Finite Group Databasebeta

← #11736959

Subgroups — #11736959

Order 896.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90244C211yesmaximal, characteristic
2#90232C211yesmaximal, characteristic, complemented
3C7×#435C211yesmaximal, characteristic, complemented
4#89906C211yesmaximal, characteristic, complemented
5#89905C211yesmaximal, characteristic, complemented
6#89886C211yesmaximal, characteristic, complemented
7#89885C211yesmaximal, characteristic, complemented
8#89620C211yesmaximal, characteristic, complemented
9#89601C211yesmaximal, characteristic, complemented
10#89509C211yesmaximal, characteristic, complemented
11#89507C211yesmaximal, characteristic, complemented
12#89505C211yesmaximal, characteristic, complemented
13#89505C211yesmaximal, characteristic, complemented
14#89488C211yesmaximal, characteristic, complemented
15#89485C211yesmaximal, characteristic, complemented
16C2×#1157C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×C7×M16C2211yescharacteristic
19C22×C7×C8C2211yescharacteristic
20C4×C7×D4C2211yescharacteristic
21#6503—21no—
22#6498—21no—
23#6493C2211yescharacteristic
24#6493C2211yescharacteristic
25#6489C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27#6487C2211yescharacteristic
28#6487C2211yescharacteristic
29C2×Dic28C2211yescharacteristic
30C2×#1132C2211yescharacteristic
31C2×#1132C2211yescharacteristic
32#6456—21no—
33#6452—21no—
34#6447—21no—
35#6447—21no—
36#6442—21no—
37#6440—21no—
38#6437C2211yescharacteristic
39#6433C2211yescharacteristic
40C4×Dic14C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42C7×C4⋊C8C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45C7×#97C2211yescharacteristic
46#6395C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49#6395C2211yescharacteristic
50#6395C2211yescharacteristic
51#6395C2211yescharacteristic
52#6392C2211yescharacteristic
53#6391C2211yescharacteristic
54#6391C2211yescharacteristic
55#6390C2211yescharacteristic
56#6390C2211yescharacteristic
57#6390C2211yescharacteristic
58#6390C2211yescharacteristic
59#3317—71nomaximal, Sylow
60C2×C7×D4D411yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C22×Dic7—21no—
68#1157—22no—
69#1157—22no—
70#1157—41no—
71#1157—41no—
72C2×D28—21no—
73C2×D28—21no—
74C2×D28C2311yescharacteristic
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×Dic14C2311yescharacteristic
79C2×Dic14C2311yescharacteristic
80C7×M16—41no—
81C2×C7×C8—41no—
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C2×C7×C8C2311yescharacteristic
85C2×C7×C8C2311yescharacteristic
86C7×C4⋊C4D411yescharacteristic
87C7×C22⋊C4D411yescharacteristic
88C7×C22⋊C4D411yescharacteristic
89C42×C7C2311yescharacteristic
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7—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—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109Dic28—22no—
110#1132—41no—
111#1132—41no—
112#552—71no—
113#550—71no—
114#490—71no—
115#482—71no—
116#481—71no—
117#479—71no—
118#468—71no—
119#465—71no—
120#460—71no—
121#452—71no—
122#452—71no—
123#441—71no—
124#435D711yescharacteristic, complemented
125#417—71no—
126#416—71no—
127C23×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C7×D4—41no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7C2×D411yescharacteristic
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2×D411yescharacteristic
139C2×C4×C7C2×D411yescharacteristic
140C2×C4×C7C2411yescharacteristic
141C22⋊D7—42no—
142C22⋊D7—42no—
143C22⋊D7—81no—
144C22⋊D7—81no—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—42no—
152D28—21no—
153D28—21no—
154D28—21no—
155D28—21no—
156D28—42no—
157D28—81no—
158C4×D7—42no—
159C4×D7—42no—
160Dic14—21no—
161Dic14—21no—
162Dic14—21no—
163Dic14#4112yes—
164C7×C8—21no—
165C7×C8—21no—
166C7×C8—21no—
167C7×C8—21no—
168C2×#41—71no—
169C2×#41—71no—
170C2×Q16—71no—
171C2×SD16—71no—
172C2×SD16—71no—
173C2×M16C2×D711yescharacteristic
174C22×C8C2×D711yescharacteristic
175#127—71no—
