Finite Group Databasebeta

← #14218322

Subgroups — #14218322

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
2C7×#2921C211yesmaximal, characteristic, complemented
3C2×#89769C211yesmaximal, characteristic, complemented
4#11736812C218yesmaximal, complemented
5#11736811C218yesmaximal, complemented
6#11736490C211yesmaximal, characteristic, complemented
7#11736489C211yesmaximal, characteristic, complemented
8#11736487C211yesmaximal, characteristic, complemented
9#11736486C211yesmaximal, characteristic, complemented
10#11736072C212yesmaximal, complemented
11#11736071C212yesmaximal, complemented
12#11735965C212yesmaximal, complemented
13C4×#6470C212yesmaximal, complemented
14C2×C7×#117C2211yescharacteristic
15#90482C2212yes—
16C22×#1136C2211yescharacteristic
17C23×C7⋊C8C2211yescharacteristic
18C2×#6470C2212yes—
19C2×#6470C2212yes—
20#90226C412yes—
21#90225C414yes—
22#90219C2218yes—
23#90209C412yes—
24#90206C414yes—
25#90185C414yes—
26#90168C2211yescharacteristic
27C4×#1157C2212yes—
28#90161C2211yescharacteristic
29C7×#433C2218yes—
30C7×#423C2211yescharacteristic
31C7×#423C2211yescharacteristic
32C7×#407C2212yes—
33C4×C7×M16C2212yes—
34#89901C2212yes—
35#89901C2212yes—
36#89894C2212yes—
37#89894C2212yes—
38#89891C2212yes—
39#89891C2212yes—
40#89769C2214yes—
41#89769C2214yes—
42C2×#6377C2211yescharacteristic
43C2×#6377C2211yescharacteristic
44C2×C4×C7⋊C8C2211yescharacteristic
45C2×C4×C7⋊C8C2211yescharacteristic
46#89610C2214yescomplemented
47#89608C2218yescomplemented
48#89606C2214yescomplemented
49#89603C2214yescomplemented
50#89602C2214yescomplemented
51#89600C2218yescomplemented
52#89502C2218yes—
53#89501C2218yes—
54#89500C2218yes—
55(C22⋊D7)⋊C8C2218yes—
56#89480C2214yescomplemented
57#89480C2214yescomplemented
58#89477C2218yescomplemented
59#89462C2218yescomplemented
60C4×#1131C2214yescomplemented
61C4×C8×D7C2214yescomplemented
62#60206—71nomaximal, Sylow
63C23×C4×C7C2311yescharacteristic
64C2×#1157C2312yes—
65C2×C7×M16C2312yes—
66C2×C7×M16C2312yes—
67C7×#117C2×C412yes—
68C7×#117C2×C412yes—
69C7×#117C2314yes—
70C2×C7×C4⋊C4C2×C412yes—
71C2×C7×C22⋊C4C2×C412yes—
72C2×C42×C7C2311yescharacteristic
73C2×C42×C7C2311yescharacteristic
74#6516C2×C414yes—
75#6493C2×C414yes—
76#6493C2×C414yes—
77#6492C2×C414yes—
78C4×C22⋊D7C2×C418yes—
79C4×C22⋊D7C2318yes—
80#6489C2312yes—
81#6487C2×C414yes—
82#6487C2×C414yes—
83C2×#1136—24no—
84C2×#1136C2314yes—
85C22×C7⋊C8—24no—
86C22×C7⋊C8C2311yescharacteristic
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C22×C7⋊C8C2312yes—
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92#6470—28no—
93#6470—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×C4⋊C8C2314yes—
118C7×C4⋊C8C2314yes—
119C7×C22⋊C8C2318yes—
120C7×#97C2314yes—
121C4×C7×C8C2314yes—
122(C2×D7)⋊C8C2318yes—
123(C2×D7)⋊C8C2318yes—
124#6389C2314yes—
125#6389C2314yes—
126Dic7⋊C8C2318yes—
127Dic7⋊C8C2318yes—
128C8×Dic7C2314yes—
129C8×Dic7C2314yes—
130#6377C2314yes—
131#6377C2314yes—
132C4×C7⋊C8C2312yes—
133C4×C7⋊C8C2312yes—
134C4×C7⋊C8C2312yes—
135C4×C7⋊C8C2312yes—
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#2921D711yescharacteristic, complemented
147C4×#131—72no—
148C2×#405—71no—
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—24no—
169C7×M16—24no—
170C2×C7×C8C2414yes—
171C2×C7×C8C2414yes—
172C7×C4⋊C4C22×C414yes—
173C7×C4⋊C4C22×C414yes—
174C7×C22⋊C4C22×C414yes—
175C7×C22⋊C4C22×C414yes—
176C42×C7C22×C412yes—
177C42×C7C22×C411yescharacteristic
178C42×C7C22×C411yescharacteristic
179C42×C7C2412yes—
180C42×C7C2412yes—
181C22⋊Dic7C22×C418yes—
182C22⋊Dic7C22×C414yes—
183(C2×D7)⋊C4C22×C418yes—
184(C2×D7)⋊C4C22×C418yes—
185(C2×D7)⋊C4C22×C418yes—
186C4⋊Dic7C22×C414yes—
187C4⋊Dic7C22×C414yes—
188C4⋊Dic7C22×C414yes—
189Dic7⋊C4C22×C418yes—
190Dic7⋊C4C22×C418yes—
191Dic7⋊C4C22×C418yes—
192C4×Dic7C22×C418yes—
193C4×Dic7C2414yes—
