Finite Group Databasebeta

← #31897532

Subgroups — #31897532

Order 18176.

598 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588102C211yesmaximal, characteristic
2#22586570C211yesmaximal, characteristic
3#22586218C212yesmaximal
4#22586218C212yesmaximal
5C71×#2851C211yesmaximal, characteristic
6C22×C4⋊Dic71—21no—
7C2×C22⋊(C71⋊C8)C2212yes—
8C2×#14556039—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—
25C2×C4⋊Dic71—24no—
26C2×C4⋊Dic71—42no—
27C2×C4⋊Dic71—42no—
28C22⋊(C71⋊C8)—22no—
29C22⋊(C71⋊C8)Q814yes—
30#14556039—42no—
31C71⋊#95—24no—
32C71⋊#95—42no—
33C23×C4×C71D411yescharacteristic
34C23×C4×C71D411yescharacteristic
35C23×C4×C71C2311yescharacteristic
36#125×C71—42no—
37C2×C4⋊C4×C71—21no—
38C2×C4⋊C4×C71—21no—
39C2×C4⋊C4×C71—21no—
40C2×C4⋊C4×C71—21no—
41C2×C4⋊C4×C71C2×C411yescharacteristic
42C2×C4⋊C4×C71C2×C411yescharacteristic
43C2×C4⋊C4×C71C2×C411yescharacteristic
44C2×C4⋊C4×C71C2×C411yescharacteristic
45C2×C22⋊C4×C71—21no—
46C2×C22⋊C4×C71—21no—
47C2×C42×C71—42no—
48C2×C42×C71C2×C411yescharacteristic
49C2×C42×C71C2×C411yescharacteristic
50C2×C42×C71D411yescharacteristic
51C2×C42×C71D411yescharacteristic
52#95×C71—21no—
53#95×C71—21no—
54#95×C71—21no—
55#95×C71—21no—
56#95×C71—21no—
57#95×C71—21no—
58C22×Dic71—24no—
59C22×Dic71—42no—
60C22×Dic71—42no—
61C22×Dic71—42no—
62C22×Dic71—42no—
63C22×Dic71—42no—
64C22×Dic71—42no—
65C2×C71⋊C8—24no—
66C2×C71⋊C8—42no—
67C2×C71⋊C8—42no—
68C4⋊Dic71—48no—
69C24×C71C2×D411yescharacteristic
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—22no—
73C22×C4×C71—22no—
74C22×C4×C71—21no—
75C22×C4×C71—21no—
76C22×C4×C71—21no—
77C22×C4×C71—21no—
78C22×C4×C71—21no—
79C22×C4×C71—21no—
80C22×C4×C71—41no—
81C22×C4×C71—41no—
82C22×C4×C71—41no—
83C22×C4×C71—41no—
84C22×C4×C71C22⋊C411yescharacteristic
85C22×C4×C71C22⋊C411yescharacteristic
86C22×C4×C71C22⋊C411yescharacteristic
87C22×C4×C71C22⋊C411yescharacteristic
88C22×C4×C71C22×C411yescharacteristic
89C22×C4×C71C2×D411yescharacteristic
90C22×C4×C71C2×Q812yes—
91C22×C4×C71#4112yes—
92C4⋊C4×C71—41no—
93C4⋊C4×C71—41no—
94C4⋊C4×C71—41no—
95C4⋊C4×C71—41no—
96C4⋊C4×C71—42no—
97C4⋊C4×C71—42no—
98C4⋊C4×C71—42no—
99C4⋊C4×C71—42no—
100C22⋊C4×C71—41no—
101C22⋊C4×C71—41no—
102C22⋊C4×C71—41no—
103C22⋊C4×C71—41no—
104C42×C71—22no—
105C42×C71—22no—
106C42×C71—41no—
107C42×C71—41no—
108C23×C71—21no—
109C23×C71—21no—
110C23×C71—21no—
111C23×C71—21no—
112C23×C71C2×C22⋊C411yescharacteristic
113C23×C71#12411yescharacteristic
114C23×C71C4⋊Q811yescharacteristic
115C23×C71AGL(1,ℤ/8)11yescharacteristic
116C23×C71AGL(1,ℤ/8)11yescharacteristic
117C23×C71#13711yescharacteristic
118C23×C71#13711yescharacteristic
119C2×C4×C71—24no—
120C2×C4×C71—24no—
121C2×C4×C71—21no—
122C2×C4×C71—21no—
123C2×C4×C71—21no—
124C2×C4×C71—21no—
125C2×C4×C71—21no—
126C2×C4×C71—21no—
