Finite Group Databasebeta

← #13509040

Subgroups — #13509040

Order 1792.

526 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3183C211yesmaximal, characteristic, complemented
2#11735986C211yesmaximal, characteristic, complemented
3#11729112C212yesmaximal, complemented
4#11729079C212yesmaximal, complemented
5#11728845C211yesmaximal, characteristic, complemented
6#11728844C211yesmaximal, characteristic, complemented
7#11728843C211yesmaximal, characteristic
8#11728842C211yesmaximal, characteristic, complemented
9#11728672C212yesmaximal, complemented
10#11728671C212yesmaximal, complemented
11#11727618C211yesmaximal, characteristic, complemented
12C7×#529—21no—
13#90439C2211yescharacteristic
14#90436—21no—
15C7×#499C2211yescharacteristic, complemented
16C7×#495C2211yescharacteristic, complemented
17C7×#488—21no—
18C7×#470C2211yescharacteristic
19C7×#469C2211yescharacteristic
20C7×#468—21no—
21C7×#442C2211yescharacteristic, complemented
22C7×#441C2211yescharacteristic, complemented
23C7×#440—21no—
24C4×C7×M16C2211yescharacteristic
25#89951C2212yes—
26#89919C2212yes—
27#89897C2211yescharacteristic
28C2×#6379—21no—
29#89489—21no—
30#89489C2212yes—
31#89472—21no—
32#89472C2212yes—
33#89360C2212yescomplemented
34#89358C2212yescomplemented
35#89348C2211yescharacteristic
36#89348C2211yescharacteristic
37#89283C2212yes—
38#89283C2212yes—
39#89282C2212yescomplemented
40#89282C2212yescomplemented
41#89261C2212yescomplemented
42#89258C2212yescomplemented
43#33690—71nomaximal, Sylow
44C2×#1157—21no—
45C2×C7×Q16C2311yescharacteristic
46C2×C7×SD16—21no—
47C2×C7×D8C2311yescharacteristic
48C2×C7×M16D411yescharacteristic
49C2×C7×M16C2311yescharacteristic
50C7×#126—41no—
51C7×#125—21no—
52C7×#124—21no—
53C7×#123—41no—
54C7×C22⋊Q8—21no—
55C7×#121—21no—
56C4×C7×Q8—21no—
57C4×C7×D4—21no—
58C4×C7×D4—41no—
59C7×#117—41no—
60C2×C42×C7D411yescharacteristic
61C2×#1136—21no—
62#6472—22no—
63#6471—22no—
64#6470—42no—
65D7×M16—42no—
66#6467—21no—
67#6467C2312yes—
68#6464—22no—
69#6464—22no—
70C7×#107—21no—
71C7×#106C2311yescharacteristic
72C7×#106C2311yescharacteristic
73C7×#103—21no—
74C7×#103—21no—
75C7×#102—21no—
76C7×#102—21no—
77C7×#97D411yescharacteristic
78C7×#97C2311yescharacteristic
79C4×C7×C8D411yescharacteristic
80C4×C7×C8C2311yescharacteristic
81#6403—22no—
82#6402—22no—
83#6401—22no—
84#6400—22no—
85#6393—21no—
86#6393C2312yes—
87#6386C2312yes—
88#6386C2312yes—
89#6379—22no—
90#6379—22no—
91#3183D711yescharacteristic, complemented
92#2956—71no—
93#2239—71no—
94#2238—71no—
95#2231—72no—
96#2212—72no—
97#2191—71no—
98#2190—71no—
99AGL(1,ℤ/16)—72no—
100#2180—72no—
101#2165—71no—
102C2×C7×Q8—21no—
103C2×C7×D4—21no—
104C2×C7×D4—41no—
105C22×C4×C7—21no—
106C22×C4×C7—41no—
107C22×C4×C7C2×D411yescharacteristic
108C2×C22⋊D7—41no—
109#1157—22no—
110#1157—22no—
111#1157—41no—
112#1157—41no—
113C2×D28—21no—
114C2×C4×D7—41no—
115C2×Dic14—21no—
116C7×Q16—22no—
117C7×SD16—42no—
118C7×D8—22no—
119C7×M16—22no—
120C7×M16—22no—
121C2×C7×C8C2×D411yescharacteristic
