Finite Group Databasebeta

← #13509128

Subgroups — #13509128

Order 1792.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3155C211yesmaximal, characteristic
2#11735991C211yesmaximal, characteristic, complemented
3#11729832C211yesmaximal, characteristic, complemented
4#11729831C211yesmaximal, characteristic, complemented
5#11729787C211yesmaximal, characteristic, complemented
6C2×#89357C211yesmaximal, characteristic, complemented
7#11729073C212yesmaximal, complemented
8#11729072C212yesmaximal, complemented
9#11729043C212yesmaximal, complemented
10#11729042C212yesmaximal, complemented
11#11727635C211yesmaximal, characteristic, complemented
12C2×C7×AGL(1,ℤ/8)—21no—
13C2×C7×#135C2211yescharacteristic
14C22×C7×D8C2211yescharacteristic
15C7×#535—21no—
16#90210—21no—
17C7×#496—22no—
18C7×#495C2212yes—
19C7×#493C2212yes—
20C7×#405C2211yescharacteristic
21#89893—21no—
22#89878C2211yescharacteristic
23C2×#6392C2211yescharacteristic
24C2×#6386C2211yescharacteristic
25C22×C7⋊C16C2211yescharacteristic
26#89474—22no—
27#89455C2212yescomplemented
28#89454C2212yescomplemented
29#89359C2211yescharacteristic, complemented
30#89359C2211yescharacteristic, complemented
31#89359C2212yescomplemented
32#89357C2212yes—
33#89357C2212yes—
34#89357C2212yes—
35#89357C2212yes—
36#89357C2211yescharacteristic, complemented
37#89357C2212yescomplemented
38#89357C2211yescharacteristic, complemented
39#89256C2212yescomplemented
40C4×C7⋊C16C2212yescomplemented
41#33702—71nomaximal, Sylow
42C2×C7×#41—21no—
43C22×C7×D4—21no—
44C7×AGL(1,ℤ/8)—44no—
45C7×#135—22no—
46C7×#135—22no—
47C2×C7×Q16C2311yescharacteristic
48C2×C7×SD16—22no—
49C2×C7×SD16—21no—
50C2×C7×D8—22no—
51C2×C7×D8—21no—
52C2×C7×D8—22no—
53C2×C7×D8—22no—
54C2×C7×D8C2312yes—
55C2×C7×D8C2311yescharacteristic
56C2×C7×M16D411yescharacteristic
57C22×C7×C8C2311yescharacteristic
58C7×#127—22no—
59C7×#124—22no—
60C7×#121—41no—
61C7×#121—41no—
62C7×#120—42no—
63C7×#117D411yescharacteristic
64#6489—21no—
65C2×C4⋊Dic7—21no—
66#6453—41no—
67#6451—41no—
68#6441—42no—
69#6433—22no—
70#6432—22no—
71C7×#97D412yes—
72C4×C7×C8C2312yes—
73#6392—21no—
74#6392—22no—
75#6392C2312yes—
76#6392C2311yescharacteristic
77#6392C2311yescharacteristic
78#6391—21no—
79#6391—22no—
80#6386—22no—
81C2×C7⋊C16—22no—
82C2×C7⋊C16C2311yescharacteristic
83C2×C7⋊C16C2311yescharacteristic
84C2×C7⋊C16C2312yes—
85#3164—71no—
86#3155D711yescharacteristic
87#2244—72no—
88#2243—72no—
89#2242—72no—
90#2240—72no—
91#2140—71no—
92#2139—71no—
93#2138—71no—
94C2×#357—71no—
95#2108—71no—
96C24×C7—41no—
97C7×#41—42no—
98C7×#41—42no—
99C2×C7×Q8—21no—
100C2×C7×D4—22no—
101C2×C7×D4—21no—
102C2×C7×D4—41no—
103C2×C7×D4—42no—
104C2×C7×D4—41no—
105C2×C7×D4—42no—
106C2×C7×D4—42no—
107C2×C7×D4—42no—
108C22×C4×C7—41no—
109C22×C4×C7C2×D411yescharacteristic
110C22×Dic7—41no—
111C2×Dic14—42no—
112C7×Q16—21no—
113C7×Q16—21no—
114C7×SD16—42no—
115C7×SD16—44no—
116C7×D8—22no—
117C7×D8—22no—
118C7×D8—21no—
119C7×D8—21no—
120C7×D8—42no—
