Finite Group Databasebeta

← #11741059

Subgroups — #11741059

Order 896.

420 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
1C7×#553C211yesmaximal, characteristic, complemented
2#90241C211yesmaximal, characteristic, complemented
3#89840C211yesmaximal, characteristic, complemented
4#89838C211yesmaximal, characteristic, complemented
5#89813C211yesmaximal, characteristic, complemented
6#89810C211yesmaximal, characteristic, complemented
7#89809C211yesmaximal, characteristic, complemented
8#89808C211yesmaximal, characteristic, complemented
9#89788C211yesmaximal, characteristic, complemented
10C4×#1141C211yesmaximal, characteristic, complemented
11D4×C7⋊C8C211yesmaximal, characteristic, complemented
12#89775C211yesmaximal, characteristic, complemented
13#89774C211yesmaximal, characteristic, complemented
14#89742C211yesmaximal, characteristic, complemented
15C2×#6383C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C7×#126—21no—
19C7×#125C2211yescharacteristic, complemented
20C7×#123—21no—
21C7×#121—21no—
22C7×#121C2211yescharacteristic
23C7×#121C2211yescharacteristic, complemented
24C4×C7×D4—21no—
25C4×C7×D4C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic, complemented
27C7×#117C2211yescharacteristic, complemented
28C2×C7×C4⋊C4C2211yescharacteristic
29#6500—21no—
30C2×#1142C2211yescharacteristic, complemented
31C2×#1141C2211yescharacteristic
32C2×#1141C2211yescharacteristic
33#6493C2211yescharacteristic
34#6487C2211yescharacteristic
35C22×C7⋊C8C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic, complemented
37#6457—21no—
38#6447—21no—
39D7×C22⋊C4—21no—
40#6438C2211yescharacteristic
41C4×D28C2211yescharacteristic
42#6406C2211yescharacteristic, complemented
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic, complemented
45#6384C2211yescharacteristic, complemented
46#6384C2211yescharacteristic, complemented
47#6384C2211yescharacteristic, complemented
48#6383C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic, complemented
55#6381C2211yescharacteristic, complemented
56#6381C2211yescharacteristic, complemented
57C4⋊(C7⋊C8)C2211yescharacteristic, complemented
58C4×C7⋊C8C2211yescharacteristic, complemented
59#3297—71nomaximal, Sylow
60C23×D7—21no—
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4D411yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71#1157—22no—
72#1157—41no—
73C2×D28—22no—
74C2×D28—21no—
75C2×D28—41no—
76C2×D28C2311yescharacteristic
77C2×D28C2311yescharacteristic
78C2×C4×D7—21no—
79C2×C4×D7—21no—
80C2×Dic14C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4D411yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C22⋊C4—21no—
89C7×C22⋊C4—21no—
90C7×C22⋊C4—21no—
91C7×C22⋊C4—21no—
92C7×C22⋊C4D411yescharacteristic
93C7×C22⋊C4D411yescharacteristic
94C42×C7—21no—
95C42×C7C2311yescharacteristic
96C22⋊Dic7—21no—
97#1142—41no—
98#1141—22no—
99#1141—41no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102(C2×D7)⋊C4—21no—
103(C2×D7)⋊C4—21no—
104C4⋊Dic7C2311yescharacteristic
105Dic7⋊C4—21no—
106C2×C7⋊C8—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#553D711yescharacteristic, complemented
113#546—71no—
114#488—71no—
115#484—71no—
116#482—71no—
117#478—71no—
118#466—71no—
119#465—71no—
120#463—71no—
121#449—71no—
122#447—71no—
123C4×D8—71no—
124C8×D4—71no—
125#416—71no—
126C2×#102—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C23×C7C2×D411yescharacteristic
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—42no—
133C22×D7—42no—
134C22×D7—81no—
135C7×D4—22no—
136C7×D4—21no—
137C7×D4—21no—
138C7×D4—42no—
139C7×D4—81no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—81no—
148C2×C4×C7—81no—
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2411yescharacteristic
154C22⋊D7—42no—
155C22⋊D7—81no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158D28—21no—
159D28—21no—
160D28—21no—
161D28—42no—
162D28#4112yes—
163C4×D7—42no—
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—
176C22×C8—71no—
177C22×C8—71no—
178#126—21no—
179#125C2×D711yescharacteristic, complemented
180#124—71no—
181#123—21no—
182#123—141no—
183C22⋊Q8—71no—
184#121—21no—
185#121—71no—
186#121—141no—
187#121C2×D711yescharacteristic
188#121C2×D711yescharacteristic, complemented
