Finite Group Databasebeta

← #11736538

Subgroups — #11736538

Order 896.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90226C211yesmaximal, characteristic, complemented
2#90225C211yesmaximal, characteristic, complemented
3C7×#424C211yesmaximal, characteristic, complemented
4#89907C211yesmaximal, characteristic, complemented
5#89906C211yesmaximal, characteristic, complemented
6#89887C211yesmaximal, characteristic, complemented
7#89886C211yesmaximal, characteristic, complemented
8#89620C211yesmaximal, characteristic, complemented
9#89619C211yesmaximal, characteristic, complemented
10#89618C211yesmaximal, characteristic, complemented
11#89616C211yesmaximal, characteristic, complemented
12#89616C211yesmaximal, characteristic, complemented
13#89615C211yesmaximal, characteristic, complemented
14#89614C211yesmaximal, characteristic, complemented
15#89601C211yesmaximal, characteristic, complemented
16C2×C7×M16C2211yescharacteristic
17C22×C7×C8C2211yescharacteristic
18C7×#117C2211yescharacteristic
19#6493C2211yescharacteristic
20#6493C2211yescharacteristic
21#6493C2211yescharacteristic
22#6487C2211yescharacteristic
23C2×D56C2211yescharacteristic
24C2×D56C2211yescharacteristic
25C2×#1132C2211yescharacteristic
26C2×#1132C2211yescharacteristic
27#6461—21no—
28#6459—21no—
29#6457—21no—
30#6457—21no—
31#6448—21no—
32#6447—21no—
33#6447—21no—
34#6446—21no—
35#6438C2211yescharacteristic
36#6437C2211yescharacteristic
37C4×D28C2211yescharacteristic
38C4×D28C2211yescharacteristic
39C4×D28C2211yescharacteristic
40#6433C2211yescharacteristic
41#6432C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×C4⋊C8C2211yescharacteristic
45C7×C4⋊C8C2211yescharacteristic
46C7×C4⋊C8C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52#6395C2211yescharacteristic
53#6392C2211yescharacteristic
54#6392C2211yescharacteristic
55#6391C2211yescharacteristic
56#6391C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3257—71nomaximal, Sylow
60C22×C4×C7C2311yescharacteristic
61C2×C22⋊D7—21no—
62C2×C22⋊D7—21no—
63C2×C22⋊D7—21no—
64C2×D28—21no—
65C2×D28—21no—
66C2×D28C2311yescharacteristic
67C2×D28C2311yescharacteristic
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7—21no—
71C2×C4×D7—21no—
72C2×Dic14—21no—
73C2×Dic14C2311yescharacteristic
74C7×M16—41no—
75C2×C7×C8—41no—
76C2×C7×C8C2311yescharacteristic
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4D411yescharacteristic
81C7×C4⋊C4D411yescharacteristic
82C7×C22⋊C4D411yescharacteristic
83C7×C22⋊C4D411yescharacteristic
84C42×C7C2311yescharacteristic
85C42×C7C2311yescharacteristic
86C22⋊Dic7—21no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104C4×Dic7—21no—
105D56—41no—
106D56—41no—
107#1132—41no—
108#1132—41no—
109#542—71no—
110#540—71no—
111#479—71no—
112#478—71no—
113#476—71no—
114#474—71no—
115#469—71no—
116#468—71no—
117#466—71no—
118#465—71no—
119#463—71no—
120#463—71no—
121#460—71no—
122#459—71no—
123#424D711yescharacteristic, complemented
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7C2×D411yescharacteristic
131C2×C4×C7C2×D411yescharacteristic
132C2×C4×C7C2×D411yescharacteristic
133C2×C4×C7C2×D411yescharacteristic
134C2×C4×C7C2×D411yescharacteristic
135C2×C4×C7C2411yescharacteristic
136C22⋊D7—81no—
137C22⋊D7—81no—
138C22⋊D7—81no—
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144D28—21no—
145D28—21no—
146D28—21no—
147D28—21no—
148D28—21no—
149D28—21no—
150D28—81no—
151D28—81no—
152C4×D7—81no—
153C4×D7—81no—
154C4×D7—81no—
155Dic14—21no—
156Dic14—21no—
157Dic14—81no—
158C7×C8—21no—
159C7×C8—21no—
160C7×C8—21no—
161C7×C8—21no—
162C2×SD16—71no—
163C2×SD16—71no—
164C2×D8—71no—
165C2×D8—71no—
166C2×M16C2×D711yescharacteristic
167C22×C8C2×D711yescharacteristic
168C4⋊Q8—71no—
169#127—71no—
170#126—141no—
171#125—71no—
172#124—71no—
