Finite Group Databasebeta

← #11733313

Subgroups — #11733313

Order 896.

546 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×#6460C211yesmaximal, characteristic, complemented
2C2×#6446C211yesmaximal, characteristic
3C7×#388C211yesmaximal, characteristic, complemented
4#89758C211yesmaximal, characteristic, complemented
5#89728C211yesmaximal, characteristic
6#89728C211yesmaximal, characteristic
7#89725C211yesmaximal, characteristic, complemented
8#89723C211yesmaximal, characteristic, complemented
9#89719C211yesmaximal, characteristic, complemented
10C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
11#89708C211yesmaximal, characteristic, complemented
12#89436C211yesmaximal, characteristic, complemented
13D7×#95C211yesmaximal, characteristic, complemented
14#89432C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C42×C7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37#6460—24no—
38#6446—24no—
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#2488—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—22no—
63C2×C22⋊D7—22no—
64C2×C22⋊D7—22no—
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C2×C4×D7—22no—
73C2×C4×D7—22no—
74C2×C4×D7—22no—
75C2×C4×D7—22no—
76C2×C4×D7—22no—
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—41no—
80C2×C4×D7—41no—
81C2×Dic14—22no—
82C2×Dic14—22no—
83C2×Dic14—41no—
84C7×C4⋊C4—22no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4—22no—
87C7×C22⋊C4—22no—
88C42×C7—22no—
89C22⋊Dic7—22no—
90C22⋊Dic7—41no—
91C22⋊Dic7—41no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4D414yes—
95C4⋊Dic7—22no—
96C4⋊Dic7—22no—
97C4⋊Dic7—22no—
98Dic7⋊C4—22no—
99C4×Dic7—41no—
100C2×#123—71no—
101C2×C22⋊Q8—71no—
102#400—71no—
103#400—71no—
104#397—71no—
105#393—71no—
106#391—71no—
107#388—71no—
108#388D711yescharacteristic, complemented
109#386—71no—
110#386—71no—
111#382—71no—
112#381—71no—
113C4×C22⋊C4—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×C7C2411yescharacteristic
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7C2×D412yes—
128C22×D7#4112yes—
129C22×D7#4112yes—
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—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7C2×D412yes—
146C2×C4×C7#4112yes—
147C2×C4×C7#4112yes—
148C22⋊D7—44no—
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×Dic7C2×D412yes—
170C4×D7—44no—
171C4×D7—44no—
172C4×D7—44no—
173Dic14—44no—
174C22×Q8—71no—
175C22×D4—71no—
176C23×C4—71no—
177C23×C4—71no—
178C23×C4—71no—
179#123—144no—
180C22⋊Q8—144no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4C2×D711yescharacteristic
186C2×C22⋊C4—71no—
187C2×C22⋊C4—71no—
188C2×C22⋊C4—71no—
189C2×C22⋊C4—71no—
190C2×C22⋊C4—71no—
191C2×C22⋊C4—71no—
192C2×C22⋊C4C2×D711yescharacteristic
193C2×C22⋊C4C2×D711yescharacteristic
194C2×C22⋊C4C2×D711yescharacteristic
195C2×C42—71no—
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#95C2×D711yescharacteristic
210#95C2×D711yescharacteristic
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7C22×D411yescharacteristic
219C22×C7C2×#4111yescharacteristic
220C22×C7C2×#4111yescharacteristic
221C22×C7C2×#4111yescharacteristic
222C22×C7C2×#4111yescharacteristic
223C22×C7H(5,𝔽2)11yescharacteristic
224C22×C72−1+411yescharacteristic
225C2×D7—22no—
226C2×D7—22no—
227C2×D7—22no—
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—22no—
231C2×D7#12414yes—
232C4×C7—22no—
233C4×C7—22no—
234C4×C7—22no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238Dic7—22no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7—41no—
245C24—71no—
246C24C22×D711yescharacteristic
247C2×Q8—142no—
248C2×Q8—142no—
249C2×Q8—281no—
250C2×D4—142no—
251C2×D4—142no—
252C2×D4—142no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—142no—
261C22×C4—142no—
262C22×C4—142no—
263C22×C4—142no—
264C22×C4—142no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—281no—
268C22×C4—281no—
269C22×C4C22×D711yescharacteristic
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—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—22no—
283C22⋊C4—74no—
284C22⋊C4—142no—
285C22⋊C4—142no—
286C22⋊C4—142no—
287C22⋊C4—281no—
288C22⋊C4—281no—
289C42—22no—
290C42—281no—
291C2×C7—81no—
292C2×C7C2×#12411yescharacteristic
293C2×C7#52911yescharacteristic
294C2×C7#53911yescharacteristic
295C2×C7#54611yescharacteristic
296C2×C7#54711yescharacteristic
297C2×C7#55111yescharacteristic
298C2×C7#55211yescharacteristic
299D7—24no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—72no—
308C23—72no—
309C23—72no—
310C23—141no—
311C23—141no—
312C23—141no—
313C23—141no—
314C23C23×D711yescharacteristic
315Q8—284no—
316D4—284no—
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—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—72no—
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—284no—
354C2×C4—284no—
355C2×C4—284no—
356C2×C4#115712yes—
357C2×C4D4×D712yes—
358C2×C4#115912yes—
359C7#248811yescharacteristic, complemented, Sylow
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—74no—
368C22—142no—
369C22—142no—
370C22—142no—
371C22—142no—
372C22—142no—
373C22—142no—
374C22C2×#115711yescharacteristic
375C22C2×D4×D711yescharacteristic
376C22C2×#115911yescharacteristic
377C22D7×#4111yescharacteristic
378C22D7×#4111yescharacteristic
379C22#655311yescharacteristic
380C22#655411yescharacteristic
381C4—22no—
382C4—22no—
383C4—22no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—142no—
388C4—281no—
389C4—281no—
390C4—281no—
391C4—281no—
392C4—281no—
393C4—281no—
394C2—81no—
395C2—144no—
396C2#9022011yescharacteristic
397C2#9024111yescharacteristic
398C2#9024811yescharacteristic
399C2#9030211yescharacteristic
400C2#9034911yescharacteristic
401C2D7×#12411yescharacteristic
402C2#9039811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC22×C22⋊D7C22
hypercenter#388D7
hypocenterC7#2488
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
3. C7×#388C2semidirect
4. #89758C2semidirect
7. #89725C2semidirect
8. #89723C2semidirect
9. #89719C2semidirect
10. C4×(C2×D7)⋊C4C2semidirect
11. #89708C2semidirect
12. #89436C2semidirect
13. D7×#95C2semidirect
14. #89432C2semidirect
15. #89418C2semidirect
108. #388D7semidirect
359. C7#2488semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64462
C7×#3882
#897582
#897282
#897252
#897232
#897192
C4×(C2×D7)⋊C42
#897082
#894362
D7×#952
#894322
#894182
#24887