Finite Group Databasebeta

← #11738778

Subgroups — #11738778

Order 896.

402 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#90303C211yesmaximal, characteristic
2#90204C211yesmaximal, characteristic, complemented
3C7×#481C211yesmaximal, characteristic, complemented
4#89814C211yesmaximal, characteristic, complemented
5#89738C211yesmaximal, characteristic, complemented
6#89643C211yesmaximal, characteristic, complemented
7#89640C211yesmaximal, characteristic, complemented
8#89639C211yesmaximal, characteristic, complemented
9#89624C211yesmaximal, characteristic, complemented
10#89548C211yesmaximal, characteristic, complemented
11#89542C211yesmaximal, characteristic, complemented
12#89540C211yesmaximal, characteristic, complemented
13#89539C211yesmaximal, characteristic, complemented
14#89505C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×D4×D7C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×#121C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic, complemented
20#6501—21no—
21#6500—21no—
22#6499C2211yescharacteristic
23C2×#1142C2211yescharacteristic
24#6492—21no—
25C4×C22⋊D7C2211yescharacteristic
26#6489—21no—
27#6487C2211yescharacteristic, complemented
28C2×#1136C2211yescharacteristic, complemented
29C2×Dic28C2211yescharacteristic, complemented
30C2×#1132C2211yescharacteristic
31C2×#1131C2211yescharacteristic
32#6461—21no—
33#6460C2211yescharacteristic
34#6460C2211yescharacteristic
35#6456C2211yescharacteristic
36#6455C2211yescharacteristic
37#6453C2211yescharacteristic, complemented
38#6452—21no—
39#6450C2211yescharacteristic, complemented
40#6446—21no—
41D7×C22⋊C4—21no—
42C7×#106C2211yescharacteristic, complemented
43C7×#106C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C7×C22⋊C8C2211yescharacteristic, complemented
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6390C2211yescharacteristic, complemented
51Dic7⋊C8C2211yescharacteristic, complemented
52C8×Dic7C2211yescharacteristic, complemented
53#6384C2211yescharacteristic, complemented
54#6384C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58#6381C2211yescharacteristic, complemented
59#3323—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7D411yescharacteristic
68D4×D7—22no—
69D4×D7—41no—
70#1157—22no—
71#1157—41no—
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7C2311yescharacteristic
76C2×Dic14C2311yescharacteristic
77C2×Dic14C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7D411yescharacteristic
89#1142—41no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7—21no—
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106#1136—41no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109Dic28—41no—
110#1132—22no—
111#1131—41no—
112#552—71no—
113#546—71no—
114#488—71no—
115#485—71no—
116#481D711yescharacteristic, complemented
117#478—71no—
118#470—71no—
119#468—71no—
120#464—71no—
121#452—71no—
122#450—71no—
123C4×SD16—71no—
124#436—71no—
125#418—71no—
126#417—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—42no—
131C22×D7—42no—
132C22×D7—81no—
133C22×D7C2×D411yescharacteristic
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—81no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—42no—
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2411yescharacteristic
145C22⋊D7—41no—
146C22⋊D7—42no—
147C22⋊D7—42no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154D28—21no—
155D28#4112yes—
156C4×D7—21no—
157C4×D7—21no—
158C4×D7—42no—
159C4×D7—81no—
160Dic14—21no—
161Dic14—21no—
162Dic14—21no—
163Dic14—21no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C22×D4—71no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×SD16—71no—
173C2×M16—71no—
174C2×M16—71no—
175#126—141no—
176#125—71no—
177#125—141no—
178#124—71no—
179#123—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183C22⋊Q8—71no—
184#121—141no—
185#121C2×D711yescharacteristic
186#120—141no—
187C4×Q8—71no—
188C4×D4—71no—
189C4×D4—71no—
190#117—71no—
191#117—141no—
192C2×C4⋊C4C2×D711yescharacteristic, complemented