173#124—141no—
174#123—141no—
175#123—141no—
176#123—141no—
177C22⋊Q8—71no—
178C22⋊Q8—141no—
179#121—71no—
180#121—71no—
181#121—71no—
182#121—141no—
183#121—141no—
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—71no—
187C4×D4—71no—
188#117C2×D711yescharacteristic
189#107—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8C2×D711yescharacteristic
194C4⋊C8C2×D711yescharacteristic
195C4⋊C8C2×D711yescharacteristic
196C4⋊C8C2×D711yescharacteristic
197#103—71no—
198#103—71no—
199#102—71no—
200#102—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7C22×D411yescharacteristic
209C2×D7—41no—
210C2×D7—41no—
211C2×D7—41no—
212C2×D7—41no—
213C2×D7—41no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C2×D7—41no—
218C4×C7—21no—
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7H(5,𝔽2)11yescharacteristic
224C4×C72−1+411yescharacteristic
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230C2×Q8—71no—
231C2×Q8—141no—
232C2×D4—71no—
233C2×D4—71no—
234C2×D4—71no—
235C2×D4—141no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4—141no—
239C2×D4—141no—
240C22×C4—141no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244SD16—281no—
245SD16—281no—
246D8—281no—
247D8—281no—
248M16—41no—
249C2×C8—41no—
250C2×C8C22×D711yescharacteristic
251C2×C8C22×D711yescharacteristic
252C2×C8C22×D711yescharacteristic
253C2×C8C22×D711yescharacteristic
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—71no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4D2811yescharacteristic
264C4⋊C4D2811yescharacteristic
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4D2811yescharacteristic
275C22⋊C4D2811yescharacteristic
276C42—141no—
277C42C22×D711yescharacteristic
278C42C22×D711yescharacteristic
279C2×C7—41no—
280C2×C7#53711yescharacteristic
281C2×C7#57611yescharacteristic
282C2×C7#57711yescharacteristic
283D7—81no—
284D7—81no—
285D7—81no—
286C23—141no—
287C23—141no—
288C23—141no—
289C23C2×D2811yescharacteristic
290Q8—141no—
291Q8—141no—
292Q8—561no—
293D4—141no—
294D4—141no—
295D4—141no—
296D4—141no—
297D4—141no—
298D4—141no—
299D4—561no—
300D4—561no—
301D4—561no—
302D4—561no—
303D4—561no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—561no—
312C2×C4—561no—
313C2×C4—561no—
314C2×C4C2×D2811yescharacteristic
315C2×C4C2×D2811yescharacteristic
316C2×C4C2×D2811yescharacteristic
317C2×C4C2×D2811yescharacteristic
318C2×C4C2×D2811yescharacteristic
319C2×C4C23×D711yescharacteristic
320C8—21no—
321C8—21no—
322C8—21no—
323C8—21no—
324C7#325711yescharacteristic, complemented, Sylow
325C22—21no—
326C22—21no—
327C22—21no—
328C22—281no—
329C22—281no—
330C22—281no—
331C22—281no—
332C22—281no—
333C22—281no—
334C22—281no—
335C22—281no—
336C22—281no—
337C22C22×D2811yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—281no—
344C4—281no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4#654811yescharacteristic
349C4#655111yescharacteristic
350C2—41no—
351C2—561no—
352C2—561no—
353C2—561no—
354C2#9020111yescharacteristic
355C2#9044211yescharacteristic
356C2#9044311yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#424C2
socleC22×C7C22×D4
hypercenter#424D7
hypocenterC7#3257
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #90226C2semidirect
2. #90225C2semidirect
3. C7×#424C2semidirect
4. #89907C2semidirect
5. #89906C2semidirect
6. #89887C2semidirect
7. #89886C2semidirect
8. #89620C2semidirect
9. #89619C2semidirect
10. #89618C2semidirect
11. #89616C2semidirect
12. #89616C2semidirect
13. #89615C2semidirect
14. #89614C2semidirect
15. #89601C2semidirect
123. #424D7semidirect
324. C7#3257semidirect

Maximal subgroups

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

subgroupindex
#902262
#902252
C7×#4242
#899072
#899062
#898872
#898862
#896202
#896192
#896182
#896162
#896152
#896142
#896012
#32577