Finite Group Databasebeta

← #11738424

Subgroups — #11738424

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
1#90424C211yesmaximal, characteristic, complemented
2#90252C211yesmaximal, characteristic, complemented
3C7×#474C211yesmaximal, characteristic, complemented
4#89863C211yesmaximal, characteristic, complemented
5#89788C211yesmaximal, characteristic, complemented
6#89662C211yesmaximal, characteristic, complemented
7#89659C211yesmaximal, characteristic, complemented
8#89658C211yesmaximal, characteristic, complemented
9#89647C211yesmaximal, characteristic, complemented
10#89617C211yesmaximal, characteristic, complemented
11#89611C211yesmaximal, characteristic, complemented
12#89559C211yesmaximal, characteristic, complemented
13#89552C211yesmaximal, characteristic, complemented
14#89547C211yesmaximal, characteristic
15#89539C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C4×C7×D4C2211yescharacteristic
18#6510—21no—
19#6509—21no—
20#6501C2211yescharacteristic
21#6499C2211yescharacteristic
22C2×#1142C2211yescharacteristic
23C2×#1141C2211yescharacteristic
24#6492—21no—
25#6487—21no—
26C2×Dic28C2211yescharacteristic
27C2×#1132C2211yescharacteristic
28C2×#1131C2211yescharacteristic
29C2×C8×D7C2211yescharacteristic
30#6461—21no—
31#6460C2211yescharacteristic
32#6460C2211yescharacteristic
33#6458C2211yescharacteristic
34#6457—21no—
35#6456C2211yescharacteristic
36#6451C2211yescharacteristic
37#6450C2211yescharacteristic
38#6448—21no—
39#6442—21no—
40#6438C2211yescharacteristic
41#6435C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#107C2211yescharacteristic
45C7×C4⋊C8C2211yescharacteristic
46C7×#102C2211yescharacteristic
47C7×#102C2211yescharacteristic
48#6395C2211yescharacteristic
49#6390C2211yescharacteristic
50Dic7⋊C8C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3262—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×D28—21no—
65C2×D28C2311yescharacteristic
66C2×C4×D7—21no—
67C2×C4×D7C2311yescharacteristic
68C2×Dic14—21no—
69C2×Dic14C2311yescharacteristic
70C2×Dic14C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4—21no—
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C7×C22⋊C4—21no—
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82#1142—41no—
83#1141—41no—
84(C2×D7)⋊C4—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4D411yescharacteristic
91(C2×D7)⋊C4D411yescharacteristic
92C4⋊Dic7—21no—
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4D411yescharacteristic
99Dic7⋊C4D411yescharacteristic
100C4×Dic7—21no—
101C4×Dic7—21no—
102C4×Dic7C2311yescharacteristic
103C2×C7⋊C8C2311yescharacteristic
104C2×C7⋊C8C2311yescharacteristic
105Dic28—41no—
106#1132—41no—
107#1131—41no—
108C8×D7—41no—
109#542—71no—
110#542—71no—
111#477—71no—
112#477—71no—
113#474—71no—
114#474D711yescharacteristic, complemented
115#468—71no—
116#468—71no—
117#467—71no—
118#466—71no—
119#464—71no—
120#464—71no—
121#463—71no—
122#463—71no—
123#424—71no—
124C23×C7—21no—
125C22×D7—21no—
126C22×D7C2×D411yescharacteristic
127C7×Q8—81no—
128C7×D4—21no—
129C7×D4—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—81no—
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2×D411yescharacteristic
136C2×C4×C7C2411yescharacteristic
137C22⋊D7—81no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7C2×D411yescharacteristic
142C2×Dic7C2×D411yescharacteristic
143C2×Dic7C2×D411yescharacteristic
144D28—21no—
145D28—21no—
146D28—81no—
147C4×D7—21no—
148C4×D7—21no—
149C4×D7—81no—
150Dic14—21no—
151Dic14—21no—
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—141no—