194#1136—28no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—24no—
198C2×C7⋊C8—24no—
199C2×C7⋊C8#4114yes—
200C2×C7⋊C8#4114yes—
201C2×C7⋊C8C2412yes—
202C2×C7⋊C8C2412yes—
203C2×C7⋊C8C2411yescharacteristic
204C2×C7⋊C8C2412yes—
205C2×C7⋊C8C2411yescharacteristic
206#1131—28no—
207#1131—28no—
208C8×D7—28no—
209C8×D7—28no—
210C2×#131—72no—
211C2×#131—72no—
212C22×M16—71no—
213C23×C8—71no—
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—74no—
229#436—78no—
230#435—78no—
231#435—78no—
232#435—78no—
233C8×D4—74no—
234C8×D4—78no—
235#433—78no—
236#433C2×D718yes—
237#432—78no—
238#424—72no—
239#424—72no—
240#423C2×D711yescharacteristic
241#423C2×D711yescharacteristic
242#408—72no—
243#408—72no—
244#407C2×D712yes—
245#405—72no—
246#405—72no—
247#405—74no—
248#405—74no—
249#405—74no—
250#405—74no—
251C4×M16—74no—
252C4×M16C2×D712yes—
253C2×#97—71no—
254C2×#97—71no—
255C2×C4×C8—71no—
256C2×C4×C8—71no—
257C2×C4×C8—74no—
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—24no—
289C7×C8—24no—
290C7⋊C8—22no—
291C7⋊C8—22no—
292C7⋊C8C2×#4112yes—
293C7⋊C8C2×#4114yes—
294C7⋊C8C2×#4112yes—
295C2×#41—72no—
296C23×C4C22×D711yescharacteristic
297#131—148no—
298#131—148no—
299C2×M16—74no—
300C2×M16—74no—
301C2×M16—74no—
302C2×M16—144no—
303C2×M16C22×D712yes—
304C2×M16C22×D712yes—
305C22×C8—71no—
306C22×C8—71no—
307C22×C8—71no—
308C22×C8—72no—
309C22×C8—72no—
310C22×C8—71no—
311C22×C8—74no—
312C22×C8—74no—
313C22×C8—144no—
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⋊C8C22×D714yes—
352C4⋊C8C22×D714yes—
353C22⋊C8—78no—
354C22⋊C8—78no—
355C22⋊C8C22×D718yes—
356#97—74no—
357#97—74no—
358#97—74no—
359#97—74no—
360#97C22×D714yes—
361C4×C8—74no—
362C4×C8—74no—
363C4×C8—72no—
364C4×C8—72no—
365C4×C8—72no—
366C4×C8—72no—
367C4×C8C22×D714yes—
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—24no—
405M16—24no—
406M16—148no—
407M16—148no—
408M16—148no—
409C2×C8—74no—
410C2×C8—74no—
411C2×C8—71no—
412C2×C8—72no—
413C2×C8—71no—
414C2×C8—72no—
415C2×C8—72no—
416C2×C8—148no—
417C2×C8—148no—
418C2×C8—144no—
419C2×C8—144no—
420C2×C8—142no—
421C2×C8—142no—
422C2×C8C23×D714yes—
423C2×C8C23×D714yes—
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×C4D7×#4112yes—
479C2×C4D7×#4112yes—
480C2×C4C24×D711yescharacteristic
481C8—24no—
482C8—24no—
483C8—72no—
484C8—74no—
485C8—72no—
486C8—142no—
487C8—142no—
488C7#6020611yescharacteristic, complemented, Sylow
489C22—22no—
490C22—22no—
491C22—24no—
492C22—44no—
493C22—144no—
494C22—148no—
495C22#9021312yes—
496C22C23×C4×D711yescharacteristic
497C4—22no—
498C4—24no—
499C4—22no—
500C4—22no—
501C4—144no—
502C4—148no—
503C4#9021312yes—
504C4C2×D7×#4111yescharacteristic
505C4C2×D7×#4111yescharacteristic
506C2—22no—
507C2—42no—
508C2—284no—
509C2C2×#9021311yescharacteristic
510C2#1174423412yes—
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2921C2
prime radical#11742806C2
cosocleC2×C4×C7C25
hypercenter#2921D7
hypocenterC7#60206
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. C7×#2921C2semidirect
3. C2×#89769C2semidirect
4. #11736812C2semidirect
5. #11736811C2semidirect
6. #11736490C2semidirect
7. #11736489C2semidirect
8. #11736487C2semidirect
9. #11736486C2semidirect
10. #11736072C2semidirect
11. #11736071C2semidirect
12. #11735965C2semidirect
13. C4×#6470C2semidirect
46. #89610C22semidirect
47. #89608C22semidirect
48. #89606C22semidirect
49. #89603C22semidirect
50. #89602C22semidirect
51. #89600C22semidirect
56. #89480C22semidirect
57. #89480C22semidirect
58. #89477C22semidirect
59. #89462C22semidirect
60. C4×#1131C22semidirect
61. C4×C8×D7C22semidirect
146. #2921D7semidirect
488. C7#60206semidirect

Maximal subgroups

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

subgroupindex
#117428062
C7×#29212
C2×#897692
#117368122
#117368112
#117364902
#117364892
#117364872
#117364862
#117360722
#117360712
#117359652
C4×#64702
#602067