Finite Group Databasebeta

← #31897527

Subgroups — #31897527

Order 18176.

566 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊(C71⋊C8)C211yesmaximal, characteristic
2#22586567C211yesmaximal, characteristic
3#22586218C212yesmaximal
4#22586218C212yesmaximal
5C71×#2851C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7C2×C4×C71⋊C8—21no—
8C2×C71⋊#95—21no—
9C2×C71⋊#95—21no—
10C2×C71⋊#95C2212yes—
11#17985754—21no—
12C2×#125×C71C411yescharacteristic, complemented
13C22×C42×C71C2211yescharacteristic
14#389×C71C411yescharacteristic, complemented
15#389×C71C411yescharacteristic, complemented
16#387×C71—21no—
17#387×C71—21no—
18#385×C71—21no—
19#385×C71—21no—
20#382×C71C411yescharacteristic, complemented
21#380×C71C2211yescharacteristic
22#380×C71C2211yescharacteristic
23C23×Dic71—22no—
24C22×C71⋊C8—22no—
25C22⋊(C71⋊C8)—22no—
26C22⋊(C71⋊C8)Q814yes—
27C4×C71⋊C8—42no—
28C71⋊#95—24no—
29C71⋊#95—24no—
30C71⋊#95—42no—
31C23×C4×C71D411yescharacteristic
32C23×C4×C71D411yescharacteristic
33C23×C4×C71C2311yescharacteristic
34#125×C71—42no—
35C2×C4⋊C4×C71—21no—
36C2×C4⋊C4×C71—21no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71C2×C411yescharacteristic
40C2×C4⋊C4×C71C2×C411yescharacteristic
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C22⋊C4×C71—21no—
44C2×C22⋊C4×C71—21no—
45C2×C42×C71—42no—
46C2×C42×C71C2×C411yescharacteristic
47C2×C42×C71C2×C411yescharacteristic
48C2×C42×C71D411yescharacteristic
49C2×C42×C71D411yescharacteristic
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56C22×Dic71—24no—
57C22×Dic71—42no—
58C22×Dic71—42no—
59C22×Dic71—42no—
60C22×Dic71—42no—
61C22×Dic71—42no—
62C22×Dic71—42no—
63C2×C71⋊C8—24no—
64C2×C71⋊C8—42no—
65C2×C71⋊C8—42no—
66C24×C71C2×D411yescharacteristic
67C22×C4×C71—22no—
68C22×C4×C71—22no—
69C22×C4×C71—22no—
70C22×C4×C71—22no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—21no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71—41no—
78C22×C4×C71—41no—
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22⋊C411yescharacteristic
82C22×C4×C71C22⋊C411yescharacteristic
83C22×C4×C71C22×C411yescharacteristic
84C22×C4×C71C2×D411yescharacteristic
85C22×C4×C71C2×Q812yes—
86C22×C4×C71#4112yes—
87C4⋊C4×C71—41no—
88C4⋊C4×C71—41no—
89C4⋊C4×C71—41no—
90C4⋊C4×C71—41no—
91C4⋊C4×C71—42no—
92C4⋊C4×C71—42no—
93C4⋊C4×C71—42no—
94C4⋊C4×C71—42no—
95C22⋊C4×C71—42no—
96C22⋊C4×C71—42no—
97C42×C71—22no—
98C42×C71—22no—
99C42×C71—41no—
100C42×C71—41no—
101C23×C71—21no—
102C23×C71—21no—
103C23×C71—21no—
104C23×C71—21no—
105C23×C71C2×C22⋊C411yescharacteristic
106C23×C71#12411yescharacteristic
107C23×C71C4⋊Q811yescharacteristic
108C23×C71#13512yes—
109C23×C71#13512yes—
110C2×C4×C71—21no—
111C2×C4×C71—21no—
112C2×C4×C71—21no—
113C2×C4×C71—21no—
114C2×C4×C71—21no—
115C2×C4×C71—21no—
116C2×C4×C71—21no—
117C2×C4×C71—21no—
118C2×C4×C71—21no—
119C2×C4×C71—21no—
120C2×C4×C71—21no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—42no—
125C2×C4×C71—42no—
126C2×C4×C71—44no—
127C2×C4×C71—44no—
128C2×C4×C71—41no—
129C2×C4×C71—41no—
130C2×C4×C71—41no—
131C2×C4×C71—41no—
132C2×C4×C71—41no—
133C2×C4×C71—41no—
134C2×C4×C71—41no—
135C2×C4×C71—41no—
136C2×C4×C71—41no—
137C2×C4×C71—41no—
138C2×C4×C71—41no—
139C2×C4×C71—41no—
140C2×C4×C71#9912yes—
141C2×C4×C71#10412yes—
142C2×C4×C71C4×Q812yes—
143C2×C4×C71C22⋊Q812yes—
144C2×Dic71—44no—
145C2×Dic71—44no—
146C2×Dic71—44no—
147C2×Dic71—82no—
148C2×Dic71—82no—
149C2×Dic71—82no—
150C2×Dic71—82no—
151C71⋊C8—44no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—21no—
156C22×C71—22no—
157C22×C71—22no—
158C22×C71—22no—
159C22×C71—22no—
160C22×C71#39111yescharacteristic
161C22×C71C2×#9911yescharacteristic
162C22×C71#41011yescharacteristic
163C22×C71#41611yescharacteristic
164C22×C71#41611yescharacteristic
165C22×C71C2×#10411yescharacteristic
166C22×C71#42111yescharacteristic
167C22×C71#48812yes—
168C22×C71#49912yes—
169C4×C71—22no—
170C4×C71—22no—
171C4×C71—22no—
172C4×C71—22no—
173C4×C71—41no—
174C4×C71—41no—
175C4×C71—81no—
