Finite Group Databasebeta

← #11739799

Subgroups — #11739799

Order 896.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C7×D4C211yesmaximal, characteristic
2C2×#6498C211yesmaximal, characteristic
3#89992C211yesmaximal, characteristic
4#89983C211yesmaximal, characteristic, complemented
5#89982C211yesmaximal, characteristic, complemented
6#89982C211yesmaximal, characteristic, complemented
7C2×C7⋊#95C211yesmaximal, characteristic, complemented
8#89754C211yesmaximal, characteristic, complemented
9#89752C211yesmaximal, characteristic, complemented
10#89724C211yesmaximal, characteristic, complemented
11#89724C211yesmaximal, characteristic, complemented
12#89715C211yesmaximal, characteristic, complemented
13#89714C211yesmaximal, characteristic, complemented
14#89705C211yesmaximal, characteristic, complemented
15#89705C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C4×C7×D4—24no—
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C42×C7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32#6498—24no—
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C7⋊#95C2211yescharacteristic
40C7⋊#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2419—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C24×C7C2311yescharacteristic
56C2×C7×D4—22no—
57C2×C7×D4—22no—
58C2×C7×D4D414yes—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×Dic7—22no—
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7—22no—
74C22×Dic7—22no—
75C22×Dic7—41no—
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C7×C4⋊C4—22no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4—22no—
87C42×C7—22no—
88C22⋊Dic7—22no—
89C22⋊Dic7—22no—
90C22⋊Dic7—22no—
91C22⋊Dic7—22no—
92C22⋊Dic7—22no—
93C22⋊Dic7—41no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—41no—
99Dic7⋊C4—41no—
100C2×#123—71no—
101C2×C4×D4D711yescharacteristic
102#400—71no—
103#398—71no—
104#397—71no—
105#397—71no—
106#387—71no—
107#387—71no—
108#386—71no—
109#385—71no—
110#385—71no—
111#381—71no—
112#381—71no—
113#380—71no—
114C2×#95—71no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7C2×D412yes—
126C23×C7C2×D412yes—
127C23×C7#4112yes—
128C23×C7#4112yes—
129C23×C7C2411yescharacteristic
130C7×D4—28no—
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—21no—
143C2×C4×C7—21no—
144C2×C4×C7—44no—
145C2×C4×C7—44no—
146C2×C4×C7C2×D412yes—
147C2×C4×C7#4112yes—
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—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—44no—
172C2×Dic7—44no—
173C2×Dic7#4112yes—
174C22×D4C2×D711yescharacteristic
175C23×C4—71no—
176C23×C4—71no—
177C23×C4C2×D711yescharacteristic
178C23×C4C2×D711yescharacteristic
179#123—144no—
180C4×D4—24no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4C2×D711yescharacteristic
188C2×C22⋊C4—71no—
189C2×C22⋊C4—71no—
190C2×C22⋊C4—71no—
191C2×C22⋊C4—71no—
192C2×C22⋊C4—71no—
193C2×C22⋊C4—71no—
194C2×C22⋊C4C2×D711yescharacteristic
195C2×C22⋊C4C2×D711yescharacteristic
196C2×C42C2×D711yescharacteristic
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211C22×C7—22no—
212C22×C7—22no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7#12314yes—
225C22×C7C22×D411yescharacteristic
226C22×C7C2×#4111yescharacteristic
227C22×C7C2×#4111yescharacteristic
228C22×C7C2×#4111yescharacteristic
229C22×C7C2×#4111yescharacteristic
230C22×C7H(5,𝔽2)11yescharacteristic
231C22×C72−1+411yescharacteristic
232C4×C7—22no—
233C4×C7—22no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—41no—
237Dic7—22no—
238Dic7—41no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7—41no—
245C24C22×D711yescharacteristic
246C24C22×D711yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4C22⋊D714yes—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—142no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—281no—
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—142no—
277C4⋊C4—142no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—281no—
281C4⋊C4—281no—
282C22⋊C4—22no—
283C22⋊C4—22no—
284C22⋊C4—142no—
285C22⋊C4—142no—
286C22⋊C4—142no—
287C22⋊C4—142no—
288C22⋊C4—142no—
289C22⋊C4—281no—
290C42—22no—
291C2×C7—24no—
292C2×C7—42no—
293C2×C7C2×#12311yescharacteristic
294C2×C7#52911yescharacteristic
295C2×C7#53911yescharacteristic
296C2×C7#54611yescharacteristic
297C2×C7#54711yescharacteristic
298C2×C7#55111yescharacteristic
299C2×C7#55211yescharacteristic
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23#115712yes—
311C23#115712yes—
312C23C2×C22⋊D712yes—
313C23C2×C22⋊D712yes—
314C23C23×D711yescharacteristic
315D4—28no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—72no—
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—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—284no—
356C2×C4—284no—
357C2×C4#115912yes—
358C2×C4C2×C22⋊D712yes—
359C7#241911yescharacteristic, complemented, Sylow
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22#649214yes—
374C22C2×#115711yescharacteristic
375C22C2×#115711yescharacteristic
376C22C2×#115911yescharacteristic
377C22#654811yescharacteristic
378C22D7×#4111yescharacteristic
379C22#655411yescharacteristic
380C22C22×C22⋊D711yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—142no—
387C4—281no—
388C4—281no—
389C4—281no—
390C4—281no—
391C4—281no—
392C4—281no—
393C4—281no—
394C2—24no—
395C2—42no—
396C2#9022911yescharacteristic
397C2#9023211yescharacteristic
398C2#9024711yescharacteristic
399C2#9025511yescharacteristic
400C2C2×#649211yescharacteristic
401C2#9049711yescharacteristic
402C2#9051511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C4×C7×D4C2
prime radicalC22×C7×D4C22
hypercenterC2×C4×D4D7
hypocenterC7#2419
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
4. #89983C2semidirect
5. #89982C2semidirect
6. #89982C2semidirect
7. C2×C7⋊#95C2semidirect
8. #89754C2semidirect
9. #89752C2semidirect
10. #89724C2semidirect
11. #89724C2semidirect
12. #89715C2semidirect
13. #89714C2semidirect
14. #89705C2semidirect
15. #89705C2semidirect
359. C7#2419semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C7×D42
C2×#64982
#899922
#899832
#899822
C2×C7⋊#952
#897542
#897522
#897242
#897152
#897142
#897052
#24197