Finite Group Databasebeta

← #11739003

Subgroups — #11739003

Order 896.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90424C211yesmaximal, characteristic, complemented
2#90410C211yesmaximal, characteristic, complemented
3C7×#486C211yesmaximal, characteristic, complemented
4#89852C211yesmaximal, characteristic, complemented
5#89846C211yesmaximal, characteristic, complemented
6#89557C211yesmaximal, characteristic, complemented
7#89554C211yesmaximal, characteristic, complemented
8#89552C211yesmaximal, characteristic, complemented
9#89550C211yesmaximal, characteristic, complemented
10#89547C211yesmaximal, characteristic, complemented
11#89530C211yesmaximal, characteristic, complemented
12#89529C211yesmaximal, characteristic, complemented
13#89528C211yesmaximal, characteristic, complemented
14#89469C211yesmaximal, characteristic, complemented
15#89461C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#127C2211yescharacteristic
18#6510—21no—
19#6509—21no—
20#6502—21no—
21#6502—21no—
22#6501C2211yescharacteristic
23#6501C2211yescharacteristic
24#6498—21no—
25#6498—21no—
26D4×Dic7C2211yescharacteristic
27D4×Dic7C2211yescharacteristic
28C2×#1142C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31C2×#1141C2211yescharacteristic
32C2×#1131C2211yescharacteristic
33C2×C8×D7C2211yescharacteristic
34#6461—21no—
35#6460C2211yescharacteristic
36#6458C2211yescharacteristic
37#6457—21no—
38#6456C2211yescharacteristic
39#6450C2211yescharacteristic
40#6438C2211yescharacteristic
41#6435C2211yescharacteristic
42#6433C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C7×#102C2211yescharacteristic
47C4×C7×C8C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3179—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68C22×Dic7—21no—
69C2×D28—21no—
70C2×D28C2311yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7C2311yescharacteristic
73C2×Dic14C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C7×C4⋊C4—21no—
77C7×C4⋊C4C2311yescharacteristic
78C7×C4⋊C4C2311yescharacteristic
79C42×C7C2311yescharacteristic
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84#1142—22no—
85#1142—22no—
86#1141—22no—
87#1141—22no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4D411yescharacteristic
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108C8×D7—22no—
109#542—71no—
110#540—71no—
111#501—71no—
112#496—71no—
113#488—71no—
114#486D711yescharacteristic, complemented
115#469—71no—
116#468—71no—
117#466—71no—
118#465—71no—
119#442—71no—
120#440—71no—
121C4×SD16—71no—
122C4×D8—71no—
123#405—71no—
124C23×C7—21no—
125C23×C7—21no—
126C22×D7—21no—
127C22×D7C2×D411yescharacteristic
128C7×Q8—81no—
129C7×D4—21no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—81no—
134C7×D4—81no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C22⋊D7—81no—
141C22⋊D7—81no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—81no—
146C2×Dic7—81no—
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150D28—21no—
151D28—21no—
152D28—81no—
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—81no—
156Dic14—21no—
157Dic14—21no—
158C7×C8—21no—
159C7×C8—21no—
160C7⋊C8#4112yes—
161C7⋊C8#4112yes—
162C2×SD16—71no—
163C2×SD16—71no—
164C2×D8—71no—
165C2×D8—71no—
166C2×M16—71no—
167C22×C8—71no—
168C4⋊Q8C2×D711yescharacteristic
169#127C2×D711yescharacteristic
170#126—141no—
171#125—71no—
172#124—71no—
173#124—141no—
174#123—141no—
175#123—141no—
176#123—141no—
177C22⋊Q8—71no—
178C22⋊Q8—141no—
179#121—71no—
180#121—71no—
181#121—71no—
182#121—141no—
183#121—141no—
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—71no—
187C4×D4—71no—
188#117—71no—
189#107—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193#103—71no—
194#103—71no—
195#102—71no—
196#102—71no—
197#102C2×D711yescharacteristic
198#102C2×D711yescharacteristic
199#102C2×D711yescharacteristic
200#102C2×D711yescharacteristic
201#97—71no—
202#97—71no—
203C4×C8—71no—
204C4×C8C2×D711yescharacteristic
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7C22×D411yescharacteristic
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7C2×#4111yescharacteristic
223C4×C7C2×#4111yescharacteristic
224Dic7—21no—
225Dic7—21no—
226Dic7—21no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230C2×Q8—21no—
231C2×Q8—71no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—71no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4C22×D711yescharacteristic
239C2×D4C22×D711yescharacteristic
240C22×C4—71no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4—141no—
244SD16—142no—
245SD16—142no—
246D8—142no—
247D8—142no—
248M16—142no—
249C2×C8—71no—
250C2×C8—71no—
251C2×C8—142no—
252C2×C8C22×D711yescharacteristic
253C2×C8C22×D711yescharacteristic
254C4⋊C4—21no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4C22×D711yescharacteristic
264C4⋊C4C22×D711yescharacteristic
265C22⋊C4—71no—
266C22⋊C4—71no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C42—71no—
277C42—141no—
278C42C22×D711yescharacteristic
279C2×C7—81no—
280C2×C7—81no—
281C2×C7#53211yescharacteristic
282C2×C7#57611yescharacteristic
283C2×C7#57711yescharacteristic
284D7—41no—
285D7—81no—
286C23—21no—
287C23—21no—
288C23—71no—
289C23—141no—
290Q8—81no—
291Q8—141no—
292Q8—141no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—81no—
298D4—81no—
299D4—141no—
300D4—141no—
301D4—561no—
302D4—561no—
303D4—561no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—71no—
307C2×C4—71no—
308C2×C4—71no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—561no—
315C2×C4—561no—
316C2×C4—561no—
317C2×C4D4×D711yescharacteristic
318C2×C4D4×D711yescharacteristic
319C2×C4C23×D711yescharacteristic
320C8—21no—
321C8—21no—
322C8—72no—
323C8—72no—
324C7#317911yescharacteristic, complemented, Sylow
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—141no—
332C22—141no—
333C22—141no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—141no—
343C4—141no—
344C4—141no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4D7×#4111yescharacteristic
349C4D7×#4111yescharacteristic
350C2—81no—
351C2—81no—
352C2—281no—
353C2—561no—
354C2#9035911yescharacteristic
355C2#9044811yescharacteristic
356C2#9045311yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#486C2
socleC22×C7C22×D4
hypercenter#486D7
hypocenterC7#3179
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
1. #90424C2semidirect
2. #90410C2semidirect
3. C7×#486C2semidirect
4. #89852C2semidirect
5. #89846C2semidirect
6. #89557C2semidirect
7. #89554C2semidirect
8. #89552C2semidirect
9. #89550C2semidirect
10. #89547C2semidirect
11. #89530C2semidirect
12. #89529C2semidirect
13. #89528C2semidirect
14. #89469C2semidirect
15. #89461C2semidirect
114. #486D7semidirect
324. C7#3179semidirect

Maximal subgroups

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

subgroupindex
#904242
#904102
C7×#4862
#898522
#898462
#895572
#895542
#895522
#895502
#895472
#895302
#895292
#895282
#894692
#894612
#31797