Finite Group Databasebeta

← #13339029

Subgroups — #13339029

Order 1792.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#461C211yesmaximal, characteristic, complemented
2#11740660C211yesmaximal, characteristic, complemented
3#11739752C211yesmaximal, characteristic
4#11739746C211yesmaximal, characteristic, complemented
5#11729527C211yesmaximal, characteristic, complemented
6#11728793C214yesmaximal, complemented
7#11728791C214yesmaximal, complemented
8#11728185C211yesmaximal, characteristic, complemented
9#11728176C211yesmaximal, characteristic, complemented
10C22×C7×SD16—21no—
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C2×#6495—21no—
14#90168C2211yescharacteristic
15C7×#461—22no—
16C7×#461C2214yes—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×#103—21no—
19C2×C22⋊(C7⋊C8)—22no—
20#89860C2212yescomplemented
21#89859—22no—
22#89850C2212yescomplemented
23#89791C2212yes—
24#89788—22no—
25#89787C2212yes—
26D4×C7⋊C8—22no—
27D4×C7⋊C8C2214yes—
28#89739—22no—
29#89738—21no—
30#89697C2211yescharacteristic
31#89697C2211yescharacteristic
32#89696C2211yescharacteristic
33#89342C2214yescomplemented
34#89280C2212yescomplemented
35#89277C2212yescomplemented
36#89274C2214yescomplemented
37#19392—71nomaximal, Sylow
38C22×C7×Q8—21no—
39C22×C7×D4D411yescharacteristic
40C23×C4×C7—21no—
41C2×#1157—21no—
42C2×C7×SD16—24no—
43C2×C7×SD16—44no—
44C22×C7×C8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8—21no—
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8C2312yes—
49C4×C7×D4—21no—
50C4×C7×D4—21no—
51C4×C7×D4D412yes—
52C4×C7×D4C2312yes—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C7×C22⋊C4—21no—
57C2×C42×C7C2311yescharacteristic
58C2×#1142—22no—
59#6495—44no—
60C2×#1141—22no—
61#6493—21no—
62#6493—21no—
63C2×#1136—21no—
64C22×C7⋊C8—24no—
65#6438—21no—
66#6437C2311yescharacteristic
67C4×D28—21no—
68C4×D28—21no—
69#6432C2311yescharacteristic
70C7×C4⋊C8—21no—
71C7×C4⋊C8C2312yes—
72C7×#103—24no—
73C22⋊(C7⋊C8)—21no—
74C22⋊(C7⋊C8)—24no—
75C22⋊(C7⋊C8)D412yes—
76#6384—22no—
77#6383—22no—
78#6383—24no—
79#6381—24no—
80C4⋊(C7⋊C8)—21no—
81C4⋊(C7⋊C8)D412yes—
82C4⋊(C7⋊C8)C2312yes—
83C4⋊(C7⋊C8)C2312yes—
84C4×C7⋊C8—21no—
85C4×C7⋊C8C2312yes—
86#3226—71no—
87#3036—71no—
88C2×#461D711yescharacteristic, complemented
89#2972—71no—
90#1681—74no—
91#1673—74no—
92#1571—71no—
93#1546—71no—
94#1540—71no—
95C24×C7—21no—
96C2×C7×Q8—22no—
97C2×C7×Q8—41no—
98C2×C7×Q8—42no—
99C2×C7×Q8—42no—
100C2×C7×D4—21no—
101C2×C7×D4—22no—
102C2×C7×D4D814yes—
103C2×C7×D4C2×D412yes—
104C22×C4×C7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—41no—
108C22×C4×C7—82no—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7#4111yescharacteristic
112C2×C22⋊D7—41no—
113#1157—42no—
114#1157—42no—
115C2×D28—21no—
116C2×D28—21no—
117C2×D28—21no—
118C2×C4×D7—41no—
119C2×Dic14—21no—
120C7×SD16—48no—
121C2×C7×C8—22no—
