Finite Group Databasebeta

← #11741086

Subgroups — #11741086

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
1C7×#554C211yesmaximal, characteristic, complemented
2#90243C211yesmaximal, characteristic
3#89866C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89824C211yesmaximal, characteristic, complemented
6#89822C211yesmaximal, characteristic, complemented
7#89820C211yesmaximal, characteristic, complemented
8#89819C211yesmaximal, characteristic, complemented
9#89802C211yesmaximal, characteristic, complemented
10#89800C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic, complemented
12#89743C211yesmaximal, characteristic, complemented
13C2×#6384C211yesmaximal, characteristic, complemented
14#89740C211yesmaximal, characteristic, complemented
15#89738C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C7×C4⋊Q8C2211yescharacteristic
19C7×#125—21no—
20C7×C22⋊Q8—21no—
21C7×C22⋊Q8—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×C22⋊Q8C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4—21no—
26C4×C7×D4C2211yescharacteristic
27C2×C7×C4⋊C4C2211yescharacteristic
28C2×C7×C4⋊C4C2211yescharacteristic
29C2×#1144C2211yescharacteristic
30C2×#1144C2211yescharacteristic
31C2×#1143C2211yescharacteristic
32#6502—21no—
33#6493C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6459—21no—
38#6448—21no—
39(C22⋊D7)⋊C4—21no—
40#6438C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6409C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6384C2211yescharacteristic
50#6384C2211yescharacteristic
51#6383C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6382C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3316—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8C2311yescharacteristic
62C2×C7×D4D411yescharacteristic
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C2×C22⋊D7—21no—
68C2×C22⋊D7—21no—
69C22×Dic7—21no—
70#1157—22no—
71#1157—41no—
72C2×D28C2311yescharacteristic
73C2×C4×D7—21no—
74C2×Dic14—22no—
75C2×Dic14—21no—
76C2×Dic14—41no—
77C2×Dic14C2311yescharacteristic
78C2×Dic14C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4D411yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C4⋊C4C2311yescharacteristic
89C7×C22⋊C4—21no—
90C7×C22⋊C4—21no—
91C7×C22⋊C4D411yescharacteristic
92C7×C22⋊C4D411yescharacteristic
93C42×C7—21no—
94C42×C7C2311yescharacteristic
95C22⋊Dic7—21no—
96#1144—22no—
97#1144—41no—
98#1143—41no—
99(C2×D7)⋊C4—21no—
100(C2×D7)⋊C4—21no—
101(C2×D7)⋊C4—21no—
102C4⋊Dic7C2311yescharacteristic
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106#1136—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#554D711yescharacteristic, complemented
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#451—71no—
123#441—71no—
124#435—71no—
125#417—71no—
126C2×#103—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—81no—
133C7×D4—22no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—81no—
143C2×C4×C7—81no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2411yescharacteristic
149C22⋊D7—42no—
150C22⋊D7—42no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—42no—
155D28—21no—
156D28—21no—
157C4×D7—42no—
158Dic14—21no—
159Dic14—21no—
160Dic14—21no—
161Dic14—42no—
162Dic14#4112yes—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C2×#41—71no—
168C22×Q8—71no—
169C2×Q16—71no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×M16—71no—
173C22×C8—71no—
174C4⋊Q8C2×D711yescharacteristic
175#125—21no—
176#124—71no—
177#124—141no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8C2×D711yescharacteristic
183C22⋊Q8C2×D711yescharacteristic
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×Q8C2×D711yescharacteristic
188C4×D4—21no—
189C4×D4—141no—
190C4×D4C2×D711yescharacteristic
191C2×C4⋊C4C2×D711yescharacteristic
192C2×C4⋊C4C2×D711yescharacteristic
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#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207C22⋊C8—71no—
208C22⋊C8—71no—
209#97—71no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7#13712yes—
216C22×C7C22×D411yescharacteristic
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—21no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7C2×#4111yescharacteristic
229C4×C7H(5,𝔽2)11yescharacteristic
230Dic7—22no—
231Dic7—41no—
232Dic7—41no—
233#41—142no—
234#41—281no—
235C2×Q8—21no—
236C2×Q8—71no—
237C2×Q8—71no—
238C2×Q8—142no—
239C2×Q8—141no—
240C2×Q8—281no—
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—141no—
244C2×D4—141no—
245C2×D4C22⋊D711yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4C22×D711yescharacteristic
251C22×C4C22×D711yescharacteristic
252Q16—142no—
253Q16—281no—
254SD16—281no—
255M16—281no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8—71no—
259C2×C8—71no—
260C2×C8—281no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—71no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4C22⋊D711yescharacteristic
271C4⋊C4C22×D711yescharacteristic
272C4⋊C4C22×D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4C22⋊D711yescharacteristic
281C22⋊C4C22⋊D711yescharacteristic
282C42—21no—
283C42—141no—
284C42C22×D711yescharacteristic
285C2×C7—22no—
286C2×C7—41no—
287C2×C7#54611yescharacteristic
288C2×C7C2×#13711yescharacteristic
289C2×C7#57711yescharacteristic
290D7—81no—
291C23—141no—
292C23C2×C22⋊D711yescharacteristic
293C23C2×C22⋊D711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—72no—
297Q8—81no—
298Q8—141no—
299Q8—141no—
300Q8—141no—
301Q8—282no—
302D4—22no—
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—21no—
314C2×C4—21no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—282no—
321C2×C4—282no—
322C2×C4C2×C22⋊D711yescharacteristic
323C2×C4C2×C22⋊D711yescharacteristic
324C2×C4C2×C22⋊D711yescharacteristic
325C2×C4C2×C22⋊D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—141no—
328C8—141no—
329C8—141no—
330C8—141no—
331C7#331611yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—281no—
338C22—281no—
339C22—281no—
340C22#650512yes—
341C22C22×C22⋊D711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—142no—
351C4—281no—
352C4—281no—
353C4D7×#4111yescharacteristic
354C4#655311yescharacteristic
355C2—22no—
356C2—41no—
357C2—561no—
358C2C2×#650511yescharacteristic
359C2#9052011yescharacteristic
360C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#554C2
socleC22×C7C22×D4
prime radical#90243C2
hypercenter#554D7
hypocenterC7#3316
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #554

Complements

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

NH ≅ G/Ntype
1. C7×#554C2semidirect
3. #89866C2semidirect
4. #89861C2semidirect
5. #89824C2semidirect
6. #89822C2semidirect
7. #89820C2semidirect
8. #89819C2semidirect
9. #89802C2semidirect
10. #89800C2semidirect
11. #89783C2semidirect
12. #89743C2semidirect
13. C2×#6384C2semidirect
14. #89740C2semidirect
15. #89738C2semidirect
112. #554D7semidirect
331. C7#3316semidirect

Maximal subgroups

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

subgroupindex
C7×#5542
#902432
#898662
#898612
#898242
#898222
#898202
#898192
#898022
#898002
#897832
#897432
C2×#63842
#897402
#897382
#33167