Finite Group Databasebeta

← #13339025

Subgroups — #13339025

Order 1792.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3036C211yesmaximal, characteristic
2#11740716C211yesmaximal, characteristic, complemented
3#11739746C211yesmaximal, characteristic, complemented
4C2×#89780C211yesmaximal, characteristic, complemented
5#11729558C211yesmaximal, characteristic, complemented
6#11728790C214yesmaximal, complemented
7#11728760C214yesmaximal, complemented
8#11728261C211yesmaximal, characteristic, complemented
9#11728247C211yesmaximal, characteristic, complemented
10C2×C7×AGL(1,ℤ/8)—21no—
11C7×#532C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C2×#6432C2211yescharacteristic
14C7×#463—22no—
15C7×#461C2212yes—
16C7×#459C2212yes—
17C7×#423C2211yescharacteristic
18C7×#417—21no—
19C2×#6406—21no—
20C2×C22⋊(C7⋊C8)—22no—
21#89851C2212yescomplemented
22#89842C2212yescomplemented
23#89806—22no—
24D4×C7⋊C8—22no—
25D4×C7⋊C8C2214yes—
26#89780—22no—
27#89780—22no—
28#89780C2214yes—
29C2×#6382—21no—
30#89697C2211yescharacteristic
31#89697C2211yescharacteristic
32C2×C4⋊(C7⋊C8)C2211yescharacteristic
33#89342C2212yescomplemented
34#89339C2212yescomplemented
35#89276C2214yescomplemented
36#89275C2214yescomplemented
37#19391—71nomaximal, Sylow
38C2×C7×#41—21no—
39C22×C7×D4D411yescharacteristic
40C23×C4×C7—21no—
41C22×Dic14—21no—
42C7×AGL(1,ℤ/8)—44no—
43C2×C7×SD16—22no—
44C2×C7×D8—22no—
45C2×C7×M16—21no—
46C7×C4⋊Q8C2311yescharacteristic
47C7×#127C2311yescharacteristic
48C7×#124—21no—
49C7×#121—21no—
50C7×#121—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4D412yes—
56C4×C7×D4C2312yes—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C42×C7C2311yescharacteristic
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7—21no—
62C22×C7⋊C8—21no—
63C22×C7⋊C8—24no—
64#6432—21no—
65#6432—21no—
66#6432—21no—
67#6432C2312yes—
68C7×C4⋊C8—21no—
69C7×C4⋊C8C2312yes—
70C7×#103—22no—
71C7×#102—22no—
72#6406—24no—
73#6406—24no—
74C22⋊(C7⋊C8)—21no—
75C22⋊(C7⋊C8)—24no—
76C22⋊(C7⋊C8)D412yes—
77#6382—24no—
78#6382—24no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)D412yes—
81C4⋊(C7⋊C8)C2312yes—
82C4⋊(C7⋊C8)C2312yes—
83C4×C7⋊C8—21no—
84C4×C7⋊C8C2312yes—
85#3227—71no—
86C2×#476—71no—
87#3036D711yescharacteristic
88#2972—71no—
89#1677—74no—
90#1671—74no—
91#1570—71no—
92#1548—71no—
93#1540—71no—
94C24×C7—21no—
95C7×#41—42no—
96C7×#41—42no—
97C2×C7×Q8—21no—
98C2×C7×D4—21no—
99C2×C7×D4—22no—
100C2×C7×D4—21no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—41no—
104C2×C7×D4SD1614yes—
105C2×C7×D4C2×D412yes—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—41no—
110C22×C4×C7—82no—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7#4111yescharacteristic
114C22×Dic7—41no—
115C2×Dic14—22no—
116C2×Dic14—41no—
117C2×Dic14—82no—
118C7×SD16—44no—
119C7×D8—44no—
120C7×M16—42no—
121C2×C7×C8—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4C2×D412yes—
126C7×C22⋊C4—22no—
127C7×C22⋊C4—22no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C42×C7C2×D411yescharacteristic
131C42×C7C2×D411yescharacteristic
132C42×C7#4111yescharacteristic
133C42×C7C2411yescharacteristic
134C4⋊Dic7—22no—
135C4⋊Dic7—22no—
136C4⋊Dic7—41no—
137C4⋊Dic7—41no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—22no—
140C2×C7⋊C8—42no—
141C2×C7⋊C8—84no—
142C2×C7⋊C8C2×D414yes—
143C2×AGL(1,ℤ/8)—21no—
144#532C2×D711yescharacteristic
145C2×C4⋊Q8—71no—
146C2×C4×D4C2×D711yescharacteristic
147#498—72no—
148#486—72no—
149#481—142no—
150#476—74no—
151#476—142no—
152#476—142no—
153#463—22no—
154#461C2×D712yes—
155#459C2×D712yes—
156C8×D4—74no—
157C8×D4—142no—
158#431—71no—
159#431—71no—
160C2×#106—141no—
161#423C2×D711yescharacteristic
162C2×C4⋊C8—71no—
163#417—21no—
164C2×#102—141no—
165C2×C22⋊C8—142no—
166#334—74no—
167#333—74no—
168#332—72no—
169#331—72no—
170C23×C7—22no—
171C23×C7—24no—
172C23×C7—41no—
173C23×C7—81no—
174C23×C7—81no—
175C23×C7#12111yescharacteristic
176C7×Q8—41no—
177C7×Q8—41no—
178C7×D4—22no—
179C7×D4—24no—
180C7×D4—41no—
181C7×D4—41no—
182C7×D4—42no—
183C7×D4—81no—
184C7×D4—82no—
185C7×D4C2×SD1614yes—
186C2×C4×C7—22no—
187C2×C4×C7—22no—
188C2×C4×C7—41no—
189C2×C4×C7—44no—
190C2×C4×C7—41no—
191C2×C4×C7—82no—
192C2×C4×C7—81no—
193C2×C4×C7—82no—
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7C2×SD1612yes—
198C2×C4×C7#13512yes—
199C2×C4×C7C22×D411yescharacteristic