176#127—141no—
177#126—141no—
178#125—71no—
179#125—141no—
180#123—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183#121—71no—
184#121—71no—
185#121—141no—
186#121—141no—
187C4×Q8—71no—
188C4×Q8—71no—
189C4×D4—141no—
190C4×D4C2×D711yescharacteristic
191#117—71no—
192#117—141no—
193C2×C4⋊C4—71no—
194#107—71no—
195#106—71no—
196#106—71no—
197C4⋊C8C2×D711yescharacteristic
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#102—71no—
203#102—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208C22⋊C8C2×D711yescharacteristic
209C22⋊C8C2×D711yescharacteristic
210#97C2×D711yescharacteristic
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7C22×D411yescharacteristic
218C2×D7—41no—
219C2×D7—41no—
220C2×D7—41no—
221C2×D7—41no—
222C2×D7—41no—
223C2×D7—41no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7C2×#4111yescharacteristic
229C4×C7H(5,𝔽2)11yescharacteristic
230Dic7—22no—
231Dic7—41no—
232Dic7—41no—
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236#41—142no—
237#41—142no—
238#41—281no—
239#41—281no—
240C2×Q8—71no—
241C2×Q8—71no—
242C2×D4—71no—
243C2×D4—71no—
244C2×D4—141no—
245C2×D4—141no—
246C2×D4—141no—
247C2×D4—141no—
248C2×D4—141no—
249C2×D4—141no—
250C2×D4D2811yescharacteristic
251C22×C4—141no—
252C22×C4—141no—
253C22×C4—141no—
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256Q16—142no—
257SD16—281no—
258SD16—281no—
259M16—41no—
260C2×C8—41no—
261C2×C8C22×D711yescharacteristic
262C2×C8C22×D711yescharacteristic
263C2×C8C22×D711yescharacteristic
264C2×C8C22×D711yescharacteristic
265C4⋊C4—71no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4—141no—
275C4⋊C4—141no—
276C4⋊C4D2811yescharacteristic
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C22⋊C4—141no—
282C22⋊C4D2811yescharacteristic
283C22⋊C4D2811yescharacteristic
284C42—141no—
285C42—141no—
286C42—141no—
287C42C22×D711yescharacteristic
288C2×C7—41no—
289C2×C7—41no—
290C2×C7#54611yescharacteristic
291C2×C7#57511yescharacteristic
292C2×C7#57711yescharacteristic
293D7—81no—
294D7—81no—
295C23—141no—
296C23—141no—
297C23C2×D2811yescharacteristic
298C23C2×D2811yescharacteristic
299Q8—72no—
300Q8—141no—
301Q8—141no—
302Q8—141no—
303D4—41no—
304D4—141no—
305D4—141no—
306D4—141no—
307D4—141no—
308D4—282no—
309D4—282no—
310D4—282no—
311D4—561no—
312D4—561no—
313D4—561no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—282no—
325C2×C4—282no—
326C2×C4—282no—
327C2×C4C2×D2811yescharacteristic
328C2×C4C2×D2811yescharacteristic
329C2×C4C2×D2811yescharacteristic
330C2×C4C2×D2811yescharacteristic
331C2×C4C23×D711yescharacteristic
332C8—21no—
333C8—21no—
334C8—21no—
335C8—21no—
336C7#331711yescharacteristic, complemented, Sylow
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—281no—
344C22—281no—
345C22—281no—
346C22—281no—
347C22—281no—
348C22—281no—
349C22C22×D2811yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—142no—
355C4—281no—
356C4—281no—
357C4—281no—
358C4—281no—
359C4—281no—
360C4#654811yescharacteristic
361C4D7×#4111yescharacteristic
362C2—41no—
363C2—41no—
364C2—561no—
365C2—561no—
366C2#9024511yescharacteristic
367C2#9043911yescharacteristic
368C2#9044211yescharacteristic
subgroupG/H
centerC22C22×D28
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#435C2
socleC22×C7C22×D4
prime radical#90244C2
hypercenter#435D7
hypocenterC7#3317
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
2. #90232C2semidirect
3. C7×#435C2semidirect
4. #89906C2semidirect
5. #89905C2semidirect
6. #89886C2semidirect
7. #89885C2semidirect
8. #89620C2semidirect
9. #89601C2semidirect
10. #89509C2semidirect
11. #89507C2semidirect
12. #89505C2semidirect
13. #89505C2semidirect
14. #89488C2semidirect
15. #89485C2semidirect
124. #435D7semidirect
336. C7#3317semidirect

Maximal subgroups

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

subgroupindex
#902442
#902322
C7×#4352
#899062
#899052
#898862
#898852
#896202
#896012
#895092
#895072
#895052
#894882
#894852
#33177