Finite Group Databasebeta

← #13510390

Subgroups — #13510390

Order 1792.

748 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×D4×D8C211yesmaximal, characteristic, complemented
2C2×#89919C212yesmaximal, complemented
3#11736891C211yesmaximal, characteristic, complemented
4#11729789C212yesmaximal, complemented
5#11729048C211yesmaximal, characteristic, complemented
6#11729044C211yesmaximal, characteristic, complemented
7#11728830C212yesmaximal, complemented
8#11728827C212yesmaximal, complemented
9C4×#6401C211yesmaximal, characteristic, complemented
10#11728643C211yesmaximal, characteristic, complemented
11#11728612C211yesmaximal, characteristic, complemented
12C22×C7×D8C2212yes—
13C7×D42—21no—
14C2×#6467C2212yescomplemented
15#90244—21no—
16C7×#493C2211yescharacteristic
17C7×#466C2212yes—
18C7×#459—21no—
19C7×#447—22no—
20C4×C7×D8C2211yescharacteristic
21C7×C8×D4C2211yescharacteristic
22C2×#6401C2211yescharacteristic
23C2×#6401C2212yescomplemented
24#89919—28no—
25C2×#6400C2212yes—
26C2×#6400C2212yescomplemented
27C2×#6400C2212yescomplemented
28#89887C2212yescomplemented
29C2×#6386C2212yescomplemented
30#89620—21no—
31#89506—22no—
32C4×Dic28C2211yescharacteristic, complemented
33#89467C2211yescharacteristic, complemented
34#89357C2211yescharacteristic
35#89357C2212yescomplemented
36C22⋊(C7⋊C16)C2212yescomplemented
37#89287C2211yescharacteristic, complemented
38#89286C2212yescomplemented
39#89285C2211yescharacteristic, complemented
40C4⋊(C7⋊C16)C2211yescharacteristic, complemented
41C4×C7⋊C16C2211yescharacteristic, complemented
42#33840—71nomaximal, Sylow
43C22×C7×D4—22no—
44C22×C7×D4—41no—
45C2×#1157—22no—
46C2×C7×D8—24no—
47C2×C7×D8—22no—
48C2×C7×D8—21no—
49C2×C7×D8—42no—
50C2×C7×D8C2312yes—
51C2×C7×D8C2311yescharacteristic
52C22×C7×C8C2312yes—
53C7×#127—21no—
54C7×#121—22no—
55C7×#121—41no—
56C7×#120—42no—
57C4×C7×D4—21no—
58C4×C7×D4D411yescharacteristic
59#6503—41no—
60#6493—22no—
61C2×Dic28C2311yescharacteristic
62#6467—24no—
63#6467—42no—
64C2×D56—21no—
65C2×D56C2312yes—
66C2×#1132—22no—
67#6456—41no—
68#6447—42no—
69#6437—21no—
70C4×Dic14—21no—
71C7×#107C2311yescharacteristic
72C7×C4⋊C8D411yescharacteristic
73C7×#102—22no—
74C7×#102—21no—
75C7×C22⋊C8D412yes—
76C4×C7×C8C2311yescharacteristic
77#6401—24no—
78#6401D414yescomplemented
79#6400—24no—
80#6400—24no—
81#6400—42no—
82#6395—22no—
83#6392C2311yescharacteristic
84#6390—21no—
85#6386—24no—
86C2×C7⋊C16C2311yescharacteristic
87C2×C7⋊C16C2312yes—
88C2×C7⋊C16C2311yescharacteristic
89C2×#509—72no—
90#3287—71no—
91D4×D8D711yescharacteristic, complemented
92#2246—71no—
93#2218—72no—
94#2207—71no—
95#2206—71no—
96#2186—72no—
97#2184—72no—
98C4×SD32—71no—
99#2169—71no—
100C24×C7—42no—
101C2×C7×D4—21no—
102C2×C7×D4—22no—
103C2×C7×D4—44no—
104C2×C7×D4—42no—
105C2×C7×D4—42no—
106C2×C7×D4—41no—
107C2×C7×D4—41no—
108C2×C7×D4—82no—
109C2×C7×D4—84no—
110C2×C7×D4D811yescharacteristic
111C22×C4×C7—41no—
112C22×C4×C7C2×D412yes—
113C2×C22⋊D7—42no—
114C2×C22⋊D7—42no—
115#1157—44no—
116#1157—82no—
117C2×D28—22no—
118C2×D28—41no—
119C2×D28—41no—
120C2×C4×D7—42no—
121C2×Dic14—21no—
122C7×D8—22no—
123C7×D8—21no—
124C7×D8—22no—
125C7×D8—44no—
126C7×D8—81no—
127C7×D8C2×D412yes—
128C2×C7×C8—22no—
129C2×C7×C8—22no—
130C2×C7×C8C2×D411yescharacteristic
131C2×C7×C8C2×D412yes—
132C2×C7×C8C2411yescharacteristic
133C7×C4⋊C4—21no—
134C7×C4⋊C4D811yescharacteristic
135C7×C22⋊C4—41no—
136C7×C22⋊C4—42no—
137C7×C22⋊C4D812yes—
138C42×C7C2×D411yescharacteristic
139(C2×D7)⋊C4—42no—
140C4⋊Dic7—21no—
141Dic7⋊C4—41no—
142C4×Dic7—41no—
143Dic28—21no—
144Dic28C2×D412yescomplemented
145D56—22no—
146D56—22no—
147D56—81no—
148#1132—44no—
149C7⋊C16—22no—
150C7⋊C16—21no—
151C7⋊C16C2×D412yescomplemented
152C2×#135—72no—
153C22×D8C2×D712yes—
154#550—141no—
155D42—21no—
156#509—148no—
157C2×SD32—71no—
158C2×SD32—72no—
159C2×D16—72no—
160C2×D16—72no—
161C2×D16—72no—
162C2×M32—72no—
163#493—71no—
164#493C2×D711yescharacteristic
165#466—72no—
166#466C2×D712yes—
167#460—141no—
168#459—21no—
169#449—142no—
170#447—22no—
171C4×Q16—71no—
172C4×D8C2×D711yescharacteristic
173C8×D4C2×D711yescharacteristic
174#367—71no—
175C4⋊C16—71no—
176#358—71no—
177#357—71no—
178#357—72no—
179#357—72no—
180C22⋊C16—72no—
181C4×C16—71no—
182C23×C7—41no—
183C23×C7—42no—
184C23×C7—84no—
