Finite Group Databasebeta

← #14218332

Subgroups — #14218332

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#11743304C211yesmaximal, characteristic
2C7×#2932C211yesmaximal, characteristic, complemented
3#11741240C212yesmaximal, complemented
4Dic7×#131C212yesmaximal, complemented
5#11740720C211yesmaximal, characteristic, complemented
6#11736900C218yesmaximal, complemented
7#11736899C218yesmaximal, complemented
8#11736489C211yesmaximal, characteristic, complemented
9#11736484C211yesmaximal, characteristic, complemented
10#11736100C212yesmaximal, complemented
11#11736094C212yesmaximal, complemented
12C2×#89480C211yesmaximal, characteristic, complemented
13#11735925C211yesmaximal, characteristic, complemented
14C2×C7×#131C2212yes—
15C7×#532C2211yescharacteristic
16#90519C2212yes—
17#41×Dic7C2212yes—
18C2×#6511C2212yes—
19C22×#1131C2211yescharacteristic
20C22×C8×D7C2211yescharacteristic
21#90422C412yes—
22#90410C412yes—
23#90373C414yes—
24#90308C414yes—
25#90293C414yes—
26#90229C2218yes—
27C2×#6435C2211yescharacteristic
28#90161C2211yescharacteristic
29C7×#435C2218yescomplemented
30C7×#423C2211yescharacteristic
31C7×#408C2212yes—
32C2×C7×#97C2211yescharacteristic
33#90006C2212yes—
34#89900C2218yes—
35#89898C2212yes—
36#89894C2212yes—
37#89894C2214yescomplemented
38#89891C2212yes—
39M16×Dic7C2212yes—
40M16×Dic7C2214yescomplemented
41#89883C2218yes—
42#89874C2212yes—
43#89873C2212yes—
44#89873C2214yescomplemented
45C2×#6389C2211yescharacteristic
46C2×C8×Dic7C2211yescharacteristic
47C2×C8×Dic7C2214yescomplemented
48#89784C2214yescomplemented
49D4×C7⋊C8C2214yescomplemented
50#89696C2211yescharacteristic
51C2×C4×C7⋊C8C2211yescharacteristic
52#89612C2218yes—
53#89603C2214yescomplemented
54#89602C2214yescomplemented
55#89500C2218yes—
56(C22⋊D7)⋊C8C2218yes—
57#89492C2218yescomplemented
58#89491C2218yescomplemented
59#89480C2214yes—
60#89480C2214yes—
61#89473C2218yescomplemented
62#60206—71nomaximal, Sylow
63C2×C7×#41C2312yes—
64C22×C4×D7C2311yescharacteristic
65C7×#131—28no—
66C2×C7×M16C2312yes—
67C2×C7×M16C2314yes—
68C22×C7×C8C2314yes—
69C22×C7×C8C2311yescharacteristic
70C22×C7×C8C2311yescharacteristic
71C7×C4⋊Q8C2×C411yescharacteristic
72C7×#127C2×C411yescharacteristic
73C7×#124C2×C412yes—
74C7×#121C2×C414yes—
75C4×C7×D4C2314yes—
76C2×C42×C7C2311yescharacteristic
77#6511—28no—
78#6509C2×C414yes—
79Q8×Dic7C2×C414yes—
80#6502C2×C418yes—
81#6501C2×C414yes—
82#6500C2×C414yes—
83#6498C2×C418yes—
84#6498C2×C414yes—
85D4×Dic7C2×C414yes—
86D4×Dic7C2×C418yes—
87C4×C22⋊D7C2318yes—
88C2×(C2×D7)⋊C4C2×C412yes—
89#6489C2312yes—
90#6489C2314yes—
91#6487C2×C414yes—
92C2×Dic7⋊C4C2×C412yes—
93C2×C4×Dic7C2311yescharacteristic
94C2×C4×Dic7C2314yes—
95C2×#1136C2312yes—
96C2×#1136C2314yes—
97C22×C7⋊C8C2314yes—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C2×#1131—24no—
101C2×#1131C2314yes—
102C2×C8×D7—24no—
103C2×C8×D7C2312yes—
104C2×C8×D7C2312yes—
105#6461C2×C418yes—
106#6452C2×C418yes—
107#6449C2×C418yes—
108#6448C2×C418yes—
109#6442C2×C418yes—
110#6441C2×C418yes—
111#6435C2×C412yes—
112#6435C2×C412yes—
113#6435C2314yes—
114#6433C2×C412yes—
115#6432C2×C412yes—
116C4×Dic14C2314yes—
117C7×C4⋊C8C2314yes—
118C7×C22⋊C8C2318yes—
119C7×#97C2314yes—
120C22⋊(C7⋊C8)C2318yes—
121(C2×D7)⋊C8C2318yes—
122#6389C2314yes—
123#6389C2314yes—
124#6389C2314yes—
125#6389C2314yes—
126Dic7⋊C8C2318yes—
127Dic7⋊C8C2318yes—
128C8×Dic7C2314yes—
129C8×Dic7C2314yes—
130C8×Dic7C2312yes—
131C8×Dic7C2312yes—
