Finite Group Databasebeta

← #11738051

Subgroups — #11738051

Order 896.

400 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#124C211yesmaximal, characteristic
2#90269C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic, complemented
4#89962C211yesmaximal, characteristic, complemented
5#89958C211yesmaximal, characteristic, complemented
6#89941C211yesmaximal, characteristic
7#89939C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89800C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89607C211yesmaximal, characteristic, complemented
12#89580C211yesmaximal, characteristic, complemented
13#89567C211yesmaximal, characteristic, complemented
14#89548C211yesmaximal, characteristic
15#89537C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic, complemented
19C2×C7×SD16C2211yescharacteristic
20C7×#124C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic, complemented
22#6510C2211yescharacteristic
23#6509C2211yescharacteristic
24#6507C2211yescharacteristic, complemented
25C2×#1144C2211yescharacteristic, complemented
26C2×#1144C2211yescharacteristic, complemented
27C2×#1143C2211yescharacteristic
28#6499C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×Dic28C2211yescharacteristic, complemented
31C2×#1132C2211yescharacteristic
32#6460C2211yescharacteristic
33#6459—21no—
34#6452—21no—
35#6451C2211yescharacteristic, complemented
36#6448—21no—
37#6448—21no—
38D7×C22⋊C4—21no—
39D7×C22⋊C4—21no—
40#6438C2211yescharacteristic
41#6435—21no—
42C42×D7C2211yescharacteristic
43C4×Dic14—21no—
44C4×Dic14C2211yescharacteristic, complemented
45C7×C4⋊C8C2211yescharacteristic, complemented
46C7×#103C2211yescharacteristic, complemented
47C7×#102C2211yescharacteristic
48#6409C2211yescharacteristic, complemented
49#6406C2211yescharacteristic
50#6395C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic
53#6390C2211yescharacteristic, complemented
54#6389C2211yescharacteristic, complemented
55#6389C2211yescharacteristic, complemented
56#6384C2211yescharacteristic, complemented
57#6383C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic, complemented
59#3215—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C2×C22⋊D7—21no—
65Q8×D7—41no—
66Q8×D7—41no—
67D4×D7—41no—
68D4×D7—41no—
69C2×D28C2311yescharacteristic
70C2×C4×D7—21no—
71C2×C4×D7D411yescharacteristic
72C2×C4×D7D411yescharacteristic
73C2×C4×D7C2311yescharacteristic
74C2×Dic14—21no—
75C2×Dic14—21no—
76C2×Dic14C2311yescharacteristic
77C2×Dic14C2311yescharacteristic
78C7×Q16—41no—
79C7×SD16—41no—
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C7×C22⋊C4—21no—
86C42×C7—21no—
87C42×C7C2311yescharacteristic
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90#1144—41no—
91#1144—41no—
92#1143—41no—
93#1142—41no—
94(C2×D7)⋊C4—21no—
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—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107C4×Dic7—21no—
108C4×Dic7D411yescharacteristic
109C4×Dic7D411yescharacteristic
110C4×Dic7C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112C2×C7⋊C8C2311yescharacteristic
113Dic28—41no—
114#1132—41no—
115#533—71no—
116C2×#124—71no—
117#497—71no—
118#497—71no—
119#489—71no—
120#489—71no—
121#464—71no—
122#464—71no—
123#464—71no—
124#464D711yescharacteristic, complemented
125#451—71no—
126#451—71no—
127#450—71no—
128#450—71no—
129#432—71no—
130C23×C7—21no—
131C22×D7—21no—
132C22×D7—42no—
133C22×D7—42no—
134C22×D7—42no—
135C22×D7C2×D411yescharacteristic
136C7×Q8—21no—
137C7×Q8—21no—
138C7×Q8—21no—
139C7×Q8—21no—
140C7×D4—21no—
141C7×D4—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C22⋊D7—42no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155D28—21no—
156D28—21no—
157C4×D7—22no—
158C4×D7—21no—
159C4×D7—21no—
160C4×D7—41no—
161C4×D7—42no—
162Dic14—21no—
163Dic14—21no—
164Dic14—21no—
165Dic14—21no—
166Dic14—42no—
167Dic14—42no—
168C7×C8—21no—
169C7×C8—21no—
170C7⋊C8—21no—
171C7⋊C8—21no—
172C22×Q8—71no—
173C22×D4—71no—
174C2×Q16—71no—
175C2×Q16—71no—
176C2×Q16—71no—
177C2×Q16C2×D711yescharacteristic, complemented
178C2×SD16—71no—
179C2×SD16—71no—
180C2×SD16—71no—
181C2×SD16C2×D711yescharacteristic
182C4⋊Q8—71no—
183C4⋊Q8—71no—
184#125—141no—
185#124—71no—
186#124—71no—
187#124—71no—
188#124—141no—
189#124—141no—
