Finite Group Databasebeta

← #11737237

Subgroups — #11737237

Order 896.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90269C211yesmaximal, characteristic, complemented
2#90239C211yesmaximal, characteristic
3C7×#440C211yesmaximal, characteristic, complemented
4#89947C211yesmaximal, characteristic, complemented
5#89946C211yesmaximal, characteristic, complemented
6#89798C211yesmaximal, characteristic, complemented
7#89790C211yesmaximal, characteristic, complemented
8#89658C211yesmaximal, characteristic, complemented
9#89646C211yesmaximal, characteristic, complemented
10#89584C211yesmaximal, characteristic, complemented
11#89564C211yesmaximal, characteristic, complemented
12#89552C211yesmaximal, characteristic, complemented
13#89532C211yesmaximal, characteristic, complemented
14#89482C211yesmaximal, characteristic, complemented
15#89479C211yesmaximal, characteristic, complemented
16C2×#1157—21no—
17C2×C7×SD16C2211yescharacteristic
18C4×C7×Q8C2211yescharacteristic
19C4×C7×D4C2211yescharacteristic
20#6509C2211yescharacteristic
21#6507C2211yescharacteristic
22C2×#1143C2211yescharacteristic
23#6503C2211yescharacteristic
24#6501C2211yescharacteristic
25C2×#1142C2211yescharacteristic
26C4×C22⋊D7—21no—
27C2×#1132C2211yescharacteristic
28C2×#1131C2211yescharacteristic
29C2×#1131C2211yescharacteristic
30#6460C2211yescharacteristic
31#6459—21no—
32#6458C2211yescharacteristic
33#6452—21no—
34#6451C2211yescharacteristic
35#6451C2211yescharacteristic
36#6448—21no—
37#6447—21no—
38C4×D28C2211yescharacteristic
39#6435—21no—
40C42×D7C2211yescharacteristic
41C4×Dic14—21no—
42C4×Dic14C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#103C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C7×#97C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6395C2211yescharacteristic
50#6392C2211yescharacteristic
51#6390C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53C8×Dic7C2211yescharacteristic
54#6384C2211yescharacteristic
55#6383C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6377C2211yescharacteristic
59#3182—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66#1157—44no—
67C2×D28—21no—
68C2×D28C2311yescharacteristic
69C2×C4×D7—21no—
70C2×C4×D7D411yescharacteristic
71C2×C4×D7D411yescharacteristic
72C2×C4×D7C2311yescharacteristic
73C2×Dic14—21no—
74C2×Dic14—21no—
75C2×Dic14C2311yescharacteristic
76C7×SD16—22no—
77C2×C7×C8C2311yescharacteristic
78C2×C7×C8C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4C2311yescharacteristic
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C42×C7—21no—
84C42×C7C2311yescharacteristic
85C22⋊Dic7—21no—
86#1143—22no—
87#1142—22no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92C4⋊Dic7—21no—
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×Dic7D411yescharacteristic
103C4×Dic7D411yescharacteristic
104C4×Dic7C2311yescharacteristic
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1132—22no—
108#1131—22no—
109#1131—22no—
110#533—71no—
111#532—71no—
112#500—71no—
113#492—71no—
114#487—71no—
115#486—71no—
116#468—71no—
117#468—71no—
118#468—71no—
119#468—71no—
120#440—71no—
121#440—71no—
122#440—71no—
123#440D711yescharacteristic, complemented
124C4×M16—71no—
125C23×C7—21no—
126C22×D7—21no—
127C22×D7C2×D411yescharacteristic
128C7×Q8—21no—
129C7×Q8—21no—
130C7×D4—21no—
131C7×D4—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—42no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2411yescharacteristic
139C22⋊D7—42no—
140C22⋊D7—42no—
141C22⋊D7—42no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147C2×Dic7C2×D411yescharacteristic
148D28—21no—
149D28—21no—
150D28—42no—
151C4×D7—21no—
152C4×D7—21no—
153C4×D7—22no—
154C4×D7—22no—
155C4×D7—42no—
156Dic14—21no—
157Dic14—21no—
158Dic14—42no—
159Dic14—42no—
160C7×C8—21no—
161C7×C8#4112yescomplemented
162C7⋊C8—21no—
163C7⋊C8#4112yescomplemented
164C2×#41—141no—
165C2×SD16—71no—
166C2×SD16—71no—
167C2×SD16—71no—
168C2×SD16C2×D711yescharacteristic
169C2×M16—71no—
170C2×M16—71no—
171C4⋊Q8—71no—
172C4⋊Q8—71no—
