Finite Group Databasebeta

← #11738293

Subgroups — #11738293

Order 896.

408 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90291C211yesmaximal, characteristic, complemented
2#90287C211yesmaximal, characteristic, complemented
3C7×#469C211yesmaximal, characteristic, complemented
4#89926C211yesmaximal, characteristic, complemented
5D8×Dic7C211yesmaximal, characteristic, complemented
6#89906C211yesmaximal, characteristic, complemented
7M16×Dic7C211yesmaximal, characteristic, complemented
8#89814C211yesmaximal, characteristic, complemented
9#89811C211yesmaximal, characteristic, complemented
10#89628C211yesmaximal, characteristic, complemented
11#89624C211yesmaximal, characteristic, complemented
12#89537C211yesmaximal, characteristic, complemented
13#89536C211yesmaximal, characteristic, complemented
14#89530C211yesmaximal, characteristic, complemented
15#89528C211yesmaximal, characteristic, complemented
16C2×#1159—21no—
17C2×C7×D8C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C7×#121C2211yescharacteristic
20C7×#121C2211yescharacteristic
21#6503C2211yescharacteristic
22#6502—21no—
23#6499C2211yescharacteristic
24#6498—21no—
25D4×Dic7—21no—
26D4×Dic7C2211yescharacteristic
27D4×Dic7C2211yescharacteristic
28C2×#1142C2211yescharacteristic
29C2×#1141C2211yescharacteristic
30#6493C2211yescharacteristic
31#6487C2211yescharacteristic
32C2×C4×Dic7C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C2×#1132C2211yescharacteristic
35#6456C2211yescharacteristic
36#6453C2211yescharacteristic
37#6451C2211yescharacteristic
38#6450C2211yescharacteristic
39#6448—21no—
40(C22⋊D7)⋊C4—21no—
41#6442—21no—
42#6440—21no—
43C7×#106C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49#6392C2211yescharacteristic
50#6390C2211yescharacteristic
51#6389C2211yescharacteristic
52#6389C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54C8×Dic7C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6382C2211yescharacteristic
58#6381C2211yescharacteristic
59#3171—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C22×Dic7—21no—
69C22×Dic7—21no—
70C22×Dic7D411yescharacteristic
71C22×Dic7D411yescharacteristic
72#1159—44no—
73C2×D28C2311yescharacteristic
74C2×C4×D7—21no—
75C2×Dic14—21no—
76C2×Dic14C2311yescharacteristic
77C7×D8—22no—
78C7×M16—22no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C7×C22⋊C4—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90#1142—22no—
91#1141—22no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C4×Dic7D411yescharacteristic
103C4×Dic7D411yescharacteristic
104C4×Dic7C2311yescharacteristic
105C4×Dic7C2311yescharacteristic
106#1136—22no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#1132—22no—
110#532—71no—
111#529—71no—
112#501—71no—
113#496—71no—
114#490—71no—
115#489—71no—
116#469—71no—
117#469D711yescharacteristic, complemented
118#468—71no—
119#468—71no—
120C4×SD16—71no—
121C4×SD16—71no—
122C4×D8—71no—
123C4×D8—71no—
124C4×M16—71no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7C2×D411yescharacteristic
128C22×D7—21no—
129C7×D4—21no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—42no—
134C7×D4—42no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C22⋊D7—42no—
142C22⋊D7—42no—
143C22⋊D7—42no—
144C2×Dic7—22no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—41no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155D28—21no—
156D28—21no—
157C4×D7—42no—
158Dic14—21no—
159Dic14—21no—
160Dic14—42no—
161C7×C8—21no—
162C7×C8#4112yescomplemented
163C7⋊C8—21no—
164C7⋊C8#4112yescomplemented
165C2×#41—141no—
166C2×SD16—71no—
167C2×SD16—71no—
168C2×D8—71no—
169C2×D8C2×D711yescharacteristic
170C2×M16—71no—
171C2×M16C2×D711yescharacteristic
172C4⋊Q8—71no—
173#127—71no—