193C2×C22⋊C4—141no—
194#107—71no—
195#106C2×D711yescharacteristic, complemented
196#106C2×D711yescharacteristic, complemented
197C4⋊C8—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102C2×D711yescharacteristic
207#102C2×D711yescharacteristic
208C22⋊C8—71no—
209C22⋊C8C2×D711yescharacteristic, complemented
210C4×C8—71no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7C22×D411yescharacteristic
218C2×D7—21no—
219C2×D7—22no—
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—84no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7C2×#4111yescharacteristic
230C4×C7H(5,𝔽2)11yescharacteristic
231Dic7—21no—
232Dic7—21no—
233Dic7—21no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237C24—141no—
238#41—142no—
239#41—281no—
240C2×Q8—71no—
241C2×Q8—71no—
242C2×D4—21no—
243C2×D4—71no—
244C2×D4—71no—
245C2×D4—141no—
246C2×D4—142no—
247C2×D4—141no—
248C2×D4—281no—
249C2×D4C22×D711yescharacteristic
250C22×C4—21no—
251C22×C4—71no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255Q16—281no—
256SD16—142no—
257SD16—281no—
258M16—281no—
259M16—281no—
260C2×C8—71no—
261C2×C8—71no—
262C2×C8C22×D711yescharacteristic
263C2×C8C22×D711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4C22×D711yescharacteristic
274C4⋊C4C22×D711yescharacteristic
275C4⋊C4C22×D711yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—71no—
278C22⋊C4—71no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C42—71no—
287C42—141no—
288C42—141no—
289C2×C7—41no—
290C2×C7—81no—
291C2×C7#54611yescharacteristic
292C2×C7#57511yescharacteristic
293C2×C7#57711yescharacteristic
294D7—41no—
295D7—42no—
296C23—21no—
297C23—71no—
298C23—141no—
299C23—282no—
300C23—282no—
301C23—561no—
302C23D4×D711yescharacteristic
303Q8—141no—
304Q8—141no—
305Q8—141no—
306Q8—141no—
307D4—21no—
308D4—21no—
309D4—72no—
310D4—81no—
311D4—141no—
312D4—281no—
313D4—282no—
314D4—282no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—42no—
321C2×C4—71no—
322C2×C4—71no—
323C2×C4—71no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—282no—
330C2×C4—561no—
331C2×C4D4×D711yescharacteristic
332C2×C4C23×D711yescharacteristic
333C8—21no—
334C8—21no—
335C8—141no—
336C8—141no—
337C7#332311yescharacteristic, complemented, Sylow
338C22—21no—
339C22—21no—
340C22—21no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—141no—
345C22—141no—
346C22—141no—
347C22—142no—
348C22—281no—
349C22—281no—
350C22—564no—
351C22C2×D4×D711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—141no—
357C4—141no—
358C4—141no—
359C4—281no—
360C4—281no—
361C4—281no—
362C4D7×#4111yescharacteristic
363C4#655311yescharacteristic
364C2—41no—
365C2—81no—
366C2—281no—
367C2—282no—
368C2#9034511yescharacteristic
369C2#9045311yescharacteristic
370C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#481C2
socleC22×C7C22×D4
prime radical#90303C2
hypercenter#481D7
hypocenterC7#3323
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. #90204C2semidirect
3. C7×#481C2semidirect
4. #89814C2semidirect
5. #89738C2semidirect
6. #89643C2semidirect
7. #89640C2semidirect
8. #89639C2semidirect
9. #89624C2semidirect
10. #89548C2semidirect
11. #89542C2semidirect
12. #89540C2semidirect
13. #89539C2semidirect
14. #89505C2semidirect
15. #89501C2semidirect
19. C2×C7×C4⋊C4C22semidirect
27. #6487C22semidirect
28. C2×#1136C22semidirect
29. C2×Dic28C22semidirect
37. #6453C22semidirect
39. #6450C22semidirect
42. C7×#106C22semidirect
43. C7×#106C22semidirect
46. C7×C22⋊C8C22semidirect
50. #6390C22semidirect
51. Dic7⋊C8C22semidirect
52. C8×Dic7C22semidirect
53. #6384C22semidirect
54. #6384C22semidirect
55. #6384C22semidirect
58. #6381C22semidirect
116. #481D7semidirect
192. C2×C4⋊C4C2×D7semidirect
195. #106C2×D7semidirect
196. #106C2×D7semidirect
209. C22⋊C8C2×D7semidirect
337. C7#3323semidirect

Maximal subgroups

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

subgroupindex
#903032
#902042
C7×#4812
#898142
#897382
#896432
#896402
#896392
#896242
#895482
#895422
#895402
#895392
#895052
#895012
#33237