122C2×C7×C8—42no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—22no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4C2×D412yes—
129C7×C22⋊C4—22no—
130C7×C22⋊C4—22no—
131C42×C7C2×D411yescharacteristic
132C42×C7C2×D411yescharacteristic
133C42×C7#4111yescharacteristic
134C42×C7C2411yescharacteristic
135#1142—44no—
136#1141—44no—
137(C2×D7)⋊C4—41no—
138(C2×D7)⋊C4—41no—
139C4⋊Dic7—21no—
140C4⋊Dic7—21no—
141#1136—42no—
142C2×C7⋊C8—22no—
143C2×C7⋊C8—22no—
144C2×C7⋊C8—84no—
145C2×C7⋊C8C2×D414yes—
146C2×AGL(1,ℤ/8)—141no—
147C22×SD16—21no—
148#532—71no—
149C2×C4⋊Q8C2×D711yescharacteristic
150C2×C4×D4C2×D711yescharacteristic
151#500—72no—
152#486—72no—
153#480—142no—
154#474—142no—
155#463—142no—
156#461—22no—
157#461—72no—
158#461C2×D714yes—
159#459—72no—
160C8×D4—74no—
161C8×D4—142no—
162#431—71no—
163#431—71no—
164#423—71no—
165C2×C4⋊C8C2×D711yescharacteristic
166#417—141no—
167C2×#103—21no—
168C2×C22⋊C8—142no—
169#335—74no—
170#332—74no—
171#332—72no—
172#331—72no—
173C23×C7—22no—
174C23×C7—24no—
175C23×C7—81no—
176C23×C7—81no—
177C23×C7#12111yescharacteristic
178C22×D7—41no—
179C7×Q8—42no—
180C7×Q8—42no—
181C7×Q8—82no—
182C7×D4—22no—
183C7×D4—24no—
184C7×D4C2×D814yes—
185C2×C4×C7—22no—
186C2×C4×C7—22no—
187C2×C4×C7—41no—
188C2×C4×C7—44no—
189C2×C4×C7—42no—
190C2×C4×C7—82no—
191C2×C4×C7—81no—
192C2×C4×C7—82no—
193C2×C4×C7#12111yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7C2×D812yes—
197C2×C4×C7#13512yes—
198C2×C4×C7C22×D411yescharacteristic
199C2×C4×C7C22×D411yescharacteristic
200C2×C4×C7C2×#4111yescharacteristic
201C22⋊D7—82no—
202C2×Dic7—41no—
203D28—41no—
204D28—41no—
205D28—42no—
206D28—81no—
207C4×D7—82no—
208Dic14—41no—
209Dic14—41no—
210C7×C8—42no—
211C7⋊C8—24no—
212C7⋊C8—42no—
213C7⋊C8—42no—
214C2×#41—141no—
215C22×Q8—21no—
216C22×D4C22⋊D711yescharacteristic
217C23×C4—21no—
218AGL(1,ℤ/8)—284no—
219C2×SD16—24no—
220C2×SD16—44no—
221C2×SD16—142no—
222C2×D8—142no—
223C2×M16—141no—
224C22×C8—21no—
225C22×C8—144no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—71no—
230C4⋊Q8C22×D712yes—
231#127—71no—
232#124—141no—
233#121—141no—
234#121—141no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4—141no—
238C4×D4—141no—
239C4×D4C22⋊D712yes—
240C4×D4C22×D712yes—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C22⋊D711yescharacteristic
244C2×C22⋊C4—21no—
245C2×C42C22×D711yescharacteristic
246#107—144no—
247C4⋊C8—21no—
248C4⋊C8—72no—
249C4⋊C8—72no—
250C4⋊C8—72no—
251C4⋊C8—141no—
252C4⋊C8C22×D712yes—
253#103—24no—
254#103—142no—
255#102—144no—
256#102—142no—
257C22⋊C8—72no—
258C22⋊C8—144no—
259C22⋊C8—141no—
260C4×C8—72no—
261C4×C8—141no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—24no—
265C22×C7—42no—
266C22×C7—42no—
267C22×C7—84no—
268C22×C7#46612yes—
269C22×C7C2×#12111yescharacteristic
270C2×D7—81no—
271C2×D7—81no—
272C2×D7—81no—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—21no—
276C4×C7—42no—
277C4×C7—82no—
278C4×C7#46612yes—
279C4×C7C22×D811yescharacteristic
