Finite Group Databasebeta

← #11737708

Subgroups — #11737708

Order 896.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90521C211yesmaximal, characteristic, complemented
2#90334C211yesmaximal, characteristic, complemented
3C7×#452C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic, complemented
5#89963C211yesmaximal, characteristic, complemented
6#89957C211yesmaximal, characteristic, complemented
7#89948C211yesmaximal, characteristic, complemented
8#89935C211yesmaximal, characteristic, complemented
9#89818C211yesmaximal, characteristic, complemented
10#89591C211yesmaximal, characteristic, complemented
11#89567C211yesmaximal, characteristic, complemented
12#89558C211yesmaximal, characteristic, complemented
13#89532C211yesmaximal, characteristic, complemented
14#89504C211yesmaximal, characteristic, complemented
15#89492C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic, complemented
19C2×C7×SD16C2211yescharacteristic, complemented
20C7×C22⋊Q8C2211yescharacteristic
21#6510C2211yescharacteristic
22#6510C2211yescharacteristic
23#6510C2211yescharacteristic, complemented
24Q8×Dic7C2211yescharacteristic, complemented
25C2×#1143C2211yescharacteristic
26C2×#1143C2211yescharacteristic
27C2×#1143C2211yescharacteristic, complemented
28#6503C2211yescharacteristic
29#6503C2211yescharacteristic
30#6500—21no—
31#6500—21no—
32C2×#1141C2211yescharacteristic, complemented
33#6493—21no—
34#6493—21no—
35#6493C2211yescharacteristic
36#6489C2211yescharacteristic
37C2×D56C2211yescharacteristic
38C2×D56C2211yescharacteristic
39#6461—21no—
40#6460—21no—
41#6456C2211yescharacteristic, complemented
42#6451C2211yescharacteristic, complemented
43#6448—21no—
44#6446—21no—
45C7×#103C2211yescharacteristic, complemented
46C7×#102C2211yescharacteristic, complemented
47C7×C22⋊C8C2211yescharacteristic
48#6409C2211yescharacteristic, complemented
49#6406C2211yescharacteristic, complemented
50C22⋊(C7⋊C8)C2211yescharacteristic
51#6395C2211yescharacteristic
52#6395C2211yescharacteristic
53#6389C2211yescharacteristic, complemented
54Dic7⋊C8C2211yescharacteristic, complemented
55Dic7⋊C8C2211yescharacteristic, complemented
56C8×Dic7C2211yescharacteristic, complemented
57#6383C2211yescharacteristic
58#6383C2211yescharacteristic
59#3224—71nomaximal, Sylow
60C23×D7—21no—
61C7×#41—41no—
62C7×#41—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C2×C7×D4—21no—
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7—21no—
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7—21no—
72C2×C22⋊D7—21no—
73C2×C22⋊D7—21no—
74C2×D28—21no—
75C2×D28—41no—
76C2×D28—41no—
77C2×D28C2311yescharacteristic
78C2×D28C2311yescharacteristic
79C2×D28C2311yescharacteristic
80C2×C4×D7—21no—
81C2×Dic14—21no—
82C7×Q16—41no—
83C7×SD16—41no—
84C2×C7×C8C2311yescharacteristic
85C2×C7×C8C2311yescharacteristic
86C7×C4⋊C4—21no—
87C7×C4⋊C4C2311yescharacteristic
88C7×C22⋊C4—21no—
89C22⋊Dic7D411yescharacteristic
90C22⋊Dic7D411yescharacteristic
91#1143—41no—
92#1143—41no—
93#1143—41no—
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)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103C4⋊Dic7—21no—
