Finite Group Databasebeta

← #11737520

Subgroups — #11737520

Order 896.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90518C211yesmaximal, characteristic, complemented
2#90294C211yesmaximal, characteristic, complemented
3C7×#449C211yesmaximal, characteristic
4#90011C211yesmaximal, characteristic, complemented
5#89938C211yesmaximal, characteristic, complemented
6SD16×Dic7C211yesmaximal, characteristic, complemented
7#89925C211yesmaximal, characteristic, complemented
8#89924C211yesmaximal, characteristic, complemented
9#89806C211yesmaximal, characteristic, complemented
10#89563C211yesmaximal, characteristic, complemented
11#89561C211yesmaximal, characteristic, complemented
12#89533C211yesmaximal, characteristic, complemented
13#89530C211yesmaximal, characteristic, complemented
14#89493C211yesmaximal, characteristic, complemented
15#89492C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C2×C7×D8C2211yescharacteristic
19C7×#121C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23#6499—21no—
24#6499C2211yescharacteristic
25#6498—21no—
26D4×Dic7—21no—
27D4×Dic7—21no—
28D4×Dic7C2211yescharacteristic
29D4×Dic7C2211yescharacteristic
30D4×Dic7C2211yescharacteristic
31C2×#1142C2211yescharacteristic
32#6489—21no—
33#6489C2211yescharacteristic
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7C2211yescharacteristic
36#6487C2211yescharacteristic
37#6451C2211yescharacteristic
38#6450C2211yescharacteristic
39#6442—21no—
40#6441—21no—
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49#6391C2211yescharacteristic
50#6391C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6382C2211yescharacteristic
58#6382C2211yescharacteristic
59#3125—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—22no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7—21no—
68C22×C4×C7C2311yescharacteristic
69C22×Dic7—21no—
70C22×Dic7—21no—
71C22×Dic7—21no—
72C22×Dic7—21no—
73C2×Dic14—21no—
74C2×Dic14C2311yescharacteristic
75C7×SD16—22no—
76C7×D8—22no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4C2311yescharacteristic
80C7×C22⋊C4—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7D411yescharacteristic
88C22⋊Dic7D411yescharacteristic
89#1144—22no—
90#1142—22no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7D411yescharacteristic
95C4⋊Dic7D411yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106C4×Dic7C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#542—71no—
110#520—71no—
111#483—71no—
112#481—71no—
113#476—71no—
114#475—71no—
115#464—71no—
116#463—71no—
117#452—71no—
118#449D711yescharacteristic
119#442—71no—
120#441—71no—
121C4×SD16—71no—
122C4×SD16—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7C2×D411yescharacteristic
127C7×Q8—21no—
128C7×Q8#4112yes—
129C7×D4—24no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—81no—
134C7×D4#4112yes—
135C2×C4×C7—24no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—81no—
147C2×Dic7—81no—
148C2×Dic7—81no—
149C2×Dic7—81no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154Dic14—21no—
155Dic14—21no—
156Dic14—81no—
157C7×C8—21no—
158C7×C8—21no—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C2×#41C2×D711yescharacteristic
162C2×Q16—71no—
163C2×SD16—71no—
164C2×SD16C2×D711yescharacteristic
165C2×D8C2×D711yescharacteristic
166C4⋊Q8—71no—
167#126—141no—
168#124—71no—
169#124—141no—
170#123—141no—
171C22⋊Q8—71no—
172C22⋊Q8—141no—
173#121C2×D711yescharacteristic
174C4×Q8—71no—
175C4×Q8—71no—
176C4×Q8—71no—
177C4×D4—71no—
178C4×D4—71no—
179C4×D4—71no—
180C4×D4—141no—
181C4×D4—141no—
182#117—71no—
183#117—141no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—141no—
186#106—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191C4⋊C8—71no—
192#103—71no—
193#103—71no—
194#103—71no—
195#103C2×D711yescharacteristic
196#102—71no—
197#102—71no—
198#102—71no—
199#102C2×D711yescharacteristic
200C22⋊C8—71no—
201C22⋊C8C2×D711yescharacteristic
202#97—71no—
203C4×C8—71no—
204C22×C7—21no—
205C22×C7—22no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×D411yescharacteristic
214C4×C7—21no—
215C4×C7—22no—
216C4×C7—41no—
217C4×C7C2×#4111yescharacteristic
218C4×C7C2×#4111yescharacteristic
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—21no—
223Dic7—41no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227#41—22no—
228#41—22no—
229C2×Q8—71no—
230C2×Q8—141no—
231C2×Q8C22×D711yescharacteristic
232C2×D4—21no—
233C2×D4—21no—
234C2×D4C22×D711yescharacteristic
235C2×D4C22×D711yescharacteristic
236C22×C4—21no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4C22×D711yescharacteristic
242Q16—142no—
243SD16—22no—
244SD16—142no—
245D8—22no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8C22×D711yescharacteristic
249C2×C8C22×D711yescharacteristic
250C4⋊C4—71no—
251C4⋊C4—71no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4C22×D711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—71no—
264C22⋊C4—71no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C42—71no—
272C42—71no—
273C42—141no—
274C42—141no—
275C42—141no—
276C2×C7—41no—
277C2×C7—42no—
278C2×C7—81no—
279C2×C7#52511yescharacteristic
280C2×C7#57711yescharacteristic
281C2×C7#57711yescharacteristic
282C23—21no—
283C23—21no—
284C23D4×D711yescharacteristic
285Q8—21no—
286Q8—141no—
287Q8—141no—
288Q8—561no—
289Q8#115912yes—
290D4—24no—
291D4—21no—
292D4—21no—
293D4—21no—
294D4—81no—
295D4#115912yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—71no—
302C2×C4—71no—
303C2×C4—71no—
304C2×C4—71no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—561no—
310C2×C4—561no—
311C2×C4—561no—
312C2×C4—561no—
313C2×C4D4×D711yescharacteristic
314C2×C4C23×D711yescharacteristic
315C8—21no—
316C8—21no—
317C8—141no—
318C8—141no—
319C7#312511yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—141no—
334C4—141no—
335C4—141no—
336C4—141no—
337C4—281no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4C2×#115911yescharacteristic
342C4C2×#115911yescharacteristic
343C2—41no—
344C2—42no—
345C2—81no—
346C2#9027711yescharacteristic
347C2#9046111yescharacteristic
348C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#449C2
socleC22×C7C22×D4
hypercenter#449D7
hypocenterC7#3125
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #90518C2semidirect
2. #90294C2semidirect
4. #90011C2semidirect
5. #89938C2semidirect
6. SD16×Dic7C2semidirect
7. #89925C2semidirect
8. #89924C2semidirect
9. #89806C2semidirect
10. #89563C2semidirect
11. #89561C2semidirect
12. #89533C2semidirect
13. #89530C2semidirect
14. #89493C2semidirect
15. #89492C2semidirect
319. C7#3125semidirect

Maximal subgroups

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

subgroupindex
#905182
#902942
C7×#4492
#900112
#899382
SD16×Dic72
#899252
#899242
#898062
#895632
#895612
#895332
#895302
#894932
#894922
#31257