Finite Group Databasebeta

← #11738752

Subgroups — #11738752

Order 896.

398 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90290C211yesmaximal, characteristic
2#90202C211yesmaximal, characteristic, complemented
3C7×#480C211yesmaximal, characteristic, complemented
4#89814C211yesmaximal, characteristic, complemented
5#89738C211yesmaximal, characteristic, complemented
6#89658C211yesmaximal, characteristic, complemented
7#89654C211yesmaximal, characteristic, complemented
8#89647C211yesmaximal, characteristic, complemented
9#89645C211yesmaximal, characteristic, complemented
10#89555C211yesmaximal, characteristic, complemented
11#89543C211yesmaximal, characteristic, complemented
12#89534C211yesmaximal, characteristic, complemented
13#89532C211yesmaximal, characteristic, complemented
14#89505C211yesmaximal, characteristic, complemented
15#89501C211yesmaximal, characteristic, complemented
16C2×#1159C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×#121C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20#6503—21no—
21#6503C2211yescharacteristic
22#6502—21no—
23C2×#1142C2211yescharacteristic
24C4×C22⋊D7—21no—
25C4×C22⋊D7C2211yescharacteristic
26#6487—21no—
27#6487C2211yescharacteristic
28C2×#1136C2211yescharacteristic
29C2×Dic28C2211yescharacteristic
30C2×#1132C2211yescharacteristic
31C2×#1131C2211yescharacteristic
32#6460C2211yescharacteristic
33#6459—21no—
34#6458C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36#6452—21no—
37#6451C2211yescharacteristic
38#6450C2211yescharacteristic
39#6450C2211yescharacteristic
40#6447—21no—
41(C22⋊D7)⋊C4—21no—
42C7×#107C2211yescharacteristic
43C7×#107C2211yescharacteristic
44C7×#102C2211yescharacteristic
45C7×#102C2211yescharacteristic
46C7×C22⋊C8C2211yescharacteristic
47#6406C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50#6390C2211yescharacteristic
51Dic7⋊C8C2211yescharacteristic
52C8×Dic7C2211yescharacteristic
53#6384C2211yescharacteristic
54#6384C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58#6381C2211yescharacteristic
59#3333—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4C2311yescharacteristic
62C22×C4×C7—21no—
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—21no—
65C2×C22⋊D7—21no—
66C2×C22⋊D7—21no—
67C2×C22⋊D7—21no—
68C2×C22⋊D7D411yescharacteristic
69C22×Dic7—21no—
70#1159—22no—
71#1159—41no—
72#1157—22no—
73#1157—41no—
74C2×D28—21no—
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7C2311yescharacteristic
79C2×Dic14—21no—
80C2×Dic14C2311yescharacteristic
81C2×Dic14C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C7×C4⋊C4—21no—
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C22⋊C4—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7D411yescharacteristic
91#1142—41no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4D411yescharacteristic
95C4⋊Dic7—21no—
96C4⋊Dic7C2311yescharacteristic
97Dic7⋊C4—21no—
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4D411yescharacteristic
103C4×Dic7—21no—
104C4×Dic7—21no—
105C4×Dic7C2311yescharacteristic
106#1136—41no—
107C2×C7⋊C8C2311yescharacteristic
108C2×C7⋊C8C2311yescharacteristic
109Dic28—22no—
110#1132—41no—
111#1131—41no—
112#554—71no—
113#550—71no—
114#486—71no—
115#480—71no—
116#480D711yescharacteristic, complemented
117#477—71no—
118#468—71no—
119#468—71no—
120#460—71no—
121#452—71no—
122#452—71no—
123C4×Q16—71no—
124#436—71no—
125#417—71no—
126#417—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C22×D7C2×D411yescharacteristic
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—81no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—81no—
140C2×C4×C7C2×D411yescharacteristic
141C2×C4×C7C2411yescharacteristic
142C22⋊D7—22no—
143C22⋊D7—42no—
144C22⋊D7—42no—
145C22⋊D7—42no—
146C22⋊D7—81no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—42no—
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C2×Dic7C2×D411yescharacteristic
154D28—21no—
155D28—21no—
156D28—81no—
157C4×D7—21no—
158C4×D7—21no—
159C4×D7—42no—
160C4×D7—81no—
161Dic14—21no—
162Dic14—21no—
163Dic14—21no—
164Dic14—81no—
165Dic14#4112yes—
166C7×C8—21no—
167C7×C8—21no—
168C7⋊C8—21no—
169C7⋊C8—21no—
170C2×#41—71no—
171C2×#41—71no—
