Finite Group Databasebeta

← #13510370

Subgroups — #13510370

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 414 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#11736881C211yesmaximal, characteristic, complemented
4#11729790C212yesmaximal, complemented
5#11729047C211yesmaximal, characteristic, complemented
6#11729045C211yesmaximal, characteristic, complemented
7#11728830C212yesmaximal, complemented
8#11728828C212yesmaximal, complemented
9#11728646C211yesmaximal, characteristic, complemented
10C4×#6400C211yesmaximal, characteristic, complemented
11#11728612C211yesmaximal, characteristic, complemented
12C22×C7×D8C2212yes—
13C7×D42—21no—
14C2×#6467C2212yescomplemented
15#90242—21no—
16C7×#493C2211yescharacteristic
17C7×#466C2212yes—
18C7×#459—21no—
19C7×#447—22no—
20C4×C7×D8C2211yescharacteristic
21C7×C8×D4C2211yescharacteristic
22C2×#6401C2212yes—
23C2×#6401C2212yescomplemented
24C2×#6401C2212yescomplemented
25#89919—28no—
26C2×#6400C2211yescharacteristic
27C2×#6400C2212yescomplemented
28#89888C2212yescomplemented
29C2×#6386C2212yescomplemented
30#89613—21no—
31#89505—22no—
32#89471C2211yescharacteristic, complemented
33C4×D56C2211yescharacteristic, complemented
34#89357C2211yescharacteristic
35#89357C2212yescomplemented
36C22⋊(C7⋊C16)C2212yescomplemented
37#89287C2212yescomplemented
38#89286C2211yescharacteristic, complemented
39#89284C2211yescharacteristic, complemented
40C4⋊(C7⋊C16)C2211yescharacteristic, complemented
41C4×C7⋊C16C2211yescharacteristic, complemented
42#33838—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#6501—41no—
60#6487—22no—
61C2×Dic28—21no—
62C2×Dic28C2312yes—
63#6467—24no—
64#6467—42no—
65C2×D56C2311yescharacteristic
66C2×#1132—22no—
67D7×C4⋊C4—41no—
68#6446—42no—
69C4×D28—21no—
70#6432—21no—
71C7×#107C2311yescharacteristic
72C7×C4⋊C8D411yescharacteristic
73C7×#102—22no—
74C7×#102—21no—
75C7×C22⋊C8D412yes—
76C4×C7×C8C2311yescharacteristic
77#6401—24no—
78#6401—24no—
79#6401—42no—
80#6400—24no—
81#6400D414yescomplemented
82#6395—21no—
83#6392C2311yescharacteristic
84#6390—22no—
85#6386—24no—
86C2×C7⋊C16C2311yescharacteristic
87C2×C7⋊C16C2312yes—
88C2×C7⋊C16C2311yescharacteristic
89C2×#509—72no—
90#3283—71no—
91D4×D8D711yescharacteristic, complemented
92#2248—71no—
93#2220—72no—
94#2208—71no—
95#2206—71no—
96#2187—72no—
97#2186—72no—
98C4×D16—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—
114#1157—44no—
115#1157—82no—
116C2×D28—21no—
117C2×C4×D7—41no—
118C2×C4×D7—42no—
119C2×Dic14—22no—
120C7×D8—22no—
121C7×D8—21no—
122C7×D8—22no—
123C7×D8—44no—
124C7×D8—81no—
125C7×D8C2×D412yes—
126C2×C7×C8—22no—
127C2×C7×C8—22no—
128C2×C7×C8C2×D411yescharacteristic
129C2×C7×C8C2×D412yes—
130C2×C7×C8C2411yescharacteristic
131C7×C4⋊C4—21no—
132C7×C4⋊C4D811yescharacteristic
133C7×C22⋊C4—41no—
134C7×C22⋊C4—42no—
135C7×C22⋊C4D812yes—
136C42×C7C2×D411yescharacteristic
137C22⋊Dic7—42no—