174#126—141no—
175#125—71no—
176#124—71no—
177#124—141no—
178#123—141no—
179C22⋊Q8—71no—
180#121—71no—
181#121—141no—
182#121C2×D711yescharacteristic
183#121C2×D711yescharacteristic
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—71no—
187C4×D4—71no—
188C4×D4—141no—
189C4×D4—141no—
190#117—141no—
191C2×C42—71no—
192#107—71no—
193#107—71no—
194#106—71no—
195#106C2×D711yescharacteristic
196#103—71no—
197#103—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102C2×D711yescharacteristic
203#102C2×D711yescharacteristic
204#97—71no—
205#97—71no—
206C4×C8—71no—
207C4×C8—71no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7C22×D411yescharacteristic
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7C2×#4111yescharacteristic
225C4×C7C2×#4111yescharacteristic
226Dic7—21no—
227Dic7—21no—
228Dic7—21no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7AGL(1,ℤ/8)12yes—
233#41—284no—
234C2×Q8—71no—
235C2×Q8—141no—
236C2×D4—21no—
237C2×D4—21no—
238C2×D4—71no—
239C2×D4—141no—
240C2×D4—141no—
241C2×D4—141no—
242C2×D4C22×D711yescharacteristic
243C2×D4C22×D711yescharacteristic
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—141no—
247C22×C4—141no—
248C22×C4—141no—
249C22×C4C22×D711yescharacteristic
250SD16—142no—
251SD16—142no—
252D8—22no—
253D8—142no—
254M16—22no—
255M16—142no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8C22×D711yescharacteristic
259C2×C8C22×D711yescharacteristic
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4C22×D711yescharacteristic
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C42—71no—
279C42—71no—
280C42—71no—
281C42—71no—
282C42—141no—
283C2×C7—41no—
284C2×C7—81no—
285C2×C7—81no—
286C2×C7#53211yescharacteristic
287C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
288C2×C7#57511yescharacteristic
289D7—81no—
290C23—21no—
291C23—21no—
292C23—141no—
293C23D4×D711yescharacteristic
294Q8—141no—
295Q8—141no—
296Q8—282no—
297D4—21no—
298D4—21no—
299D4—21no—
300D4—21no—
301D4—42no—
302D4—42no—
303D4—141no—
304D4—141no—
305D4—282no—
306D4—282no—
307D4—282no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—71no—
313C2×C4—71no—
314C2×C4—71no—
315C2×C4—71no—
316C2×C4—142no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—282no—
321C2×C4—282no—
322C2×C4—282no—
323C2×C4—281no—
324C2×C4D4×D711yescharacteristic
325C2×C4C23×D711yescharacteristic
326C8—21no—
327C8—72no—
328C8—141no—
329C8#115912yescomplemented
330C7#317111yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—281no—
341C22—281no—
342C22—281no—
343C22C2×D4×D711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—72no—
348C4—141no—
349C4—141no—
350C4—141no—
351C4—281no—
352C4—281no—
353C4—281no—
354C4C2×#115911yescharacteristic
355C4D7×#4111yescharacteristic
356C2—41no—
357C2—81no—
358C2—81no—
359C2—561no—
360C2#9028711yescharacteristic
361C2D7×AGL(1,ℤ/8)11yescharacteristic
362C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#469C2
socleC22×C7C22×D4
hypercenter#469D7
hypocenterC7#3171
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #90291C2semidirect
2. #90287C2semidirect
3. C7×#469C2semidirect
4. #89926C2semidirect
5. D8×Dic7C2semidirect
6. #89906C2semidirect
7. M16×Dic7C2semidirect
8. #89814C2semidirect
9. #89811C2semidirect
10. #89628C2semidirect
11. #89624C2semidirect
12. #89537C2semidirect
13. #89536C2semidirect
14. #89530C2semidirect
15. #89528C2semidirect
117. #469D7semidirect
162. C7×C8#41semidirect
164. C7⋊C8#41semidirect
329. C8#1159semidirect
330. C7#3171semidirect

Maximal subgroups

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

subgroupindex
#902912
#902872
C7×#4692
#899262
D8×Dic72
#899062
M16×Dic72
#898142
#898112
#896282
#896242
#895372
#895362
#895302
#895282
#31717