Finite Group Databasebeta

← #11738066

Subgroups — #11738066

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#90367C211yesmaximal, characteristic
2#90272C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic, complemented
4#89961C211yesmaximal, characteristic, complemented
5#89959C211yesmaximal, characteristic, complemented
6#89946C211yesmaximal, characteristic, complemented
7#89941C211yesmaximal, characteristic, complemented
8#89828C211yesmaximal, characteristic, complemented
9C4×#1143C211yesmaximal, characteristic, complemented
10#89619C211yesmaximal, characteristic, complemented
11#89610C211yesmaximal, characteristic, complemented
12#89586C211yesmaximal, characteristic, complemented
13#89575C211yesmaximal, characteristic, complemented
14#89549C211yesmaximal, characteristic, complemented
15#89542C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×D4×D7C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic, complemented
19C2×C7×SD16C2211yescharacteristic
20C7×#124C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic, complemented
22#6509C2211yescharacteristic
23#6509C2211yescharacteristic
24#6509C2211yescharacteristic, complemented
25C2×#1143C2211yescharacteristic
26#6501C2211yescharacteristic
27C2×#1131C2211yescharacteristic
28C2×#1131C2211yescharacteristic, complemented
29#6457—21no—
30#6455C2211yescharacteristic
31D7×C4⋊C4C2211yescharacteristic, complemented
32#6452—21no—
33#6447—21no—
34#6446—21no—
35#6445—21no—
36D7×C22⋊C4—21no—
37#6439—21no—
38C4×D28C2211yescharacteristic
39C4×D28C2211yescharacteristic
40#6435—21no—
41#6433C2211yescharacteristic, complemented
42C7×C4⋊C8C2211yescharacteristic, complemented
43C7×#103C2211yescharacteristic, complemented
44C7×#102C2211yescharacteristic
45#6409C2211yescharacteristic, complemented
46#6409C2211yescharacteristic, complemented
47#6409C2211yescharacteristic, complemented
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic
53(C2×D7)⋊C8C2211yescharacteristic, complemented
54#6392C2211yescharacteristic, complemented
55#6391C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic, complemented
58C4×C7⋊C8C2211yescharacteristic, complemented
59#3323—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66#1161—22no—
67#1161—41no—
68D4×D7—22no—
69D4×D7—41no—
70C2×D28—21no—
71C2×D28D411yescharacteristic
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7C2311yescharacteristic
77C2×C4×D7C2311yescharacteristic
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—
89#1143—22no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4D411yescharacteristic
96(C2×D7)⋊C4D411yescharacteristic
97C4⋊Dic7D411yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107C4×Dic7—21no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1131—41no—
111#1131—41no—
112#552—71no—
113#546—71no—
114#488—71no—
115#485—71no—
116#481—71no—
117#478—71no—
118#470—71no—
119#468—71no—
120#464D711yescharacteristic, complemented
121#452—71no—
122#450—71no—
123C4×SD16—71no—
124#436—71no—
125#418—71no—
126#417—71no—
127C23×C7—21no—
128C22×D7—21no—
129C22×D7—42no—
130C22×D7—42no—
131C22×D7—81no—
132C22×D7C2×D411yescharacteristic
133C22×D7C2×D411yescharacteristic
134C7×Q8—21no—
135C7×Q8—21no—
136C7×Q8—21no—
137C7×Q8—21no—
138C7×D4—21no—
139C7×D4—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—42no—
147C22⋊D7—81no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153D28—21no—
154D28—41no—
155D28—42no—
156D28#4112yes—
157C4×D7—21no—
158C4×D7—21no—
159C4×D7—21no—
160C4×D7—21no—
161C4×D7—42no—
162C4×D7—42no—
163C4×D7—81no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—71no—
169C22×D4—71no—
170C2×Q16C2×D711yescharacteristic, complemented
171C2×SD16—71no—
172C2×SD16C2×D711yescharacteristic
173C2×M16—71no—
174C2×M16—71no—
175#126—141no—
176#125—71no—
177#125—141no—
178#124C2×D711yescharacteristic
179#123—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183C22⋊Q8—71no—
184#121—71no—
185#121—141no—
186#120—141no—
187C4×Q8C2×D711yescharacteristic, complemented
188C4×D4—71no—
