Finite Group Databasebeta

← #11736917

Subgroups — #11736917

Order 896.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90256C211yesmaximal, characteristic
2#90234C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89906C211yesmaximal, characteristic, complemented
5#89893C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic, complemented
7C2×#6392C211yesmaximal, characteristic, complemented
8#89618C211yesmaximal, characteristic, complemented
9#89601C211yesmaximal, characteristic, complemented
10#89508C211yesmaximal, characteristic, complemented
11#89507C211yesmaximal, characteristic, complemented
12#89495C211yesmaximal, characteristic, complemented
13#89494C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89474C211yesmaximal, characteristic, complemented
16C2×C7×M16C2211yescharacteristic
17C22×C7×C8C2211yescharacteristic
18C4×C7×D4C2211yescharacteristic
19#6502—21no—
20D4×Dic7—21no—
21#6493C2211yescharacteristic
22#6493C2211yescharacteristic
23#6489C2211yescharacteristic
24C2×C4⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×#1132C2211yescharacteristic
30#6461—21no—
31#6453—21no—
32#6446—21no—
33(C22⋊D7)⋊C4—21no—
34#6441—21no—
35#6441—21no—
36C4×D28C2211yescharacteristic
37#6433C2211yescharacteristic
38#6432C2211yescharacteristic
39C4×Dic14C2211yescharacteristic
40C7×C4⋊C8C2211yescharacteristic
41C7×C22⋊C8C2211yescharacteristic
42C7×C22⋊C8C2211yescharacteristic
43C7×#97C2211yescharacteristic
44#6395C2211yescharacteristic
45#6395C2211yescharacteristic
46#6395C2211yescharacteristic
47#6392C2211yescharacteristic
48#6392C2211yescharacteristic
49#6392C2211yescharacteristic
50#6392C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6391C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3346—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C22×Dic7—21no—
66C22×Dic7—21no—
67C22×Dic7—21no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×Dic14—21no—
71C2×Dic14C2311yescharacteristic
72C7×M16—22no—
73C2×C7×C8—22no—
74C2×C7×C8C2311yescharacteristic
75C2×C7×C8C2311yescharacteristic
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C7×C4⋊C4D411yescharacteristic
79C7×C22⋊C4D411yescharacteristic
80C7×C22⋊C4D411yescharacteristic
81C42×C7C2311yescharacteristic
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4—21no—
88C4⋊Dic7—21no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105#1132—22no—
106#554—71no—
107#553—71no—
108#501—71no—
109#485—71no—
110#484—71no—
111#481—71no—
112#480—71no—
113#479—71no—
114#474—71no—
115#468—71no—
116#465—71no—
117#440—71no—
118#435D711yescharacteristic, complemented
119#428—71no—
120C2×#107—71no—
121C23×C7C2×D411yescharacteristic
122C23×C7C2×D411yescharacteristic
123C22×D7—21no—
124C7×D4—22no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7C2×D411yescharacteristic
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2411yescharacteristic
134C22⋊D7—42no—
135C22⋊D7—81no—
136C2×Dic7—21no—
137C2×Dic7—21no—
138C2×Dic7—21no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—81no—
144C2×Dic7—81no—
145C2×Dic7—81no—
146D28—21no—
147D28—21no—
148C4×D7—81no—
149Dic14—21no—
150Dic14—21no—
151Dic14—81no—
152C7×C8—21no—
153C7×C8—21no—
154C7×C8—21no—
155C7×C8#4112yes—
156C2×SD16—71no—
157C2×M16C2×D711yescharacteristic
158C22×C8C2×D711yescharacteristic
159C4⋊Q8—71no—
160#126—141no—
161#125—71no—
162#125—141no—
163#123—141no—
164C22⋊Q8—71no—
165C22⋊Q8—71no—
166C22⋊Q8—141no—
167C22⋊Q8—141no—
168#121—71no—
169#121—71no—
170#121—141no—
171C4×Q8—71no—
172C4×D4—71no—
173C4×D4—141no—
174C4×D4—141no—
175C4×D4C2×D711yescharacteristic
