Finite Group Databasebeta

← #11737256

Subgroups — #11737256

Order 896.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#1161C211yesmaximal, characteristic, complemented
2#90239C211yesmaximal, characteristic, complemented
3C7×#440C211yesmaximal, characteristic, complemented
4#89944C211yesmaximal, characteristic, complemented
5#89935C211yesmaximal, characteristic, complemented
6#89798C211yesmaximal, characteristic, complemented
7#89786C211yesmaximal, characteristic, complemented
8#89662C211yesmaximal, characteristic, complemented
9#89657C211yesmaximal, characteristic, complemented
10#89583C211yesmaximal, characteristic, complemented
11#89563C211yesmaximal, characteristic, complemented
12#89558C211yesmaximal, characteristic, complemented
13#89551C211yesmaximal, characteristic, complemented
14#89483C211yesmaximal, characteristic, complemented
15#89477C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1157—21no—
18C2×C7×SD16C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20C4×C7×D4C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6503C2211yescharacteristic
24#6501C2211yescharacteristic
25C2×#1141C2211yescharacteristic
26C4×C22⋊D7—21no—
27C2×D56C2211yescharacteristic
28#6458C2211yescharacteristic
29#6456—21no—
30#6456C2211yescharacteristic
31#6455—21no—
32#6455C2211yescharacteristic
33#6451C2211yescharacteristic
34#6448—21no—
35#6447—21no—
36C4×D28—21no—
37C4×D28C2211yescharacteristic
38C4×D28C2211yescharacteristic
39C42×D7—21no—
40C42×D7C2211yescharacteristic
41C7×#107C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×#97C2211yescharacteristic
45#6409C2211yescharacteristic
46#6406C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6391C2211yescharacteristic
52Dic7⋊C8C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54#6383C2211yescharacteristic
55#6383C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3115—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66#1161—22no—
67#1161—22no—
68#1157—44no—
69C2×D28—21no—
70C2×D28—21no—
71C2×D28C2311yescharacteristic
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7D411yescharacteristic
78C2×C4×D7D411yescharacteristic
79C2×C4×D7C2311yescharacteristic
80C2×Dic14—21no—
81C7×SD16—22no—
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C7×C4⋊C4—21no—
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C22⋊C4—21no—
88C42×C7—21no—
89C42×C7C2311yescharacteristic
90C22⋊Dic7—21no—
91#1143—22no—
92#1141—22no—
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—
98C4⋊Dic7—21no—
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7D411yescharacteristic
105C4×Dic7D411yescharacteristic
106C4×Dic7C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109D56—22no—
110#532—71no—
111C4×#41—71no—
112#482—71no—
113#482—71no—
114#475—71no—
115#474—71no—
116#460—71no—
117#459—71no—
118#449—71no—
119#449—71no—
120#442—71no—
121#442—71no—
122#440—71no—
123#440D711yescharacteristic, complemented
124#432—71no—
125C23×C7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7C2×D411yescharacteristic
129C7×Q8—21no—
130C7×Q8#4112yescomplemented
131C7×D4—21no—
132C7×D4—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—42no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C22⋊D7—42no—
141C22⋊D7—42no—
142C22⋊D7—42no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148D28—24no—
149D28—21no—
150D28—21no—
151D28—21no—
152D28—42no—
153D28#4112yescomplemented
154C4×D7—24no—
155C4×D7—21no—
156C4×D7—21no—
157C4×D7—22no—
158C4×D7—41no—
159C4×D7—42no—
160C4×D7—42no—
161C4×D7—42no—
162Dic14—42no—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C2×#41—71no—
168C2×#41—141no—
169C2×SD16—71no—
170C2×SD16C2×D711yescharacteristic
171C2×D8—71no—
172C2×D8—71no—
173C4⋊Q8—71no—
174#127—71no—
175#124—141no—
176#121—71no—
177#121—71no—
178#121—141no—
179C4×Q8—71no—
180C4×Q8C2×D711yescharacteristic
181C4×D4—71no—
182C4×D4—71no—