138(C2×D7)⋊C4—41no—
139C4⋊Dic7—21no—
140C4⋊Dic7—41no—
141C4⋊Dic7—41no—
142Dic7⋊C4—42no—
143Dic28—22no—
144Dic28—22no—
145Dic28—81no—
146D56—21no—
147D56C2×D412yescomplemented
148#1132—44no—
149C7⋊C16—22no—
150C7⋊C16—21no—
151C7⋊C16C2×D412yescomplemented
152C2×#135—72no—
153C22×D8C2×D712yes—
154#547—141no—
155D42—21no—
156#509—148no—
157C2×SD32—72no—
158C2×SD32—72no—
159C2×SD32—72no—
160C2×D16—71no—
161C2×D16—72no—
162C2×M32—72no—
163#494—71no—
164#493C2×D711yescharacteristic
165#467—72no—
166#466C2×D712yes—
167#461—141no—
168#459—21no—
169#452—142no—
170#447—22no—
171C4×D8—71no—
172C4×D8C2×D711yescharacteristic
173C8×D4C2×D711yescharacteristic
174#366—71no—
175C4⋊C16—71no—
176#358—72no—
177#357—71no—
178#357—72no—
179#357—71no—
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—41no—
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×D812yes—
203C2×C4×C7C2×D811yescharacteristic
204C2×C4×C7C2×D811yescharacteristic
205C2×C4×C7C22×D411yescharacteristic
206C22⋊D7—84no—
207C2×Dic7—41no—
208C2×Dic7—42no—
209D28—22no—
210D28—41no—
211C4×D7—84no—
212C4×D7—82no—
213Dic14—42no—
214Dic14—42no—
215C7×C8—22no—
216C7×C8—21no—
217C7×C8C22×D411yescharacteristic
218C7×C8H(5,𝔽2)11yescharacteristic
219C2×#41—142no—
220C22×D4—22no—
221C22×D4—41no—
222#135—144no—
223#135—282no—
224C2×Q16—72no—
225C2×Q16—141no—
226C2×SD16—142no—
227C2×D8—24no—
228C2×D8—22no—
229C2×D8—21no—
230C2×D8—42no—
231C2×D8—71no—
232C2×D8C22×D712yes—
233C2×D8C22×D711yescharacteristic
234C22×C8C22×D712yes—
235C4⋊Q8—141no—
236#127—21no—
237#123—282no—
238C22⋊Q8—142no—
239#121—22no—
240#121—41no—
241#121—281no—
242#120—42no—
243C4×D4—21no—
244C4×D4—141no—
245C4×D4C22⋊D711yescharacteristic
246C2×C4⋊C4—281no—
247SD32—144no—
248SD32—144no—
249SD32—282no—
250D16—74no—
251D16—144no—
252M32—144no—
253C2×C16—71no—
254C2×C16—71no—
255C2×C16—72no—
256#107—71no—
257#107C22×D711yescharacteristic
258C4⋊C8C22⋊D711yescharacteristic
259#103—142no—
260#102—21no—
261#102—22no—
262#102—141no—
263C22⋊C8C22⋊D712yes—
264C4×C8C22×D711yescharacteristic
265C22×C7—22no—
266C22×C7—22no—
267C22×C7—22no—
268C22×C7—42no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7—82no—
272C22×C7—82no—
273C22×C7—82no—
274C22×C7—168no—
275C22×C7C22×D811yescharacteristic
276C2×D7—42no—
277C2×D7—81no—
278C2×D7—81no—
279C4×C7—21no—
280C4×C7—22no—
281C4×C7—81no—
282C4×C7#50912yescomplemented
283C4×C7D4211yescharacteristic
284C4×C7#57611yescharacteristic
285Dic7—81no—
286Dic7—82no—
287C24—42no—
288#41—284no—
289#41—562no—
290C2×Q8—142no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—44no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—82no—
299C2×D4—84no—
300C2×D4—141no—
301C2×D4—282no—
302C2×D4#114111yescharacteristic
303C22×C4—41no—
304C22×C4—281no—
305C22×C4—282no—
306C22×C4C2×C22⋊D712yes—
307Q16—142no—
308Q16—142no—
309Q16—561no—
310SD16—284no—
311D8—22no—
312D8—21no—
313D8—22no—
314D8—44no—
315D8—72no—
