Finite Group Databasebeta

← #11737643

Subgroups — #11737643

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90508C211yesmaximal, characteristic
2#90326C211yesmaximal, characteristic, complemented
3C7×#451C211yesmaximal, characteristic, complemented
4C2×#6409C211yesmaximal, characteristic, complemented
5#89961C211yesmaximal, characteristic, complemented
6#89959C211yesmaximal, characteristic, complemented
7#89957C211yesmaximal, characteristic, complemented
8#89956C211yesmaximal, characteristic, complemented
9#89820C211yesmaximal, characteristic, complemented
10#89577C211yesmaximal, characteristic, complemented
11#89575C211yesmaximal, characteristic, complemented
12#89567C211yesmaximal, characteristic, complemented
13#89563C211yesmaximal, characteristic, complemented
14#89507C211yesmaximal, characteristic, complemented
15#89502C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×#1161C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×Q16C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21#6510—21no—
22#6510C2211yescharacteristic
23#6509—21no—
24#6509C2211yescharacteristic
25#6509C2211yescharacteristic
26Q8×Dic7C2211yescharacteristic
27Q8×Dic7C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29#6493—21no—
30#6493C2211yescharacteristic
31C4×C22⋊D7—21no—
32C4×C22⋊D7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C22×C7⋊C8C2211yescharacteristic
35C2×#1131C2211yescharacteristic
36#6460—21no—
37D7×C4⋊C4C2211yescharacteristic
38#6452—21no—
39#6451C2211yescharacteristic
40(C22⋊D7)⋊C4—21no—
41#6441—21no—
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6409C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52(C2×D7)⋊C8C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6389C2211yescharacteristic
56Dic7⋊C8C2211yescharacteristic
57#6383C2211yescharacteristic
58#6382C2211yescharacteristic
59#3316—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—22no—
62C2×C7×Q8—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C2×C7×Q8C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7D411yescharacteristic
70C22×Dic7—21no—
71#1161—22no—
72#1161—41no—
73C2×D28—21no—
74C2×D28C2311yescharacteristic
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7C2311yescharacteristic
78C2×Dic14—21no—
79C7×Q16—22no—
80C7×Q16—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C22⋊Dic7D411yescharacteristic
87#1143—41no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8—41no—
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1131—41no—
112#554—71no—
113#548—71no—
114#489—71no—
115#483—71no—
116#481—71no—
117#475—71no—
118#470—71no—
119#465—71no—
120#464—71no—
121#452—71no—
122#451D711yescharacteristic, complemented
123#441—71no—
124#435—71no—
125#417—71no—
126C2×#103—71no—
127C23×C7C2×D411yescharacteristic
128C22×D7—21no—
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×Q8—21no—
135C7×Q8—42no—
136C7×Q8#4112yes—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—42no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—22no—
147C22⋊D7—42no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—81no—
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155D28—21no—
156D28—21no—
157D28—42no—
158C4×D7—21no—
159C4×D7—21no—
160C4×D7—42no—
161C4×D7—81no—
162Dic14—81no—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C2×#41—71no—
168C22×Q8C2×D711yescharacteristic
169C2×Q16C2×D711yescharacteristic
170C2×Q16C2×D711yescharacteristic
171C2×SD16—71no—
172C2×M16—71no—
173C22×C8—71no—
174C4⋊Q8—71no—
175#125—141no—
176#124—71no—
177#124—141no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—141no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C22⋊Q8C2×D711yescharacteristic
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×Q8—71no—
188C4×D4—71no—
189C4×D4—141no—
190C4×D4—141no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193#106—71no—
194#106—71no—
195#106—71no—
196C4⋊C8—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204#102—71no—
205#102—71no—
206#102—71no—
207C22⋊C8—71no—
208C22⋊C8C2×D711yescharacteristic
209#97—71no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7#13712yes—
213C22×C7C22×D411yescharacteristic
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—22no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7C2×#4111yescharacteristic
226C4×C7H(5,𝔽2)11yescharacteristic
227Dic7—21no—
228Dic7—21no—
229Dic7—21no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233#41—142no—
234#41—281no—
235C2×Q8—21no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—141no—
239C2×Q8C22×D711yescharacteristic
240C2×Q8C22×D711yescharacteristic
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—71no—
244C2×D4—141no—
245C2×D4—141no—
246C22×C4—21no—
247C22×C4—71no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4C22×D711yescharacteristic
252Q16—22no—
253Q16—41no—
254SD16—281no—
255M16—281no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8—281no—
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—71no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—71no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—71no—
276C22⋊C4—71no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C42—71no—
283C42—141no—
284C42—141no—
285C2×C7—22no—
286C2×C7#54611yescharacteristic
287C2×C7C2×#13711yescharacteristic
288C2×C7#57711yescharacteristic
289D7—41no—
290D7—81no—
291C23—71no—
292C23—141no—
293C23D4×D711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—42no—
300Q8—561no—
301Q8#115912yes—
302D4—142no—
303D4—141no—
304D4—141no—
305D4—282no—
306D4—282no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—71no—
315C2×C4—71no—
316C2×C4—71no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—282no—
323C2×C4—561no—
324C2×C4—561no—
325C2×C4D4×D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—141no—
330C8—141no—
331C7#331611yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—141no—
335C22—141no—
336C22—141no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22#648112yes—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—141no—
348C4—141no—
349C4—141no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4C2×#115911yescharacteristic
354C4#654811yescharacteristic
355C2—22no—
356C2—281no—
357C2—561no—
358C2#9028211yescharacteristic
359C2C2×#648111yescharacteristic
360C2#9046911yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#451C2
socleC22×C7C22×D4
prime radical#90508C2
hypercenter#451D7
hypocenterC7#3316
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #90326C2semidirect
3. C7×#451C2semidirect
4. C2×#6409C2semidirect
5. #89961C2semidirect
6. #89959C2semidirect
7. #89957C2semidirect
8. #89956C2semidirect
9. #89820C2semidirect
10. #89577C2semidirect
11. #89575C2semidirect
12. #89567C2semidirect
13. #89563C2semidirect
14. #89507C2semidirect
15. #89502C2semidirect
122. #451D7semidirect
331. C7#3316semidirect

Maximal subgroups

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

subgroupindex
#905082
#903262
C7×#4512
C2×#64092
#899612
#899592
#899572
#899562
#898202
#895772
#895752
#895672
#895632
#895072
#895022
#33167