Finite Group Databasebeta

← #31913420

Subgroups — #31913420

Order 18176.

540 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17985789C212yesmaximal
2#22588332C211yesmaximal, characteristic
3#22588328C211yesmaximal, characteristic
4#22586458C212yesmaximal
5C4×#123×C71C211yesmaximal, characteristic
6#17986023—21no—
7C2×#14556045C2212yes—
8C2×#14556043C2212yes—
9#17985859—21no—
10#17985789—24no—
11#17985768—21no—
12#17985766—21no—
13C2×#123×C71C2211yescharacteristic
14C2×C4×D4×C71C2211yescharacteristic
15C22×C42×C71C411yescharacteristic, complemented
16#388×C71—21no—
17#387×C71—21no—
18#386×C71C2211yescharacteristic
19#385×C71C411yescharacteristic, complemented
20#381×C71C411yescharacteristic, complemented
21C4×C4⋊C4×C71—21no—
22C4×C22⋊C4×C71—21no—
23C4×C22⋊C4×C71C411yescharacteristic, complemented
24C2×#11765849—22no—
25C2×C4⋊Dic71—41no—
26C2×C22⋊Dic71—22no—
27C4⋊(C71⋊C8)—41no—
28C4⋊(C71⋊C8)—41no—
29#14556045—22no—
30#14556045—22no—
31#14556043—22no—
32#14556043—22no—
33C71⋊#95—41no—
34C22×D4×C71C2311yescharacteristic
35C23×C4×C71C2×C411yescharacteristic
36C23×C4×C71C2×C411yescharacteristic
37C23×C4×C71C2×C411yescharacteristic
38C23×C4×C71D411yescharacteristic
39#123×C71—21no—
40#123×C71—21no—
41#123×C71D412yes—
42#123×C71Q812yes—
43C4×D4×C71—41no—
44C4×D4×C71—41no—
45C2×C4⋊C4×C71—21no—
46C2×C4⋊C4×C71—21no—
47C2×C4⋊C4×C71D411yescharacteristic
48C2×C22⋊C4×C71—21no—
49C2×C22⋊C4×C71—21no—
50C2×C22⋊C4×C71C2×C411yescharacteristic
51C2×C22⋊C4×C71C2×C411yescharacteristic
52C2×C22⋊C4×C71C2×C411yescharacteristic
53C2×C22⋊C4×C71D411yescharacteristic
54C2×C42×C71—21no—
55C2×C42×C71—21no—
56C2×C42×C71—21no—
57C2×C42×C71—41no—
58C2×C42×C71—41no—
59C2×C42×C71D411yescharacteristic
60#95×C71—21no—
61#95×C71—21no—
62#95×C71—21no—
63#95×C71—21no—
64#95×C71—21no—
65C22×Dic71—42no—
66#11765849—42no—
67#11765849—42no—
68C2×C71⋊C8—42no—
69C4⋊Dic71—82no—
70C22⋊Dic71—42no—
71C22⋊Dic71—42no—
72C24×C71C22×C411yescharacteristic
73C24×C71C2×D411yescharacteristic
74C2×D4×C71—41no—
75C2×D4×C71—41no—
76C2×D4×C71C2×D411yescharacteristic
77C2×D4×C71C2×Q811yescharacteristic
78C2×D4×C71#4111yescharacteristic
79C2×D4×C71#4111yescharacteristic
80C22×C4×C71—21no—
81C22×C4×C71—21no—
82C22×C4×C71—21no—
83C22×C4×C71—21no—
84C22×C4×C71—21no—
85C22×C4×C71—21no—
86C22×C4×C71—21no—
87C22×C4×C71—21no—
88C22×C4×C71—21no—
89C22×C4×C71—42no—
90C22×C4×C71—42no—
91C22×C4×C71—42no—
92C22×C4×C71—41no—
93C22×C4×C71—41no—
94C22×C4×C71—41no—
95C22×C4×C71C22⋊C411yescharacteristic
96C22×C4×C71C22⋊C411yescharacteristic
97C22×C4×C71C22⋊C411yescharacteristic
98C22×C4×C71C22⋊C411yescharacteristic
99C22×C4×C71C4⋊C412yescomplemented
100C22×C4×C71C2×D411yescharacteristic
101C4⋊C4×C71—22no—
102C4⋊C4×C71—22no—
103C4⋊C4×C71—41no—
104C4⋊C4×C71—42no—
105C22⋊C4×C71—21no—
106C22⋊C4×C71—22no—
107C22⋊C4×C71—22no—
108C22⋊C4×C71—41no—
109C22⋊C4×C71—41no—
110C22⋊C4×C71—42no—
111C22⋊C4×C71—41no—
112C22⋊C4×C71C4⋊C412yescomplemented
113C42×C71—21no—
114C42×C71—21no—
115C42×C71—42no—
116C42×C71—42no—
117C42×C71—41no—
118C42×C71—41no—
119C23×C71—21no—
120C23×C71—21no—
121C23×C71—21no—
122C23×C71—21no—
123C23×C71—41no—
