Finite Group Databasebeta

← #11738178

Subgroups — #11738178

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#90307C211yesmaximal, characteristic, complemented
2#90303C211yesmaximal, characteristic, complemented
3C7×#466C211yesmaximal, characteristic, complemented
4#89928C211yesmaximal, characteristic, complemented
5#89925C211yesmaximal, characteristic, complemented
6#89885C211yesmaximal, characteristic, complemented
7#89883C211yesmaximal, characteristic, complemented
8#89809C211yesmaximal, characteristic, complemented
9#89807C211yesmaximal, characteristic, complemented
10#89662C211yesmaximal, characteristic, complemented
11#89660C211yesmaximal, characteristic, complemented
12#89557C211yesmaximal, characteristic, complemented
13#89552C211yesmaximal, characteristic, complemented
14#89542C211yesmaximal, characteristic, complemented
15#89537C211yesmaximal, characteristic, complemented
16C2×D4×D7C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×C7×D8C2211yescharacteristic
19C22×C7×C8C2211yescharacteristic, complemented
20C7×#121C2211yescharacteristic
21C7×#121C2211yescharacteristic, complemented
22#6502—21no—
23#6501—21no—
24#6501C2211yescharacteristic
25#6500—21no—
26#6499—21no—
27#6499C2211yescharacteristic
28D4×Dic7C2211yescharacteristic, complemented
29C2×#1142C2211yescharacteristic, complemented
30C2×#1141C2211yescharacteristic
31C4×C22⋊D7C2211yescharacteristic
32#6489C2211yescharacteristic, complemented
33C2×#1131C2211yescharacteristic
34#6460C2211yescharacteristic
35#6456C2211yescharacteristic
36#6455C2211yescharacteristic
37#6451C2211yescharacteristic, complemented
38#6448—21no—
39#6446—21no—
40D7×C22⋊C4—21no—
41#6440—21no—
42C7×#107C2211yescharacteristic, complemented
43C7×#102C2211yescharacteristic
44C7×#102C2211yescharacteristic, complemented
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47C22⋊(C7⋊C8)C2211yescharacteristic, complemented
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6390C2211yescharacteristic, complemented
52#6390C2211yescharacteristic, complemented
53#6389C2211yescharacteristic, complemented
54Dic7⋊C8C2211yescharacteristic, complemented
55#6384C2211yescharacteristic, complemented
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic, complemented
58#6381C2211yescharacteristic, complemented
59#3319—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C2×C22⋊D7D411yescharacteristic
70C22×Dic7—21no—
71D4×D7—22no—
72D4×D7—41no—
73#1157—22no—
74#1157—41no—
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7C2311yescharacteristic
79C2×Dic14—21no—
80C2×Dic14C2311yescharacteristic
81C7×D8—41no—
82C2×C7×C8—41no—
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C22⋊C4—21no—
88C7×C22⋊C4—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7—21no—
93C22⋊Dic7—21no—
94C22⋊Dic7D411yescharacteristic
95#1142—41no—
96#1141—22no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4D411yescharacteristic
101C4⋊Dic7—21no—
102C4⋊Dic7C2311yescharacteristic
103C4⋊Dic7C2311yescharacteristic
104Dic7⋊C4—21no—
105Dic7⋊C4D411yescharacteristic
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1131—41no—
112#555—71no—
113#546—71no—
114#489—71no—
115#485—71no—
116#482—71no—
117#474—71no—
118#468—71no—
119#466D711yescharacteristic, complemented
120#463—71no—
121#449—71no—
122#447—71no—
123#442—71no—
124#435—71no—
125#418—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7—42no—
132C22×D7—42no—
133C22×D7—81no—
134C22×D7C2×D411yescharacteristic
135C7×D4—21no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—21no—
139C7×D4—81no—
140C7×D4—81no—
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—41no—
148C22⋊D7—42no—
149C22⋊D7—42no—
150C22⋊D7—81no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—42no—
155C2×Dic7C2×D411yescharacteristic
156C2×Dic7C2×D411yescharacteristic
157C2×Dic7C2×D411yescharacteristic
158D28—21no—
159D28#4112yes—
160C4×D7—21no—
161C4×D7—21no—
162C4×D7—42no—
163C4×D7—81no—
164Dic14—21no—
165Dic14—21no—
166Dic14—81no—
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×D8C2×D711yescharacteristic
176C2×M16—71no—
177C22×C8C2×D711yescharacteristic, complemented
178C4⋊Q8—71no—
179#124—71no—
180#124—141no—