185C23×C7—84no—
186C23×C7—162no—
187C23×C7C2×D812yes—
188C22×D7—42no—
189C7×D4—22no—
190C7×D4—22no—
191C7×D4—42no—
192C7×D4—41no—
193C7×D4—42no—
194C7×D4—84no—
195C7×D4—84no—
196C7×D4—82no—
197C7×D4—161no—
198C2×C4×C7—22no—
199C2×C4×C7—22no—
200C2×C4×C7—41no—
201C2×C4×C7—82no—
202C2×C4×C7C2×D811yescharacteristic
203C2×C4×C7C2×D812yes—
204C2×C4×C7C2×D811yescharacteristic
205C2×C4×C7C22×D411yescharacteristic
206C22⋊D7—84no—
207C22⋊D7—162no—
208C2×Dic7—41no—
209C2×Dic7—41no—
210D28—42no—
211D28—42no—
212D28—82no—
213D28—161no—
214C4×D7—84no—
215Dic14—22no—
216Dic14—41no—
217C7×C8—22no—
218C7×C8—21no—
219C7×C8C22×D411yescharacteristic
220C7×C8H(5,𝔽2)11yescharacteristic
221C2×#41—142no—
222C22×D4—22no—
223C22×D4—41no—
224#135—144no—
225#135—282no—
226C2×Q16—71no—
227C2×SD16—142no—
228C2×D8—24no—
229C2×D8—22no—
230C2×D8—21no—
231C2×D8—42no—
232C2×D8—72no—
233C2×D8—141no—
234C2×D8C22×D712yes—
235C2×D8C22×D711yescharacteristic
236C22×C8C22×D712yes—
237#127—21no—
238#127—141no—
239#127—281no—
240#121—22no—
241#121—41no—
242#121—142no—
243#121—282no—
244#120—42no—
245C4×Q8—141no—
246C4×D4—21no—
247C4×D4—281no—
248C4×D4C22⋊D711yescharacteristic
249SD32—74no—
250SD32—144no—
251D16—144no—
252D16—144no—
253D16—282no—
254M32—144no—
255C2×C16—71no—
256C2×C16—71no—
257C2×C16—72no—
258#107—71no—
259#107C22×D711yescharacteristic
260C4⋊C8C22⋊D711yescharacteristic
261#103—141no—
262#102—21no—
263#102—22no—
264#102—142no—
265C22⋊C8C22⋊D712yes—
266C4×C8C22×D711yescharacteristic
267C22×C7—22no—
268C22×C7—22no—
269C22×C7—22no—
270C22×C7—42no—
271C22×C7—81no—
272C22×C7—81no—
273C22×C7—82no—
274C22×C7—82no—
275C22×C7—82no—
276C22×C7—168no—
277C22×C7C22×D811yescharacteristic
278C2×D7—82no—
279C2×D7—82no—
280C2×D7—82no—
281C4×C7—21no—
282C4×C7—22no—
283C4×C7—81no—
284C4×C7#50912yescomplemented
285C4×C7D4211yescharacteristic
286C4×C7#57611yescharacteristic
287Dic7—42no—
288Dic7—81no—
289C24—42no—
290#41—284no—
291#41—562no—
292C2×Q8—141no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—84no—
302C2×D4—142no—
303C2×D4—282no—
304C2×D4—282no—
305C2×D4—281no—
306C2×D4—281no—
307C2×D4#114111yescharacteristic
308C22×C4—41no—
309C22×C4—282no—
310C22×C4C2×C22⋊D712yes—
311Q16—72no—
312Q16—141no—
313SD16—284no—
314D8—22no—
315D8—21no—
316D8—22no—
317D8—44no—
318D8—81no—
319D8—142no—
320D8—142no—
321D8—561no—
322D8D4×D712yes—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8C2×C22⋊D711yescharacteristic
326C2×C8C2×C22⋊D712yes—
327C2×C8C23×D711yescharacteristic
328C4⋊C4—21no—
329C4⋊C4—141no—
330C4⋊C4—281no—
331C4⋊C4#114111yescharacteristic
332C22⋊C4—41no—
333C22⋊C4—42no—
334C22⋊C4—282no—
335C22⋊C4#114112yes—
336C42—281no—
337C42C2×C22⋊D711yescharacteristic
338C16—72no—
339C16—142no—
340C16—141no—
341C2×C7—42no—
342C2×C7—82no—
343C2×C7—162no—
344C2×C7D4×D811yescharacteristic
345C2×C7C2×#50911yescharacteristic
346C2×C7#341511yescharacteristic
347D7—162no—
348C23—41no—
349C23—42no—
350C23—84no—
351C23—84no—
352C23—162no—
353C23—282no—
354C23C2×#114112yes—
355Q8—142no—
356Q8—281no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—41no—
361D4—42no—
362D4—84no—
363D4—84no—
364D4—82no—
365D4—161no—
366D4—282no—
367D4—282no—
368D4—564no—
369D4—562no—
370D4—1121no—
371D4—1122no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—82no—
376C2×C4—281no—
377C2×C4—281no—
378C2×C4—564no—
379C2×C4C2×#114111yescharacteristic
380C2×C4C2×#114112yes—
381C2×C4C2×#114111yescharacteristic
382C2×C4C22×C22⋊D711yescharacteristic
383C8—22no—
384C8—21no—
385C8C2×D4×D711yescharacteristic
386C8#654811yescharacteristic
387C7#3384011yescharacteristic, complemented, Sylow
388C22—22no—
389C22—22no—
390C22—22no—
391C22—42no—
392C22—81no—
393C22—81no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—168no—
398C22—562no—
399C22—562no—
400C22—562no—
401C22C22×#114111yescharacteristic
402C4—22no—
403C4—21no—
404C4—81no—
405C4—282no—
406C4—561no—
407C4#8991912yescomplemented
408C4D4×C22⋊D711yescharacteristic
409C4#9052611yescharacteristic
410C2—42no—
411C2—82no—
412C2—162no—
413C2—1122no—
414C2D4×#114111yescharacteristic
415C2C2×#8991911yescharacteristic
416C2#1174145211yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×D4×D8C2
socleC22×C7C22×D8
hypercenterD4×D8D7
hypocenterC7#33840
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×D8