104C4⋊Dic7D411yescharacteristic
105C4⋊Dic7D411yescharacteristic
106C4⋊Dic7C2311yescharacteristic
107Dic7⋊C4—21no—
108C4×Dic7—21no—
109C4×Dic7C2311yescharacteristic
110C4×Dic7C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112C2×C7⋊C8C2311yescharacteristic
113D56—41no—
114D56—41no—
115#542—71no—
116#535—71no—
117#496—71no—
118#495—71no—
119#489—71no—
120#486—71no—
121#464—71no—
122#463—71no—
123#460—71no—
124#459—71no—
125#452D711yescharacteristic, complemented
126#450—71no—
127#449—71no—
128#447—71no—
129#433—71no—
130C23×C7—21no—
131C23×C7C2×D411yescharacteristic
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—42no—
136C22×D7—81no—
137C22×D7—81no—
138C7×Q8—21no—
139C7×Q8—21no—
140C7×Q8—21no—
141C7×Q8—21no—
142C7×D4—21no—
143C7×D4—21no—
144C7×D4—42no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—42no—
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2411yescharacteristic
153C22⋊D7—81no—
154C22⋊D7—81no—
155C22⋊D7—81no—
156C22⋊D7—81no—
157C22⋊D7—81no—
158C2×Dic7—21no—
159C2×Dic7C2×D411yescharacteristic
160C2×Dic7C2×D411yescharacteristic
161C2×Dic7C2×D411yescharacteristic
162C2×Dic7C2×D411yescharacteristic
163D28—21no—
164D28—21no—
165D28—21no—
166D28—21no—
167D28—21no—
168D28—21no—
169D28—42no—
170C4×D7—81no—
171Dic14—81no—
172C7×C8—21no—
173C7×C8—21no—
174C7⋊C8—21no—
175C7⋊C8—21no—
176C2×#41C2×D711yescharacteristic
177C22×D4—71no—
178C2×Q16C2×D711yescharacteristic, complemented
179C2×SD16—71no—
180C2×SD16—71no—
181C2×SD16—71no—
182C2×SD16C2×D711yescharacteristic, complemented
183C2×D8—71no—
184C2×D8—71no—
185C2×D8—71no—
186C4⋊Q8—71no—
187#127—71no—
188#127—71no—
189#126—141no—
190#124—71no—
191#124—71no—
192#124—71no—
193#124—141no—
194#123—141no—
195C22⋊Q8—141no—
196C22⋊Q8C2×D711yescharacteristic
197#121—71no—
198#121—141no—
199#121—141no—
200#120—141no—
201#120—141no—
202C4×Q8—71no—
203C4×D4—71no—
204#117—71no—
205C4⋊C8—71no—
206C4⋊C8—71no—
207#103—71no—
208#103C2×D711yescharacteristic, complemented
209#102—71no—
210#102—71no—
211#102—71no—
212#102—71no—
213#102—71no—
214#102C2×D711yescharacteristic, complemented
215C22⋊C8—71no—
216C22⋊C8C2×D711yescharacteristic
217#97—71no—
218C4×C8—71no—
219C22×C7—21no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7C22×D411yescharacteristic
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—84no—
236C4×C7—21no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7H(5,𝔽2)11yescharacteristic
241C4×C7H(5,𝔽2)11yescharacteristic
242Dic7—21no—
243Dic7—21no—
244Dic7—21no—
245Dic7—21no—
246Dic7—41no—
247C24—141no—
248#41—41no—
249#41—41no—
250C2×Q8—141no—
251C2×Q8C22×D711yescharacteristic
252C2×Q8C22×D711yescharacteristic
253C2×D4—21no—
254C2×D4—71no—
255C2×D4—71no—
256C2×D4—71no—
257C2×D4—141no—
258C2×D4—141no—
259C2×D4—141no—
260C2×D4—141no—
261C2×D4—141no—
262C2×D4—141no—
263C2×D4—281no—
264C2×D4—281no—
265C2×D4C22×D711yescharacteristic
266C22×C4—21no—
267C22×C4—141no—
268C22×C4C22×D711yescharacteristic
269Q16—41no—
270SD16—41no—
271SD16—281no—
272SD16—281no—
273SD16—281no—
274D8—281no—
275D8—281no—
276D8—281no—
277C2×C8—71no—
