Finite Group Databasebeta

← #11740596

Subgroups — #11740596

Order 896.

354 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×#529C211yesmaximal, characteristic, complemented
2#90170C211yesmaximal, characteristic, complemented
3#89838C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89829C211yesmaximal, characteristic, complemented
6#89826C211yesmaximal, characteristic, complemented
7#89824C211yesmaximal, characteristic, complemented
8#89821C211yesmaximal, characteristic, complemented
9#89814C211yesmaximal, characteristic, complemented
10#89811C211yesmaximal, characteristic
11#89800C211yesmaximal, characteristic, complemented
12#89794C211yesmaximal, characteristic, complemented
13#89788C211yesmaximal, characteristic, complemented
14#89781C211yesmaximal, characteristic, complemented
15#89696C211yesmaximal, characteristic, complemented
16C7×#126—21no—
17C7×#125C2211yescharacteristic
18C7×#124C2211yescharacteristic
19C7×#123—21no—
20C7×C22⋊Q8C2211yescharacteristic
21C7×#121C2211yescharacteristic
22C4×C7×Q8C2211yescharacteristic
23C4×C7×D4—21no—
24C4×C7×D4C2211yescharacteristic
25C7×#117—21no—
26C2×C42×C7C2211yescharacteristic
27C2×#1144C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1141C2211yescharacteristic
31#6493C2211yescharacteristic
32#6492—21no—
33C4×C22⋊D7—21no—
34#6487C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C2×#1136C2211yescharacteristic
37#6439—21no—
38#6438C2211yescharacteristic
39C4×D28C2211yescharacteristic
40#6435—21no—
41#6433C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6406C2211yescharacteristic
46#6406C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6383C2211yescharacteristic
50#6383C2211yescharacteristic
51#6382C2211yescharacteristic
52#6382C2211yescharacteristic
53#6381C2211yescharacteristic
54#6381C2211yescharacteristic
55C4⋊(C7⋊C8)C2211yescharacteristic
56C4⋊(C7⋊C8)C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3256—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7D411yescharacteristic
65C22×C4×C7D411yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C2×C22⋊D7—21no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×Dic14C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4—21no—
74C7×C4⋊C4—21no—
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C7×C22⋊C4—21no—
79C7×C22⋊C4—21no—
80C7×C22⋊C4—21no—
81C7×C22⋊C4—21no—
82C42×C7—21no—
83C42×C7D411yescharacteristic
84C42×C7D411yescharacteristic
85C42×C7C2311yescharacteristic
86C42×C7C2311yescharacteristic
87C22⋊Dic7—21no—
88#1144—41no—
89#1143—41no—
90#1142—41no—
91#1141—41no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103#1136—41no—
104#1136—41no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109#529—71no—
110#529D711yescharacteristic, complemented
111#479—71no—
112#479—71no—
113#478—71no—
114#478—71no—
115#470—71no—
116#469—71no—
117#468—71no—
118#468—71no—
119#464—71no—
120#464—71no—
121#463—71no—
122#463—71no—
123#423—71no—
124C23×C7—21no—
125C23×C7C2×D411yescharacteristic
126C22×D7—21no—
127C7×Q8—21no—
128C7×Q8—21no—
129C7×D4—21no—
130C7×D4—21no—
131C7×D4—42no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2×D411yescharacteristic
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—42no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150D28—21no—
151D28—21no—
152C4×D7—42no—
153Dic14—21no—
154Dic14—21no—
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—
164C2×M16—71no—
165#126—21no—
166#126—141no—
167#125—71no—
168#125C2×D711yescharacteristic
169#124—71no—
170#124C2×D711yescharacteristic
171#123—21no—
172#123—141no—
173C22⋊Q8—71no—
174C22⋊Q8C2×D711yescharacteristic
175#121—71no—
176#121C2×D711yescharacteristic
177C4×Q8—71no—
178C4×Q8C2×D711yescharacteristic
179C4×D4—21no—
180C4×D4—71no—
181C4×D4—141no—
182C4×D4C2×D711yescharacteristic
183#117—21no—
184#117—141no—
185C2×C42C2×D711yescharacteristic
186#107—71no—
187#107—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—71no—
226C2×Q8C22×D711yescharacteristic
227C2×D4—21no—
228C2×D4—71no—
229C2×D4—141no—
230C2×D4C22×D711yescharacteristic
231C22×C4—21no—
232C22×C4—141no—
233C22×C4C22⋊D711yescharacteristic
234C22×C4C22⋊D711yescharacteristic
235C22×C4C22×D711yescharacteristic
236Q16—281no—
237SD16—281no—
238SD16—281no—
239D8—281no—
240M16—281no—
241M16—281no—
242C2×C8—71no—
243C2×C8—71no—
244C2×C8—71no—
245C2×C8—71no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—71no—
251C4⋊C4—71no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
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—
268C42—21no—
269C42—141no—
270C42C22⋊D711yescharacteristic
271C42C22⋊D711yescharacteristic
272C42C22×D711yescharacteristic
273C42C22×D711yescharacteristic
274C2×C7—41no—
275C2×C7—81no—
276C2×C7#53711yescharacteristic
277C2×C7#57511yescharacteristic
278C2×C7#57511yescharacteristic
279D7—81no—
280C23—21no—
281C23—141no—
282C23C2×C22⋊D711yescharacteristic
283Q8—21no—
284Q8—21no—
285Q8—141no—
286Q8—141no—
287D4—21no—
288D4—21no—
289D4—42no—
290D4—141no—
291D4—141no—
292D4—282no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—282no—
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#325611yescharacteristic, 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#9051311yescharacteristic
343C2#9051311yescharacteristic
344C2#9052411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#529C2
socleC22×C7C22×D4
prime radical#89811C2
hypercenter#529D7
hypocenterC7#3256
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C7×#529C2semidirect
2. #90170C2semidirect
3. #89838C2semidirect
4. #89835C2semidirect
5. #89829C2semidirect
6. #89826C2semidirect
7. #89824C2semidirect
8. #89821C2semidirect
9. #89814C2semidirect
11. #89800C2semidirect
12. #89794C2semidirect
13. #89788C2semidirect
14. #89781C2semidirect
15. #89696C2semidirect
110. #529D7semidirect
315. C7#3256semidirect

Maximal subgroups

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

subgroupindex
C7×#5292
#901702
#898382
#898352
#898292
#898262
#898242
#898212
#898142
#898112
#898002
#897942
#897882
#897812
#896962
#32567