Finite Group Databasebeta

← #11737813

Subgroups — #11737813

Order 896.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90410C211yesmaximal, characteristic
2#90274C211yesmaximal, characteristic, complemented
3C7×#460C211yesmaximal, characteristic, complemented
4#89946C211yesmaximal, characteristic, complemented
5#89943C211yesmaximal, characteristic, complemented
6#89934C211yesmaximal, characteristic, complemented
7#89934C211yesmaximal, characteristic, complemented
8#89847C211yesmaximal, characteristic, complemented
9#89794C211yesmaximal, characteristic, complemented
10#89611C211yesmaximal, characteristic, complemented
11#89601C211yesmaximal, characteristic, complemented
12#89552C211yesmaximal, characteristic, complemented
13#89550C211yesmaximal, characteristic, complemented
14#89538C211yesmaximal, characteristic, complemented
15#89535C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×SD16C2211yescharacteristic
18C7×#127C2211yescharacteristic
19C4×C7×Q8C2211yescharacteristic
20#6509C2211yescharacteristic
21#6507C2211yescharacteristic
22#6502—21no—
23#6502—21no—
24#6501C2211yescharacteristic
25#6501C2211yescharacteristic
26#6498—21no—
27#6498—21no—
28D4×Dic7C2211yescharacteristic
29D4×Dic7C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31C2×#1142C2211yescharacteristic
32C2×#1131C2211yescharacteristic
33C2×C8×D7C2211yescharacteristic
34#6461—21no—
35#6460C2211yescharacteristic
36#6453C2211yescharacteristic
37#6452—21no—
38#6439—21no—
39#6435C2211yescharacteristic
40#6433—21no—
41#6433C2211yescharacteristic
42C4×Dic14C2211yescharacteristic
43C7×C4⋊C8C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50#6392C2211yescharacteristic
51#6391C2211yescharacteristic
52#6390C2211yescharacteristic
53#6390C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56#6382C2211yescharacteristic
57#6381C2211yescharacteristic
58C4⋊(C7⋊C8)C2211yescharacteristic
59#3260—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4—21no—
62C2×C7×D4—21no—
63C2×C7×D4C2311yescharacteristic
64C2×C7×D4C2311yescharacteristic
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68C22×Dic7—21no—
69C2×C4×D7C2311yescharacteristic
70C2×Dic14C2311yescharacteristic
71C7×SD16—41no—
72C7×SD16—41no—
73C2×C7×C8C2311yescharacteristic
74C2×C7×C8C2311yescharacteristic
75C7×C4⋊C4—21no—
76C7×C4⋊C4C2311yescharacteristic
77C42×C7—21no—
78C42×C7C2311yescharacteristic
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83#1142—41no—
84#1142—41no—
85(C2×D7)⋊C4—21no—
86(C2×D7)⋊C4—21no—
87(C2×D7)⋊C4D411yescharacteristic
88(C2×D7)⋊C4D411yescharacteristic
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7C2311yescharacteristic
92C4⋊Dic7C2311yescharacteristic
93C4⋊Dic7C2311yescharacteristic
94Dic7⋊C4—21no—
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4D411yescharacteristic
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—41no—
108C8×D7—41no—
109#541—71no—
110#540—71no—
111#479—71no—
112#479—71no—
113#478—71no—
114#478—71no—
115#468—71no—
116#468—71no—
117#465—71no—
118#465—71no—
119#461—71no—
120#461—71no—
121#460—71no—
122#460D711yescharacteristic, complemented
123#424—71no—
124C23×C7—21no—
125C23×C7—21no—
126C22×D7C2×D411yescharacteristic
127C7×Q8—21no—
128C7×Q8—21no—
129C7×D4—21no—
130C7×D4—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—81no—
134C7×D4—81no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2411yescharacteristic
140C22⋊D7—81no—
141C22⋊D7—81no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—21no—
146C2×Dic7—81no—
147C2×Dic7—81no—
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150C2×Dic7C2×D411yescharacteristic
151C4×D7—21no—
152C4×D7—21no—
153Dic14—21no—
154Dic14—21no—
155C7×C8—21no—
156C7×C8—21no—
157C7⋊C8—21no—
158C7⋊C8—21no—
159C2×SD16—71no—
