Finite Group Databasebeta

← #11738126

Subgroups — #11738126

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90321C211yesmaximal, characteristic, complemented
2#90309C211yesmaximal, characteristic, complemented
3C7×#465C211yesmaximal, characteristic, complemented
4#89944C211yesmaximal, characteristic, complemented
5#89937C211yesmaximal, characteristic, complemented
6C2×#6395C211yesmaximal, characteristic, complemented
7C8×C22⋊D7C211yesmaximal, characteristic, complemented
8#89816C211yesmaximal, characteristic, complemented
9#89808C211yesmaximal, characteristic, complemented
10#89643C211yesmaximal, characteristic, complemented
11#89638C211yesmaximal, characteristic, complemented
12#89576C211yesmaximal, characteristic, complemented
13#89572C211yesmaximal, characteristic, complemented
14#89547C211yesmaximal, characteristic, complemented
15#89528C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic, complemented
19C22×C7×C8C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21C7×#121C2211yescharacteristic
22#6510C2211yescharacteristic
23#6502—21no—
24#6501C2211yescharacteristic, complemented
25D4×Dic7—21no—
26D4×Dic7C2211yescharacteristic, complemented
27C2×#1141C2211yescharacteristic
28C2×#1141C2211yescharacteristic
29C4×C22⋊D7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×C8×D7C2211yescharacteristic, complemented
32#6458—21no—
33#6458C2211yescharacteristic
34#6457—21no—
35#6456C2211yescharacteristic
36#6455C2211yescharacteristic, complemented
37#6453C2211yescharacteristic, complemented
38#6449—21no—
39#6445—21no—
40D7×C22⋊C4—21no—
41#6442—21no—
42C7×#106C2211yescharacteristic, complemented
43C7×#103C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic, complemented
45#6409C2211yescharacteristic, complemented
46#6409C2211yescharacteristic, complemented
47C22⋊(C7⋊C8)C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic, complemented
53Dic7⋊C8C2211yescharacteristic, complemented
54C8×Dic7C2211yescharacteristic, complemented
55#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6381C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3297—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7D411yescharacteristic
67C22×Dic7—21no—
68C22×Dic7—21no—
69#1161—22no—
70#1161—41no—
71C2×D28—22no—
72C2×D28—21no—
73C2×D28—21no—
74C2×D28—21no—
75C2×D28—41no—
76C2×D28C2311yescharacteristic
77C2×D28C2311yescharacteristic
78C2×C4×D7—21no—
79C2×C4×D7—21no—
80C2×C4×D7C2311yescharacteristic
81C7×SD16—41no—
82C2×C7×C8—41no—
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C7×C4⋊C4—21no—
86C7×C4⋊C4C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C22⋊C4—21no—
89C7×C22⋊C4—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7D411yescharacteristic
93#1141—22no—
94#1141—41no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4D411yescharacteristic
101C4⋊Dic7—21no—
102C4⋊Dic7—21no—
103C4⋊Dic7C2311yescharacteristic
104C4⋊Dic7C2311yescharacteristic
105Dic7⋊C4—21no—
106Dic7⋊C4D411yescharacteristic
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C8×D7—41no—
112#553—71no—
113#546—71no—
114#488—71no—
115#484—71no—
116#482—71no—
117#478—71no—
118#466—71no—
119#465D711yescharacteristic, complemented
120#463—71no—
121#449—71no—
122#447—71no—
123C4×D8—71no—
124C8×D4—71no—
125#416—71no—
126C2×#102—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—42no—
132C22×D7—42no—
133C22×D7—81no—
134C22×D7C2×D411yescharacteristic
135C7×Q8—21no—
136C7×Q8—21no—
137C7×D4—21no—
138C7×D4—21no—
139C7×D4—42no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C22⋊D7—22no—
148C22⋊D7—81no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—81no—
152C2×Dic7—81no—
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156D28—21no—
157D28—21no—
158D28—21no—
159D28—42no—
160D28—42no—
161D28—81no—
162D28#4112yes—
163C4×D7—21no—
164C4×D7—21no—
165C4×D7—42no—
166C4×D7—81no—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C22×D4—71no—
173C2×SD16C2×D711yescharacteristic, complemented
174C2×D8—71no—
175C2×D8—71no—
176C22×C8—71no—
177C22×C8C2×D711yescharacteristic
178#126—141no—
179#125—71no—
180#124—71no—
181#123—141no—
182#123—141no—
183C22⋊Q8C2×D711yescharacteristic
184#121—71no—
185#121—71no—
186#121—141no—
187#121—141no—
188#121C2×D711yescharacteristic
