Finite Group Databasebeta

← #11740914

Subgroups — #11740914

Order 896.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#542C211yesmaximal, characteristic, complemented
2#90225C211yesmaximal, characteristic, complemented
3#89866C211yesmaximal, characteristic, complemented
4#89856C211yesmaximal, characteristic, complemented
5#89829C211yesmaximal, characteristic, complemented
6#89826C211yesmaximal, characteristic, complemented
7#89824C211yesmaximal, characteristic, complemented
8#89820C211yesmaximal, characteristic, complemented
9#89813C211yesmaximal, characteristic, complemented
10#89811C211yesmaximal, characteristic
11#89800C211yesmaximal, characteristic, complemented
12#89792C211yesmaximal, characteristic, complemented
13#89788C211yesmaximal, characteristic, complemented
14#89780C211yesmaximal, characteristic, complemented
15#89771C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×#126—21no—
18C7×#124—21no—
19C7×#124C2211yescharacteristic
20C7×#123—21no—
21C7×C22⋊Q8—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×#121C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4C2211yescharacteristic
26C7×#117C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6493C2211yescharacteristic
32#6487C2211yescharacteristic
33C2×#1136C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35#6461—21no—
36#6459—21no—
37#6448—21no—
38#6446—21no—
39#6438C2211yescharacteristic
40C4×D28C2211yescharacteristic
41#6432C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6382C2211yescharacteristic
54#6382C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3263—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C2×C22⋊D7—21no—
67C2×D28C2311yescharacteristic
68C2×C4×D7—21no—
69C2×Dic14—21no—
70C2×Dic14C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4D411yescharacteristic
75C7×C4⋊C4D411yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C4⋊C4C2311yescharacteristic
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C7×C22⋊C4—21no—
83C7×C22⋊C4D411yescharacteristic
84C7×C22⋊C4D411yescharacteristic
85C42×C7—21no—
86C42×C7C2311yescharacteristic
87C42×C7C2311yescharacteristic
88C22⋊Dic7—21no—
89#1144—41no—
90#1143—41no—
91#1142—41no—
92#1141—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4—21no—
97C4⋊Dic7—21no—
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103#1136—41no—
104C2×C7⋊C8—41no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#542—71no—
110#542D711yescharacteristic, complemented
111#476—71no—
112#476—71no—
113#475—71no—
114#475—71no—
115#470—71no—
116#469—71no—
117#465—71no—
118#465—71no—
119#464—71no—
120#464—71no—
121#463—71no—
122#463—71no—
123#424—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×Q8—81no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—81no—
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×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2411yescharacteristic
145C22⋊D7—81no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149D28—21no—
150D28—21no—
151C4×D7—81no—
152Dic14—21no—
153Dic14—21no—
154Dic14—81no—
155C7⋊C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C7⋊C8—21no—
159C2×Q16—71no—
160C2×SD16—71no—
161C2×SD16—71no—
162C2×D8—71no—
163C2×M16—71no—
164C22×C8—71no—
165C4⋊Q8—71no—
166C4⋊Q8C2×D711yescharacteristic
167#126—21no—
168#126—141no—
169#124—21no—
170#124—71no—
171#124—141no—
172#124C2×D711yescharacteristic
173#123—21no—
174#123—141no—
175C22⋊Q8—21no—
176C22⋊Q8—71no—
177C22⋊Q8—141no—
178C22⋊Q8C2×D711yescharacteristic
179#121—71no—
180#121C2×D711yescharacteristic
181C4×Q8—71no—
182C4×Q8C2×D711yescharacteristic
183C4×D4—71no—
184C4×D4C2×D711yescharacteristic
185#117C2×D711yescharacteristic
186#106—71no—
187#106—71no—
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191C4⋊C8—71no—
192C4⋊C8—71no—
193C4⋊C8—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#102—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—21no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7C22×D411yescharacteristic
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—21no—
216C4×C7—21no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7H(5,𝔽2)11yescharacteristic
221C4×C72−1+411yescharacteristic
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—21no—
226C2×Q8—71no—
227C2×Q8—141no—
228C2×Q8C22×D711yescharacteristic
229C2×D4—21no—
230C2×D4—71no—
231C2×D4—141no—
232C2×D4C22×D711yescharacteristic
233C22×C4—21no—
234C22×C4—141no—
235C22×C4C22×D711yescharacteristic
236Q16—281no—
237SD16—281no—
238SD16—281no—
239D8—281no—
240M16—281no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8—71no—
244C2×C8—71no—
245C2×C8—281no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4C22⋊D711yescharacteristic
255C4⋊C4C22⋊D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4C22⋊D711yescharacteristic
269C22⋊C4C22⋊D711yescharacteristic
270C42—21no—
271C42—141no—
272C42C22×D711yescharacteristic
273C42C22×D711yescharacteristic
274C2×C7—41no—
275C2×C7—81no—
276C2×C7#53711yescharacteristic
277C2×C7#57711yescharacteristic
278C2×C7#57711yescharacteristic
279D7—81no—
280C23—21no—
281C23—141no—
282C23C2×C22⋊D711yescharacteristic
283Q8—21no—
284Q8—21no—
285Q8—81no—
286Q8—141no—
287Q8—141no—
288Q8—561no—
289D4—21no—
290D4—21no—
291D4—81no—
292D4—141no—
293D4—141no—
294D4—561no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—81no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—561no—
305C2×C4C2×C22⋊D711yescharacteristic
306C2×C4C2×C22⋊D711yescharacteristic
307C2×C4C2×C22⋊D711yescharacteristic
308C2×C4C2×C22⋊D711yescharacteristic
309C2×C4C2×C22⋊D711yescharacteristic
310C2×C4C23×D711yescharacteristic
311C8—141no—
312C8—141no—
313C8—141no—
314C8—141no—
315C7#326311yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—281no—
323C22—281no—
324C22—281no—
325C22C22×C22⋊D711yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4#655311yescharacteristic
338C4#655411yescharacteristic
339C2—41no—
340C2—81no—
341C2—561no—
342C2#9052411yescharacteristic
343C2#9052711yescharacteristic
344C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#542C2
socleC22×C7C22×D4
prime radical#89811C2
hypercenter#542D7
hypocenterC7#3263
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. C7×#542C2semidirect
2. #90225C2semidirect
3. #89866C2semidirect
4. #89856C2semidirect
5. #89829C2semidirect
6. #89826C2semidirect
7. #89824C2semidirect
8. #89820C2semidirect
9. #89813C2semidirect
11. #89800C2semidirect
12. #89792C2semidirect
13. #89788C2semidirect
14. #89780C2semidirect
15. #89771C2semidirect
110. #542D7semidirect
315. C7#3263semidirect

Maximal subgroups

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

subgroupindex
C7×#5422
#902252
#898662
#898562
#898292
#898262
#898242
#898202
#898132
#898112
#898002
#897922
#897882
#897802
#897712
#32637