121C7×D8—44no—
122C7×M16SD1614yes—
123C2×C7×C8C2×D411yescharacteristic
124C2×C7×C8C2×D412yes—
125C2×C7×C8#4112yes—
126C2×C7×C8#4112yes—
127C2×C7×C8C2411yescharacteristic
128C7×C4⋊C4D811yescharacteristic
129C7×C4⋊C4D811yescharacteristic
130C7×C22⋊C4—42no—
131C7×C22⋊C4—42no—
132C7×C22⋊C4D812yes—
133C42×C7C2×D412yes—
134C22⋊Dic7—41no—
135C4⋊Dic7—21no—
136C4⋊Dic7—22no—
137C4⋊Dic7—21no—
138C4⋊Dic7—42no—
139C4⋊Dic7—41no—
140Dic7⋊C4—42no—
141Dic7⋊C4—42no—
142C4×Dic7—41no—
143C7⋊C16—21no—
144C7⋊C16—21no—
145C7⋊C16—22no—
146C2×AGL(1,ℤ/8)—21no—
147C2×#135C2×D711yescharacteristic
148C22×D8C2×D711yescharacteristic
149#544—141no—
150#535—21no—
151C2×M32—71no—
152C22×C16—71no—
153#501—142no—
154#500—72no—
155#499—72no—
156#496—22no—
157#495C2×D712yes—
158#493C2×D712yes—
159#428—141no—
160#427—71no—
161C2×#107—71no—
162#405C2×D711yescharacteristic
163#358—71no—
164#358—71no—
165#358—72no—
166#357—72no—
167#357—72no—
168#357—72no—
169#357—72no—
170#357—71no—
171#357—71no—
172#357—72no—
173#346—72no—
174C4×C16—72no—
175C23×C7—42no—
176C23×C7—41no—
177C23×C7—82no—
178C23×C7—161no—
179C23×C7—161no—
180C23×C7C2×D811yescharacteristic
181C7×Q8—41no—
182C7×Q8—41no—
183C7×D4—42no—
184C7×D4—41no—
185C7×D4—41no—
186C7×D4—42no—
187C7×D4—82no—
188C7×D4—82no—
189C7×D4—162no—
190C7×D4—162no—
191C2×C4×C7—41no—
192C2×C4×C7—82no—
193C2×C4×C7#12412yes—
194C2×C4×C7C2×D812yes—
195C2×C4×C7C2×D812yes—
196C2×C4×C7C2×D811yescharacteristic
197C2×C4×C7C2×SD1612yes—
198C2×C4×C7C22×D411yescharacteristic
199C2×Dic7—41no—
200C2×Dic7—41no—
201C2×Dic7—42no—
202C2×Dic7—161no—
203Dic14—162no—
204C7×C8#12412yes—
205C7×C8C2×SD1614yes—
206C7×C8C2×#4111yescharacteristic
207C7×C8C2×#4111yescharacteristic
208C2×#41—21no—
209C22×D4—21no—
210AGL(1,ℤ/8)—44no—
211#135—22no—
212#135—22no—
213C2×Q16C22×D711yescharacteristic
214C2×SD16—21no—
215C2×SD16—22no—
216C2×D8—22no—
217C2×D8—22no—
218C2×D8—21no—
219C2×D8—22no—
220C2×D8C22×D712yes—
221C2×D8C22×D711yescharacteristic
222C2×M16C22⋊D711yescharacteristic
223C22×C8C22×D711yescharacteristic
224C4⋊Q8—142no—
225C4⋊Q8—281no—
226#127—22no—
227#125—142no—
228#125—281no—
229#124—22no—
230C22⋊Q8—282no—
231#121—41no—
232#121—41no—
233#120—42no—
234#117—141no—
235#117C22⋊D711yescharacteristic
236C2×C4⋊C4—141no—
237M32—142no—
238C2×C16—71no—
239C2×C16—72no—
240C2×C16—71no—
241C2×C16—142no—
242#107—72no—
243#107—71no—
244#107—71no—
245#107—141no—
246#107—142no—
247#106—141no—
248#106—142no—
249#97C22⋊D712yes—
250C4×C8C22×D712yes—
251C22×C7—21no—
252C22×C7—21no—
253C22×C7—81no—
254C22×C7—82no—
255C22×C7—81no—
256C22×C7—82no—
257C22×C7—81no—
258C22×C7—82no—
259C22×C7—164no—
260C22×C7#48612yes—
261C22×C7C22×D811yescharacteristic
262C4×C7—22no—
263C4×C7—22no—
264C4×C7—81no—
265C4×C7#48612yes—
266C4×C7C2×#12411yescharacteristic
267C4×C7C22×SD1611yescharacteristic
268Dic7—81no—
269Dic7—82no—
270Dic7—81no—
271C24—41no—
272#41—42no—
273#41—42no—
274C2×Q8—21no—
275C2×Q8—282no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C22×C4—41no—
