Finite Group Databasebeta

← #14218329

Subgroups — #14218329

Order 1792.

1878 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11742806C211yesmaximal, characteristic
2C2×C7×#405C211yesmaximal, characteristic, complemented
3#11741201C212yesmaximal, complemented
4#11741179C212yesmaximal, complemented
5#11739654C211yesmaximal, characteristic, complemented
6#11736002C218yesmaximal, complemented
7#11736001C218yesmaximal, complemented
8#11735936C211yesmaximal, characteristic, complemented
9#11735934C212yesmaximal, complemented
10#11735925C211yesmaximal, characteristic, complemented
11#11735903C211yesmaximal, characteristic, complemented
12C4×#6464C212yesmaximal, complemented
13#11735890C211yesmaximal, characteristic, complemented
14C22×C7×M16C2211yescharacteristic
15C23×C7×C8C2211yescharacteristic
16C2×C7×#117C2211yescharacteristic
17#90482C2212yes—
18C2×#6464C2212yes—
19C2×#6464C2212yes—
20#90226C412yes—
21#90225C414yes—
22#90219C2218yes—
23#90209C412yes—
24#90206C414yes—
25#90185C414yes—
26#90168C2211yescharacteristic
27C4×#1157C2212yes—
28#90161C2211yescharacteristic
29C7×#405C2214yes—
30C7×#405C2214yes—
31C2×C7×#97C2211yescharacteristic
32C2×C7×#97C2211yescharacteristic
33C2×C4×C7×C8C2211yescharacteristic
34C2×C4×C7×C8C2211yescharacteristic
35#89979C2212yes—
36#89900C2218yes—
37#89899C2218yes—
38#89883C2218yes—
39#89882C2212yes—
40#89882C2212yes—
41C8×C22⋊D7C2218yes—
42#89874C2212yes—
43#89874C2212yes—
44#89873C2212yes—
45#89873C2212yes—
46#89772C2218yes—
47#89696C2211yescharacteristic
48#89696C2211yescharacteristic
49C4×#1136C2212yes—
50#89481C2218yescomplemented
51#89478C2218yescomplemented
52#89477C2218yescomplemented
53#89473C2218yescomplemented
54#89461C2214yescomplemented
55#89461C2214yescomplemented
56#89460C2214yescomplemented
57C4×#1131C2214yescomplemented
58C8×D28C2214yescomplemented
59C4×C8×D7C2214yescomplemented
60#89451C2214yescomplemented
61C8×Dic14C2214yescomplemented
62#60206—71nomaximal, Sylow
63C23×C4×C7C2311yescharacteristic
64C2×#1157C2312yes—
65C2×C7×M16—24no—
66C2×C7×M16C2314yes—
67C22×C7×C8—24no—
68C22×C7×C8C2311yescharacteristic
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2312yes—
71C22×C7×C8C2312yes—
72C22×C7×C8C2311yescharacteristic
73C22×C7×C8C2311yescharacteristic
74C7×#117C2×C412yes—
75C7×#117C2×C412yes—
76C7×#117C2314yes—
77C2×C7×C4⋊C4C2×C412yes—
78C2×C7×C22⋊C4C2×C412yes—
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81#6516C2×C414yes—
82#6493C2×C414yes—
83#6493C2×C414yes—
84#6492C2×C414yes—
85C4×C22⋊D7C2×C418yes—
86C4×C22⋊D7C2318yes—
87#6489C2312yes—
88#6487C2×C414yes—
89#6487C2×C414yes—
90C2×#1136C2312yes—
91C2×#1136C2312yes—
92#6464—28no—
93#6464—28no—
94C2×#1131C2314yes—
95C2×#1131C2314yes—
96C2×C8×D7C2314yes—
97C2×C8×D7C2314yes—
98#6461C2×C418yes—
99#6459C2×C418yes—
100#6457C2×C418yes—
101#6452C2×C418yes—
102#6448C2×C418yes—
103#6447C2×C418yes—
104#6446C2×C418yes—
105#6442C2×C418yes—
106#6438C2×C412yes—
107#6437C2×C411yescharacteristic
108C4×D28C2×C414yes—
