Finite Group Databasebeta

← #11738735

Subgroups — #11738735

Order 896.

414 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#90303C211yesmaximal, characteristic, complemented
2#90205C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89811C211yesmaximal, characteristic, complemented
5#89738C211yesmaximal, characteristic, complemented
6#89663C211yesmaximal, characteristic, complemented
7#89660C211yesmaximal, characteristic, complemented
8#89658C211yesmaximal, characteristic, complemented
9#89644C211yesmaximal, characteristic, complemented
10#89559C211yesmaximal, characteristic, complemented
11#89545C211yesmaximal, characteristic, complemented
12#89542C211yesmaximal, characteristic, complemented
13#89540C211yesmaximal, characteristic, complemented
14#89506C211yesmaximal, 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×#1141C2211yescharacteristic
24#6493—21no—
25#6493C2211yescharacteristic, complemented
26C4×C22⋊D7—21no—
27C4×C22⋊D7C2211yescharacteristic
28C2×#1136C2211yescharacteristic, complemented
29C2×D56C2211yescharacteristic
30C2×#1132C2211yescharacteristic, complemented
31C2×#1131C2211yescharacteristic
32#6461—21no—
33#6460C2211yescharacteristic
34#6458C2211yescharacteristic
35#6457—21no—
36#6456C2211yescharacteristic
37#6456C2211yescharacteristic, complemented
38#6455C2211yescharacteristic
39#6453C2211yescharacteristic, complemented
40#6446—21no—
41D7×C22⋊C4—21no—
42C7×#107C2211yescharacteristic, complemented
43C7×#107C2211yescharacteristic, 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#6383C2211yescharacteristic
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic, complemented
58#6382C2211yescharacteristic, complemented
59#3322—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⋊D7—21no—
68C2×C22⋊D7D411yescharacteristic
69D4×D7—22no—
70D4×D7—41no—
71#1157—22no—
72#1157—41no—
73C2×D28—21no—
74C2×D28C2311yescharacteristic
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7—21no—
79C2×C4×D7C2311yescharacteristic
80C2×Dic14C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C22⋊C4—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7D411yescharacteristic
91#1141—41no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4D411yescharacteristic
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4D411yescharacteristic
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106#1136—41no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109D56—22no—
110#1132—41no—
111#1131—41no—
112#553—71no—
113#546—71no—
114#488—71no—
115#485—71no—
116#480D711yescharacteristic, complemented
117#478—71no—
118#469—71no—
119#468—71no—
120#463—71no—
121#449—71no—
122#447—71no—
123C4×D8—71no—
124#436—71no—
125#418—71no—
126#417—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×D4—21no—
136C7×D4—21no—
137C7×D4—81no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—81no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—41no—
147C22⋊D7—42no—
148C22⋊D7—42no—
149C22⋊D7—42no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155D28—21no—
156D28—21no—
157D28—21no—
158D28—81no—
159D28#4112yes—
160C4×D7—21no—
161C4×D7—21no—
162C4×D7—42no—
163C4×D7—81no—
164C4×D7—81no—
165Dic14—21no—
166Dic14—21no—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C22×D4—71no—
173C2×SD16—71no—
174C2×D8—71no—
175C2×D8—71no—
176C2×M16—71no—
177C2×M16—71no—
178#126—141no—
179#125—71no—
180#124—71no—
181#123—141no—
182#123—141no—
183C22⋊Q8—71no—
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⋊C4C2×D711yescharacteristic, complemented
196C2×C22⋊C4—141no—
197#107C2×D711yescharacteristic, complemented
198#107C2×D711yescharacteristic, complemented
199#106—71no—
200C4⋊C8—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102—71no—
209#102C2×D711yescharacteristic
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8C2×D711yescharacteristic, complemented
213C4×C8—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
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×Q8—71no—
246C2×D4—21no—
247C2×D4—71no—
248C2×D4—71no—
249C2×D4—71no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—142no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—281no—
256C2×D4C22×D711yescharacteristic
257C22×C4—21no—
258C22×C4—71no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4—141no—
262C22×C4C22×D711yescharacteristic
263SD16—281no—
264D8—142no—
265D8—281no—
266M16—281no—
267M16—281no—
268C2×C8—71no—
269C2×C8—71no—
270C2×C8C22×D711yescharacteristic
271C2×C8C22×D711yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4C22×D711yescharacteristic
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—71no—
283C22⋊C4—71no—
284C22⋊C4—141no—
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—41no—
295C2×C7—81no—
296C2×C7#54611yescharacteristic
297C2×C7#57511yescharacteristic
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—141no—
311Q8—141no—
312D4—21no—
313D4—21no—
314D4—72no—
315D4—81no—
316D4—141no—
317D4—141no—
318D4—141no—
319D4—281no—
320D4—282no—
321D4—282no—
322D4—282no—
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—81no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—282no—
338C2×C4—561no—
339C2×C4—561no—
340C2×C4D4×D711yescharacteristic
341C2×C4C23×D711yescharacteristic
342C8—21no—
343C8—21no—
344C8—141no—
345C8—141no—
346C7#332211yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—141no—
354C22—141no—
355C22—141no—
356C22—142no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22—281no—
362C22—564no—
363C22C2×D4×D711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—141no—
369C4—141no—
370C4—141no—
371C4—281no—
372C4—281no—
373C4D7×#4111yescharacteristic
374C4#655311yescharacteristic
375C2—41no—
376C2—81no—
377C2—281no—
378C2—282no—
379C2—561no—
380C2#9034511yescharacteristic
381C2#9044811yescharacteristic
382C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
hypercenter#480D7
hypocenterC7#3322
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #90303C2semidirect
2. #90205C2semidirect
3. C7×#480C2semidirect
4. #89811C2semidirect
5. #89738C2semidirect
6. #89663C2semidirect
7. #89660C2semidirect
8. #89658C2semidirect
9. #89644C2semidirect
10. #89559C2semidirect
11. #89545C2semidirect
12. #89542C2semidirect
13. #89540C2semidirect
14. #89506C2semidirect
15. #89501C2semidirect
19. C2×C7×C4⋊C4C22semidirect
25. #6493C22semidirect
28. C2×#1136C22semidirect
30. C2×#1132C22semidirect
37. #6456C22semidirect
39. #6453C22semidirect
42. C7×#107C22semidirect
43. C7×#107C22semidirect
46. C7×C22⋊C8C22semidirect
50. #6390C22semidirect
51. Dic7⋊C8C22semidirect
52. C8×Dic7C22semidirect
53. #6384C22semidirect
54. #6384C22semidirect
57. #6383C22semidirect
58. #6382C22semidirect
116. #480D7semidirect
195. C2×C4⋊C4C2×D7semidirect
197. #107C2×D7semidirect
198. #107C2×D7semidirect
212. C22⋊C8C2×D7semidirect
346. C7#3322semidirect

Maximal subgroups

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

subgroupindex
#903032
#902052
C7×#4802
#898112
#897382
#896632
#896602
#896582
#896442
#895592
#895452
#895422
#895402
#895062
#895012
#33227