Finite Group Databasebeta

← #11734402

Subgroups — #11734402

Order 896.

468 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#394C211yesmaximal, characteristic
2#89992C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic
4#89750C211yesmaximal, characteristic, complemented
5#89749C211yesmaximal, characteristic, complemented
6#89724C211yesmaximal, characteristic, complemented
7#89723C211yesmaximal, characteristic, complemented
8#89722C211yesmaximal, characteristic, complemented
9#89720C211yesmaximal, characteristic, complemented
10#89717C211yesmaximal, characteristic, complemented
11#89715C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
14C7⋊#401C211yesmaximal, characteristic, complemented
15#89425C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×C4×Dic7C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
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#2576—71nomaximal, Sylow
52C24×C7C2311yescharacteristic
53C24×C7C2311yescharacteristic
54C2×C7×D4—22no—
55C2×C7×D4—22no—
56C2×C7×D4—41no—
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×Dic7—22no—
63C22×Dic7—22no—
64C22×Dic7—22no—
65C22×Dic7—41no—
66C22×Dic7—41no—
67C22×Dic7—41no—
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C7×C4⋊C4—41no—
77C7×C22⋊C4—22no—
78C7×C22⋊C4—22no—
79C7×C22⋊C4—22no—
80C7×C22⋊C4—22no—
81C22⋊Dic7—22no—
82C22⋊Dic7—22no—
83C22⋊Dic7—22no—
84C22⋊Dic7—41no—
85C22⋊Dic7—41no—
86C22⋊Dic7—41no—
87C4⋊Dic7—22no—
88C4⋊Dic7—41no—
89Dic7⋊C4—22no—
90Dic7⋊C4—41no—
91C4×Dic7—22no—
92C4×Dic7—41no—
93C4×Dic7—41no—
94#401—71no—
95#397—71no—
96#396—71no—
97#394D711yescharacteristic
98#388—71no—
99#388—71no—
100#387—71no—
101#387—71no—
102#386—71no—
103#386—71no—
104#385—71no—
105#382—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—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7#4112yes—
122C23×C7#4112yes—
123C23×C7C2411yescharacteristic
124C7×D4—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#4112yes—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
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—44no—
163C2×Dic7—44no—
164C2×Dic7#4112yes—
165C2×Dic7#4112yes—
166C22×D4C2×D711yescharacteristic
167C23×C4—71no—
168C23×C4—71no—
169C2×C4⋊C4—71no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
173C2×C4⋊C4C2×D711yescharacteristic
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×C22⋊C4C2×D711yescharacteristic
184C2×C42—71no—
185C2×C42—71no—
186C2×C42—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#95—71no—
200#95C2×D711yescharacteristic
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7C2×#4111yescharacteristic
216C22×C7C2×#4111yescharacteristic
217C22×C7C2×#4111yescharacteristic
218C22×C7C2×#4111yescharacteristic
219C22×C7C2×#4111yescharacteristic
220C22×C7H(5,𝔽2)11yescharacteristic
221C22×C7H(5,𝔽2)11yescharacteristic
222C4×C7—22no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227Dic7—22no—
228Dic7—22no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233Dic7—41no—
234Dic7—41no—
235C24C22×D711yescharacteristic
236C24C22×D711yescharacteristic
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—41no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—142no—
249C22×C4—142no—
250C22×C4—142no—
251C22×C4—281no—
252C22×C4—281no—
253C22×C4—281no—
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256C22×C4C22×D711yescharacteristic
257C22×C4C22×D711yescharacteristic
258C22×C4C22×D711yescharacteristic
259C4⋊C4—41no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—281no—
263C4⋊C4—281no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—142no—
269C22⋊C4—142no—
270C22⋊C4—142no—
271C22⋊C4—281no—
272C22⋊C4—281no—
273C22⋊C4—281no—
274C42—142no—
275C42—281no—
276C42—281no—
277C2×C7—42no—
278C2×C7—42no—
279C2×C7#52911yescharacteristic
280C2×C7#52911yescharacteristic
281C2×C7#53811yescharacteristic
282C2×C7#55111yescharacteristic
283C2×C7#55111yescharacteristic
284C2×C7#55311yescharacteristic
285C2×C7#55311yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23#115912yes—
299C23#115912yes—
300C23C23×D711yescharacteristic
301D4—44no—
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—21no—
315C2×C4—21no—
316C2×C4—72no—
317C2×C4—72no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
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—284no—
341C2×C4—284no—
342C2×C4#115912yes—
343C7#257611yescharacteristic, complemented, Sylow
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22C2×#115911yescharacteristic
359C22C2×#115911yescharacteristic
360C22C2×#115911yescharacteristic
361C22#654811yescharacteristic
362C22D7×#4111yescharacteristic
363C22D7×#4111yescharacteristic
364C22#655311yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—142no—
371C4—142no—
372C4—281no—
373C4—281no—
374C4—281no—
375C4—281no—
376C4—281no—
377C4—281no—
378C2—42no—
379C2—42no—
380C2#9027811yescharacteristic
381C2#9029111yescharacteristic
382C2#9029211yescharacteristic
383C2#9030911yescharacteristic
384C2#9034011yescharacteristic
385C2#9034911yescharacteristic
386C2#9036911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC22×C7×D4C22
hypercenter#394D7
hypocenterC7#2576
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
4. #89750C2semidirect
5. #89749C2semidirect
6. #89724C2semidirect
7. #89723C2semidirect
8. #89722C2semidirect
9. #89720C2semidirect
10. #89717C2semidirect
11. #89715C2semidirect
12. C22⋊C4×Dic7C2semidirect
13. C22⋊C4×Dic7C2semidirect
14. C7⋊#401C2semidirect
15. #89425C2semidirect
343. C7#2576semidirect

Maximal subgroups

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

subgroupindex
C7×#3942
#899922
#897502
#897492
#897242
#897232
#897222
#897202
#897172
#897152
C22⋊C4×Dic72
C7⋊#4012
#894252
#25767