181#124—141no—
182#123—141no—
183C22⋊Q8—71no—
184#121—71no—
185#121—141no—
186#121—141no—
187#121C2×D711yescharacteristic
188#121C2×D711yescharacteristic, complemented
189#120—141no—
190C4×D4—71no—
191C4×D4—71no—
192C4×D4—71no—
193#117—71no—
194#117—71no—
195#117—141no—
196C2×C22⋊C4—141no—
197#107—71no—
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, complemented
211C22⋊C8—71no—
212C22⋊C8—71no—
213#97—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7C22×D411yescharacteristic
224C2×D7—21no—
225C2×D7—22no—
226C2×D7—21no—
227C2×D7—21no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—84no—
231C4×C7—21no—
232C4×C7—41no—
233C4×C7—41no—
234C4×C7C2×#4111yescharacteristic
235C4×C7H(5,𝔽2)11yescharacteristic
236Dic7—21no—
237Dic7—21no—
238Dic7—21no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242C24—141no—
243#41—142no—
244#41—281no—
245C2×Q8—71no—
246C2×Q8—141no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—71no—
250C2×D4—71no—
251C2×D4—141no—
252C2×D4—142no—
253C2×D4—141no—
254C2×D4—141no—
255C2×D4—281no—
256C2×D4C22×D711yescharacteristic
257C2×D4C22×D711yescharacteristic
258C22×C4—71no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4—141no—
262C22×C4C22×D711yescharacteristic
263SD16—281no—
264D8—41no—
265D8—142no—
266M16—281no—
267C2×C8—41no—
268C2×C8—71no—
269C2×C8—71no—
270C2×C8C22×D711yescharacteristic
271C2×C8C22×D711yescharacteristic
272C4⋊C4—71no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4C22×D711yescharacteristic
278C4⋊C4C22×D711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—71no—
282C22⋊C4—71no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C42—71no—
292C42—141no—
293C42—141no—
294C2×C7—41no—
295C2×C7—81no—
296C2×C7—81no—
297C2×C7#54611yescharacteristic
298C2×C7#57511yescharacteristic
299C2×C7#57611yescharacteristic
300D7—41no—
301D7—42no—
302C23—21no—
303C23—21no—
304C23—71no—
305C23—141no—
306C23—282no—
307C23—282no—
308C23—561no—
309C23D4×D711yescharacteristic
310Q8—141no—
311Q8—141no—
312Q8—561no—
313D4—21no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—72no—
318D4—81no—
319D4—81no—
320D4—141no—
321D4—281no—
322D4—282no—
323D4—282no—
324D4—561no—
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—141no—
337C2×C4—282no—
338C2×C4—282no—
339C2×C4—561no—
340C2×C4D4×D711yescharacteristic
341C2×C4C23×D711yescharacteristic
342C8—21no—
343C8—21no—
344C8—141no—
345C8—141no—
346C7#331911yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—141no—
357C22—141no—
358C22—141no—
359C22—142no—
360C22—281no—
361C22—281no—
362C22—564no—
363C22C2×D4×D711yescharacteristic
364C4—21no—
365C4—41no—
366C4—41no—
367C4—141no—
368C4—141no—
369C4—141no—
370C4—281no—
371C4—281no—
372C4—281no—
373C4#654811yescharacteristic
374C4D7×#4111yescharacteristic
375C2—41no—
376C2—81no—
377C2—81no—
378C2—281no—
379C2—282no—
380C2#9030111yescharacteristic
381C2#9044811yescharacteristic
382C2#9045911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#466C2
socleC22×C7C22×D4
hypercenter#466D7
hypocenterC7#3319
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #90307C2semidirect
2. #90303C2semidirect
3. C7×#466C2semidirect
4. #89928C2semidirect
5. #89925C2semidirect
6. #89885C2semidirect
7. #89883C2semidirect
8. #89809C2semidirect
9. #89807C2semidirect
10. #89662C2semidirect
11. #89660C2semidirect
12. #89557C2semidirect
13. #89552C2semidirect
14. #89542C2semidirect
15. #89537C2semidirect
19. C22×C7×C8C22semidirect
21. C7×#121C22semidirect
28. D4×Dic7C22semidirect
29. C2×#1142C22semidirect
32. #6489C22semidirect
37. #6451C22semidirect
42. C7×#107C22semidirect
44. C7×#102C22semidirect
47. C22⋊(C7⋊C8)C22semidirect
51. #6390C22semidirect
52. #6390C22semidirect
53. #6389C22semidirect
54. Dic7⋊C8C22semidirect
55. #6384C22semidirect
57. #6382C22semidirect
58. #6381C22semidirect
119. #466D7semidirect
177. C22×C8C2×D7semidirect
188. #121C2×D7semidirect
198. #107C2×D7semidirect
210. #102C2×D7semidirect
346. C7#3319semidirect

Maximal subgroups

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

subgroupindex
#903072
#903032
C7×#4662
#899282
#899252
#898852
#898832
#898092
#898072
#896622
#896602
#895572
#895522
#895422
#895372
#33197