Finite Group Databasebeta

← #11739390

Subgroups — #11739390

Order 896.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90424C211yesmaximal, characteristic
2#90420C211yesmaximal, characteristic, complemented
3C7×#500C211yesmaximal, characteristic, complemented
4#89861C211yesmaximal, characteristic, complemented
5#89852C211yesmaximal, characteristic, complemented
6#89663C211yesmaximal, characteristic, complemented
7#89662C211yesmaximal, characteristic, complemented
8#89660C211yesmaximal, characteristic, complemented
9#89657C211yesmaximal, characteristic
10#89656C211yesmaximal, characteristic, complemented
11#89654C211yesmaximal, characteristic
12#89649C211yesmaximal, characteristic, complemented
13#89647C211yesmaximal, characteristic, complemented
14#89469C211yesmaximal, characteristic, complemented
15#89462C211yesmaximal, characteristic, complemented
16C7×C4⋊Q8C2211yescharacteristic
17C7×C4⋊Q8C2211yescharacteristic
18#6510—21no—
19#6509—21no—
20#6509—21no—
21#6507—21no—
22#6461—21no—
23#6460C2211yescharacteristic
24#6459—21no—
25#6458C2211yescharacteristic
26#6457—21no—
27#6456C2211yescharacteristic
28#6455C2211yescharacteristic
29D7×C4⋊C4C2211yescharacteristic
30#6453C2211yescharacteristic
31#6452—21no—
32#6451C2211yescharacteristic
33#6450C2211yescharacteristic
34#6438C2211yescharacteristic
35#6435C2211yescharacteristic
36#6433C2211yescharacteristic
37C7×#107C2211yescharacteristic
38C7×#107C2211yescharacteristic
39C7×#107C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C4×C7×C8C2211yescharacteristic
42#6395C2211yescharacteristic
43#6395C2211yescharacteristic
44(C2×D7)⋊C8C2211yescharacteristic
45(C2×D7)⋊C8C2211yescharacteristic
46#6390C2211yescharacteristic
47#6390C2211yescharacteristic
48Dic7⋊C8C2211yescharacteristic
49Dic7⋊C8C2211yescharacteristic
50#6384C2211yescharacteristic
51#6384C2211yescharacteristic
52#6383C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3243—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—21no—
62C2×D28—21no—
63C2×D28C2311yescharacteristic
64C2×C4×D7—21no—
65C2×C4×D7—21no—
66C2×C4×D7C2311yescharacteristic
67C2×Dic14—21no—
68C2×Dic14C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C7×C4⋊C4—21no—
72C7×C4⋊C4—21no—
73C7×C4⋊C4C2311yescharacteristic
74C7×C4⋊C4C2311yescharacteristic
75C7×C4⋊C4C2311yescharacteristic
76C7×C4⋊C4C2311yescharacteristic
77C42×C7C2311yescharacteristic
78(C2×D7)⋊C4—21no—
79(C2×D7)⋊C4—21no—
80(C2×D7)⋊C4—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4D411yescharacteristic
85(C2×D7)⋊C4D411yescharacteristic
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7C2311yescharacteristic
89C4⋊Dic7C2311yescharacteristic
90Dic7⋊C4—21no—
91Dic7⋊C4—21no—
92Dic7⋊C4—21no—
93Dic7⋊C4—21no—
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4D411yescharacteristic
97Dic7⋊C4D411yescharacteristic
98C4×Dic7—21no—
99C4×Dic7—21no—
100C4×Dic7C2311yescharacteristic
101C2×C7⋊C8C2311yescharacteristic
102C2×C7⋊C8C2311yescharacteristic
103#544—71no—
104#542—71no—
105#501—71no—
106#500D711yescharacteristic, complemented
107#489—71no—
108#488—71no—
109#485—71no—
110#484—71no—
111#482—71no—
112#480—71no—
113#479—71no—
114#478—71no—
115#477—71no—
116#474—71no—
117#433—71no—
118C22×D7—21no—
119C22×D7C2×D411yescharacteristic
120C7×Q8—81no—
121C7×Q8—81no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2×D411yescharacteristic
128C2×C4×C7C2411yescharacteristic
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7C2×D411yescharacteristic
133C2×Dic7C2×D411yescharacteristic
134C2×Dic7C2×D411yescharacteristic
135D28—21no—
136D28—21no—
137D28—81no—
138C4×D7—21no—
139C4×D7—21no—
140C4×D7—81no—
141C4×D7—81no—
142Dic14—21no—
143Dic14—21no—
144Dic14—81no—
145C7×C8—21no—
146C7×C8—21no—
147C7⋊C8—21no—