124C23×C71—41no—
125C23×C71—41no—
126C23×C71—41no—
127C23×C71C2×C22⋊C411yescharacteristic
128C23×C71C2×C4⋊C411yescharacteristic
129C23×C71#11711yescharacteristic
130C23×C71#12111yescharacteristic
131C23×C71C22⋊Q811yescharacteristic
132D4×C71—41no—
133D4×C71—41no—
134D4×C71—41no—
135D4×C71—41no—
136C2×C4×C71—21no—
137C2×C4×C71—21no—
138C2×C4×C71—22no—
139C2×C4×C71—22no—
140C2×C4×C71—22no—
141C2×C4×C71—22no—
142C2×C4×C71—21no—
143C2×C4×C71—21no—
144C2×C4×C71—21no—
145C2×C4×C71—42no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—42no—
149C2×C4×C71—41no—
150C2×C4×C71—42no—
151C2×C4×C71—41no—
152C2×C4×C71—41no—
153C2×C4×C71—41no—
154C2×C4×C71—42no—
155C2×C4×C71—42no—
156C2×C4×C71—41no—
157C2×C4×C71—41no—
158C2×C4×C71—41no—
159C2×C4×C71—42no—
160C2×C4×C71—42no—
161C2×C4×C71—41no—
162C2×C4×C71—41no—
163C2×C4×C71—41no—
164C2×C4×C71—81no—
165C2×C4×C71#9911yescharacteristic
166C2×C4×C71#9911yescharacteristic
167C2×C4×C71#12111yescharacteristic
168C2×C4×C71C22⋊Q811yescharacteristic
169C2×Dic71—82no—
170C2×Dic71—82no—
171C2×Dic71—82no—
172C71⋊C8—82no—
173C22×C71—21no—
174C22×C71—21no—
175C22×C71—21no—
176C22×C71—21no—
177C22×C71—21no—
178C22×C71—21no—
179C22×C71—41no—
180C22×C71—41no—
181C22×C71—41no—
182C22×C71—41no—
183C22×C71—42no—
184C22×C71—41no—
185C22×C71—41no—
186C22×C71—41no—
187C22×C71—41no—
188C22×C71—81no—
189C22×C71#38011yescharacteristic
190C22×C71C2×#9911yescharacteristic
191C22×C71#41011yescharacteristic
192C4×C71—21no—
193C4×C71—21no—
194C4×C71—21no—
195C4×C71—42no—
196C4×C71—42no—
197C4×C71—41no—
198C4×C71—41no—
199C4×C71—41no—
200C4×C71—42no—
201C4×C71—81no—
202C4×C71#35212yescomplemented
203Dic71—162no—
204#12794—711nomaximal, Sylow
205C2×C71—21no—
206C2×C71—21no—
207C2×C71—41no—
208C2×C71—41no—
209C2×C71—41no—
210C2×C71—41no—
211C2×C71#179911yescharacteristic
212C2×C71C2×#35211yescharacteristic
213C2×C71#212011yescharacteristic
214C4×#123D7111yescharacteristic
215C2×#352—712no—
216#1800—711no—
217#1799—711no—
218#1390—712no—
219C71#1279411yescharacteristic, complemented, Sylow
220C2×#123C2×D7111yescharacteristic
221C2×C4×D4C2×D7111yescharacteristic
222C22×C42Dic7111yescharacteristic, complemented
223#423—1421no—
224C2×#100—712no—
225C2×#99—712no—
226#388—21no—
227#387—21no—
228#386C2×D7111yescharacteristic
229#385Dic7111yescharacteristic, complemented
230#381Dic7111yescharacteristic, complemented
231#380—1421no—
232C4×C4⋊C4—21no—
233C4×C22⋊C4—21no—
234C4×C22⋊C4Dic7111yescharacteristic, complemented
235#352—1424no—
236#343—1421no—
237#342—1421no—
238C22×D4C22×D7111yescharacteristic
239C23×C4C2×Dic7111yescharacteristic
240C23×C4C2×Dic7111yescharacteristic
241C23×C4C2×Dic7111yescharacteristic
242C23×C4C22⋊D7111yescharacteristic
243C2×M16—1422no—
244#123—21no—
245#123—21no—
246#123Dic14212yes—
247#123D28412yes—
248C4×D4—41no—
249C4×D4—41no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—2841no—
253C2×C4⋊C4C22⋊D7111yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—1422no—
257C2×C22⋊C4C2×Dic7111yescharacteristic
258C2×C22⋊C4C2×Dic7111yescharacteristic
259C2×C22⋊C4C2×Dic7111yescharacteristic