172C2×Q16—71no—
173C2×SD16—71no—
174C2×SD16—71no—
175C2×M16—71no—
176C2×M16—71no—
177C4⋊Q8—71no—
178#127—71no—
179#127—141no—
180#125—141no—
181C22⋊Q8—71no—
182C22⋊Q8—71no—
183C22⋊Q8—141no—
184C22⋊Q8—141no—
185#121—71no—
186#121—141no—
187#121—141no—
188#121C2×D711yescharacteristic
189C4×Q8—71no—
190C4×Q8—71no—
191C4×D4—71no—
192C4×D4—141no—
193C4×D4—141no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4C2×D711yescharacteristic
196#107—71no—
197#107C2×D711yescharacteristic
198#107C2×D711yescharacteristic
199C4⋊C8—71no—
200#103—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#102—71no—
205#102—71no—
206#102—71no—
207#102—71no—
208#102C2×D711yescharacteristic
209#102C2×D711yescharacteristic
210C22⋊C8—71no—
211C22⋊C8C2×D711yescharacteristic
212C4×C8—71no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7C22×D411yescharacteristic
220C2×D7—21no—
221C2×D7—21no—
222C2×D7—21no—
223C2×D7—41no—
224C2×D7—41no—
225C2×D7—41no—
226C4×C7—21no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7C2×#4111yescharacteristic
231C4×C7H(5,𝔽2)11yescharacteristic
232Dic7—22no—
233Dic7—21no—
234Dic7—21no—
235Dic7—41no—
236Dic7—41no—
237Dic7#13712yes—
238#41—142no—
239#41—142no—
240#41—281no—
241#41—281no—
242C2×Q8—71no—
243C2×Q8—71no—
244C2×Q8—141no—
245C2×D4—21no—
246C2×D4—71no—
247C2×D4—71no—
248C2×D4—141no—
249C2×D4—141no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—141no—
253C2×D4C22×D711yescharacteristic
254C22×C4—21no—
255C22×C4—71no—
256C22×C4—141no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4C22×D711yescharacteristic
260Q16—142no—
261SD16—281no—
262SD16—281no—
263M16—281no—
264M16—281no—
265C2×C8—71no—
266C2×C8—71no—
267C2×C8C22×D711yescharacteristic
268C2×C8C22×D711yescharacteristic
269C4⋊C4—21no—
270C4⋊C4—71no—
271C4⋊C4—71no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4—141no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4C22×D711yescharacteristic
279C4⋊C4C22×D711yescharacteristic
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—71no—
283C22⋊C4—71no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C42—71no—
288C42—141no—
289C42—141no—
290C2×C7—41no—
291C2×C7—81no—
292C2×C7#54611yescharacteristic
293C2×C7C2×#13711yescharacteristic
294C2×C7#57611yescharacteristic
295D7—41no—
296D7—81no—
297C23—21no—
298C23—71no—
299C23—141no—
300C23D4×D711yescharacteristic
301Q8—72no—
302Q8—141no—
303Q8—141no—
304Q8—141no—
305Q8—561no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—142no—
310D4—141no—
311D4—141no—
312D4—282no—
313D4—282no—
314D4—282no—
315D4—561no—
316D4—561no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—71no—
323C2×C4—71no—
324C2×C4—71no—
325C2×C4—81no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—282no—
332C2×C4—282no—
333C2×C4—561no—
334C2×C4D4×D711yescharacteristic
335C2×C4C23×D711yescharacteristic
336C8—21no—
337C8—21no—
338C8—141no—
339C8—141no—
340C7#333311yescharacteristic, complemented, Sylow
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—141no—
348C22—141no—
349C22—141no—
350C22—281no—
351C22—281no—
352C22—281no—
353C22C2×D4×D711yescharacteristic
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—72no—
359C4—141no—
360C4—141no—
361C4—142no—
362C4—281no—
363C4—281no—
364C4D7×#4111yescharacteristic
365C4#655311yescharacteristic
366C2—41no—
367C2—81no—
368C2—281no—
369C2—561no—
370C2#9034511yescharacteristic
371C2#9044811yescharacteristic
372C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#480C2
socleC22×C7C22×D4
prime radical#90290C2
hypercenter#480D7
hypocenterC7#3333
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #90202C2semidirect
3. C7×#480C2semidirect
4. #89814C2semidirect
5. #89738C2semidirect
6. #89658C2semidirect
7. #89654C2semidirect
8. #89647C2semidirect
9. #89645C2semidirect
10. #89555C2semidirect
11. #89543C2semidirect
12. #89534C2semidirect
13. #89532C2semidirect
14. #89505C2semidirect
15. #89501C2semidirect
116. #480D7semidirect
340. C7#3333semidirect

Maximal subgroups

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

subgroupindex
#902902
#902022
C7×#4802
#898142
#897382
#896582
#896542
#896472
#896452
#895552
#895432
#895342
#895322
#895052
#895012
#33337