Finite Group Databasebeta

← #31890834

Subgroups — #31890834

Order 18176.

544 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22588408C211yesmaximal, characteristic
2#22586567C211yesmaximal, characteristic
3#22586218C212yesmaximal
4#22586218C212yesmaximal
5C71×#2283C211yesmaximal, characteristic
6C2×C22⋊(C71⋊C8)C2212yes—
7C2×C71⋊#95—21no—
8C2×C71⋊#95—21no—
9C2×C71⋊#95C2212yes—
10#17985754—21no—
11#17985754—21no—
12C22×C42×C71C2211yescharacteristic
13#383×C71—21no—
14#382×C71C411yescharacteristic, complemented
15#382×C71C411yescharacteristic, complemented
16#381×C71—21no—
17#380×C71C2211yescharacteristic
18#380×C71C2211yescharacteristic
19C4×C4⋊C4×C71C411yescharacteristic, complemented
20C4×C4⋊C4×C71C411yescharacteristic, complemented
21C4×C22⋊C4×C71—21no—
22#376×C71—21no—
23C23×Dic71—22no—
24C22×C71⋊C8—22no—
25C22⋊(C71⋊C8)—24no—
26C71⋊#95—24no—
27C71⋊#95—24no—
28C71⋊#95—42no—
29C23×C4×C71D411yescharacteristic
30C23×C4×C71D411yescharacteristic
31C23×C4×C71C2311yescharacteristic
32C2×C4⋊C4×C71C2×C411yescharacteristic
33C2×C4⋊C4×C71C2×C411yescharacteristic
34C2×C4⋊C4×C71C2×C411yescharacteristic
35C2×C4⋊C4×C71C2×C411yescharacteristic
36C2×C22⋊C4×C71—21no—
37C2×C22⋊C4×C71—21no—
38C2×C42×C71—24no—
39C2×C42×C71—21no—
40C2×C42×C71—21no—
41C2×C42×C71C2×C411yescharacteristic
42C2×C42×C71C2×C411yescharacteristic
43C2×C42×C71D411yescharacteristic
44C2×C42×C71D411yescharacteristic
45#95×C71—21no—
46#95×C71—21no—
47#95×C71—21no—
48#95×C71—21no—
49#95×C71—21no—
50#95×C71—21no—
51#95×C71—21no—
52#95×C71—21no—
53C22×Dic71—24no—
54C22×Dic71—42no—
55C22×Dic71—42no—
56C22×Dic71—42no—
57C22×Dic71—42no—
58C22×Dic71—42no—
59C22×Dic71—42no—
60C2×C71⋊C8—42no—
61C2×C71⋊C8—42no—
62C2×C71⋊C8—42no—
63C24×C71C2×D411yescharacteristic
64C22×C4×C71—22no—
65C22×C4×C71—22no—
66C22×C4×C71—22no—
67C22×C4×C71—21no—
68C22×C4×C71—21no—
69C22×C4×C71—22no—
70C22×C4×C71—21no—
71C22×C4×C71—21no—
72C22×C4×C71—21no—
73C22×C4×C71—21no—
74C22×C4×C71—41no—
75C22×C4×C71—41no—
76C22×C4×C71—41no—
77C22×C4×C71C22⋊C411yescharacteristic
78C22×C4×C71C22⋊C411yescharacteristic
79C22×C4×C71C22⋊C411yescharacteristic
80C22×C4×C71C22⋊C411yescharacteristic
81C22×C4×C71C22×C411yescharacteristic
82C22×C4×C71C2×D411yescharacteristic
83C22×C4×C71#4114yes—
84C4⋊C4×C71—41no—
85C4⋊C4×C71—41no—
86C4⋊C4×C71—41no—
87C4⋊C4×C71—41no—
88C22⋊C4×C71—42no—
89C22⋊C4×C71—41no—
90C22⋊C4×C71—41no—
91C42×C71—22no—
92C42×C71—22no—
93C42×C71—22no—
94C42×C71—22no—
95C42×C71—42no—
96C42×C71—42no—
97C23×C71—21no—
98C23×C71—21no—
99C23×C71—21no—
100C23×C71—21no—
101C23×C71C2×C22⋊C411yescharacteristic
102C23×C71#12312yes—
103C23×C71#13512yes—
104C23×C71AGL(1,ℤ/8)11yescharacteristic
105C23×C71#13711yescharacteristic
106C2×C4×C71—21no—
107C2×C4×C71—21no—
108C2×C4×C71—21no—
109C2×C4×C71—21no—
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—44no—
122C2×C4×C71—44no—
123C2×C4×C71—42no—
124C2×C4×C71—42no—
125C2×C4×C71—41no—
126C2×C4×C71—41no—
127C2×C4×C71—41no—
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#10412yes—
138C2×C4×C71#11712yes—
139C2×C4×C71#12312yes—
140C2×Dic71—44no—
141C2×Dic71—44no—
142C2×Dic71—44no—
143C2×Dic71—82no—
144C2×Dic71—82no—
145C2×Dic71—82no—
146C2×Dic71—82no—
147C71⋊C8—82no—
148C22×C71—22no—
149C22×C71—22no—
150C22×C71—22no—
151C22×C71—21no—
152C22×C71—21no—
153C22×C71—21no—
154C22×C71—21no—
155C22×C71—22no—
156C22×C71#38111yescharacteristic
157C22×C71#41011yescharacteristic
158C22×C71#41011yescharacteristic
159C22×C71#41611yescharacteristic
160C22×C71#41711yescharacteristic
161C22×C71C2×#10411yescharacteristic
162C22×C71#42111yescharacteristic
163C22×C71#48212yes—
164C22×C71#48512yes—
165C4×C71—22no—
166C4×C71—22no—
167C4×C71—22no—
168C4×C71—41no—
169C4×C71—41no—
170C4×C71—41no—
171C4×C71—81no—
172C4×C71—81no—
173C4×C71—81no—
174C4×C71—81no—
175Dic71—84no—
176#8714—711nomaximal, Sylow
177C2×C71—22no—
178C2×C71—22no—
