Finite Group Databasebeta

← #11736345

Subgroups — #11736345

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90522C211yesmaximal, characteristic
2#90202C211yesmaximal, characteristic, complemented
3C7×#417C211yesmaximal, characteristic, complemented
4#90011C211yesmaximal, characteristic, complemented
5#89909C211yesmaximal, characteristic, complemented
6#89899C211yesmaximal, characteristic, complemented
7#89743C211yesmaximal, characteristic, complemented
8#89580C211yesmaximal, characteristic, complemented
9D7×#103C211yesmaximal, characteristic, complemented
10#89563C211yesmaximal, characteristic, complemented
11#89561C211yesmaximal, characteristic, complemented
12#89550C211yesmaximal, characteristic, complemented
13#89549C211yesmaximal, characteristic, complemented
14#89539C211yesmaximal, characteristic, complemented
15#89537C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20Q8×Dic7C2211yescharacteristic
21C2×#1144C2211yescharacteristic
22#6501—21no—
23#6501C2211yescharacteristic
24#6499—21no—
25#6499C2211yescharacteristic
26C2×#1142C2211yescharacteristic
27C4×C22⋊D7—21no—
28C4×C22⋊D7—21no—
29C4×C22⋊D7C2211yescharacteristic
30#6487—21no—
31#6487—21no—
32#6487C2211yescharacteristic
33#6487C2211yescharacteristic
34C2×Dic28C2211yescharacteristic
35C2×C8×D7C2211yescharacteristic
36#6460C2211yescharacteristic
37#6459—21no—
38D7×C4⋊C4C2211yescharacteristic
39#6452—21no—
40#6451C2211yescharacteristic
41#6450C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48C22⋊(C7⋊C8)C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6391C2211yescharacteristic
51#6390C2211yescharacteristic
52#6390C2211yescharacteristic
53#6389C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55#6384C2211yescharacteristic
56#6384C2211yescharacteristic
57#6382C2211yescharacteristic
58#6382C2211yescharacteristic
59#3316—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4C2311yescharacteristic
65C22×C4×C7—21no—
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7D411yescharacteristic
70Q8×D7—22no—
71Q8×D7—41no—
72C2×C4×D7—21no—
73C2×C4×D7—21no—
74C2×C4×D7C2311yescharacteristic
75C2×Dic14—21no—
76C2×Dic14—21no—
77C2×Dic14C2311yescharacteristic
78C2×Dic14C2311yescharacteristic
79C7×M16—41no—
80C2×C7×C8C2311yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C7×C4⋊C4—21no—
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7D411yescharacteristic
91#1144—22no—
92#1142—41no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98C4⋊Dic7C2311yescharacteristic
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4D411yescharacteristic
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110Dic28—41no—
111C8×D7—41no—
112#554—71no—
113#548—71no—
114#489—71no—
115#483—71no—
116#481—71no—
117#475—71no—
118#470—71no—
119#465—71no—
120#464—71no—
121#452—71no—
122#451—71no—
123#441—71no—
124#435—71no—
125#417D711yescharacteristic, complemented
126C2×#103—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8#4112yes—
132C7×D4—21no—
133C7×D4—21no—
134C7×D4—42no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—42no—
141C2×C4×C7—81no—
142C2×C4×C7C2×D411yescharacteristic
143C2×C4×C7C2411yescharacteristic
144C22⋊D7—22no—
145C22⋊D7—42no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—42no—
156C4×D7—81no—
157Dic14—21no—
158Dic14—21no—
159Dic14—21no—
160Dic14—21no—
161Dic14—42no—
162Dic14—81no—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C2×#41C2×D711yescharacteristic
168C22×Q8—71no—
169C2×Q16—71no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×M16C2×D711yescharacteristic
173C22×C8—71no—
174C4⋊Q8—71no—
175#125—141no—
176#124—71no—
177#124—141no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183C22⋊Q8—141no—
184#121—71no—
185#121—141no—
186C4×Q8—71no—
187C4×Q8—71no—
188C4×D4—71no—
189C4×D4—141no—
190C4×D4—141no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4C2×D711yescharacteristic
193#106—71no—
194#106—71no—
195#106—71no—
196C4⋊C8—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204#102—71no—
205#102C2×D711yescharacteristic
206#102C2×D711yescharacteristic
207C22⋊C8—71no—
208C22⋊C8—71no—
209#97—71no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7C22×D411yescharacteristic
217C2×D7—21no—
218C2×D7—21no—
219C2×D7#13712yes—
220C4×C7—22no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7C2×#4111yescharacteristic
225C4×C7H(5,𝔽2)11yescharacteristic
226Dic7—21no—
227Dic7—21no—
228Dic7—21no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233#41—22no—
234#41—41no—
235C2×Q8—71no—
236C2×Q8—71no—
237C2×Q8—141no—
238C2×Q8—141no—
239C2×Q8—142no—
240C2×Q8—281no—
241C2×Q8C22×D711yescharacteristic
242C2×D4—21no—
243C2×D4—71no—
244C2×D4—141no—
245C2×D4C22×D711yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—71no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4C22×D711yescharacteristic
252Q16—142no—
253Q16—281no—
254SD16—281no—
255M16—41no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8—281no—
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—71no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4C22×D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—71no—
275C22⋊C4—71no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C42—71no—
283C42—141no—
284C42—141no—
285C2×C7—41no—
286C2×C7—81no—
287C2×C7#54611yescharacteristic
288C2×C7C2×#13711yescharacteristic
289C2×C7#57711yescharacteristic
290D7—22no—
291C23—21no—
292C23—71no—
293C23D4×D711yescharacteristic
294Q8—21no—
295Q8—141no—
296Q8—141no—
297Q8—141no—
298Q8—141no—
299Q8—282no—
300Q8—561no—
301Q8#115712yes—
302D4—21no—
303D4—21no—
304D4—42no—
305D4—142no—
306D4—282no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—71no—
314C2×C4—71no—
315C2×C4—71no—
316C2×C4—81no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—282no—
324C2×C4—561no—
325C2×C4D4×D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—141no—
330C8—141no—
331C7#331611yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—72no—
339C22—141no—
340C22—141no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—141no—
347C4—141no—
348C4—141no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4C2×#115711yescharacteristic
354C4#654811yescharacteristic
355C2—41no—
356C2—81no—
357C2—142no—
358C2#9018111yescharacteristic
359C2#9046611yescharacteristic
360C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#417C2
socleC22×C7C22×D4
prime radical#90522C2
hypercenter#417D7
hypocenterC7#3316
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. #90202C2semidirect
3. C7×#417C2semidirect
4. #90011C2semidirect
5. #89909C2semidirect
6. #89899C2semidirect
7. #89743C2semidirect
8. #89580C2semidirect
9. D7×#103C2semidirect
10. #89563C2semidirect
11. #89561C2semidirect
12. #89550C2semidirect
13. #89549C2semidirect
14. #89539C2semidirect
15. #89537C2semidirect
125. #417D7semidirect
331. C7#3316semidirect

Maximal subgroups

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

subgroupindex
#905222
#902022
C7×#4172
#900112
#899092
#898992
#897432
#895802
D7×#1032
#895632
#895612
#895502
#895492
#895392
#895372
#33167