Finite Group Databasebeta

← #11734245

Subgroups — #11734245

Order 896.

476 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#393C211yesmaximal, characteristic, complemented
2#90003C211yesmaximal, characteristic, complemented
3#90002C211yesmaximal, characteristic, complemented
4#89733C211yesmaximal, characteristic
5#89730C211yesmaximal, characteristic
6#89728C211yesmaximal, characteristic
7#89717C211yesmaximal, characteristic, complemented
8#89715C211yesmaximal, characteristic, complemented
9C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
10#89441C211yesmaximal, characteristic, complemented
11#89438C211yesmaximal, characteristic, complemented
12#89436C211yesmaximal, characteristic, complemented
13#89432C211yesmaximal, characteristic, complemented
14#89423C211yesmaximal, characteristic, complemented
15#89423C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×C4×Dic7C2211yescharacteristic
33C2×C4×Dic7C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35C7×#95C2211yescharacteristic
36C7×#95C2211yescharacteristic
37C7×#95C2211yescharacteristic
38C7⋊#95C2211yescharacteristic
39C7⋊#95C2211yescharacteristic
40C7⋊#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51#2572—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C23×D7C2311yescharacteristic
54C2×C7×Q8—22no—
55C2×C7×Q8—41no—
56C2×C7×Q8—41no—
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—22no—
64C2×C22⋊D7—22no—
65C2×C22⋊D7—41no—
66C22×Dic7—22no—
67C22×Dic7—22no—
68C22×Dic7—41no—
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C2×C4×D7—22no—
77C2×C4×D7—22no—
78C2×C4×D7—41no—
79C7×C22⋊C4—22no—
80C7×C22⋊C4—41no—
81C7×C22⋊C4—41no—
82C22⋊Dic7—22no—
83C22⋊Dic7—22no—
84(C2×D7)⋊C4—22no—
85(C2×D7)⋊C4—22no—
86(C2×D7)⋊C4—22no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89C4⋊Dic7—41no—
90Dic7⋊C4—41no—
91C4×Dic7—22no—
92C4×Dic7—22no—
93C4×Dic7—41no—
94#400—71no—
95#394—71no—
96#393—71no—
97#393D711yescharacteristic, complemented
98#391—71no—
99#388—71no—
100#388—71no—
101#386—71no—
102#386—71no—
103#383—71no—
104#383—71no—
105#381—71no—
106#381—71no—
107C4×C22⋊C4—71no—
108C4×C22⋊C4—71no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7#4112yes—
116C23×C7C2411yescharacteristic
117C22×D7—21no—
118C22×D7—21no—
119C22×D7—21no—
120C22×D7—21no—
121C22×D7—21no—
122C22×D7—21no—
123C22×D7#4112yes—
124C7×Q8—44no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7#4112yes—
143C22⋊D7—44no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—44no—
164C2×Dic7#4112yes—
165C2×Dic7#4112yes—
166C4×D7—44no—
167C22×Q8C2×D711yescharacteristic
168C22×D4—71no—
169C23×C4—71no—
170C23×C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
173C2×C22⋊C4—71no—
174C2×C22⋊C4—71no—
175C2×C22⋊C4—71no—
176C2×C22⋊C4—71no—
177C2×C22⋊C4—71no—
178C2×C22⋊C4—71no—
179C2×C22⋊C4—71no—
180C2×C22⋊C4C2×D711yescharacteristic
181C2×C22⋊C4C2×D711yescharacteristic
182C2×C22⋊C4C2×D711yescharacteristic
183C2×C42—71no—
184C2×C42—71no—
185C2×C42—71no—
186#95—71no—
187#95—71no—
188#95—71no—
189#95—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95C2×D711yescharacteristic
200#95C2×D711yescharacteristic
201#95C2×D711yescharacteristic
202C22×C7—22no—
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7C2×#4111yescharacteristic
210C22×C7C2×#4111yescharacteristic
211C22×C7C2×#4111yescharacteristic
212C22×C7C2×#4111yescharacteristic
213C22×C7C2×#4111yescharacteristic
214C22×C7H(5,𝔽2)11yescharacteristic
215C22×C7H(5,𝔽2)11yescharacteristic
216C2×D7—22no—
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C4×C7—22no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229Dic7—22no—
230Dic7—22no—
231Dic7—41no—
232Dic7—41no—
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236C24—71no—
237C24C22×D711yescharacteristic
238C2×Q8—22no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×D4—142no—
242C2×D4—142no—
243C2×D4—281no—
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—71no—
249C22×C4—71no—
250C22×C4—71no—
251C22×C4—142no—
252C22×C4—142no—
253C22×C4—142no—
254C22×C4—142no—
255C22×C4—281no—
256C22×C4—281no—
257C22×C4C22×D711yescharacteristic
258C22×C4C22×D711yescharacteristic
259C22×C4C22×D711yescharacteristic
260C22×C4C22×D711yescharacteristic
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C4⋊C4—281no—
264C4⋊C4—281no—
265C22⋊C4—22no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—142no—
269C22⋊C4—142no—
270C22⋊C4—142no—
271C22⋊C4—142no—
272C22⋊C4—142no—
273C22⋊C4—281no—
274C22⋊C4—281no—
275C42—142no—
276C42—142no—
277C42—281no—
278C2×C7—42no—
279C2×C7#52911yescharacteristic
280C2×C7#52911yescharacteristic
281C2×C7#53811yescharacteristic
282C2×C7#55111yescharacteristic
283C2×C7#55111yescharacteristic
284C2×C7#55111yescharacteristic
285C2×C7#55511yescharacteristic
286D7—42no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—72no—
294C23—141no—
295C23—141no—
296C23—141no—
297C23—141no—
298C23—141no—
299C23—141no—
300C23#116112yes—
301C23C23×D711yescharacteristic
302Q8—44no—
303D4—284no—
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—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—72no—
322C2×C4—72no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—284no—
343C2×C4—284no—
344C2×C4#115912yes—
345C7#257211yescharacteristic, complemented, Sylow
346C22—22no—
347C22—22no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—142no—
354C22—142no—
355C22—142no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22C2×#115911yescharacteristic
361C22#654811yescharacteristic
362C22#654811yescharacteristic
363C22C2×#116111yescharacteristic
364C22D7×#4111yescharacteristic
365C22D7×#4111yescharacteristic
366C22D7×#4111yescharacteristic
367C4—22no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—142no—
374C4—142no—
375C4—281no—
376C4—281no—
377C4—281no—
378C4—281no—
379C4—281no—
380C2—42no—
381C2—282no—
382C2#9028411yescharacteristic
383C2#9031511yescharacteristic
384C2#9032811yescharacteristic
385C2#9032811yescharacteristic
386C2#9036111yescharacteristic
387C2#9037011yescharacteristic
388C2#9037111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#393C2
prime radicalC22×C22⋊D7C22
hypercenter#393D7
hypocenterC7#2572
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C7×#393C2semidirect
2. #90003C2semidirect
3. #90002C2semidirect
7. #89717C2semidirect
8. #89715C2semidirect
9. C22⋊C4×Dic7C2semidirect
10. #89441C2semidirect
11. #89438C2semidirect
12. #89436C2semidirect
13. #89432C2semidirect
14. #89423C2semidirect
15. #89423C2semidirect
97. #393D7semidirect
345. C7#2572semidirect

Maximal subgroups

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

subgroupindex
C7×#3932
#900032
#900022
#897332
#897302
#897282
#897172
#897152
C22⋊C4×Dic72
#894412
#894382
#894362
#894322
#894232
#25727