278C2×C8—71no—
279C2×C8C22×D711yescharacteristic
280C2×C8C22×D711yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—71no—
283C4⋊C4—71no—
284C4⋊C4—71no—
285C4⋊C4—141no—
286C4⋊C4—141no—
287C4⋊C4C22×D711yescharacteristic
288C22⋊C4—21no—
289C22⋊C4—71no—
290C22⋊C4—71no—
291C22⋊C4—141no—
292C22⋊C4—141no—
293C22⋊C4—141no—
294C22⋊C4—141no—
295C22⋊C4—141no—
296C22⋊C4—141no—
297C22⋊C4—141no—
298C22⋊C4—141no—
299C42—71no—
300C42—71no—
301C42—141no—
302C2×C7—41no—
303C2×C7—81no—
304C2×C7#53511yescharacteristic
305C2×C7#57611yescharacteristic
306C2×C7#57711yescharacteristic
307D7—81no—
308D7—81no—
309D7—81no—
310C23—21no—
311C23—141no—
312C23—141no—
313C23—141no—
314C23—282no—
315C23—561no—
316C23—561no—
317C23D4×D711yescharacteristic
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—561no—
323D4—21no—
324D4—21no—
325D4—42no—
326D4—141no—
327D4—141no—
328D4—141no—
329D4—141no—
330D4—141no—
331D4—141no—
332D4—282no—
333D4—561no—
334D4—561no—
335D4—561no—
336D4—561no—
337D4—561no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—71no—
345C2×C4—71no—
346C2×C4—71no—
347C2×C4—71no—
348C2×C4—141no—
349C2×C4—561no—
350C2×C4D4×D711yescharacteristic
351C2×C4C23×D711yescharacteristic
352C8—21no—
353C8—21no—
354C8—141no—
355C8—141no—
356C7#322411yescharacteristic, complemented, Sylow
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—281no—
364C22—281no—
365C22—281no—
366C22—281no—
367C22—281no—
368C22—281no—
369C22—281no—
370C22—281no—
371C22—281no—
372C22—564no—
373C22C2×D4×D711yescharacteristic
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—141no—
379C4—141no—
380C4—141no—
381C4—141no—
382C4—281no—
383C4#654811yescharacteristic
384C4#654811yescharacteristic
385C2—41no—
386C2—81no—
387C2—561no—
388C2—561no—
389C2—561no—
390C2#9028311yescharacteristic
391C2#9046011yescharacteristic
392C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#452C2
socleC22×C7C22×D4
hypercenter#452D7
hypocenterC7#3224
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #90521C2semidirect
2. #90334C2semidirect
3. C7×#452C2semidirect
4. #90010C2semidirect
5. #89963C2semidirect
6. #89957C2semidirect
7. #89948C2semidirect
8. #89935C2semidirect
9. #89818C2semidirect
10. #89591C2semidirect
11. #89567C2semidirect
12. #89558C2semidirect
13. #89532C2semidirect
14. #89504C2semidirect
15. #89492C2semidirect
18. C2×C7×Q16C22semidirect
19. C2×C7×SD16C22semidirect
23. #6510C22semidirect
24. Q8×Dic7C22semidirect
27. C2×#1143C22semidirect
32. C2×#1141C22semidirect
41. #6456C22semidirect
42. #6451C22semidirect
45. C7×#103C22semidirect
46. C7×#102C22semidirect
48. #6409C22semidirect
49. #6406C22semidirect
53. #6389C22semidirect
54. Dic7⋊C8C22semidirect
55. Dic7⋊C8C22semidirect
56. C8×Dic7C22semidirect
125. #452D7semidirect
178. C2×Q16C2×D7semidirect
182. C2×SD16C2×D7semidirect
208. #103C2×D7semidirect
214. #102C2×D7semidirect
356. C7#3224semidirect

Maximal subgroups

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

subgroupindex
#905212
#903342
C7×#4522
#900102
#899632
#899572
#899482
#899352
#898182
#895912
#895672
#895582
#895322
#895042
#894922
#32247