148C7⋊C8—21no—
149C4⋊Q8—71no—
150C4⋊Q8—141no—
151C4⋊Q8C2×D711yescharacteristic
152C4⋊Q8C2×D711yescharacteristic
153#126—141no—
154#125—71no—
155#125—71no—
156#125—141no—
157#124—71no—
158#124—141no—
159#123—141no—
160C22⋊Q8—71no—
161C22⋊Q8—141no—
162C22⋊Q8—141no—
163C22⋊Q8—141no—
164#121—71no—
165C4×Q8—71no—
166C4×D4—71no—
167#117—71no—
168#117—71no—
169C2×C4⋊C4—71no—
170#107—71no—
171#107—71no—
172#107C2×D711yescharacteristic
173#107C2×D711yescharacteristic
174#107C2×D711yescharacteristic
175#107C2×D711yescharacteristic
176#106—71no—
177#106—71no—
178C4⋊C8—71no—
179C4⋊C8—71no—
180#103—71no—
181#103—71no—
182#103—71no—
183#103—71no—
184#102—71no—
185#102—71no—
186#102—71no—
187#102—71no—
188C22⋊C8—71no—
189C22⋊C8—71no—
190#97—71no—
191C4×C8C2×D711yescharacteristic
192C22×C7C22×D411yescharacteristic
193C2×D7—21no—
194C2×D7—21no—
195C2×D7—21no—
196C2×D7—41no—
197C2×D7—41no—
198C2×D7—41no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—41no—
202C4×C7—41no—
203C4×C7—41no—
204C4×C7—41no—
205C4×C72−1+411yescharacteristic
206C4×C72−1+411yescharacteristic
207Dic7—21no—
208Dic7—21no—
209Dic7—21no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—71no—
216C2×Q8—141no—
217C2×D4—71no—
218C2×D4—141no—
219C22×C4—71no—
220C22×C4—141no—
221C22×C4—141no—
222C2×C8—71no—
223C2×C8—71no—
224C2×C8C22×D711yescharacteristic
225C2×C8C22×D711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—71no—
229C4⋊C4—71no—
230C4⋊C4—71no—
231C4⋊C4—71no—
232C4⋊C4—141no—
233C4⋊C4—141no—
234C4⋊C4—141no—
235C4⋊C4—141no—
236C4⋊C4—141no—
237C4⋊C4—141no—
238C4⋊C4—141no—
239C4⋊C4—141no—
240C4⋊C4C22×D711yescharacteristic
241C4⋊C4C22×D711yescharacteristic
242C4⋊C4C22×D711yescharacteristic
243C4⋊C4C22×D711yescharacteristic
244C22⋊C4—71no—
245C22⋊C4—71no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4—141no—
251C22⋊C4—141no—
252C42—71no—
253C42—141no—
254C42—141no—
255C42C22×D711yescharacteristic
256C2×C7#53611yescharacteristic
257C2×C7#57611yescharacteristic
258C2×C7#57811yescharacteristic
259D7—41no—
260D7—81no—
261C23—71no—
262C23—141no—
263Q8—81no—
264Q8—81no—
265Q8—141no—
266Q8—141no—
267Q8—561no—
268D4—141no—
269D4—141no—
270D4—561no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—71no—
276C2×C4—71no—
277C2×C4—71no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—141no—
281C2×C4—141no—
282C2×C4—141no—
283C2×C4—561no—
284C2×C4—561no—
285C2×C4D4×D711yescharacteristic
286C2×C4D4×D711yescharacteristic
287C2×C4C23×D711yescharacteristic
288C8—21no—
289C8—21no—
290C8—141no—
291C8—141no—
292C7#324311yescharacteristic, complemented, Sylow
293C22—141no—
294C22—141no—
295C22—141no—
296C22—281no—
297C22—281no—
298C22—281no—
299C22C2×D4×D711yescharacteristic
300C4—21no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—141no—
307C4—141no—
308C4—141no—
309C4—281no—
310C4—281no—
311C4—281no—
312C4#655111yescharacteristic
313C4#655111yescharacteristic
314C2—281no—
315C2—561no—
316C2#9041511yescharacteristic
317C2#9044811yescharacteristic
318C2#9045711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#500C2
socleC22×C7C22×D4
prime radical#6458C22
hypercenter#500D7
hypocenterC7#3243
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
2. #90420C2semidirect
3. C7×#500C2semidirect
4. #89861C2semidirect
5. #89852C2semidirect
6. #89663C2semidirect
7. #89662C2semidirect
8. #89660C2semidirect
10. #89656C2semidirect
12. #89649C2semidirect
13. #89647C2semidirect
14. #89469C2semidirect
15. #89462C2semidirect
106. #500D7semidirect
292. C7#3243semidirect

Maximal subgroups

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

subgroupindex
#904242
#904202
C7×#5002
#898612
#898522
#896632
#896622
#896602
#896572
#896562
#896542
#896492
#896472
#894692
#894622
#32437