160C2×SD16—71no—
161C2×SD16C2×D711yescharacteristic
162C2×SD16C2×D711yescharacteristic
163C2×M16—71no—
164C22×C8—71no—
165C4⋊Q8—71no—
166#127C2×D711yescharacteristic
167#126—141no—
168#126—141no—
169#125—71no—
170#125—71no—
171#125—141no—
172#125—141no—
173#123—141no—
174#123—141no—
175C22⋊Q8—71no—
176C22⋊Q8—71no—
177#121—71no—
178#121—71no—
179#121—141no—
180#121—141no—
181C4×Q8—71no—
182C4×Q8C2×D711yescharacteristic
183C4×D4—71no—
184C4×D4—71no—
185#117—71no—
186#107—71no—
187#107—71no—
188#106—71no—
189#106—71no—
190C4⋊C8—71no—
191C4⋊C8—71no—
192C4⋊C8—71no—
193C4⋊C8C2×D711yescharacteristic
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#102—71no—
199#102—71no—
200#102C2×D711yescharacteristic
201#102C2×D711yescharacteristic
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7C22×D411yescharacteristic
209C2×D7—21no—
210C2×D7—21no—
211C2×D7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—41no—
215C4×C7—41no—
216C4×C7H(5,𝔽2)11yescharacteristic
217C4×C72−1+411yescharacteristic
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—71no—
226C2×Q8C22×D711yescharacteristic
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—141no—
230C2×D4—141no—
231C2×D4C22×D711yescharacteristic
232C2×D4C22×D711yescharacteristic
233C22×C4—71no—
234C22×C4—141no—
235C22×C4—141no—
236SD16—41no—
237SD16—41no—
238SD16—281no—
239SD16—281no—
240M16—281no—
241C2×C8—71no—
242C2×C8—71no—
243C2×C8—281no—
244C2×C8C22×D711yescharacteristic
245C2×C8C22×D711yescharacteristic
246C4⋊C4—21no—
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—71no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4C22×D711yescharacteristic
262C22⋊C4—71no—
263C22⋊C4—71no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—21no—
271C42—71no—
272C42—141no—
273C42C22×D711yescharacteristic
274C2×C7—81no—
275C2×C7—81no—
276C2×C7#53711yescharacteristic
277C2×C7#57711yescharacteristic
278C2×C7#57711yescharacteristic
279D7—41no—
280C23—21no—
281C23—21no—
282C23—71no—
283Q8—21no—
284Q8—21no—
285Q8—141no—
286Q8—141no—
287D4—21no—
288D4—21no—
289D4—21no—
290D4—21no—
291D4—81no—
292D4—81no—
293D4—561no—
294D4—561no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—71no—
298C2×C4—71no—
299C2×C4—71no—
300C2×C4—141no—
301C2×C4—141no—
302C2×C4—141no—
303C2×C4—141no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—561no—
307C2×C4—561no—
308C2×C4D4×D711yescharacteristic
309C2×C4D4×D711yescharacteristic
310C2×C4C23×D711yescharacteristic
311C8—21no—
312C8—21no—
313C8—141no—
314C8—141no—
315C7#326011yescharacteristic, complemented, Sylow
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—141no—
323C22—141no—
324C22—141no—
325C22C2×D4×D711yescharacteristic
326C4—21no—
327C4—21no—
328C4—41no—
329C4—41no—
330C4—141no—
331C4—141no—
332C4—141no—
333C4—281no—
334C4—281no—
335C4—281no—
336C4—281no—
337C4#654811yescharacteristic
338C4#655411yescharacteristic
339C2—81no—
340C2—81no—
341C2—281no—
342C2#9029911yescharacteristic
343C2#9045311yescharacteristic
344C2#9046611yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#460C2
socleC22×C7C22×D4
prime radical#90410C2
hypercenter#460D7
hypocenterC7#3260
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #90274C2semidirect
3. C7×#460C2semidirect
4. #89946C2semidirect
5. #89943C2semidirect
6. #89934C2semidirect
7. #89934C2semidirect
8. #89847C2semidirect
9. #89794C2semidirect
10. #89611C2semidirect
11. #89601C2semidirect
12. #89552C2semidirect
13. #89550C2semidirect
14. #89538C2semidirect
15. #89535C2semidirect
122. #460D7semidirect
315. C7#3260semidirect

Maximal subgroups

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

subgroupindex
#904102
#902742
C7×#4602
#899462
#899432
#899342
#898472
#897942
#896112
#896012
#895522
#895502
#895382
#895352
#32607