200C2×C4×C7C22×D411yescharacteristic
201C2×C4×C7C2×#4111yescharacteristic
202C2×Dic7—42no—
203C2×Dic7—82no—
204Dic14—82no—
205Dic14—82no—
206C7×C8—42no—
207C7⋊C8—24no—
208C7⋊C8—42no—
209C7⋊C8—42no—
210C2×#41—21no—
211C22×Q8—141no—
212C22×D4C22⋊D711yescharacteristic
213C23×C4—21no—
214AGL(1,ℤ/8)—44no—
215C2×SD16—22no—
216C2×D8—22no—
217C2×M16—21no—
218C22×C8—144no—
219C22×C8—141no—
220C4⋊Q8—72no—
221C4⋊Q8—141no—
222C4⋊Q8—141no—
223C4⋊Q8—141no—
224C4⋊Q8C22×D711yescharacteristic
225#127C22×D711yescharacteristic
226#124—21no—
227#121—21no—
228#121—21no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4C22⋊D712yes—
234C4×D4C22×D712yes—
235C2×C4⋊C4—141no—
236C2×C4⋊C4—141no—
237C2×C4⋊C4C22⋊D711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C42C22×D711yescharacteristic
240#106—144no—
241#106—144no—
242C4⋊C8—21no—
243C4⋊C8—72no—
244C4⋊C8—72no—
245C4⋊C8—72no—
246C4⋊C8—141no—
247C4⋊C8C22×D712yes—
248#103—22no—
249#102—22no—
250#102—144no—
251#102—144no—
252C22⋊C8—72no—
253C22⋊C8—144no—
254C22⋊C8—141no—
255C4×C8—72no—
256C4×C8—141no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—24no—
260C22×C7—42no—
261C22×C7—42no—
262C22×C7—84no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7#46512yes—
267C22×C7C2×#12111yescharacteristic
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—42no—
272C4×C7—81no—
273C4×C7#46512yes—
274C4×C7C22×SD1611yescharacteristic
275C4×C7C2×#13511yescharacteristic
276C4×C7#57712yes—
277Dic7—82no—
278C24—21no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—142no—
283C2×Q8—281no—
284C2×Q8—562no—
285C2×D4—21no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4#114214yes—
292C2×D4C2×C22⋊D712yes—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—82no—
298C22×C4—281no—
299C22×C4D4×D711yescharacteristic
300C22×C4#115911yescharacteristic
301C22×C4C2×C22⋊D711yescharacteristic
302SD16—44no—
303D8—44no—
304M16—42no—
305C2×C8—22no—
306C2×C8—74no—
307C2×C8—142no—
308C2×C8—142no—
309C2×C8—282no—
310C2×C8—564no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—142no—
315C4⋊C4—142no—
316C4⋊C4—281no—
317C4⋊C4—281no—
318C4⋊C4C2×C22⋊D712yes—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42D4×D711yescharacteristic
324C42#115911yescharacteristic
325C42C2×C22⋊D711yescharacteristic
326C42C23×D711yescharacteristic
327C2×C7—22no—
328C2×C7—44no—
329C2×C7—161no—
330C2×C7#304211yescharacteristic
331C2×C7C2×#46511yescharacteristic
332C2×C7#306811yescharacteristic
333C23—22no—
334C23—24no—
335C23—41no—
336C23—81no—
337C23—81no—
338C23#650211yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—562no—
342Q8—562no—
343D4—22no—
344D4—24no—
345D4—42no—
346D4—41no—
347D4—41no—
348D4—81no—
349D4—82no—
350D4C2×#114214yes—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—82no—
359C2×C4—282no—
360C2×C4—562no—
361C2×C4C2×#114212yes—
362C2×C4#650211yescharacteristic
363C2×C4#650211yescharacteristic
364C2×C4#650211yescharacteristic
365C2×C4#651312yes—
366C2×C4C2×D4×D711yescharacteristic
367C2×C4C2×#115911yescharacteristic
368C2×C4C22×C22⋊D711yescharacteristic
369C8—42no—
370C8—144no—
371C8—282no—
372C8—282no—
373C7#1939111yescharacteristic, complemented, Sylow
374C22—21no—
375C22—21no—
376C22—24no—
377C22—42no—
378C22—42no—
379C22—84no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#8981312yes—
384C22C2×#650211yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—42no—
389C4—81no—
390C4—562no—
391C4#8981312yes—
392C4#9046612yes—
393C4C22×#114211yescharacteristic
394C4C2×#651311yescharacteristic
395C2—22no—
396C2—44no—
397C2—161no—
398C2C2×#8981311yescharacteristic
399C2#1174138011yescharacteristic
400C2#1174144711yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3036C2
socleC22×C7C2×#121
hypercenter#3036D7
hypocenterC7#19391
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #11740716C2semidirect
3. #11739746C2semidirect
4. C2×#89780C2semidirect
5. #11729558C2semidirect
6. #11728790C2semidirect
7. #11728760C2semidirect
8. #11728261C2semidirect
9. #11728247C2semidirect
21. #89851C22semidirect
22. #89842C22semidirect
33. #89342C22semidirect
34. #89339C22semidirect
35. #89276C22semidirect
36. #89275C22semidirect
373. C7#19391semidirect

Maximal subgroups

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

subgroupindex
C7×#30362
#117407162
#117397462
C2×#897802
#117295582
#117287902
#117287602
#117282612
#117282472
#193917