Complements

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

NH ≅ G/Ntype
1. C7×D4×D8C2semidirect
2. C2×#89919C2semidirect
3. #11736891C2semidirect
4. #11729789C2semidirect
5. #11729048C2semidirect
6. #11729044C2semidirect
7. #11728830C2semidirect
8. #11728827C2semidirect
9. C4×#6401C2semidirect
10. #11728643C2semidirect
11. #11728612C2semidirect
14. C2×#6467C22semidirect
23. C2×#6401C22semidirect
26. C2×#6400C22semidirect
27. C2×#6400C22semidirect
28. #89887C22semidirect
29. C2×#6386C22semidirect
32. C4×Dic28C22semidirect
33. #89467C22semidirect
35. #89357C22semidirect
36. C22⋊(C7⋊C16)C22semidirect
37. #89287C22semidirect
38. #89286C22semidirect
39. #89285C22semidirect
40. C4⋊(C7⋊C16)C22semidirect
41. C4×C7⋊C16C22semidirect
78. #6401D4semidirect
91. D4×D8D7semidirect
144. Dic28C2×D4semidirect
151. C7⋊C16C2×D4semidirect
284. C4×C7#509semidirect
387. C7#33840semidirect
407. C4#89919semidirect

Maximal subgroups

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

subgroupindex
C7×D4×D82
C2×#899192
#117368912
#117297892
#117290482
#117290442
#117288302
#117288272
C4×#64012
#117286432
#117286122
#338407