183C4×D4—71no—
184C4×D4—141no—
185C4×D4—141no—
186C4×D4—141no—
187C4×D4C2×D711yescharacteristic
188#117—71no—
189#117—141no—
190C2×C42—71no—
191C2×C42—141no—
192#107—71no—
193#107C2×D711yescharacteristic
194#106—71no—
195#106—71no—
196C4⋊C8—71no—
197C4⋊C8—71no—
198#103—71no—
199#103C2×D711yescharacteristic
200#102—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205#102C2×D711yescharacteristic
206C22⋊C8—71no—
207C22⋊C8—71no—
208#97—71no—
209#97C2×D711yescharacteristic
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×D411yescharacteristic
214C2×D7—21no—
215C2×D7—22no—
216C2×D7—21no—
217C2×D7—21no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C4×C7—22no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7C2×#4111yescharacteristic
229C4×C7C2×#4111yescharacteristic
230Dic7—21no—
231Dic7—21no—
232Dic7—41no—
233Dic7—41no—
234Dic7AGL(1,ℤ/8)12yes—
235#41—142no—
236#41—142no—
237#41—284no—
238C2×Q8—141no—
239C2×Q8C22×D711yescharacteristic
240C2×D4—71no—
241C2×D4—71no—
242C2×D4—141no—
243C2×D4—141no—
244C2×D4—141no—
245C2×D4—141no—
246C2×D4—141no—
247C2×D4C22×D711yescharacteristic
248C22×C4—21no—
249C22×C4—71no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4—141no—
256SD16—22no—
257SD16—142no—
258D8—142no—
259D8—142no—
260C2×C8—71no—
261C2×C8—71no—
262C2×C8C22×D711yescharacteristic
263C2×C8C22×D711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4C22×D711yescharacteristic
270C4⋊C4C22×D711yescharacteristic
271C22⋊C4—21no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C42—21no—
279C42—71no—
280C42—71no—
281C42—71no—
282C42—141no—
283C42—141no—
284C42C22×D711yescharacteristic
285C2×C7—81no—
286C2×C7#52511yescharacteristic
287C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
288C2×C7#57511yescharacteristic
289D7—41no—
290D7—42no—
291D7—81no—
292C23—21no—
293C23—71no—
294C23—141no—
295C23—141no—
296Q8—21no—
297Q8—282no—
298Q8#115712yescomplemented
299D4—21no—
300D4—21no—
301D4—72no—
302D4—144no—
303D4—141no—
304D4—141no—
305D4—141no—
306D4—282no—
307D4—282no—
308D4—282no—
309D4—282no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—71no—
315C2×C4—71no—
316C2×C4—71no—
317C2×C4—142no—
318C2×C4—144no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—281no—
324C2×C4—282no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4D4×D711yescharacteristic
328C2×C4D4×D711yescharacteristic
329C2×C4C23×D711yescharacteristic
330C8—21no—
331C8—21no—
332C8—141no—
333C8—141no—
334C7#311511yescharacteristic, complemented, Sylow
335C22—41no—
336C22—41no—
337C22—41no—
338C22—141no—
339C22—141no—
340C22—142no—
341C22—141no—
342C22—281no—
343C22—281no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22C2×D4×D711yescharacteristic
348C4—21no—
349C4—21no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—72no—
354C4—141no—
355C4—141no—
356C4—281no—
357C4—281no—
358C4C2×#115711yescharacteristic
359C4D7×#4111yescharacteristic
360C2—81no—
361C2—281no—
362C2—282no—
363C2—561no—
364C2#9023811yescharacteristic
365C2D7×AGL(1,ℤ/8)11yescharacteristic
366C2#9046811yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
hypercenter#440D7
hypocenterC7#3115
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C4×#1161C2semidirect
2. #90239C2semidirect
3. C7×#440C2semidirect
4. #89944C2semidirect
5. #89935C2semidirect
6. #89798C2semidirect
7. #89786C2semidirect
8. #89662C2semidirect
9. #89657C2semidirect
10. #89583C2semidirect
11. #89563C2semidirect
12. #89558C2semidirect
13. #89551C2semidirect
14. #89483C2semidirect
15. #89477C2semidirect
123. #440D7semidirect
130. C7×Q8#41semidirect
153. D28#41semidirect
298. Q8#1157semidirect
334. C7#3115semidirect

Maximal subgroups

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

subgroupindex
C4×#11612
#902392
C7×#4402
#899442
#899352
#897982
#897862
#896622
#896572
#895832
#895632
#895582
#895512
#894832
#894772
#31157