260C2×C22⋊C4C22⋊D7111yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—41no—
265C2×C42—41no—
266C2×C42C22⋊D7111yescharacteristic
267C4⋊C8—2841no—
268C4⋊C8—2841no—
269#100—1422no—
270#100—1422no—
271#99—1422no—
272#99—1422no—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—2841no—
279C24C22×Dic7111yescharacteristic
280C24C2×C22⋊D7111yescharacteristic
281C2×D4—41no—
282C2×D4—41no—
283C2×D4C2×D28411yescharacteristic
284C2×D4C2×Dic14211yescharacteristic
285C2×D4#1176586511yescharacteristic
286C2×D4#1176586511yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—2842no—
303C22×C4C22⋊Dic7111yescharacteristic
304C22×C4C22⋊Dic7111yescharacteristic
305C22×C4C22⋊Dic7111yescharacteristic
306C22×C4C22⋊Dic7111yescharacteristic
307C22×C4C4⋊Dic7112yescomplemented
308C22×C4C2×C22⋊D7111yescharacteristic
309M16—2842no—
310M16—2842no—
311C2×C8—2842no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—5682no—
317C22⋊C4—21no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—42no—
323C22⋊C4—41no—
324C22⋊C4—2842no—
325C22⋊C4—2842no—
326C22⋊C4C4⋊Dic7112yescomplemented
327C42—21no—
328C42—21no—
329C42—42no—
330C42—42no—
331C42—41no—
332C42—41no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23C2×C22⋊Dic7111yescharacteristic
342C23C2×C4⋊Dic7111yescharacteristic
343C23#1455608711yescharacteristic
344C23#1455610211yescharacteristic
345C23#1455610711yescharacteristic
346D4—41no—
347D4—41no—
348D4—41no—
349D4—41no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—5682no—
380C2×C4—5682no—
381C2×C4—5682no—
382C2×C4#1455604311yescharacteristic
383C2×C4#1455604311yescharacteristic
384C2×C4#1455610211yescharacteristic
385C2×C4#1455610711yescharacteristic
386C8—5682no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—81no—
403C22#1798585911yescharacteristic
404C22C2×#1455604311yescharacteristic
405C22#1798597311yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—42no—
410C4—41no—
411C4—41no—
412C4—42no—
413C4—41no—
414C4—42no—
415C4—81no—
416C4—11362no—
417C4#1798578912yescomplemented
418C2—21no—
419C2—21no—
420C2—41no—
421C2—41no—
422C2—41no—
423C2—41no—
424C2#2258832811yescharacteristic
425C2C2×#1798578911yescharacteristic
426C2#2258998111yescharacteristic
subgroupG/H
centerC22C2×#14556043
commutatorC24×C71C22×C4
FrattiniC22×D4C22×D71
FittingC4×#123×C71C2
socleC22×C71C2×#99
prime radical, cosocleC22×D4×C71C23
hypercenterC4×#123D71
hypocenterC71#12794
LCSG ▷ C24×C71 ▷ C22×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#123

Complements

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

NH ≅ G/Ntype
15. C22×C42×C71C4semidirect
19. #385×C71C4semidirect
20. #381×C71C4semidirect
23. C4×C22⋊C4×C71C4semidirect
99. C22×C4×C71C4⋊C4semidirect
112. C22⋊C4×C71C4⋊C4semidirect
202. C4×C71#352semidirect
219. C71#12794semidirect
222. C22×C42Dic71semidirect
229. #385Dic71semidirect
230. #381Dic71semidirect
234. C4×C22⋊C4Dic71semidirect
307. C22×C4C4⋊Dic71semidirect
326. C22⋊C4C4⋊Dic71semidirect
417. C4#17985789semidirect

Maximal subgroups

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

subgroupindex
C2×#179857892
#225883322
#225883282
#225864582
C4×#123×C712
#1279471