179C2×C71#150011yescharacteristic
180C2×C71#150911yescharacteristic
181C2×C71#151211yescharacteristic
182C2×C71#152111yescharacteristic
183C2×C71#182011yescharacteristic
184C2×C71#182511yescharacteristic
185C2×C71#182611yescharacteristic
186#2283D7111yescharacteristic
187#1817—711no—
188#1434—711no—
189#1294—712no—
190#1294—712no—
191C71#871411yescharacteristic, complemented, Sylow
192C22×C42C2×D7111yescharacteristic
193C2×C22⋊C8—712no—
194#383—21no—
195#382Dic7111yescharacteristic, complemented
196#382Dic7111yescharacteristic, complemented
197#381—21no—
198#380C2×D7111yescharacteristic
199#380C2×D7111yescharacteristic
200C4×C4⋊C4Dic7111yescharacteristic, complemented
201C4×C4⋊C4Dic7111yescharacteristic, complemented
202C4×C22⋊C4—21no—
203#376—21no—
204C2×#95—712no—
205C2×#95—1421no—
206C2×#95—1421no—
207#336—1421no—
208#336—1421no—
209C23×C4—1422no—
210C23×C4C22⋊D7111yescharacteristic
211C23×C4C22⋊D7111yescharacteristic
212C23×C4C22×D7111yescharacteristic
213C22×C8—1422no—
214C2×C4⋊C4C2×Dic7111yescharacteristic
215C2×C4⋊C4C2×Dic7111yescharacteristic
216C2×C4⋊C4C2×Dic7111yescharacteristic
217C2×C4⋊C4C2×Dic7111yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—24no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C2×Dic7111yescharacteristic
224C2×C42C2×Dic7111yescharacteristic
225C2×C42C22⋊D7111yescharacteristic
226C2×C42C22⋊D7111yescharacteristic
227C22⋊C8—1424no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—2842no—
237#95—1424no—
238#95—1424no—
239C24C2×C22⋊D7111yescharacteristic
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—2842no—
254C22×C4—2842no—
255C22×C4—2842no—
256C22×C4—2842no—
257C22×C4—2842no—
258C22×C4—2842no—
259C22×C4—1424no—
260C22×C4C22⋊Dic7111yescharacteristic
261C22×C4C22⋊Dic7111yescharacteristic
262C22×C4C22⋊Dic7111yescharacteristic
263C22×C4C22⋊Dic7111yescharacteristic
264C22×C4C22×Dic7111yescharacteristic
265C22×C4C2×C22⋊D7111yescharacteristic
266C22×C4#1176586514yes—
267C2×C8—2842no—
268C2×C8—2842no—
269C2×C8—2842no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—42no—
282C42—42no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23C2×C22⋊Dic7111yescharacteristic
288C23#1455611112yes—
289C23#1455614812yes—
290C23#1455615311yescharacteristic
291C23#1455615811yescharacteristic
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—2844no—
324C2×C4—2844no—
325C2×C4—2844no—
326C2×C4—5682no—
327C2×C4—5682no—
328C2×C4—5682no—
329C2×C4—5682no—
330C2×C4#1455605512yes—
331C2×C4#1455608712yes—
332C2×C4#1455611112yes—
333C8—5682no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22#1798586211yescharacteristic
343C22#1798597311yescharacteristic
344C22#1798597311yescharacteristic
345C22#1798599511yescharacteristic
346C22#1798600011yescharacteristic
347C22C2×#1455605511yescharacteristic
348C22#1798601611yescharacteristic
349C22#1798632312yes—
350C22#1798634012yes—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—5684no—
362C2—22no—
363C2—22no—
364C2#2258682611yescharacteristic
365C2#2258686511yescharacteristic
366C2#2258687811yescharacteristic
367C2#2258691711yescharacteristic
368C2#2258841711yescharacteristic
369C2#2258843511yescharacteristic
370C2#2258843911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic71
commutatorC22×C4×C71C22×C4
FrattiniC23×C4C22×D71
FittingC71×#2283C2
socleC23×C71C2×C22⋊C4
prime radicalC24×C71C2×D4
cosocleC23×C4×C71C23
hypercenter#2283D71
hypocenterC71#8714
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
14. #382×C71C4semidirect
15. #382×C71C4semidirect
19. C4×C4⋊C4×C71C4semidirect
20. C4×C4⋊C4×C71C4semidirect
191. C71#8714semidirect
195. #382Dic71semidirect
196. #382Dic71semidirect
200. C4×C4⋊C4Dic71semidirect
201. C4×C4⋊C4Dic71semidirect

Maximal subgroups

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

subgroupindex
#225884082
#225865672
#225862182
C71×#22832
#871471