189#120—141no—
190C4×D4—21no—
191C4×D4—71no—
192C4×D4C2×D711yescharacteristic
193C4×D4C2×D711yescharacteristic, complemented
194#117C2×D711yescharacteristic, complemented
195C2×C4⋊C4C2×D711yescharacteristic
196C2×C22⋊C4—141no—
197#107—71no—
198#107—71no—
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#102—71no—
210#102—71no—
211C22⋊C8—71no—
212C22⋊C8—71no—
213C4×C8—71no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7#13512yes—
223C22×C7C22×D411yescharacteristic
224C2×D7—22no—
225C2×D7—41no—
226C2×D7—41no—
227C2×D7—41no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—84no—
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—21no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238C4×C7C2×#4111yescharacteristic
239C4×C7H(5,𝔽2)11yescharacteristic
240Dic7—41no—
241Dic7—41no—
242C24—141no—
243#41—142no—
244#41—281no—
245C2×Q8—71no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—71no—
249C2×D4—71no—
250C2×D4—142no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4—141no—
254C2×D4—281no—
255C2×D4C22⋊D711yescharacteristic
256C2×D4C22×D711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263SD16—281no—
264D8—142no—
265D8—281no—
266C2×C8—71no—
267C2×C8—71no—
268C2×C8—71no—
269C2×C8—71no—
270C2×C8—281no—
271C2×C8—281no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—71no—
276C4⋊C4—141no—
277C4⋊C4C22⋊D711yescharacteristic
278C4⋊C4C22×D711yescharacteristic
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4C22⋊D711yescharacteristic
291C22⋊C4C22⋊D711yescharacteristic
292C42—21no—
293C42C22×D711yescharacteristic
294C2×C7—22no—
295C2×C7—41no—
296C2×C7—81no—
297C2×C7#54611yescharacteristic
298C2×C7C2×#13511yescharacteristic
299C2×C7#57611yescharacteristic
300D7—42no—
301D7—81no—
302C23—21no—
303C23—141no—
304C23—141no—
305C23—282no—
306C23—282no—
307C23—561no—
308C23C2×C22⋊D711yescharacteristic
309C23C2×C22⋊D711yescharacteristic
310Q8—141no—
311Q8—141no—
312D4—22no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—72no—
317D4—81no—
318D4—141no—
319D4—141no—
320D4—141no—
321D4—282no—
322D4—282no—
323D4—561no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—282no—
336C2×C4—561no—
337C2×C4C2×C22⋊D711yescharacteristic
338C2×C4C2×C22⋊D711yescharacteristic
339C2×C4C2×C22⋊D711yescharacteristic
340C2×C4C2×C22⋊D711yescharacteristic
341C2×C4C23×D711yescharacteristic
342C8—141no—
343C8—141no—
344C8—141no—
345C8—141no—
346C7#329711yescharacteristic, complemented, Sylow
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—142no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22—281no—
360C22—281no—
361C22—564no—
362C22#651312yes—
363C22C22×C22⋊D711yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—281no—
372C4—281no—
373C4D7×#4111yescharacteristic
374C4#655311yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—282no—
379C2—561no—
380C2C2×#651311yescharacteristic
381C2#9052011yescharacteristic
382C2#9052611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#553C2
socleC22×C7C22×D4
hypercenter#553D7
hypocenterC7#3297
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. C7×#553C2semidirect
2. #90241C2semidirect
3. #89840C2semidirect
4. #89838C2semidirect
5. #89813C2semidirect
6. #89810C2semidirect
7. #89809C2semidirect
8. #89808C2semidirect
9. #89788C2semidirect
10. C4×#1141C2semidirect
11. D4×C7⋊C8C2semidirect
12. #89775C2semidirect
13. #89774C2semidirect
14. #89742C2semidirect
15. C2×#6383C2semidirect
19. C7×#125C22semidirect
23. C7×#121C22semidirect
26. C4×C7×D4C22semidirect
27. C7×#117C22semidirect
30. C2×#1142C22semidirect
36. C22×C7⋊C8C22semidirect
42. #6406C22semidirect
44. C22⋊(C7⋊C8)C22semidirect
45. #6384C22semidirect
46. #6384C22semidirect
47. #6384C22semidirect
54. #6382C22semidirect
55. #6381C22semidirect
56. #6381C22semidirect
57. C4⋊(C7⋊C8)C22semidirect
58. C4×C7⋊C8C22semidirect
112. #553D7semidirect
179. #125C2×D7semidirect
188. #121C2×D7semidirect
193. C4×D4C2×D7semidirect
194. #117C2×D7semidirect
346. C7#3297semidirect

Maximal subgroups

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

subgroupindex
C7×#5532
#902412
#898402
#898382
#898132
#898102
#898092
#898082
#897882
C4×#11412
D4×C7⋊C82
#897752
#897742
#897422
C2×#63832
#32977