173C4⋊Q8—71no—
174#127—71no—
175#125—141no—
176#124—141no—
177C22⋊Q8—71no—
178C22⋊Q8—71no—
179C22⋊Q8—141no—
180#121—71no—
181#121—71no—
182#121—141no—
183C4×Q8—71no—
184C4×Q8—141no—
185C4×Q8C2×D711yescharacteristic
186C4×D4—71no—
187C4×D4—141no—
188C4×D4C2×D711yescharacteristic
189#117—141no—
190C2×C42—71no—
191#107—71no—
192#107—71no—
193#107—71no—
194#107C2×D711yescharacteristic
195#103—71no—
196#103—71no—
197#103—71no—
198#103C2×D711yescharacteristic
199#102—71no—
200#102—71no—
201#102—71no—
202#102C2×D711yescharacteristic
203#97—71no—
204#97C2×D711yescharacteristic
205C4×C8—71no—
206C4×C8—71no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7C22×D411yescharacteristic
211C2×D7—21no—
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—41no—
215C2×D7—41no—
216C2×D7—41no—
217C4×C7—21no—
218C4×C7—21no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7C2×#4111yescharacteristic
223C4×C7C2×#4111yescharacteristic
224Dic7—21no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7AGL(1,ℤ/8)12yes—
229Dic7#13712yes—
230#41—284no—
231C2×Q8—71no—
232C2×Q8—141no—
233C2×Q8—141no—
234C2×Q8C22×D711yescharacteristic
235C2×D4—71no—
236C2×D4—141no—
237C2×D4—141no—
238C2×D4—141no—
239C2×D4—141no—
240C2×D4C22×D711yescharacteristic
241C22×C4—21no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—71no—
245C22×C4—141no—
246SD16—22no—
247SD16—142no—
248SD16—142no—
249SD16—142no—
250M16—142no—
251M16—142no—
252C2×C8—71no—
253C2×C8—71no—
254C2×C8C22×D711yescharacteristic
255C2×C8C22×D711yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—71no—
258C4⋊C4—71no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4C22×D711yescharacteristic
267C4⋊C4C22×D711yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—141no—
270C22⋊C4—141no—
271C22⋊C4—141no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C42—21no—
275C42—71no—
276C42—71no—
277C42—71no—
278C42—141no—
279C42C22×D711yescharacteristic
280C2×C7—81no—
281C2×C7#53211yescharacteristic
282C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
283C2×C7C2×#13711yescharacteristic
284D7—41no—
285D7—81no—
286C23—21no—
287C23—71no—
288C23—141no—
289Q8—21no—
290Q8—21no—
291Q8—141no—
292Q8—141no—
293Q8—282no—
294Q8—282no—
295D4—21no—
296D4—21no—
297D4—141no—
298D4—141no—
299D4—282no—
300D4—282no—
301D4—282no—
302D4—282no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—71no—
308C2×C4—71no—
309C2×C4—71no—
310C2×C4—142no—
311C2×C4—142no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—282no—
318C2×C4D4×D711yescharacteristic
319C2×C4D4×D711yescharacteristic
320C2×C4C23×D711yescharacteristic
321C8—21no—
322C8—72no—
323C8—141no—
324C8#115712yescomplemented
325C7#318211yescharacteristic, complemented, Sylow
326C22—41no—
327C22—41no—
328C22—41no—
329C22—141no—
330C22—141no—
331C22—141no—
332C22—281no—
333C22—281no—
334C22—281no—
335C22C2×D4×D711yescharacteristic
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—72no—
342C4—72no—
343C4—141no—
344C4—281no—
345C4—281no—
346C4—281no—
347C4C2×#115711yescharacteristic
348C4D7×#4111yescharacteristic
349C2—81no—
350C2—281no—
351C2—561no—
352C2#9023911yescharacteristic
353C2D7×AGL(1,ℤ/8)11yescharacteristic
354C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#440C2
socleC22×C7C22×D4
prime radical#90239C2
hypercenter#440D7
hypocenterC7#3182
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #90269C2semidirect
3. C7×#440C2semidirect
4. #89947C2semidirect
5. #89946C2semidirect
6. #89798C2semidirect
7. #89790C2semidirect
8. #89658C2semidirect
9. #89646C2semidirect
10. #89584C2semidirect
11. #89564C2semidirect
12. #89552C2semidirect
13. #89532C2semidirect
14. #89482C2semidirect
15. #89479C2semidirect
123. #440D7semidirect
161. C7×C8#41semidirect
163. C7⋊C8#41semidirect
324. C8#1157semidirect
325. C7#3182semidirect

Maximal subgroups

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

subgroupindex
#902692
#902392
C7×#4402
#899472
#899462
#897982
#897902
#896582
#896462
#895842
#895642
#895522
#895322
#894822
#894792
#31827