109C4×D28C2314yes—
110#6435C2314yes—
111C42×D7C2314yes—
112#6433C2×C412yes—
113#6432C2×C412yes—
114#6432C2×C411yescharacteristic
115C4×Dic14C2×C414yes—
116C4×Dic14C2314yes—
117C7×#97C2314yes—
118C7×#97C2314yes—
119C4×C7×C8C2312yes—
120C4×C7×C8C2312yes—
121C4×C7×C8C2312yes—
122C4×C7×C8C2312yes—
123C22⋊(C7⋊C8)C2318yes—
124(C2×D7)⋊C8C2318yes—
125(C2×D7)⋊C8C2318yes—
126#6389C2314yes—
127#6389C2314yes—
128Dic7⋊C8C2318yes—
129Dic7⋊C8C2318yes—
130C8×Dic7C2314yes—
131C8×Dic7C2314yes—
132C4⋊(C7⋊C8)C2314yes—
133C4⋊(C7⋊C8)C2314yes—
134#6377C2314yes—
135C4×C7⋊C8C2314yes—
136#3455—71no—
137#2967—78no—
138#2966—78no—
139#2961—71no—
140#2960—71no—
141#2932—71no—
142#2931—71no—
143#2930—72no—
144#2929—72no—
145#2923—72no—
146#2921—71no—
147C4×#131—72no—
148C2×#405D711yescharacteristic, complemented
149C24×C7C22×C411yescharacteristic
150C22×C4×C7—22no—
151C22×C4×C7—22no—
152C22×C4×C7C22×C412yes—
153C22×C4×C7C22×C411yescharacteristic
154C22×C4×C7C22×C412yes—
155C22×C4×C7C22×C412yes—
156C22×C4×C7C2412yes—
157C22×C4×C7C2411yescharacteristic
158C2×C22⋊D7C22×C414yes—
159C2×C22⋊D7C22×C418yes—
160#1157—24no—
161#1157—24no—
162C2×D28C22×C412yes—
163C2×D28C22×C414yes—
164C2×C4×D7C22×C418yes—
165C2×C4×D7C2414yes—
166C2×Dic14C22×C412yes—
167C2×Dic14C22×C414yes—
168C7×M16—28no—
169C2×C7×C8—22no—
170C2×C7×C8—22no—
171C2×C7×C8—24no—
172C2×C7×C8—24no—
173C2×C7×C8#4114yes—
174C2×C7×C8#4114yes—
175C2×C7×C8C2412yes—
176C2×C7×C8C2412yes—
177C2×C7×C8C2411yescharacteristic
178C2×C7×C8C2412yes—
179C2×C7×C8C2411yescharacteristic
180C7×C4⋊C4C22×C414yes—
181C7×C4⋊C4C22×C414yes—
182C7×C22⋊C4C22×C414yes—
183C7×C22⋊C4C22×C414yes—
184C42×C7C22×C412yes—
185C42×C7C22×C411yescharacteristic
186C42×C7C22×C411yescharacteristic
187C42×C7C2412yes—
188C42×C7C2412yes—
189C22⋊Dic7C22×C418yes—
190C22⋊Dic7C22×C414yes—
191(C2×D7)⋊C4C22×C418yes—
192(C2×D7)⋊C4C22×C418yes—
193(C2×D7)⋊C4C22×C418yes—
194C4⋊Dic7C22×C414yes—
195C4⋊Dic7C22×C414yes—
196C4⋊Dic7C22×C414yes—
197Dic7⋊C4C22×C418yes—
198Dic7⋊C4C22×C418yes—
199Dic7⋊C4C22×C418yes—
200C4×Dic7C22×C418yes—
201C4×Dic7C2414yes—
202#1136—24no—
203#1136—24no—
204C2×C7⋊C8C2414yes—
205C2×C7⋊C8C2414yes—
206#1131—28no—
207#1131—28no—
208C8×D7—28no—
209C8×D7—28no—
210C2×#131—72no—
211C2×#131—72no—
212C22×M16C2×D711yescharacteristic
213C23×C8C2×D711yescharacteristic
214#542—74no—
215#541—72no—
216#540—72no—
217#539—74no—
218#538—74no—
219#533—71no—
220#532—71no—
221#529—78no—
222#525—72no—
223C4×#41—72no—
224C2×#117C2×D711yescharacteristic
225#446—78no—
226#446—74no—
227C8×Q8—74no—
228#436—78no—
229#436—74no—
230#435—78no—
231#435—78no—
232#435—78no—
233C8×D4—78no—
234C8×D4—74no—
235#433—78no—
236#433—78no—
237#432—78no—
238#424—72no—
239#424—72no—
240#423—71no—
241#423—71no—
242#408—72no—
243#408—72no—
244#407—72no—
245#405—72no—
246#405—72no—
247#405—74no—
248#405—74no—
249#405C2×D714yes—
250#405C2×D714yes—
251C4×M16—72no—
252C4×M16—74no—
253C2×#97C2×D711yescharacteristic
254C2×#97C2×D711yescharacteristic
255C2×C4×C8—74no—
256C2×C4×C8C2×D711yescharacteristic
