Finite Group Databasebeta

← #11738065

Subgroups — #11738065

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#90368C211yesmaximal, characteristic
2#90270C211yesmaximal, characteristic, complemented
3C7×#464C211yesmaximal, characteristic, complemented
4#89961C211yesmaximal, characteristic, complemented
5#89958C211yesmaximal, characteristic, complemented
6#89945C211yesmaximal, characteristic, complemented
7#89943C211yesmaximal, characteristic, complemented
8#89827C211yesmaximal, characteristic, complemented
9#89802C211yesmaximal, characteristic, complemented
10#89623C211yesmaximal, characteristic, complemented
11#89608C211yesmaximal, characteristic, complemented
12#89577C211yesmaximal, characteristic, complemented
13D7×#103C211yesmaximal, characteristic, complemented
14#89549C211yesmaximal, characteristic, complemented
15#89543C211yesmaximal, characteristic, complemented
16C2×Q8×D7C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×#124C2211yescharacteristic
21C4×C7×Q8C2211yescharacteristic
22#6509C2211yescharacteristic
23#6509C2211yescharacteristic
24#6509C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26#6501C2211yescharacteristic
27C2×#1131C2211yescharacteristic
28C2×C8×D7C2211yescharacteristic
29#6459—21no—
30#6459—21no—
31D7×C4⋊C4C2211yescharacteristic
32D7×C4⋊C4C2211yescharacteristic
33#6448—21no—
34#6447—21no—
35(C22⋊D7)⋊C4—21no—
36#6441—21no—
37C4×D28—21no—
38C4×D28C2211yescharacteristic
39#6433—21no—
40#6432C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51(C2×D7)⋊C8C2211yescharacteristic
52#6391C2211yescharacteristic
53#6391C2211yescharacteristic
54#6390C2211yescharacteristic
55#6390C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6377C2211yescharacteristic
59#3316—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66Q8×D7—22no—
67Q8×D7—41no—
68#1159—22no—
69#1159—41no—
70C2×D28D411yescharacteristic
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7C2311yescharacteristic
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C2×Dic14—21no—
78C2×Dic14C2311yescharacteristic
79C7×Q16—41no—
80C7×SD16—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4C2311yescharacteristic
85C7×C22⋊C4—21no—
86C7×C22⋊C4—21no—
87C42×C7—21no—
88C42×C7C2311yescharacteristic
89C22⋊Dic7—21no—
90#1144—22no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95(C2×D7)⋊C4D411yescharacteristic
96C4⋊Dic7—21no—
97C4⋊Dic7D411yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107C4×Dic7—21no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110#1131—41no—
111C8×D7—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#464D711yescharacteristic, complemented
121#452—71no—
122#451—71no—
123#441—71no—
124#435—71no—
125#417—71no—
126C2×#103—71no—
127C23×C7—21no—
128C22×D7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×D4—21no—
135C7×D4—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—42no—
143C22⋊D7—42no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—42no—
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151D28—22no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—21no—
156C4×D7—42no—
157C4×D7—81no—
158C4×D7—81no—
159Dic14—21no—
160Dic14—42no—
161Dic14—81no—
162Dic14#4112yes—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C2×#41—71no—
168C22×Q8—71no—
169C2×Q16—71no—
170C2×Q16C2×D711yescharacteristic
171C2×SD16C2×D711yescharacteristic
172C2×M16—71no—
173C22×C8—71no—
174C4⋊Q8—71no—
175#125—141no—
176#124—141no—
177#124C2×D711yescharacteristic
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C22⋊Q8—141no—
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×Q8C2×D711yescharacteristic
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⋊C8C2×D711yescharacteristic
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103—71no—
203#103C2×D711yescharacteristic
204#102—71no—
205#102—71no—
206#102C2×D711yescharacteristic
207C22⋊C8—71no—
208C22⋊C8—71no—
209#97—71no—
210C22×C7—41no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7C22×D411yescharacteristic
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—21no—
218C2×D7—21no—
219C2×D7#13712yes—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7C2×#4111yescharacteristic
226C4×C7H(5,𝔽2)11yescharacteristic
227Dic7—22no—
228Dic7—21no—
229Dic7—21no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233#41—142no—
234#41—281no—
235C2×Q8—71no—
236C2×Q8—141no—
237C2×Q8—141no—
238C2×Q8—142no—
239C2×Q8—281no—
240C2×Q8C22×D711yescharacteristic
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—141no—
244C2×D4—141no—
245C2×D4C22×D711yescharacteristic
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4—141no—
252Q16—41no—
253Q16—142no—
254SD16—41no—
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—21no—
276C22⋊C4—71no—
277C22⋊C4—71no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C42—21no—
283C42—141no—
284C42C22×D711yescharacteristic
285C2×C7—81no—
286C2×C7#54611yescharacteristic
287C2×C7C2×#13711yescharacteristic
288C2×C7#57711yescharacteristic
289D7—22no—
290D7—41no—
291C23—21no—
292C23—71no—
293C23—71no—
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—72no—
299Q8—141no—
300Q8—282no—
301Q8—561no—
302D4—21no—
303D4—21no—
304D4—142no—
305D4—282no—
306D4—282no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—282no—
321C2×C4—282no—
322C2×C4—561no—
323C2×C4—561no—
324C2×C4D4×D711yescharacteristic
325C2×C4D4×D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—141no—
330C8—141no—
331C7#331611yescharacteristic, complemented, Sylow
332C22—41no—
333C22—41no—
334C22—41no—
335C22—72no—
336C22—141no—
337C22—141no—
338C22—141no—
339C22—141no—
340C22—141no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—141no—
348C4—141no—
349C4—142no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4#654811yescharacteristic
354C4D7×#4111yescharacteristic
355C2—81no—
356C2—142no—
357C2—281no—
358C2#9030311yescharacteristic
359C2#9046111yescharacteristic
360C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#464C2
socleC22×C7C22×D4
prime radical#90368C2
hypercenter#464D7
hypocenterC7#3316
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #90270C2semidirect
3. C7×#464C2semidirect
4. #89961C2semidirect
5. #89958C2semidirect
6. #89945C2semidirect
7. #89943C2semidirect
8. #89827C2semidirect
9. #89802C2semidirect
10. #89623C2semidirect
11. #89608C2semidirect
12. #89577C2semidirect
13. D7×#103C2semidirect
14. #89549C2semidirect
15. #89543C2semidirect
120. #464D7semidirect
331. C7#3316semidirect

Maximal subgroups

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

subgroupindex
#903682
#902702
C7×#4642
#899612
#899582
#899452
#899432
#898272
#898022
#896232
#896082
#895772
D7×#1032
#895492
#895432
#33167