Finite Group Databasebeta

← #11736809

Subgroups — #11736809

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#90225C211yesmaximal, characteristic
2#90210C211yesmaximal, characteristic, complemented
3C7×#433C211yesmaximal, characteristic, complemented
4#89622C211yesmaximal, characteristic, complemented
5#89619C211yesmaximal, characteristic, complemented
6#89618C211yesmaximal, characteristic, complemented
7#89601C211yesmaximal, characteristic, complemented
8#89509C211yesmaximal, characteristic
9#89508C211yesmaximal, characteristic
10#89495C211yesmaximal, characteristic, complemented
11#89494C211yesmaximal, characteristic, complemented
12#89486C211yesmaximal, characteristic, complemented
13#89474C211yesmaximal, characteristic, complemented
14#89469C211yesmaximal, characteristic, complemented
15#89454C211yesmaximal, characteristic, complemented
16C7×#117C2211yescharacteristic
17#6493C2211yescharacteristic
18#6489C2211yescharacteristic
19C2×C4⋊Dic7C2211yescharacteristic
20#6487C2211yescharacteristic
21#6461—21no—
22#6459—21no—
23#6453—21no—
24#6451—21no—
25#6448—21no—
26#6446—21no—
27#6441—21no—
28#6441—21no—
29#6438C2211yescharacteristic
30C4×D28C2211yescharacteristic
31#6433C2211yescharacteristic
32#6433C2211yescharacteristic
33#6432C2211yescharacteristic
34#6432C2211yescharacteristic
35#6432C2211yescharacteristic
36C4×Dic14C2211yescharacteristic
37C7×C4⋊C8C2211yescharacteristic
38C7×C4⋊C8C2211yescharacteristic
39C7×C22⋊C8C2211yescharacteristic
40C7×C22⋊C8C2211yescharacteristic
41C7×#97C2211yescharacteristic
42C4×C7×C8C2211yescharacteristic
43#6395C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6392C2211yescharacteristic
48#6392C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3243—71nomaximal, Sylow
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—21no—
62C22×Dic7—21no—
63C2×D28C2311yescharacteristic
64C2×C4×D7—21no—
65C2×Dic14—21no—
66C2×Dic14—21no—
67C2×Dic14—21no—
68C2×Dic14C2311yescharacteristic
69C2×C7×C8C2311yescharacteristic
70C2×C7×C8C2311yescharacteristic
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4D411yescharacteristic
74C7×C4⋊C4D411yescharacteristic
75C7×C22⋊C4D411yescharacteristic
76C7×C22⋊C4D411yescharacteristic
77C42×C7C2311yescharacteristic
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81(C2×D7)⋊C4—21no—
82(C2×D7)⋊C4—21no—
83(C2×D7)⋊C4—21no—
84(C2×D7)⋊C4—21no—
85C4⋊Dic7—21no—
86C4⋊Dic7—21no—
87C4⋊Dic7—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7C2311yescharacteristic
90C4⋊Dic7C2311yescharacteristic
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101C4×Dic7—21no—
102C4×Dic7—21no—
103#544—71no—
104#542—71no—
105#501—71no—
106#500—71no—
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#433D711yescharacteristic, complemented
118C23×C7C2×D411yescharacteristic
119C22×D7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7C2×D411yescharacteristic
123C2×C4×C7C2×D411yescharacteristic
124C2×C4×C7C2×D411yescharacteristic
125C2×C4×C7C2×D411yescharacteristic
126C2×C4×C7C2×D411yescharacteristic
127C2×C4×C7C2411yescharacteristic
128C22⋊D7—81no—
129C2×Dic7—21no—
130C2×Dic7—21no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—21no—
134C2×Dic7—21no—
135C2×Dic7—21no—
136C2×Dic7—81no—
137D28—21no—
138D28—21no—
139C4×D7—81no—
140Dic14—21no—
141Dic14—21no—
142Dic14—81no—
143Dic14—81no—
144Dic14—81no—
145C7×C8—21no—
146C7×C8—21no—
147C7×C8—21no—
148C7×C8—21no—
149C4⋊Q8—71no—
150C4⋊Q8—71no—
151C4⋊Q8—71no—
152C4⋊Q8—141no—
