Finite Group Databasebeta

← #11740021

Subgroups — #11740021

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
1C7×#520C211yesmaximal, characteristic, complemented
2#90225C211yesmaximal, characteristic, complemented
3#90011C211yesmaximal, characteristic, complemented
4#90010C211yesmaximal, characteristic
5#89802C211yesmaximal, characteristic, complemented
6#89800C211yesmaximal, characteristic, complemented
7C4×#1143C211yesmaximal, characteristic, complemented
8#89792C211yesmaximal, characteristic, complemented
9C4×#1142C211yesmaximal, characteristic, complemented
10#89788C211yesmaximal, characteristic, complemented
11#89786C211yesmaximal, characteristic, complemented
12#89780C211yesmaximal, characteristic, complemented
13#89772C211yesmaximal, characteristic, complemented
14#89743C211yesmaximal, characteristic, complemented
15#89740C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C4×C7×Q8C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4—21no—
20C4×C7×D4—21no—
21C4×C7×D4C2211yescharacteristic
22C4×C7×D4C2211yescharacteristic
23C7×#117—21no—
24C7×#117C2211yescharacteristic
25C2×C7×C4⋊C4—21no—
26C2×C7×C4⋊C4C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6493C2211yescharacteristic
32#6487C2211yescharacteristic
33#6461—21no—
34#6459—21no—
35#6448—21no—
36#6446—21no—
37#6438C2211yescharacteristic
38C4×D28C2211yescharacteristic
39#6432C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41#6409C2211yescharacteristic
42#6409C2211yescharacteristic
43#6406C2211yescharacteristic
44#6406C2211yescharacteristic
45C22⋊(C7⋊C8)C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6382C2211yescharacteristic
54#6382C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57#6377C2211yescharacteristic
58C4×C7⋊C8C2211yescharacteristic
59#3125—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—22no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7C2311yescharacteristic
70C2×C22⋊D7—21no—
71C2×D28C2311yescharacteristic
72C2×C4×D7—21no—
73C2×Dic14—21no—
74C2×Dic14C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4—21no—
77C7×C4⋊C4—21no—
78C7×C4⋊C4D411yescharacteristic
79C7×C4⋊C4D411yescharacteristic
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4D411yescharacteristic
85C7×C22⋊C4D411yescharacteristic
86C42×C7—21no—
87C42×C7—21no—
88C42×C7C2311yescharacteristic
89C42×C7C2311yescharacteristic
90C22⋊Dic7—21no—
91#1144—22no—
92#1143—22no—
93#1142—22no—
94#1141—22no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104C4×Dic7—21no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#542—71no—
110#520D711yescharacteristic, complemented
111#483—71no—
112#481—71no—
113#476—71no—
114#475—71no—
115#464—71no—
116#463—71no—
117#452—71no—
118#449—71no—
119#442—71no—
120#441—71no—
121C4×SD16—71no—
122C4×SD16—71no—
123#433—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C7×Q8—21no—
128C7×Q8#4112yes—
129C7×D4—24no—
130C7×D4—21no—
131C7×D4#4112yes—
132C2×C4×C7—24no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—81no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C22⋊D7—81no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151D28—21no—
152D28—21no—
153C4×D7—81no—
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×SD16—71no—
165C2×D8—71no—
166C4⋊Q8—71no—
167#126—141no—
168#124—71no—
169#124—141no—
170#123—141no—
171C22⋊Q8—71no—
172C22⋊Q8—141no—
173#121—71no—
174C4×Q8—71no—
175C4×Q8C2×D711yescharacteristic
176C4×Q8C2×D711yescharacteristic
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—71no—
180C4×D4C2×D711yescharacteristic
181C4×D4C2×D711yescharacteristic
182#117—21no—
183#117C2×D711yescharacteristic
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×D711yescharacteristic
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#103—71no—
196#102—71no—
197#102—71no—
198#102—71no—
199#102—71no—
200C22⋊C8—71no—
201C22⋊C8—71no—
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—21no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7C2×#4111yescharacteristic
223C4×C7C2×#4111yescharacteristic
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—71no—
234C2×D4—141no—
235C2×D4C22×D711yescharacteristic
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—141no—
241C22×C4C22×D711yescharacteristic
242Q16—142no—
243SD16—142no—
244SD16—142no—
245D8—142no—
246C2×C8—71no—
247C2×C8—71no—
248C2×C8—71no—
249C2×C8—71no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—71no—
254C4⋊C4—71no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4C22⋊D711yescharacteristic
259C4⋊C4C22⋊D711yescharacteristic
260C4⋊C4C22×D711yescharacteristic
261C4⋊C4C22×D711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4C22⋊D711yescharacteristic
270C22⋊C4C22⋊D711yescharacteristic
271C42—21no—
272C42—21no—
273C42—141no—
274C42C22×D711yescharacteristic
275C42C22×D711yescharacteristic
276C2×C7—41no—
277C2×C7—42no—
278C2×C7#52511yescharacteristic
279C2×C7#57711yescharacteristic
280C2×C7#57711yescharacteristic
281D7—81no—
282C23—21no—
283C23—141no—
284C23C2×C22⋊D711yescharacteristic
285Q8—21no—
286Q8—141no—
287Q8—141no—
288Q8—561no—
289Q8#115712yes—
290D4—24no—
291D4—21no—
292D4—141no—
293D4—141no—
294D4—561no—
295D4#115712yes—
296C2×C4—24no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—561no—
309C2×C4C2×C22⋊D711yescharacteristic
310C2×C4C2×C22⋊D711yescharacteristic
311C2×C4C2×C22⋊D711yescharacteristic
312C2×C4C2×C22⋊D711yescharacteristic
313C2×C4C2×C22⋊D711yescharacteristic
314C2×C4C23×D711yescharacteristic
315C8—141no—
316C8—141no—
317C8—141no—
318C8—141no—
319C7#312511yescharacteristic, complemented, Sylow
320C22—21no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—281no—
327C22—281no—
328C22—281no—
329C22C22×C22⋊D711yescharacteristic
330C4—21no—
331C4—22no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4C2×#115711yescharacteristic
342C4C2×#115711yescharacteristic
343C2—41no—
344C2—42no—
345C2—561no—
346C2#9048211yescharacteristic
347C2#9052711yescharacteristic
348C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#520C2
socleC22×C7C22×D4
prime radical#90010C2
hypercenter#520D7
hypocenterC7#3125
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. C7×#520C2semidirect
2. #90225C2semidirect
3. #90011C2semidirect
5. #89802C2semidirect
6. #89800C2semidirect
7. C4×#1143C2semidirect
8. #89792C2semidirect
9. C4×#1142C2semidirect
10. #89788C2semidirect
11. #89786C2semidirect
12. #89780C2semidirect
13. #89772C2semidirect
14. #89743C2semidirect
15. #89740C2semidirect
110. #520D7semidirect
319. C7#3125semidirect

Maximal subgroups

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

subgroupindex
C7×#5202
#902252
#900112
#900102
#898022
#898002
C4×#11432
#897922
C4×#11422
#897882
#897862
#897802
#897722
#897432
#897402
#31257