Finite Group Databasebeta

← #11737974

Subgroups — #11737974

Order 896.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90362C211yesmaximal, characteristic, complemented
2#90233C211yesmaximal, characteristic, complemented
3C7×#463C211yesmaximal, characteristic
4#89939C211yesmaximal, characteristic, complemented
5#89938C211yesmaximal, characteristic, complemented
6#89927C211yesmaximal, characteristic
7#89925C211yesmaximal, characteristic
8#89832C211yesmaximal, characteristic, complemented
9#89790C211yesmaximal, characteristic, complemented
10#89621C211yesmaximal, characteristic, complemented
11#89600C211yesmaximal, characteristic, complemented
12#89565C211yesmaximal, characteristic, complemented
13#89562C211yesmaximal, characteristic, complemented
14#89539C211yesmaximal, characteristic, complemented
15#89529C211yesmaximal, characteristic, complemented
16C2×C7×SD16C2211yescharacteristic
17C2×C7×D8C2211yescharacteristic
18C7×#124C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21#6507C2211yescharacteristic
22C2×#1144C2211yescharacteristic
23C2×#1144C2211yescharacteristic
24#6499C2211yescharacteristic
25#6499C2211yescharacteristic
26#6499C2211yescharacteristic
27D4×Dic7C2211yescharacteristic
28C2×#1142C2211yescharacteristic
29C2×#1142C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31#6489—21no—
32#6487—21no—
33C2×Dic28C2211yescharacteristic
34C2×Dic28C2211yescharacteristic
35#6450C2211yescharacteristic
36#6450C2211yescharacteristic
37#6442—21no—
38#6442—21no—
39#6442—21no—
40#6442—21no—
41#6441—21no—
42#6440—21no—
43#6432C2211yescharacteristic
44C4×Dic14C2211yescharacteristic
45C4×Dic14C2211yescharacteristic
46C7×C4⋊C8C2211yescharacteristic
47C7×#103C2211yescharacteristic
48C7×#102C2211yescharacteristic
49#6406C2211yescharacteristic
50#6390C2211yescharacteristic
51#6390C2211yescharacteristic
52#6389C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56#6384C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3402—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C2×C7×D4C2311yescharacteristic
63C22×C4×C7—21no—
64C22×Dic7—21no—
65C2×Dic14D411yescharacteristic
66C2×Dic14C2311yescharacteristic
67C2×Dic14C2311yescharacteristic
68C2×Dic14C2311yescharacteristic
69C7×SD16—41no—
70C7×D8—41no—
71C2×C7×C8C2311yescharacteristic
72C2×C7×C8C2311yescharacteristic
73C7×C4⋊C4C2311yescharacteristic
74C7×C22⋊C4—21no—
75C7×C22⋊C4—21no—
76C7×C22⋊C4—21no—
77C42×C7C2311yescharacteristic
78C22⋊Dic7—21no—
79C22⋊Dic7—21no—
80C22⋊Dic7—21no—
81C22⋊Dic7—21no—
82C22⋊Dic7—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85#1144—22no—
86#1144—41no—
87#1142—22no—
88#1142—22no—
89#1142—41no—
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7D411yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95Dic7⋊C4—21no—
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4D411yescharacteristic
101Dic7⋊C4D411yescharacteristic
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7C2311yescharacteristic
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110Dic28—41no—
111Dic28—41no—
112#556—71no—
113#556—71no—
114#497—71no—
115#497—71no—
116#495—71no—
117#464—71no—
118#464—71no—
119#463—71no—
120#463D711yescharacteristic
121#462—71no—
122#462—71no—
123#446—71no—
124#440—71no—
125#440—71no—
126C4×Q16—71no—
127C23×C7—21no—
128C23×C7—21no—
129C7×Q8—21no—
130C7×Q8—21no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—42no—
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—42no—
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148C2×Dic7C2×D411yescharacteristic
149C2×Dic7C2×D411yescharacteristic
150Dic14—21no—
151Dic14—21no—
152Dic14—21no—
153Dic14—21no—
154Dic14—21no—
155Dic14—21no—
156Dic14—41no—
157C7×C8—21no—
158C7×C8—21no—
159C7⋊C8—21no—
160C7⋊C8#4112yes—
161C2×Q16—71no—
162C2×Q16—71no—