122C2×C7×C8C2×D411yescharacteristic
123C2×C7×C8C2×D411yescharacteristic
124C2×C7×C8C2411yescharacteristic
125C7×C4⋊C4—21no—
126C7×C4⋊C4—21no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C22⋊C4—41no—
132C7×C22⋊C4—41no—
133C7×C22⋊C4—41no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—41no—
136C42×C7—21no—
137C42×C7—41no—
138C42×C7C2×D411yescharacteristic
139C42×C7C2×D411yescharacteristic
140#1136—22no—
141#1136—41no—
142C2×C7⋊C8—41no—
143Dic28—21no—
144Dic28#4112yes—
145D56—21no—
146D56#4112yes—
147#1132—22no—
148#1132—22no—
149#1131—42no—
150#1131—42no—
151C8×D7—42no—
152C8×D7—42no—
153C7⋊C16—22no—
154C7⋊C16—22no—
155#575—71no—
156#568—141no—
157#529—21no—
158#510—72no—
159#509—72no—
160#499C2×D711yescharacteristic, complemented
161#495C2×D711yescharacteristic, complemented
162#488—21no—
163#470C2×D711yescharacteristic
164#469C2×D711yescharacteristic
165#468—21no—
166#444—72no—
167#444—141no—
168#443—72no—
169#443—141no—
170#442C2×D711yescharacteristic, complemented
171#441C2×D711yescharacteristic, complemented
172#440—21no—
173#430—71no—
174C2×#104—141no—
175C4×M16C2×D711yescharacteristic
176#365—72no—
177#365—72no—
178#364—72no—
179#362—72no—
180#361—72no—
181#360—72no—
182#360—72no—
183#349—71no—
184#349—71no—
185#347—72no—
186C23×C7—41no—
187C23×C7AGL(1,ℤ/8)11yescharacteristic
188C22×D7—41no—
189C7×Q8—41no—
190C7×Q8—41no—
191C7×D4—41no—
192C7×D4—41no—
193C7×D4—82no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—21no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—81no—
202C2×C4×C7—82no—
203C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
204C2×C4×C7C22×D411yescharacteristic
205C22⋊D7—42no—
206C22⋊D7—42no—
207C2×Dic7—41no—
208D28—22no—
209D28—41no—
210C4×D7—42no—
211C4×D7—42no—
212Dic14—22no—
213Dic14—41no—
214C7×C8—21no—
215C7×C8—21no—
216C7×C8AGL(1,ℤ/8)12yes—
217C7×C8C2×#4112yes—
218C7⋊C8—42no—
219C2×#41—141no—
220#137—142no—
221AGL(1,ℤ/8)—142no—
222#135—72no—
223#135—141no—
224C2×Q16C22×D711yescharacteristic
225C2×SD16—21no—
226C2×D8C22×D711yescharacteristic
227#131—142no—
228#131—142no—
229#131—282no—
230C2×M16—141no—
231C2×M16—282no—
232C2×M16C22⋊D711yescharacteristic
233C2×M16C22×D711yescharacteristic
234#126—41no—
235#125—21no—
236#124—21no—
237#123—41no—
238C22⋊Q8—21no—
239#121—21no—
240C4×Q8—21no—
241C4×D4—21no—
242C4×D4—41no—
243#117—41no—
244C2×C42C22⋊D711yescharacteristic
245Q32—142no—
246SD32—142no—
247SD32—142no—
248D16—142no—
249M32—72no—
250M32—72no—
251#108—72no—
252#108—141no—
253#107—21no—
254#106C22×D711yescharacteristic
255#106C22×D711yescharacteristic
256#104—142no—
257#104—142no—
258#103—21no—
259#103—21no—
260#102—21no—
261#102—21no—
262#97C22⋊D711yescharacteristic
263#97C22×D711yescharacteristic
264C4×C8C22⋊D711yescharacteristic
265C4×C8C22×D711yescharacteristic
266C22×C7—21no—
267C22×C7—41no—
268C22×C7—81no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
272C2×D7—42no—
273C2×D7—161no—
274C4×C7—21no—
275C4×C7—41no—
276C4×C7—41no—
277C4×C7—81no—
278C4×C7—81no—
279C4×C7—81no—
280C4×C7#52511yescharacteristic