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#117C2×D711yescharacteristic
169C2×C4⋊C4—71no—
170#107—71no—
171#107—71no—
172#107—71no—
173#107—71no—
174#107—71no—
175#107—71no—
176#106—71no—
177#106—71no—
178C4⋊C8C2×D711yescharacteristic
179C4⋊C8C2×D711yescharacteristic
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⋊C8C2×D711yescharacteristic
189C22⋊C8C2×D711yescharacteristic
190#97C2×D711yescharacteristic
191C4×C8C2×D711yescharacteristic
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7C22×D411yescharacteristic
196C2×D7—41no—
197C2×D7—41no—
198C2×D7—41no—
199C4×C7—21no—
200C4×C7—21no—
201C4×C7—21no—
202C4×C7—21no—
203C4×C7—21no—
204C4×C72−1+411yescharacteristic
205C4×C72−1+411yescharacteristic
206Dic7—41no—
207Dic7—41no—
208Dic7—41no—
209Dic7—41no—
210Dic7—41no—
211Dic7—41no—
212Dic7—41no—
213C2×Q8—71no—
214C2×Q8—141no—
215C2×Q8—141no—
216C2×Q8—141no—
217C2×D4—71no—
218C2×D4—141no—
219C22×C4—141no—
220C22×C4—141no—
221C22×C4C22×D711yescharacteristic
222C2×C8C22×D711yescharacteristic
223C2×C8C22×D711yescharacteristic
224C2×C8C22×D711yescharacteristic
225C2×C8C22×D711yescharacteristic
226C4⋊C4—71no—
227C4⋊C4—71no—
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⋊C4—141no—
241C4⋊C4—141no—
242C4⋊C4D2811yescharacteristic
243C4⋊C4D2811yescharacteristic
244C22⋊C4—141no—
245C22⋊C4—141no—
246C22⋊C4—141no—
247C22⋊C4—141no—
248C22⋊C4—141no—
249C22⋊C4—141no—
250C22⋊C4D2811yescharacteristic
251C22⋊C4D2811yescharacteristic
252C42—141no—
253C42—141no—
254C42C22×D711yescharacteristic
255C42C22×D711yescharacteristic
256C2×C7—41no—
257C2×C7#53611yescharacteristic
258C2×C7#57611yescharacteristic
259C2×C7#57811yescharacteristic
260D7—81no—
261C23—141no—
262C23C2×D2811yescharacteristic
263Q8—141no—
264Q8—141no—
265Q8—561no—
266Q8—561no—
267Q8—561no—
268D4—141no—
269D4—141no—
270D4—561no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—141no—
274C2×C4—141no—
275C2×C4—141no—
276C2×C4—141no—
277C2×C4—141no—
278C2×C4—141no—
279C2×C4—141no—
280C2×C4—561no—
281C2×C4—561no—
282C2×C4C2×D2811yescharacteristic
283C2×C4C2×D2811yescharacteristic
284C2×C4C2×D2811yescharacteristic
285C2×C4C2×D2811yescharacteristic
286C2×C4C2×D2811yescharacteristic
287C2×C4C23×D711yescharacteristic
288C8—21no—
289C8—21no—
290C8—21no—
291C8—21no—
292C7#324311yescharacteristic, complemented, Sylow
293C22—21no—
294C22—21no—
295C22—21no—
296C22—281no—
297C22—281no—
298C22—281no—
299C22C22×D2811yescharacteristic
300C4—21no—
301C4—21no—
302C4—21no—
303C4—21no—
304C4—21no—
305C4—281no—
306C4—281no—
307C4—281no—
308C4—281no—
309C4—281no—
310C4—281no—
311C4—281no—
312C4#655411yescharacteristic
313C4#655411yescharacteristic
314C2—41no—
315C2—561no—
316C2#9021711yescharacteristic
317C2#9044311yescharacteristic
318C2#9044411yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#433C2
socleC22×C7C22×D4
prime radical#6493C22
hypercenter#433D7
hypocenterC7#3243
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
2. #90210C2semidirect
3. C7×#433C2semidirect
4. #89622C2semidirect
5. #89619C2semidirect
6. #89618C2semidirect
7. #89601C2semidirect
10. #89495C2semidirect
11. #89494C2semidirect
12. #89486C2semidirect
13. #89474C2semidirect
14. #89469C2semidirect
15. #89454C2semidirect
117. #433D7semidirect
292. C7#3243semidirect

Maximal subgroups

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

subgroupindex
#902252
#902102
C7×#4332
#896222
#896192
#896182
#896012
#895092
#895082
#894952
#894942
#894862
#894742
#894692
#894542
#32437