280C4×C7C2×#13511yescharacteristic
281C4×C7#57712yes—
282Dic7—81no—
283C24—21no—
284#41—282no—
285#41—282no—
286C2×Q8—22no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—141no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—141no—
294C2×D4—141no—
295C2×D4—141no—
296C2×D4—281no—
297C2×D4#114114yes—
298C2×D4C2×C22⋊D712yes—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—82no—
304C22×C4—281no—
305C22×C4D4×D711yescharacteristic
306C22×C4#115911yescharacteristic
307C22×C4C2×C22⋊D711yescharacteristic
308SD16—48no—
309SD16—284no—
310D8—284no—
311M16—282no—
312C2×C8—22no—
313C2×C8—42no—
314C2×C8—74no—
315C2×C8—142no—
316C2×C8—142no—
317C2×C8—564no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—141no—
324C4⋊C4—141no—
325C4⋊C4C2×C22⋊D712yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—281no—
329C22⋊C4—281no—
330C42D4×D711yescharacteristic
331C42#115911yescharacteristic
332C42C2×C22⋊D711yescharacteristic
333C42C23×D711yescharacteristic
334C2×C7—22no—
335C2×C7—44no—
336C2×C7#304211yescharacteristic
337C2×C7C2×#46611yescharacteristic
338C2×C7#306911yescharacteristic
339D7—161no—
340C23—22no—
341C23—24no—
342C23—81no—
343C23—81no—
344C23—281no—
345C23#650211yescharacteristic
346Q8—42no—
347Q8—42no—
348Q8—82no—
349Q8—281no—
350Q8—281no—
351D4—22no—
352D4—24no—
353D4—282no—
354D4—281no—
355D4—281no—
356D4—561no—
357D4—562no—
358D4C2×#114114yes—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—281no—
368C2×C4—562no—
369C2×C4C2×#114112yes—
370C2×C4#650211yescharacteristic
371C2×C4#650211yescharacteristic
372C2×C4#650211yescharacteristic
373C2×C4#651312yes—
374C2×C4C2×D4×D711yescharacteristic
375C2×C4C2×#115911yescharacteristic
376C2×C4C22×C22⋊D711yescharacteristic
377C8—42no—
378C8—144no—
379C8—282no—
380C8—282no—
381C7#1939211yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—561no—
389C22—561no—
390C22—561no—
391C22#8981012yes—
392C22C2×#650211yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—82no—
398C4—561no—
399C4#8981012yes—
400C4#9045312yes—
401C4C22×#114111yescharacteristic
402C4C2×#651311yescharacteristic
403C2—22no—
404C2—44no—
405C2—1121no—
406C2C2×#8981011yescharacteristic
407C2#1174129311yescharacteristic
408C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#461C2
socleC22×C7C2×#121
prime radical#11739752C2
hypercenterC2×#461D7
hypocenterC7#19392
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. C2×C7×#461C2semidirect
2. #11740660C2semidirect
4. #11739746C2semidirect
5. #11729527C2semidirect
6. #11728793C2semidirect
7. #11728791C2semidirect
8. #11728185C2semidirect
9. #11728176C2semidirect
20. #89860C22semidirect
22. #89850C22semidirect
33. #89342C22semidirect
34. #89280C22semidirect
35. #89277C22semidirect
36. #89274C22semidirect
88. C2×#461D7semidirect
381. C7#19392semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4612
#117406602
#117397522
#117397462
#117295272
#117287932
#117287912
#117281852
#117281762
#193927