316D8—81no—
317D8—141no—
318D8D4×D712yes—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8C2×C22⋊D711yescharacteristic
322C2×C8C2×C22⋊D712yes—
323C2×C8C23×D711yescharacteristic
324C4⋊C4—21no—
325C4⋊C4—141no—
326C4⋊C4—282no—
327C4⋊C4—281no—
328C4⋊C4—281no—
329C4⋊C4#114111yescharacteristic
330C22⋊C4—41no—
331C22⋊C4—42no—
332C22⋊C4—282no—
333C22⋊C4—281no—
334C22⋊C4#114112yes—
335C42C2×C22⋊D711yescharacteristic
336C16—72no—
337C16—142no—
338C16—141no—
339C2×C7—42no—
340C2×C7—82no—
341C2×C7—162no—
342C2×C7D4×D811yescharacteristic
343C2×C7C2×#50911yescharacteristic
344C2×C7#341611yescharacteristic
345D7—82no—
346C23—41no—
347C23—42no—
348C23—84no—
349C23—84no—
350C23—162no—
351C23—281no—
352C23C2×#114112yes—
353Q8—282no—
354Q8—282no—
355D4—22no—
356D4—22no—
357D4—42no—
358D4—41no—
359D4—42no—
360D4—84no—
361D4—84no—
362D4—82no—
363D4—142no—
364D4—161no—
365D4—281no—
366D4—564no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—82no—
371C2×C4—281no—
372C2×C4—282no—
373C2×C4—564no—
374C2×C4—562no—
375C2×C4C2×#114111yescharacteristic
376C2×C4C2×#114112yes—
377C2×C4C2×#114111yescharacteristic
378C2×C4C22×C22⋊D711yescharacteristic
379C8—22no—
380C8—21no—
381C8C2×D4×D711yescharacteristic
382C8#654811yescharacteristic
383C7#3383811yescharacteristic, complemented, Sylow
384C22—22no—
385C22—22no—
386C22—22no—
387C22—42no—
388C22—81no—
389C22—81no—
390C22—82no—
391C22—82no—
392C22—82no—
393C22—168no—
394C22—282no—
395C22—561no—
396C22—561no—
397C22C22×#114111yescharacteristic
398C4—22no—
399C4—21no—
400C4—81no—
401C4—562no—
402C4—561no—
403C4#8991912yescomplemented
404C4D4×C22⋊D711yescharacteristic
405C4#9052611yescharacteristic
406C2—42no—
407C2—82no—
408C2—162no—
409C2—562no—
410C2D4×#114111yescharacteristic
411C2C2×#8991911yescharacteristic
412C2#1174145311yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×D4×D8C2
socleC22×C7C22×D8
hypercenterD4×D8D7
hypocenterC7#33838
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. #11736881C2semidirect
4. #11729790C2semidirect
5. #11729047C2semidirect
6. #11729045C2semidirect
7. #11728830C2semidirect
8. #11728828C2semidirect
9. #11728646C2semidirect
10. C4×#6400C2semidirect
11. #11728612C2semidirect
14. C2×#6467C22semidirect
23. C2×#6401C22semidirect
24. C2×#6401C22semidirect
27. C2×#6400C22semidirect
28. #89888C22semidirect
29. C2×#6386C22semidirect
32. #89471C22semidirect
33. C4×D56C22semidirect
35. #89357C22semidirect
36. C22⋊(C7⋊C16)C22semidirect
37. #89287C22semidirect
38. #89286C22semidirect
39. #89284C22semidirect
40. C4⋊(C7⋊C16)C22semidirect
41. C4×C7⋊C16C22semidirect
81. #6400D4semidirect
91. D4×D8D7semidirect
147. D56C2×D4semidirect
151. C7⋊C16C2×D4semidirect
282. C4×C7#509semidirect
383. C7#33838semidirect
403. 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
#117368812
#117297902
#117290472
#117290452
#117288302
#117288282
#117286462
C4×#64002
#117286122
#338387