127C2×C4×C71—21no—
128C2×C4×C71—21no—
129C2×C4×C71—21no—
130C2×C4×C71—21no—
131C2×C4×C71—21no—
132C2×C4×C71—21no—
133C2×C4×C71—21no—
134C2×C4×C71—21no—
135C2×C4×C71—42no—
136C2×C4×C71—42no—
137C2×C4×C71—42no—
138C2×C4×C71—42no—
139C2×C4×C71—41no—
140C2×C4×C71—41no—
141C2×C4×C71—41no—
142C2×C4×C71—41no—
143C2×C4×C71—41no—
144C2×C4×C71—41no—
145C2×C4×C71—41no—
146C2×C4×C71—41no—
147C2×C4×C71—41no—
148C2×C4×C71—41no—
149C2×C4×C71—41no—
150C2×C4×C71—41no—
151C2×C4×C71#9912yes—
152C2×C4×C71#10412yes—
153C2×C4×C71C4×Q812yes—
154C2×C4×C71C22⋊Q812yes—
155C2×Dic71—44no—
156C2×Dic71—44no—
157C2×Dic71—44no—
158C2×Dic71—82no—
159C2×Dic71—82no—
160C2×Dic71—82no—
161C2×Dic71—82no—
162C71⋊C8—44no—
163C22×C71—21no—
164C22×C71—21no—
165C22×C71—21no—
166C22×C71—21no—
167C22×C71—21no—
168C22×C71—21no—
169C22×C71—21no—
170C22×C71—21no—
171C22×C71—21no—
172C22×C71—21no—
173C22×C71—21no—
174C22×C71—21no—
175C22×C71#39111yescharacteristic
176C22×C71C2×#9911yescharacteristic
177C22×C71#41011yescharacteristic
178C22×C71#41711yescharacteristic
179C22×C71#41711yescharacteristic
180C22×C71C2×#10411yescharacteristic
181C22×C71#42111yescharacteristic
182C22×C71#49011yescharacteristic
183C22×C71#49111yescharacteristic
184C22×C71#50111yescharacteristic
185C22×C71#50111yescharacteristic
186C4×C71—22no—
187C4×C71—22no—
188C4×C71—22no—
189C4×C71—22no—
190C4×C71—41no—
191C4×C71—41no—
192C4×C71—81no—
193C4×C71—81no—
194C4×C71—81no—
195C4×C71—81no—
196Dic71—84no—
197#9887—711nomaximal, Sylow
198C2×C71—21no—
199C2×C71—21no—
200C2×C71—21no—
201C2×C71—21no—
202C2×C71#150411yescharacteristic
203C2×C71#150911yescharacteristic
204C2×C71#151211yescharacteristic
205C2×C71#151311yescharacteristic
206C2×C71#198611yescharacteristic
207C2×C71#199211yescharacteristic
208C2×C71#199611yescharacteristic
209#2851D7111yescharacteristic
210#1749—711no—
211C4⋊#95—711no—
212#1294—712no—
213#1294—712no—
214C71#988711yescharacteristic, complemented, Sylow
215C2×#125Dic7111yescharacteristic, complemented
216C22×C4⋊C4—1421no—
217C22×C42C2×D7111yescharacteristic
218C2×C22⋊C8—712no—
219C2×#97—1421no—
220#389Dic7111yescharacteristic, complemented
221#389Dic7111yescharacteristic, complemented
222#387—21no—
223#387—21no—
224#385—21no—
225#385—21no—
226#382Dic7111yescharacteristic, complemented
227#380C2×D7111yescharacteristic
228#380C2×D7111yescharacteristic
229C2×#95—712no—
230C2×#95—1421no—
231#336—1421no—
232C23×C4—1422no—
233C23×C4C22⋊D7111yescharacteristic
234C23×C4C22⋊D7111yescharacteristic
235C23×C4C22×D7111yescharacteristic
236C22×C8—1422no—
237#125—42no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—2842no—
243C2×C4⋊C4—2842no—
244C2×C4⋊C4—1424no—
245C2×C4⋊C4C2×Dic7111yescharacteristic
246C2×C4⋊C4C2×Dic7111yescharacteristic
247C2×C4⋊C4C2×Dic7111yescharacteristic