257C2×C4×C8C2×D711yescharacteristic
258C23×C7—22no—
259C23×C7—42no—
260C23×C7C23×C412yes—
261C23×C7C23×C411yescharacteristic
262C22×D7C23×C414yes—
263C2×C4×C7—22no—
264C2×C4×C7—24no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—42no—
268C2×C4×C7—42no—
269C2×C4×C7C23×C414yes—
270C2×C4×C7C23×C412yes—
271C2×C4×C7C23×C412yes—
272C2×C4×C7C23×C412yes—
273C2×C4×C7C23×C411yescharacteristic
274C2×C4×C7C2×#4112yes—
275C2×C4×C7C2×#4112yes—
276C2×C4×C7C2511yescharacteristic
277C22⋊D7—28no—
278C22⋊D7—48no—
279C2×Dic7C23×C414yes—
280C2×Dic7C23×C418yes—
281D28—24no—
282D28—44no—
283C4×D7—24no—
284C4×D7—24no—
285C4×D7—48no—
286Dic14—24no—
287Dic14—44no—
288C7×C8—22no—
289C7×C8—22no—
290C7×C8C2×#4112yes—
291C7×C8C2×#4114yes—
292C7×C8C2×#4112yes—
293C7⋊C8—24no—
294C7⋊C8—24no—
295C2×#41—72no—
296C23×C4C22×D711yescharacteristic
297#131—148no—
298#131—148no—
299C2×M16—24no—
300C2×M16—72no—
301C2×M16—72no—
302C2×M16—74no—
303C2×M16—74no—
304C2×M16C22×D714yes—
305C22×C8—24no—
306C22×C8—74no—
307C22×C8—74no—
308C22×C8C22×D711yescharacteristic
309C22×C8C22×D712yes—
310C22×C8C22×D711yescharacteristic
311C22×C8C22×D712yes—
312C22×C8C22×D711yescharacteristic
313C22×C8C22×D711yescharacteristic
314C4⋊Q8—71no—
315C4⋊Q8—72no—
316#127—71no—
317#126—78no—
318#126—78no—
319#125—78no—
320#125—72no—
321#124—78no—
322#124—72no—
323#123—74no—
324#123—78no—
325#123—78no—
326C22⋊Q8—74no—
327C22⋊Q8—74no—
328C22⋊Q8—78no—
329#121—74no—
330#121—74no—
331#121—78no—
332#120—74no—
333C4×Q8—74no—
334C4×Q8—74no—
335C4×D4—78no—
336C4×D4—78no—
337C4×D4—74no—
338C4×D4—74no—
339#117—72no—
340#117—74no—
341#117C4×D712yes—
342#117C4×D712yes—
343#117C22×D714yes—
344C2×C4⋊C4C4×D712yes—
345C2×C22⋊C4C4×D712yes—
346C2×C42—74no—
347C2×C42C22×D711yescharacteristic
348C2×C42C22×D711yescharacteristic
349C4⋊C8—78no—
350C4⋊C8—78no—
351C4⋊C8—74no—
352C4⋊C8—74no—
353C22⋊C8—78no—
354C22⋊C8—78no—
355C22⋊C8—78no—
356#97—74no—
357#97—74no—
358#97—74no—
359#97C22×D714yes—
360#97C22×D714yes—
361C4×C8—74no—
362C4×C8—74no—
363C4×C8—74no—
364C4×C8C22×D712yes—
365C4×C8C22×D712yes—
366C4×C8C22×D712yes—
367C4×C8C22×D712yes—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—24no—
371C22×C7—44no—
372C22×C7C4×#4112yes—
373C22×C7C24×C411yescharacteristic
374C2×D7—24no—
375C2×D7—28no—
376C4×C7—22no—
377C4×C7—22no—
378C4×C7—24no—
379C4×C7—22no—
380C4×C7C4×#4112yes—
381C4×C7C22×#4111yescharacteristic
382C4×C7C22×#4111yescharacteristic
383Dic7—24no—
384Dic7—28no—
385C24C2×C4×D711yescharacteristic
386#41—144no—
387#41—144no—
388C2×Q8—72no—
389C2×Q8—74no—
390C2×D4—74no—
391C2×D4—78no—
392C2×D4—72no—
393C2×D4—74no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—74no—
397C22×C4—78no—
398C22×C4C2×C4×D711yescharacteristic
399C22×C4C2×C4×D712yes—
400C22×C4C2×C4×D712yes—
401C22×C4C2×C4×D712yes—
402C22×C4C23×D712yes—
403C22×C4C23×D711yescharacteristic
404M16—28no—
405M16—144no—
406M16—144no—
407M16—148no—
408M16—148no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—74no—
