Finite Group Databasebeta

← #11737663

Subgroups — #11737663

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#90334C211yesmaximal, characteristic, complemented
3C7×#452C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic
5#89957C211yesmaximal, characteristic, complemented
6Q16×Dic7C211yesmaximal, characteristic, complemented
7#89937C211yesmaximal, characteristic, complemented
8#89936C211yesmaximal, characteristic, complemented
9#89816C211yesmaximal, characteristic, complemented
10#89589C211yesmaximal, characteristic, complemented
11#89566C211yesmaximal, characteristic, complemented
12#89531C211yesmaximal, characteristic, complemented
13#89528C211yesmaximal, characteristic, complemented
14#89508C211yesmaximal, characteristic, complemented
15#89492C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×Q16C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C7×C22⋊Q8C2211yescharacteristic
20#6510C2211yescharacteristic
21Q8×Dic7C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24C2×#1143C2211yescharacteristic
25D4×Dic7—21no—
26D4×Dic7—21no—
27D4×Dic7C2211yescharacteristic
28D4×Dic7C2211yescharacteristic
29C2×#1141C2211yescharacteristic
30#6493C2211yescharacteristic
31#6489—21no—
32#6489C2211yescharacteristic
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7C2211yescharacteristic
35#6461—21no—
36#6460—21no—
37#6456C2211yescharacteristic
38#6451C2211yescharacteristic
39#6448—21no—
40#6446—21no—
41C7×#103C2211yescharacteristic
42C7×#102C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3124—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—22no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C22×Dic7—21no—
70C22×Dic7—21no—
71C22×Dic7—21no—
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×Dic14—21no—
75C7×Q16—22no—
76C7×SD16—22no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4C2311yescharacteristic
81C7×C22⋊C4—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7D411yescharacteristic
85C22⋊Dic7D411yescharacteristic
86#1143—22no—
87#1141—22no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7D411yescharacteristic
97C4⋊Dic7D411yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
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#484—71no—
112#480—71no—
113#477—71no—
114#474—71no—
115#464—71no—
116#463—71no—
117#452D711yescharacteristic, complemented
118#449—71no—
119#440—71no—
120#440—71no—
121C4×Q16—71no—
122C4×D8—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—21no—
130C7×Q8#4112yes—
131C7×D4—24no—
132C7×D4—21no—
133C7×D4#4112yes—
134C2×C4×C7—24no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—81no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—81no—
147C2×Dic7—81no—
148C2×Dic7—81no—
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153D28—21no—
154D28—21no—
155C4×D7—81no—
156Dic14—81no—
157C7×C8—21no—
158C7×C8—21no—
159C7⋊C8—21no—
160C7⋊C8—21no—
161C2×#41C2×D711yescharacteristic
162C2×Q16C2×D711yescharacteristic
163C2×SD16—71no—
164C2×SD16C2×D711yescharacteristic
165C2×D8—71no—
166C4⋊Q8—71no—
167#126—141no—
168#124—71no—
169#124—141no—
170#123—141no—
171C22⋊Q8—141no—
172C22⋊Q8C2×D711yescharacteristic
173#121—71no—
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#107—71no—
187#107—71no—
188#107—71no—
189#107—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×C7C22×D411yescharacteristic
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C4×C7—21no—
215C4×C7—22no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7C2×#4111yescharacteristic
219C4×C7C2×#4111yescharacteristic
220Dic7—21no—
221Dic7—21no—
222Dic7—21no—
223Dic7—21no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227#41—22no—
228#41—22no—
229C2×Q8—141no—
230C2×Q8C22×D711yescharacteristic
231C2×Q8C22×D711yescharacteristic
232C2×D4—21no—
233C2×D4—71no—
234C2×D4—141no—
235C2×D4C22×D711yescharacteristic
236C22×C4—21no—
237C22×C4—141no—
238C22×C4—141no—
239C22×C4—141no—
240C22×C4—141no—
241C22×C4C22×D711yescharacteristic
242Q16—22no—
243SD16—22no—
244SD16—142no—
245D8—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8C22×D711yescharacteristic
249C2×C8C22×D711yescharacteristic
250C4⋊C4—21no—
251C4⋊C4—71no—
252C4⋊C4—71no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—71no—
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#52511yescharacteristic
279C2×C7#57611yescharacteristic
280C2×C7#57811yescharacteristic
281D7—81no—
282C23—21no—
283C23—141no—
284C23D4×D711yescharacteristic
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—561no—
289Q8#115912yes—
290D4—24no—
291D4—21no—
292D4—141no—
293D4—141no—
294D4—561no—
295D4#115912yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—71no—
303C2×C4—71no—
304C2×C4—71no—
305C2×C4—71no—
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#312411yescharacteristic, complemented, Sylow
320C22—21no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—141no—
335C4—141no—
336C4—141no—
337C4—141no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4C2×#115911yescharacteristic
342C4C2×#115911yescharacteristic
343C2—41no—
344C2—42no—
345C2—561no—
346C2#9027711yescharacteristic
347C2#9046011yescharacteristic
348C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#452C2
socleC22×C7C22×D4
prime radical#90010C2
hypercenter#452D7
hypocenterC7#3124
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #90518C2semidirect
2. #90334C2semidirect
3. C7×#452C2semidirect
5. #89957C2semidirect
6. Q16×Dic7C2semidirect
7. #89937C2semidirect
8. #89936C2semidirect
9. #89816C2semidirect
10. #89589C2semidirect
11. #89566C2semidirect
12. #89531C2semidirect
13. #89528C2semidirect
14. #89508C2semidirect
15. #89492C2semidirect
117. #452D7semidirect
319. C7#3124semidirect

Maximal subgroups

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

subgroupindex
#905182
#903342
C7×#4522
#900102
#899572
Q16×Dic72
#899372
#899362
#898162
#895892
#895662
#895312
#895282
#895082
#894922
#31247