189#120—141no—
190C4×D4—71no—
191C4×D4—71no—
192C4×D4—71no—
193C4×D4—141no—
194#117—71no—
195C2×C4⋊C4—71no—
196C2×C22⋊C4—141no—
197#107—71no—
198#107—71no—
199#106C2×D711yescharacteristic, complemented
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103C2×D711yescharacteristic, complemented
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102—71no—
209#102—71no—
210#102C2×D711yescharacteristic, complemented
211C22⋊C8—71no—
212C22⋊C8—71no—
213C4×C8—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7#13512yes—
220C22×C7C22×D411yescharacteristic
221C2×D7—21no—
222C2×D7—22no—
223C2×D7—21no—
224C2×D7—21no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—84no—
231C4×C7—21no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7C2×#4111yescharacteristic
236C4×C7H(5,𝔽2)11yescharacteristic
237Dic7—21no—
238Dic7—21no—
239Dic7—21no—
240Dic7—41no—
241Dic7—41no—
242C24—141no—
243#41—142no—
244#41—281no—
245C2×Q8C22×D711yescharacteristic
246C2×D4—21no—
247C2×D4—71no—
248C2×D4—71no—
249C2×D4—71no—
250C2×D4—142no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—281no—
256C2×D4C22×D711yescharacteristic
257C22×C4—71no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4—141no—
262C22×C4C22×D711yescharacteristic
263SD16—41no—
264D8—142no—
265D8—281no—
266C2×C8—41no—
267C2×C8—71no—
268C2×C8—71no—
269C2×C8—281no—
270C2×C8C22×D711yescharacteristic
271C2×C8C22×D711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—71no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—71no—
284C22⋊C4—71no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—141no—
292C42—71no—
293C42—141no—
294C2×C7—22no—
295C2×C7—81no—
296C2×C7#54611yescharacteristic
297C2×C7C2×#13511yescharacteristic
298C2×C7#57611yescharacteristic
299D7—41no—
300D7—42no—
301D7—81no—
302C23—21no—
303C23—71no—
304C23—141no—
305C23—141no—
306C23—282no—
307C23—282no—
308C23—561no—
309C23D4×D711yescharacteristic
310Q8—21no—
311Q8—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—72no—
316D4—142no—
317D4—141no—
318D4—141no—
319D4—141no—
320D4—282no—
321D4—282no—
322D4—561no—
323D4—561no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—71no—
330C2×C4—71no—
331C2×C4—71no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—282no—
337C2×C4—561no—
338C2×C4—561no—
339C2×C4—561no—
340C2×C4D4×D711yescharacteristic
341C2×C4C23×D711yescharacteristic
342C8—21no—
343C8—21no—
344C8—141no—
345C8—141no—
346C7#329711yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—141no—
353C22—141no—
354C22—141no—
355C22—142no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22—564no—
362C22#647912yes—
363C22C2×D4×D711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—141no—
369C4—141no—
370C4—141no—
371C4—281no—
372C4—281no—
373C4#654811yescharacteristic
374C4D7×#4111yescharacteristic
375C2—22no—
376C2—81no—
377C2—281no—
378C2—282no—
379C2—561no—
380C2#9030111yescharacteristic
381C2C2×#647911yescharacteristic
382C2#9046011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#465C2
socleC22×C7C22×D4
hypercenter#465D7
hypocenterC7#3297
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #90321C2semidirect
2. #90309C2semidirect
3. C7×#465C2semidirect
4. #89944C2semidirect
5. #89937C2semidirect
6. C2×#6395C2semidirect
7. C8×C22⋊D7C2semidirect
8. #89816C2semidirect
9. #89808C2semidirect
10. #89643C2semidirect
11. #89638C2semidirect
12. #89576C2semidirect
13. #89572C2semidirect
14. #89547C2semidirect
15. #89528C2semidirect
18. C2×C7×SD16C22semidirect
24. #6501C22semidirect
26. D4×Dic7C22semidirect
31. C2×C8×D7C22semidirect
36. #6455C22semidirect
37. #6453C22semidirect
42. C7×#106C22semidirect
43. C7×#103C22semidirect
44. C7×#102C22semidirect
45. #6409C22semidirect
46. #6409C22semidirect
52. (C2×D7)⋊C8C22semidirect
53. Dic7⋊C8C22semidirect
54. C8×Dic7C22semidirect
57. #6381C22semidirect
58. #6381C22semidirect
119. #465D7semidirect
173. C2×SD16C2×D7semidirect
199. #106C2×D7semidirect
203. #103C2×D7semidirect
210. #102C2×D7semidirect
346. C7#3297semidirect

Maximal subgroups

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

subgroupindex
#903212
#903092
C7×#4652
#899442
#899372
C2×#63952
C8×C22⋊D72
#898162
#898082
#896432
#896382
#895762
#895722
#895472
#895282
#32977