176C4×C71—81no—
177C4×C71—81no—
178C4×C71—81no—
179Dic71—84no—
180#9886—711nomaximal, Sylow
181C2×C71—22no—
182C2×C71—22no—
183C2×C71#150011yescharacteristic
184C2×C71#150111yescharacteristic
185C2×C71#151611yescharacteristic
186C2×C71#152111yescharacteristic
187C2×C71#198611yescharacteristic
188C2×C71#199111yescharacteristic
189C2×C71#199611yescharacteristic
190#2851D7111yescharacteristic
191C4×C22⋊C8—711no—
192#1434—711no—
193#1294—712no—
194#1294—712no—
195C71#988611yescharacteristic, complemented, Sylow
196C2×#125Dic7111yescharacteristic, complemented
197C22×C42C2×D7111yescharacteristic
198C2×C22⋊C8—712no—
199C2×C4×C8—1421no—
200#389Dic7111yescharacteristic, complemented
201#389Dic7111yescharacteristic, complemented
202#387—21no—
203#387—21no—
204#385—21no—
205#385—21no—
206#382Dic7111yescharacteristic, complemented
207#380C2×D7111yescharacteristic
208#380C2×D7111yescharacteristic
209C2×#95—712no—
210C2×#95—1421no—
211C2×#95—1421no—
212#336—1421no—
213C23×C4—1422no—
214C23×C4C22⋊D7111yescharacteristic
215C23×C4C22⋊D7111yescharacteristic
216C23×C4C22×D7111yescharacteristic
217C22×C8—1422no—
218#125—42no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C2×Dic7111yescharacteristic
224C2×C4⋊C4C2×Dic7111yescharacteristic
225C2×C4⋊C4C2×Dic7111yescharacteristic
226C2×C4⋊C4C2×Dic7111yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C42—42no—
230C2×C42C2×Dic7111yescharacteristic
231C2×C42C2×Dic7111yescharacteristic
232C2×C42C22⋊D7111yescharacteristic
233C2×C42C22⋊D7111yescharacteristic
234C22⋊C8—714no—
235C22⋊C8—1422no—
236C4×C8—2842no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—2842no—
244#95—1424no—
245#95—1424no—
246C24C2×C22⋊D7111yescharacteristic
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—2842no—
260C22×C4—2842no—
261C22×C4—2842no—
262C22×C4—2842no—
263C22×C4—2842no—
264C22×C4—2842no—
265C22×C4—1424no—
266C22×C4C22⋊Dic7111yescharacteristic
267C22×C4C22⋊Dic7111yescharacteristic
268C22×C4C22⋊Dic7111yescharacteristic
269C22×C4C22⋊Dic7111yescharacteristic
270C22×C4C22×Dic7111yescharacteristic
271C22×C4C2×C22⋊D7111yescharacteristic
272C22×C4Q8×D7112yes—
273C22×C4#1176586612yes—
274C2×C8—2842no—
275C2×C8—2842no—
276C2×C8—1424no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C42—22no—
288C42—22no—
289C42—41no—
290C42—41no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23C2×C22⋊Dic7111yescharacteristic
296C23#1455611711yescharacteristic
297C23#1455612611yescharacteristic
298C23#1455614812yes—
299C23#1455614812yes—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—2844no—
331C2×C4—2844no—
332C2×C4—2844no—
333C2×C4—5682no—
334C2×C4—5682no—
335C2×C4—5682no—
336C2×C4—5682no—
337C2×C4#1455604312yes—
338C2×C4#1455605512yes—
339C2×C4Q8×Dic7112yes—
340C2×C4#1455610412yes—
341C8—2844no—
342C22—22no—
343C22—22no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—22no—
349C22—22no—
350C22#1798590611yescharacteristic
351C22C2×#1455604311yescharacteristic
352C22#1798597311yescharacteristic
353C22#1798599511yescharacteristic
354C22#1798599511yescharacteristic
355C22C2×#1455605511yescharacteristic
356C22#1798601611yescharacteristic
357C22#1798635612yes—
358C22#1798639812yes—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—5684no—
370C2—22no—
371C2—22no—
372C2#2258682611yescharacteristic
373C2#2258683111yescharacteristic
374C2#2258689611yescharacteristic
375C2#2258691711yescharacteristic
376C2#2258928511yescharacteristic
377C2#2258930611yescharacteristic
378C2#2258932611yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2851C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2851D71
hypocenterC71#9886
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C71C4semidirect
14. #389×C71C4semidirect
15. #389×C71C4semidirect
20. #382×C71C4semidirect
195. C71#9886semidirect
196. C2×#125Dic71semidirect
200. #389Dic71semidirect
201. #389Dic71semidirect
206. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊(C71⋊C8)2
#225865672
#225862182
C71×#28512
#988671