189C4×D4—71no—
190#117—71no—
191#117—141no—
192C2×C4⋊C4—71no—
193C2×C22⋊C4—141no—
194#107—71no—
195#106—71no—
196#106—71no—
197C4⋊C8C2×D711yescharacteristic, complemented
198#103—71no—
199#103—71no—
200#103—71no—
201#103C2×D711yescharacteristic, complemented
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102C2×D711yescharacteristic
208C22⋊C8—71no—
209C22⋊C8—71no—
210C4×C8—71no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7C22×D411yescharacteristic
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—22no—
218C2×D7—21no—
219C2×D7—21no—
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—84no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7C2×#4111yescharacteristic
231C4×C7H(5,𝔽2)11yescharacteristic
232Dic7—21no—
233Dic7—21no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237C24—141no—
238#41—142no—
239#41—281no—
240C2×Q8C22×D711yescharacteristic
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—71no—
244C2×D4—141no—
245C2×D4—141no—
246C2×D4—142no—
247C2×D4—141no—
248C2×D4—281no—
249C2×D4C22×D711yescharacteristic
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—141no—
255Q16—41no—
256SD16—41no—
257SD16—142no—
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—71no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4—141no—
275C4⋊C4C22×D711yescharacteristic
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—71no—
279C22⋊C4—71no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C42—21no—
287C42—141no—
288C42C22×D711yescharacteristic
289C2×C7—81no—
290C2×C7#54611yescharacteristic
291C2×C7#57511yescharacteristic
292C2×C7#57711yescharacteristic
293D7—41no—
294D7—41no—
295D7—42no—
296C23—21no—
297C23—71no—
298C23—71no—
299C23—141no—
300C23—282no—
301C23—282no—
302C23—561no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—21no—
307D4—21no—
308D4—21no—
309D4—72no—
310D4—141no—
311D4—281no—
312D4—282no—
313D4—282no—
314D4—561no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—71no—
319C2×C4—71no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—282no—
328C2×C4—282no—
329C2×C4—561no—
330C2×C4D4×D711yescharacteristic
331C2×C4D4×D711yescharacteristic
332C2×C4C23×D711yescharacteristic
333C8—21no—
334C8—21no—
335C8—141no—
336C8—141no—
337C7#332311yescharacteristic, complemented, Sylow
338C22—41no—
339C22—41no—
340C22—41no—
341C22—141no—
342C22—141no—
343C22—141no—
344C22—141no—
345C22—141no—
346C22—141no—
347C22—142no—
348C22—281no—
349C22—281no—
350C22—564no—
351C22C2×D4×D711yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—141no—
358C4—141no—
359C4—281no—
360C4—281no—
361C4—281no—
362C4#654811yescharacteristic
363C4D7×#4111yescharacteristic
364C2—81no—
365C2—281no—
366C2—281no—
367C2—282no—
368C2#9030311yescharacteristic
369C2#9045911yescharacteristic
370C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radical#90367C2
hypercenter#464D7
hypocenterC7#3323
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. #90272C2semidirect
3. C7×#464C2semidirect
4. #89961C2semidirect
5. #89959C2semidirect
6. #89946C2semidirect
7. #89941C2semidirect
8. #89828C2semidirect
9. C4×#1143C2semidirect
10. #89619C2semidirect
11. #89610C2semidirect
12. #89586C2semidirect
13. #89575C2semidirect
14. #89549C2semidirect
15. #89542C2semidirect
18. C2×C7×Q16C22semidirect
21. C4×C7×Q8C22semidirect
24. #6509C22semidirect
28. C2×#1131C22semidirect
31. D7×C4⋊C4C22semidirect
41. #6433C22semidirect
42. C7×C4⋊C8C22semidirect
43. C7×#103C22semidirect
45. #6409C22semidirect
46. #6409C22semidirect
47. #6409C22semidirect
53. (C2×D7)⋊C8C22semidirect
54. #6392C22semidirect
55. #6391C22semidirect
57. #6382C22semidirect
58. C4×C7⋊C8C22semidirect
120. #464D7semidirect
170. C2×Q16C2×D7semidirect
187. C4×Q8C2×D7semidirect
197. C4⋊C8C2×D7semidirect
201. #103C2×D7semidirect
337. C7#3323semidirect

Maximal subgroups

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

subgroupindex
#903672
#902722
C7×#4642
#899612
#899592
#899462
#899412
#898282
C4×#11432
#896192
#896102
#895862
#895752
#895492
#895422
#33237