248C2×C4⋊C4C2×Dic7111yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—42no—
252C2×C42C2×Dic7111yescharacteristic
253C2×C42C2×Dic7111yescharacteristic
254C2×C42C22⋊D7111yescharacteristic
255C2×C42C22⋊D7111yescharacteristic
256C22⋊C8—714no—
257C22⋊C8—1422no—
258#97—2842no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—2842no—
266#95—1424no—
267C24C2×C22⋊D7111yescharacteristic
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—2842no—
283C22×C4—2842no—
284C22×C4—2842no—
285C22×C4—2842no—
286C22×C4—2842no—
287C22×C4—2842no—
288C22×C4—1424no—
289C22×C4C22⋊Dic7111yescharacteristic
290C22×C4C22⋊Dic7111yescharacteristic
291C22×C4C22⋊Dic7111yescharacteristic
292C22×C4C22⋊Dic7111yescharacteristic
293C22×C4C22×Dic7111yescharacteristic
294C22×C4C2×C22⋊D7111yescharacteristic
295C22×C4Q8×D7112yes—
296C22×C4#1176586612yes—
297C2×C8—2842no—
298C2×C8—2842no—
299C2×C8—1424no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—2848no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C42—22no—
314C42—22no—
315C42—41no—
316C42—41no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C2×C22⋊Dic7111yescharacteristic
322C23#1455611711yescharacteristic
323C23#1455612611yescharacteristic
324C23#1455615311yescharacteristic
325C23#1455615311yescharacteristic
326C23#1455615811yescharacteristic
327C23#1455615811yescharacteristic
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—2844no—
361C2×C4—2844no—
362C2×C4—2844no—
363C2×C4—5682no—
364C2×C4—5682no—
365C2×C4—5682no—
366C2×C4—5682no—
367C2×C4#1455604312yes—
368C2×C4#1455605512yes—
369C2×C4Q8×Dic7112yes—
370C2×C4#1455610412yes—
371C8—2844no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22#1798590611yescharacteristic
385C22C2×#1455604311yescharacteristic
386C22#1798597311yescharacteristic
387C22#1798600011yescharacteristic
388C22#1798600011yescharacteristic
389C22C2×#1455605511yescharacteristic
390C22#1798601611yescharacteristic
391C22#1798636511yescharacteristic
392C22#1798636911yescharacteristic
393C22#1798640911yescharacteristic
394C22#1798640911yescharacteristic
395C4—22no—
396C4—22no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—5684no—
406C2—21no—
407C2—21no—
408C2—21no—
409C2—21no—
410C2#2258684411yescharacteristic
411C2#2258686511yescharacteristic
412C2#2258687811yescharacteristic
413C2#2258688311yescharacteristic
414C2#2258928511yescharacteristic
415C2#2258931011yescharacteristic
416C2#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#9887
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
214. C71#9887semidirect
215. C2×#125Dic71semidirect
220. #389Dic71semidirect
221. #389Dic71semidirect
226. #382Dic71semidirect

Maximal subgroups

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

subgroupindex
#225881022
#225865702
#225862182
C71×#28512
#988771