414C2×C8—74no—
415C2×C8—148no—
416C2×C8—148no—
417C2×C8#115714yes—
418C2×C8#115714yes—
419C2×C8C23×D712yes—
420C2×C8C23×D712yes—
421C2×C8C23×D712yes—
422C2×C8C23×D711yescharacteristic
423C2×C8C23×D711yescharacteristic
424C4⋊C4—78no—
425C4⋊C4—78no—
426C4⋊C4—78no—
427C4⋊C4—74no—
428C4⋊C4—74no—
429C4⋊C4—74no—
430C4⋊C4C2×C4×D714yes—
431C4⋊C4C2×C4×D714yes—
432C22⋊C4—74no—
433C22⋊C4—78no—
434C22⋊C4—78no—
435C22⋊C4—78no—
436C22⋊C4—78no—
437C22⋊C4C2×C4×D714yes—
438C22⋊C4C2×C4×D714yes—
439C42—78no—
440C42—74no—
441C42C2×C4×D712yes—
442C42C2×C4×D711yescharacteristic
443C42C2×C4×D711yescharacteristic
444C42C23×D712yes—
445C42C23×D712yes—
446C2×C7—22no—
447C2×C7—42no—
448C2×C7C2×C4×#4111yescharacteristic
449C2×C7#357312yes—
450D7—44no—
451C23—22no—
452C23—42no—
453C23—74no—
454C23C22×C4×D712yes—
455C23C22×C4×D711yescharacteristic
456Q8—144no—
457Q8—284no—
458D4—148no—
459D4—144no—
460D4—284no—
461D4—288no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—74no—
469C2×C4—78no—
470C2×C4—144no—
471C2×C4—144no—
472C2×C4—288no—
473C2×C4C22×C4×D714yes—
474C2×C4C22×C4×D712yes—
475C2×C4C22×C4×D712yes—
476C2×C4C22×C4×D712yes—
477C2×C4C22×C4×D711yescharacteristic
478C2×C4C2×#115712yes—
479C2×C4C2×#115712yes—
480C2×C4C24×D711yescharacteristic
481C8—22no—
482C8—22no—
483C8—144no—
484C8—144no—
485C8C2×#115712yes—
486C8C2×#115712yes—
487C8C2×#115714yes—
488C7#6020611yescharacteristic, complemented, Sylow
489C22—22no—
490C22—22no—
491C22—24no—
492C22—44no—
493C22—144no—
494C22—148no—
495C22C4×#115712yes—
496C22C23×C4×D711yescharacteristic
497C4—22no—
498C4—24no—
499C4—22no—
500C4—22no—
501C4—144no—
502C4—148no—
503C4C4×#115712yes—
504C4C22×#115711yescharacteristic
505C4C22×#115711yescharacteristic
506C2—22no—
507C2—42no—
508C2—284no—
509C2C2×C4×#115711yescharacteristic
510C2#1174422812yes—
subgroupG/H
centerC2×C8C23×D7
commutator, socleC22×C7C24×C4
FrattiniC2×C4C24×D7
FittingC2×C7×#405C2
prime radical#11742806C2
cosocleC2×C4×C7C25
hypercenterC2×#405D7
hypocenterC7#60206
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C8 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
2. C2×C7×#405C2semidirect
3. #11741201C2semidirect
4. #11741179C2semidirect
5. #11739654C2semidirect
6. #11736002C2semidirect
7. #11736001C2semidirect
8. #11735936C2semidirect
9. #11735934C2semidirect
10. #11735925C2semidirect
11. #11735903C2semidirect
12. C4×#6464C2semidirect
13. #11735890C2semidirect
50. #89481C22semidirect
51. #89478C22semidirect
52. #89477C22semidirect
53. #89473C22semidirect
54. #89461C22semidirect
55. #89461C22semidirect
56. #89460C22semidirect
57. C4×#1131C22semidirect
58. C8×D28C22semidirect
59. C4×C8×D7C22semidirect
60. #89451C22semidirect
61. C8×Dic14C22semidirect
148. C2×#405D7semidirect
488. C7#60206semidirect

Maximal subgroups

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

subgroupindex
#117428062
C2×C7×#4052
#117412012
#117411792
#117396542
#117360022
#117360012
#117359362
#117359342
#117359252
#117359032
C4×#64642
#117358902
#602067