285C22×C4—281no—
286C22×C4C2×C22⋊D711yescharacteristic
287Q16—21no—
288Q16—21no—
289SD16—44no—
290SD16—42no—
291D8—22no—
292D8—22no—
293D8—21no—
294D8—21no—
295D8—44no—
296D8—42no—
297M16#114214yes—
298C2×C8#115912yes—
299C2×C8#115912yes—
300C2×C8C2×C22⋊D712yes—
301C2×C8C2×C22⋊D711yescharacteristic
302C2×C8C23×D711yescharacteristic
303C4⋊C4—141no—
304C4⋊C4—142no—
305C4⋊C4—141no—
306C4⋊C4—282no—
307C4⋊C4—282no—
308C4⋊C4—281no—
309C4⋊C4—282no—
310C4⋊C4#114111yescharacteristic
311C4⋊C4#114111yescharacteristic
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—281no—
315C22⋊C4#114112yes—
316C42—281no—
317C42C2×C22⋊D712yes—
318C16—141no—
319C16—141no—
320C16—142no—
321C2×C7—22no—
322C2×C7—161no—
323C2×C7—162no—
324C2×C7C2×#48611yescharacteristic
325C2×C7#341511yescharacteristic
326C2×C7#341611yescharacteristic
327C23—42no—
328C23—41no—
329C23—82no—
330C23—161no—
331C23—161no—
332C23C2×#114111yescharacteristic
333Q8—41no—
334Q8—41no—
335Q8—1122no—
336D4—41no—
337D4—42no—
338D4—42no—
339D4—41no—
340D4—82no—
341D4—82no—
342D4—162no—
343D4—162no—
344C2×C4—41no—
345C2×C4—82no—
346C2×C4—281no—
347C2×C4—281no—
348C2×C4—282no—
349C2×C4—1121no—
350C2×C4C2×#114112yes—
351C2×C4C2×#114112yes—
352C2×C4C2×#114111yescharacteristic
353C2×C4C2×#114212yes—
354C2×C4#649912yes—
355C2×C4C22×C22⋊D711yescharacteristic
356C8C2×#114214yes—
357C8#649912yes—
358C8C2×#115911yescharacteristic
359C8C2×#115911yescharacteristic
360C7#3370211yescharacteristic, complemented, Sylow
361C22—21no—
362C22—21no—
363C22—81no—
364C22—82no—
365C22—81no—
366C22—82no—
367C22—81no—
368C22—82no—
369C22—164no—
370C22#8984212yes—
371C22C22×#114111yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—562no—
376C4—561no—
377C4—561no—
378C4#8984212yes—
379C4C22×#114211yescharacteristic
380C4C2×#649911yescharacteristic
381C2—22no—
382C2—162no—
383C2—161no—
384C2C2×#8984211yescharacteristic
385C2#1174145211yescharacteristic
386C2#1174145311yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
Fitting, prime radicalC7×#3155C2
socleC22×C7C22×D8
hypercenter#3155D7
hypocenterC7#33702
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3155

Complements

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

NH ≅ G/Ntype
2. #11735991C2semidirect
3. #11729832C2semidirect
4. #11729831C2semidirect
5. #11729787C2semidirect
6. C2×#89357C2semidirect
7. #11729073C2semidirect
8. #11729072C2semidirect
9. #11729043C2semidirect
10. #11729042C2semidirect
11. #11727635C2semidirect
27. #89455C22semidirect
28. #89454C22semidirect
29. #89359C22semidirect
30. #89359C22semidirect
31. #89359C22semidirect
36. #89357C22semidirect
37. #89357C22semidirect
38. #89357C22semidirect
39. #89256C22semidirect
40. C4×C7⋊C16C22semidirect
360. C7#33702semidirect

Maximal subgroups

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

subgroupindex
C7×#31552
#117359912
#117298322
#117298312
#117297872
C2×#893572
#117290732
#117290722
#117290432
#117290422
#117276352
#337027