163C2×Q16—71no—
164C2×Q16—71no—
165C2×SD16—71no—
166C2×SD16—71no—
167C2×SD16—71no—
168C2×SD16C2×D711yescharacteristic
169C2×D8C2×D711yescharacteristic
170C4⋊Q8—71no—
171C4⋊Q8—71no—
172#126—141no—
173#126—141no—
174#126—141no—
175#126—141no—
176#124—71no—
177#124—71no—
178#124—71no—
179#124C2×D711yescharacteristic
180C22⋊Q8—141no—
181C22⋊Q8—141no—
182C4×Q8—71no—
183C4×Q8—71no—
184C4×Q8—71no—
185C4×Q8—71no—
186C4×Q8—71no—
187C4×D4—71no—
188C4×D4C2×D711yescharacteristic
189#117—141no—
190#117—141no—
191#107—71no—
192C4⋊C8—71no—
193C4⋊C8—71no—
194C4⋊C8C2×D711yescharacteristic
195#103—71no—
196#103—71no—
197#103—71no—
198#103C2×D711yescharacteristic
199#102—71no—
200#102C2×D711yescharacteristic
201#97—71no—
202#97—71no—
203C4×C8—71no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7C22×D411yescharacteristic
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—41no—
214C4×C7—41no—
215C4×C7C2×#4111yescharacteristic
216C4×C7H(5,𝔽2)11yescharacteristic
217Dic7—21no—
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—41no—
222Dic7—41no—
223Dic7—41no—
224Dic7—41no—
225C2×Q8—71no—
226C2×Q8—71no—
227C2×Q8—71no—
228C2×Q8—71no—
229C2×Q8C22×D711yescharacteristic
230C2×D4C22×D711yescharacteristic
231C2×D4C22×D711yescharacteristic
232C22×C4—21no—
233C22×C4—141no—
234Q16—142no—
235Q16—281no—
236Q16—281no—
237Q16—281no—
238SD16—41no—
239SD16—142no—
240SD16—142no—
241SD16—281no—
242D8—41no—
243C2×C8—71no—
244C2×C8—71no—
245C2×C8C22×D711yescharacteristic
246C2×C8C22×D711yescharacteristic
247C4⋊C4—71no—
248C4⋊C4—71no—
249C4⋊C4—71no—
250C4⋊C4—71no—
251C4⋊C4—141no—
252C4⋊C4—141no—
253C4⋊C4—141no—
254C4⋊C4—141no—
255C4⋊C4—141no—
256C4⋊C4—141no—
257C4⋊C4—141no—
258C4⋊C4—141no—
259C4⋊C4C22×D711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—141no—
264C22⋊C4—141no—
265C22⋊C4—141no—
266C22⋊C4—141no—
267C22⋊C4—141no—
268C22⋊C4—141no—
269C22⋊C4—141no—
270C42—71no—
271C42—71no—
272C42—141no—
273C42—141no—
274C42—141no—
275C42—141no—
276C42C22×D711yescharacteristic
277C2×C7—81no—
278C2×C7—81no—
279C2×C7#55011yescharacteristic
280C2×C7#57511yescharacteristic
281C2×C7#57811yescharacteristic
282C23—21no—
283C23—21no—
284Q8—21no—
285Q8—21no—
286Q8—141no—
287Q8—141no—
288Q8—141no—
289Q8—141no—
290Q8—141no—
291Q8—141no—
292Q8—281no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—42no—
300C2×C4—71no—
301C2×C4—71no—
302C2×C4—71no—
303C2×C4—71no—
304C2×C4—141no—
305C2×C4—141no—
306C2×C4—141no—
307C2×C4—141no—
308C2×C4—282no—
309C2×C4D4×D711yescharacteristic
310C2×C4D4×D711yescharacteristic
311C2×C4C23×D711yescharacteristic
312C8—21no—
313C8—21no—
314C8—72no—
315C8—141no—
316C7#340211yescharacteristic, complemented, Sylow
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22C2×D4×D711yescharacteristic
324C4—21no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—141no—
329C4—141no—
330C4—141no—
331C4—141no—
332C4—281no—
333C4—281no—
334C4—281no—
335C4—281no—
336C4#654811yescharacteristic
337C4D7×#4111yescharacteristic
338C2—81no—
339C2—81no—
340C2#9029011yescharacteristic
341C2#9046211yescharacteristic
342C2#9046411yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#463C2
socleC22×C7C22×D4
prime radicalC2×C7×D8C22
hypercenter#463D7
hypocenterC7#3402
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #90362C2semidirect
2. #90233C2semidirect
4. #89939C2semidirect
5. #89938C2semidirect
8. #89832C2semidirect
9. #89790C2semidirect
10. #89621C2semidirect
11. #89600C2semidirect
12. #89565C2semidirect
13. #89562C2semidirect
14. #89539C2semidirect
15. #89529C2semidirect
316. C7#3402semidirect

Maximal subgroups

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

subgroupindex
#903622
#902332
C7×#4632
#899392
#899382
#899272
#899252
#898322
#897902
#896212
#896002
#895652
#895622
#895392
#895292
#34027