168#126—141no—
169#124—71no—
170#124—71no—
171#124—141no—
172#124—141no—
173#123—141no—
174#123—141no—
175C22⋊Q8—71no—
176C22⋊Q8—71no—
177C22⋊Q8—141no—
178C22⋊Q8—141no—
179#121—71no—
180#121—71no—
181C4×Q8—71no—
182C4×Q8—71no—
183C4×D4—71no—
184C4×D4C2×D711yescharacteristic
185#117—71no—
186#107—71no—
187#107—71no—
188#107C2×D711yescharacteristic
189#107C2×D711yescharacteristic
190C4⋊C8—71no—
191C4⋊C8—71no—
192C4⋊C8—71no—
193C4⋊C8C2×D711yescharacteristic
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#102—71no—
199#102—71no—
200#102C2×D711yescharacteristic
201#102C2×D711yescharacteristic
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7C22×D411yescharacteristic
206C2×D7—21no—
207C2×D7—21no—
208C2×D7—21no—
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7—41no—
217C4×C7H(5,𝔽2)11yescharacteristic
218C4×C72−1+411yescharacteristic
219Dic7—21no—
220Dic7—21no—
221Dic7—21no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—21no—
226C2×Q8—71no—
227C2×Q8—71no—
228C2×Q8—141no—
229C2×D4—71no—
230C2×D4—141no—
231C2×D4—141no—
232C2×D4C22×D711yescharacteristic
233C22×C4—21no—
234C22×C4—71no—
235C22×C4—141no—
236Q16—281no—
237SD16—281no—
238SD16—281no—
239D8—281no—
240M16—281no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8—281no—
244C2×C8C22×D711yescharacteristic
245C2×C8C22×D711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4C22×D711yescharacteristic
256C4⋊C4C22×D711yescharacteristic
257C4⋊C4C22×D711yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—71no—
260C22⋊C4—71no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—71no—
271C42—141no—
272C42—141no—
273C42C22×D711yescharacteristic
274C2×C7—81no—
275C2×C7#53711yescharacteristic
276C2×C7#57611yescharacteristic
277C2×C7#57811yescharacteristic
278D7—41no—
279D7—81no—
280C23—21no—
281C23—71no—
282C23—141no—
283Q8—81no—
284Q8—141no—
285Q8—141no—
286Q8—141no—
287Q8—141no—
288Q8—561no—
289D4—21no—
290D4—21no—
291D4—141no—
292D4—141no—
293D4—561no—
294D4—561no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—71no—
299C2×C4—71no—
300C2×C4—71no—
301C2×C4—81no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—561no—
308C2×C4D4×D711yescharacteristic
309C2×C4D4×D711yescharacteristic
310C2×C4C23×D711yescharacteristic
311C8—21no—
312C8—21no—
313C8—141no—
314C8—141no—
315C7#326211yescharacteristic, complemented, Sylow
316C22—41no—
317C22—41no—
318C22—41no—
319C22—141no—
320C22—141no—
321C22—141no—
322C22—281no—
323C22—281no—
324C22—281no—
325C22C2×D4×D711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—141no—
332C4—141no—
333C4—141no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4#655111yescharacteristic
338C4#655311yescharacteristic
339C2—81no—
340C2—281no—
341C2—561no—
342C2#9032011yescharacteristic
343C2#9044811yescharacteristic
344C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#474C2
socleC22×C7C22×D4
prime radical#89547C2
hypercenter#474D7
hypocenterC7#3262
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #90424C2semidirect
2. #90252C2semidirect
3. C7×#474C2semidirect
4. #89863C2semidirect
5. #89788C2semidirect
6. #89662C2semidirect
7. #89659C2semidirect
8. #89658C2semidirect
9. #89647C2semidirect
10. #89617C2semidirect
11. #89611C2semidirect
12. #89559C2semidirect
13. #89552C2semidirect
15. #89539C2semidirect
114. #474D7semidirect
315. C7#3262semidirect

Maximal subgroups

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

subgroupindex
#904242
#902522
C7×#4742
#898632
#897882
#896622
#896592
#896582
#896472
#896172
#896112
#895592
#895522
#895472
#895392
#32627