—22no—
238C2×C71—22no—
239C2×C71—42no—
240C2×C71—42no—
241C2×C71#150111yescharacteristic
242C2×C71#150411yescharacteristic
243C2×C71#151311yescharacteristic
244C2×C71#151611yescharacteristic
245C2×C71#181911yescharacteristic
246C2×C71#182511yescharacteristic
247C2×C71#182611yescharacteristic
248#2280D7111yescharacteristic
249#1779—711no—
250C22⋊#95—711no—
251#1294—712no—
252#1294—712no—
253C71#871511yescharacteristic, complemented, Sylow
254C24×C4C2×D7111yescharacteristic
255C2×#117Dic7111yescharacteristic, complemented
256C2×#117Dic7111yescharacteristic, complemented
257C22×C22⋊C4—1421no—
258C2×C22⋊C8—712no—
259C2×C22⋊C8—1421no—
260C2×C22⋊C8—1421no—
261#381—21no—
262#380Dic7111yescharacteristic, complemented
263#380Dic7111yescharacteristic, complemented
264#380C2×D7111yescharacteristic
265#380C2×D7111yescharacteristic
266#379—21no—
267C4×C22⋊C4—21no—
268C4×C22⋊C4—21no—
269C2×#95—712no—
270C2×#95—1421no—
271C25C22⋊D7111yescharacteristic
272C23×C4—24no—
273C23×C4—42no—
274C23×C4—42no—
275C23×C4—1422no—
276C23×C4C2×Dic7111yescharacteristic
277C23×C4C2×Dic7111yescharacteristic
278C23×C4C22⋊D7111yescharacteristic
279C23×C4C22⋊D7111yescharacteristic
280C23×C4C22⋊D7111yescharacteristic
281C23×C4C22×D7111yescharacteristic
282C22×C8—1422no—
283#117—42no—
284#117—42no—
285C2×C4⋊C4C2×Dic7111yescharacteristic
286C2×C4⋊C4C2×Dic7111yescharacteristic
287C2×C4⋊C4C2×Dic7111yescharacteristic
288C2×C4⋊C4C2×Dic7111yescharacteristic
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—2842no—
296C2×C22⋊C4—2842no—
297C2×C22⋊C4—1424no—
298C2×C42—21no—
299C2×C42—21no—
300C22⋊C8—2842no—
301C22⋊C8—2842no—
302C22⋊C8—1424no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—2842no—
308#95—1424no—
309C24—22no—
310C24—22no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—22no—
315C24—41no—
316C24—41no—
317C24C22⋊Dic7111yescharacteristic
318C24C22⋊Dic7111yescharacteristic
319C24C2×C22⋊D7111yescharacteristic
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—416no—
342C22×C4—2842no—
343C22×C4—2842no—
344C22×C4—2842no—
345C22×C4—2842no—
346C22×C4—2842no—
347C22×C4—2842no—
348C22×C4—1424no—
349C22×C4C22⋊Dic7111yescharacteristic
350C22×C4C22⋊Dic7111yescharacteristic
351C22×C4C22×Dic7111yescharacteristic
352C22×C4C2×C22⋊D7111yescharacteristic
353C22×C4#1176586514yes—
354C2×C8—2842no—
355C2×C8—2842no—
356C2×C8—2842no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—2848no—
369C42—42no—
370C42—42no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—24no—
376C23—24no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—44no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23#1455604312yes—
398C23#1455605512yes—
399C23C2×C22⋊Dic7111yescharacteristic
400C23#1455611112yes—
401C23#1455614812yes—
402C23#1455615311yescharacteristic
403C23#1455615811yescharacteristic
404C2×C4—28no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—2844no—
432C2×C4—2844no—
433C2×C4—2844no—
434C2×C4—5682no—
435C2×C4—5682no—
436C2×C4—5682no—
437C2×C4—5682no—
438C2×C4#1455604312yes—
439C2×C4#1455608712yes—
440C2×C4#1455611112yes—
441C8—5682no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—48no—
457C22—48no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22#1798586211yescharacteristic
467C22C2×#1455604311yescharacteristic
468C22C2×#1455604311yescharacteristic
469C22#1798599511yescharacteristic
470C22#1798600011yescharacteristic
471C22C2×#1455605511yescharacteristic
472C22#1798601611yescharacteristic
473C22#1798632312yes—
474C22#1798634012yes—
475C4—22no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—5684no—
484C2—22no—
485C2—22no—
486C2—42no—
487C2—42no—
488C2#2258683111yescharacteristic
489C2#2258684411yescharacteristic
490C2#2258688311yescharacteristic
491C2#2258689611yescharacteristic
492C2#2258841411yescharacteristic
493C2#2258843511yescharacteristic
494C2#2258843911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2280C2
socleC23×C71C2×C22⋊C4
prime radicalC25×C71D4
cosocleC23×C4×C71C23
hypercenter#2280D71
hypocenterC71#8715
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
3. #22586218C2semidirect
4. #22586218C2semidirect
13. C2×#117×C71C4semidirect
14. C2×#117×C71C4semidirect
16. #380×C71C4semidirect
17. #380×C71C4semidirect
253. C71#8715semidirect
255. C2×#117Dic71semidirect
256. C2×#117Dic71semidirect
262. #380Dic71semidirect
263. #380Dic71semidirect

Maximal subgroups

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

subgroupindex
#225882502
#225865632
#225862182
C71×#22802
#871571