281C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
282Dic7—42no—
283#41—142no—
284#41—142no—
285#41—281no—
286#41—281no—
287C2×Q8—21no—
288C2×Q8—141no—
289C2×D4—21no—
290C2×D4—41no—
291C2×D4—141no—
292C2×D4—281no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—281no—
296C22×C4C2×C22⋊D711yescharacteristic
297Q16—22no—
298Q16—72no—
299Q16—141no—
300SD16—42no—
301SD16—142no—
302SD16—142no—
303D8—22no—
304D8—72no—
305D8—141no—
306M16—22no—
307M16—22no—
308M16—142no—
309M16—282no—
310M16—281no—
311M16—282no—
312C2×C8—282no—
313C2×C8—281no—
314C2×C8—282no—
315C2×C8C2×C22⋊D711yescharacteristic
316C2×C8C2×C22⋊D711yescharacteristic
317C2×C8C2×C22⋊D711yescharacteristic
318C2×C8C23×D711yescharacteristic
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C42—21no—
331C42—41no—
332C42C2×C22⋊D711yescharacteristic
333C42C2×C22⋊D711yescharacteristic
334C16—142no—
335C16—142no—
336C2×C7—21no—
337C2×C7—41no—
338C2×C7—161no—
339C2×C7#310911yescharacteristic
340D7—82no—
341C23—41no—
342C23—281no—
343C23#649511yescharacteristic
344Q8—41no—
345Q8—41no—
346Q8—142no—
347Q8—281no—
348D4—41no—
349D4—41no—
350D4—82no—
351D4—142no—
352D4—281no—
353D4—282no—
354D4—282no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—282no—
365C2×C4—281no—
366C2×C4—282no—
367C2×C4#649511yescharacteristic
368C2×C4C22×C22⋊D711yescharacteristic
369C8—21no—
370C8—21no—
371C8—282no—
372C8#651212yes—
373C8D7×#4112yes—
374C7#3369011yescharacteristic, complemented, Sylow
375C22—21no—
376C22—41no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—282no—
381C22—1121no—
382C22C2×#649511yescharacteristic
383C4—21no—
384C4—41no—
385C4—41no—
386C4—81no—
387C4—81no—
388C4—81no—
389C4—282no—
390C4C2×#651211yescharacteristic
391C4#9051911yescharacteristic
392C2—21no—
393C2—41no—
394C2—161no—
395C2—562no—
396C2#1174074511yescharacteristic
subgroupG/H
centerC2#11740745
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#3183C2
socleC2×C7#3109
prime radical#11728843C2
hypercenter#3183D7
hypocenterC7#33690
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3183

Complements

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

NH ≅ G/Ntype
1. C7×#3183C2semidirect
2. #11735986C2semidirect
3. #11729112C2semidirect
4. #11729079C2semidirect
5. #11728845C2semidirect
6. #11728844C2semidirect
8. #11728842C2semidirect
9. #11728672C2semidirect
10. #11728671C2semidirect
11. #11727618C2semidirect
15. C7×#499C22semidirect
16. C7×#495C22semidirect
21. C7×#442C22semidirect
22. C7×#441C22semidirect
33. #89360C22semidirect
34. #89358C22semidirect
39. #89282C22semidirect
40. #89282C22semidirect
41. #89261C22semidirect
42. #89258C22semidirect
91. #3183D7semidirect
160. #499C2×D7semidirect
161. #495C2×D7semidirect
170. #442C2×D7semidirect
171. #441C2×D7semidirect
374. C7#33690semidirect

Maximal subgroups

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

subgroupindex
C7×#31832
#117359862
#117291122
#117290792
#117288452
#117288442
#117288432
#117288422
#117286722
#117286712
#117276182
#336907