190#124C2×D711yescharacteristic
191C22⋊Q8—71no—
192C22⋊Q8—71no—
193C22⋊Q8—141no—
194C4×Q8—71no—
195C4×Q8—141no—
196C4×Q8C2×D711yescharacteristic, complemented
197#117—141no—
198C2×C22⋊C4—141no—
199C2×C22⋊C4—141no—
200C2×C42—71no—
201C4⋊C8—71no—
202C4⋊C8C2×D711yescharacteristic, complemented
203#103—71no—
204#103—71no—
205#103—71no—
206#103C2×D711yescharacteristic, complemented
207#102—71no—
208#102—71no—
209#102—71no—
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8—71no—
213#97—71no—
214#97—71no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7C22×D411yescharacteristic
219C2×D7—21no—
220C2×D7—21no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—84no—
225C2×D7#13712yes—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7H(5,𝔽2)11yescharacteristic
232C4×C7H(5,𝔽2)11yescharacteristic
233Dic7—21no—
234Dic7—21no—
235Dic7—21no—
236Dic7—41no—
237Dic7—41no—
238Dic7—41no—
239C24—141no—
240C2×Q8—71no—
241C2×Q8—71no—
242C2×Q8—141no—
243C2×Q8—141no—
244C2×Q8—281no—
245C2×Q8—281no—
246C2×Q8C22×D711yescharacteristic
247C2×Q8C22×D711yescharacteristic
248C2×D4—71no—
249C2×D4—141no—
250C2×D4—281no—
251C2×D4—281no—
252C2×D4C22×D711yescharacteristic
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—141no—
257Q16—41no—
258Q16—281no—
259Q16—281no—
260Q16—281no—
261SD16—41no—
262SD16—281no—
263SD16—281no—
264SD16—281no—
265C2×C8—71no—
266C2×C8—71no—
267C2×C8C22×D711yescharacteristic
268C2×C8C22×D711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—71no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4—141no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4C22×D711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C42—21no—
289C42—71no—
290C42—71no—
291C42—71no—
292C42—141no—
293C42C22×D711yescharacteristic
294C2×C7—81no—
295C2×C7#53511yescharacteristic
296C2×C7C2×#13711yescharacteristic
297C2×C7#57511yescharacteristic
298D7—22no—
299D7—81no—
300C23—21no—
301C23—71no—
302C23—141no—
303C23—282no—
304C23—282no—
305C23—282no—
306Q8—21no—
307Q8—21no—
308Q8—21no—
309Q8—21no—
310Q8—141no—
311Q8—141no—
312Q8—141no—
313Q8—141no—
314Q8—282no—
315Q8—282no—
316D4—21no—
317D4—21no—
318D4—141no—
319D4—141no—
320D4—282no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—71no—
325C2×C4—71no—
326C2×C4—71no—
327C2×C4—142no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—281no—
334C2×C4—282no—
335C2×C4D4×D711yescharacteristic
336C2×C4D4×D711yescharacteristic
337C2×C4C23×D711yescharacteristic
338C8—21no—
339C8—21no—
340C8—141no—
341C8—141no—
342C7#321511yescharacteristic, complemented, Sylow
343C22—41no—
344C22—41no—
345C22—41no—
346C22—72no—
347C22—141no—
348C22—141no—
349C22—281no—
350C22—281no—
351C22—281no—
352C22—564no—
353C22C2×D4×D711yescharacteristic
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—141no—
360C4—141no—
361C4—141no—
362C4—281no—
363C4—281no—
364C4—281no—
365C4#654811yescharacteristic
366C4#655311yescharacteristic
367C2—81no—
368C2—142no—
369C2—561no—
370C2#9029811yescharacteristic
371C2#9045911yescharacteristic
372C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radicalC2×D4×D7C22
hypercenter#464D7
hypocenterC7#3215
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #90269C2semidirect
3. C7×#464C2semidirect
4. #89962C2semidirect
5. #89958C2semidirect
7. #89939C2semidirect
8. #89826C2semidirect
9. #89800C2semidirect
10. #89617C2semidirect
11. #89607C2semidirect
12. #89580C2semidirect
13. #89567C2semidirect
15. #89537C2semidirect
18. C2×C7×Q16C22semidirect
21. C4×C7×Q8C22semidirect
24. #6507C22semidirect
25. C2×#1144C22semidirect
26. C2×#1144C22semidirect
30. C2×Dic28C22semidirect
35. #6451C22semidirect
44. C4×Dic14C22semidirect
45. C7×C4⋊C8C22semidirect
46. C7×#103C22semidirect
48. #6409C22semidirect
53. #6390C22semidirect
54. #6389C22semidirect
55. #6389C22semidirect
56. #6384C22semidirect
58. C4⋊(C7⋊C8)C22semidirect
124. #464D7semidirect
177. C2×Q16C2×D7semidirect
196. C4×Q8C2×D7semidirect
202. C4⋊C8C2×D7semidirect
206. #103C2×D7semidirect
342. C7#3215semidirect

Maximal subgroups

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

subgroupindex
D7×#1242
#902692
C7×#4642
#899622
#899582
#899412
#899392
#898262
#898002
#896172
#896072
#895802
#895672
#895482
#895372
#32157