176#117—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180#107—71no—
181#107—71no—
182#107—71no—
183#107—71no—
184#107—71no—
185#107—71no—
186#106—71no—
187#106—71no—
188#106—71no—
189C4⋊C8C2×D711yescharacteristic
190#103—71no—
191#103—71no—
192#103—71no—
193#102—71no—
194#102—71no—
195#102—71no—
196C22⋊C8C2×D711yescharacteristic
197C22⋊C8C2×D711yescharacteristic
198#97C2×D711yescharacteristic
199C22×C7—21no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7#13712yes—
205C22×C7C22×D411yescharacteristic
206C2×D7—41no—
207C2×D7—41no—
208C2×D7—41no—
209C4×C7—21no—
210C4×C7—21no—
211C4×C7—21no—
212C4×C7—21no—
213C4×C7C2×#4111yescharacteristic
214C4×C72−1+411yescharacteristic
215Dic7—41no—
216Dic7—41no—
217Dic7—41no—
218Dic7—41no—
219Dic7—41no—
220Dic7—41no—
221Dic7—41no—
222C2×Q8—71no—
223C2×Q8—141no—
224C2×D4—71no—
225C2×D4—141no—
226C2×D4—141no—
227C2×D4D2811yescharacteristic
228C22×C4—141no—
229C22×C4—141no—
230C22×C4—141no—
231C22×C4—141no—
232C22×C4C22×D711yescharacteristic
233C22×C4C22×D711yescharacteristic
234SD16—142no—
235M16—22no—
236C2×C8—22no—
237C2×C8C22×D711yescharacteristic
238C2×C8C22×D711yescharacteristic
239C2×C8C22×D711yescharacteristic
240C2×C8C22×D711yescharacteristic
241C4⋊C4—71no—
242C4⋊C4—71no—
243C4⋊C4—71no—
244C4⋊C4—71no—
245C4⋊C4—71no—
246C4⋊C4—71no—
247C4⋊C4—141no—
248C4⋊C4—141no—
249C4⋊C4—141no—
250C4⋊C4—141no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4D2811yescharacteristic
257C22⋊C4—141no—
258C22⋊C4—141no—
259C22⋊C4—141no—
260C22⋊C4—141no—
261C22⋊C4—141no—
262C22⋊C4—141no—
263C22⋊C4D2811yescharacteristic
264C22⋊C4D2811yescharacteristic
265C42—141no—
266C42—141no—
267C42C22×D711yescharacteristic
268C2×C7—22no—
269C2×C7—41no—
270C2×C7#54711yescharacteristic
271C2×C7C2×#13711yescharacteristic
272C2×C7#57611yescharacteristic
273D7—81no—
274C23—141no—
275C23C2×D2811yescharacteristic
276C23C2×D2811yescharacteristic
277Q8—141no—
278Q8—141no—
279Q8—561no—
280D4—22no—
281D4—141no—
282D4—141no—
283D4—282no—
284D4—561no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—141no—
292C2×C4—141no—
293C2×C4—141no—
294C2×C4—141no—
295C2×C4—141no—
296C2×C4—561no—
297C2×C4—561no—
298C2×C4—561no—
299C2×C4—561no—
300C2×C4C2×D2811yescharacteristic
301C2×C4C2×D2811yescharacteristic
302C2×C4C2×D2811yescharacteristic
303C2×C4C2×D2811yescharacteristic
304C2×C4C23×D711yescharacteristic
305C8—21no—
306C8—21no—
307C8—21no—
308C8#115912yes—
309C7#334611yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—281no—
316C22—281no—
317C22—281no—
318C22#647212yes—
319C22C22×D2811yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—281no—
325C4—281no—
326C4—281no—
327C4—281no—
328C4—281no—
329C4—281no—
330C4—281no—
331C4C2×#115911yescharacteristic
332C4#655411yescharacteristic
333C2—22no—
334C2—41no—
335C2—561no—
336C2#9024611yescharacteristic
337C2C2×#647211yescharacteristic
338C2#9044311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radical#90256C2
hypercenter#435D7
hypocenterC7#3346
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #90234C2semidirect
3. C7×#435C2semidirect
4. #89906C2semidirect
5. #89893C2semidirect
6. #89886C2semidirect
7. C2×#6392C2semidirect
8. #89618C2semidirect
9. #89601C2semidirect
10. #89508C2semidirect
11. #89507C2semidirect
12. #89495C2semidirect
13. #89494C2semidirect
14. #89482C2semidirect
15. #89474C2semidirect
118. #435D7semidirect
309. C7#3346semidirect

Maximal subgroups

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

subgroupindex
#902562
#902342
C7×#4352
#899062
#898932
#898862
C2×#63922
#896182
#896012
#895082
#895072
#894952
#894942
#894822
#894742
#33467