132C8×Dic7C2314yes—
133C4⋊(C7⋊C8)C2314yes—
134C4×C7⋊C8C2312yes—
135C4×C7⋊C8C2312yes—
136#3455—71no—
137#2967—78no—
138#2966—78no—
139#2961—71no—
140#2960—71no—
141#2932D711yescharacteristic, complemented
142#2931—71no—
143#2930—72no—
144#2929—72no—
145#2923—72no—
146#2921—71no—
147C4×#131—72no—
148C2×#405—71no—
149C23×D7C22×C411yescharacteristic
150C7×#41—24no—
151C7×#41—24no—
152C2×C7×Q8C22×C412yes—
153C2×C7×D4C22×C414yes—
154C2×C7×D4C22×C412yes—
155C2×C7×D4C22×C414yes—
156C22×C4×C7C22×C412yes—
157C22×C4×C7C2414yes—
158C22×C4×C7C2411yescharacteristic
159C2×C22⋊D7C22×C418yes—
160C22×Dic7C22×C411yescharacteristic
161C22×Dic7C22×C418yes—
162C22×Dic7C22×C412yes—
163C2×C4×D7—22no—
164C2×C4×D7—22no—
165C2×C4×D7C22×C412yes—
166C2×C4×D7C2412yes—
167C2×Dic14C22×C414yes—
168C7×M16—24no—
169C7×M16—28no—
170C2×C7×C8—22no—
171C2×C7×C8—28no—
172C2×C7×C8#4114yes—
173C2×C7×C8C2412yes—
174C2×C7×C8C2414yes—
175C2×C7×C8C2412yes—
176C7×C4⋊C4C22×C414yes—
177C7×C22⋊C4C22×C418yes—
178C42×C7C22×C411yescharacteristic
179C42×C7C22×C411yescharacteristic
180C42×C7C2412yes—
181C22⋊Dic7C22×C418yes—
182C22⋊Dic7C22×C418yes—
183C22⋊Dic7C22×C414yes—
184(C2×D7)⋊C4C22×C414yes—
185(C2×D7)⋊C4C22×C414yes—
186(C2×D7)⋊C4C22×C418yes—
187C4⋊Dic7C22×C414yes—
188C4⋊Dic7C22×C414yes—
189C4⋊Dic7C22×C418yes—
190Dic7⋊C4C22×C418yes—
191Dic7⋊C4C22×C414yes—
192Dic7⋊C4C22×C414yes—
193Dic7⋊C4C22×C418yes—
194C4×Dic7C22×C412yes—
195C4×Dic7C22×C418yes—
196C4×Dic7C2414yes—
197C4×Dic7C2412yes—
198#1136—24no—
199#1136—28no—
200C2×C7⋊C8—22no—
201C2×C7⋊C8—28no—
202C2×C7⋊C8#4114yes—
203C2×C7⋊C8C2412yes—
204C2×C7⋊C8C2414yes—
205C2×C7⋊C8C2411yescharacteristic
206C2×C7⋊C8C2411yescharacteristic
207#1131—28no—
208C8×D7—24no—
209C8×D7—24no—
210C2×#131—72no—
211C2×#131C2×D712yes—
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#532C2×D711yescharacteristic
221#529—78no—
222#525—72no—
223C4×#41—72no—
224C2×#117—71no—
225#446—74no—
226#446—78no—
227C8×Q8—74no—
228#436—78no—
229#436—74no—
230#435—78no—
231#435—78no—
232#435C2×D718yescomplemented
233C8×D4—74no—
234C8×D4—78no—
235#433—78no—
236#433—78no—
237#432—78no—
238#424—72no—
239#424—72no—
240#423—71no—
241#423C2×D711yescharacteristic
242#408—72no—
243#408C2×D712yes—
244#407—72no—
245#405—72no—
246#405—74no—
247#405—72no—
248#405—74no—
249#405—74no—
250#405—74no—
251C4×M16—72no—
252C4×M16—74no—
253C2×#97—71no—
254C2×#97C2×D711yescharacteristic
255C2×C4×C8—71no—
256C2×C4×C8—74no—
257C2×C4×C8—71no—
258C23×C7C23×C414yes—
259C23×C7C23×C411yescharacteristic
260C22×D7—22no—
261C22×D7—42no—
262C22×D7C23×C412yes—
263C7×Q8—24no—
264C7×D4—28no—
265C7×D4—24no—
266C7×D4—44no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—42no—
270C2×C4×C7C23×C414yes—
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—48no—
278C2×Dic7—22no—
279C2×Dic7—48no—
280C2×Dic7—42no—
281C2×Dic7C23×C414yes—
282C2×Dic7C23×C418yes—
283C2×Dic7C23×C412yes—
284C4×D7—24no—
285C4×D7—22no—
286C4×D7—22no—
287Dic14—44no—
288C7×C8—22no—
289C7×C8—24no—
290C7×C8C2×#4114yes—
291C7⋊C8—22no—
292C7⋊C8—24no—
293C7⋊C8C2×#4112yes—
294C7⋊C8C2×#4112yes—
295C2×#41C22×D712yes—
296C23×C4—71no—
297#131—28no—
298#131—148no—
299C2×M16—72no—
300C2×M16—74no—
301C2×M16—74no—
302C2×M16—144no—
303C2×M16C22×D712yes—
304C2×M16C22×D714yes—
305C22×C8—71no—
306C22×C8—71no—
307C22×C8—74no—
308C22×C8—72no—
309C22×C8—72no—
310C22×C8—144no—
311C22×C8C22×D711yescharacteristic
312C22×C8C22×D714yes—
313C22×C8C22×D711yescharacteristic
314C4⋊Q8—72no—
315C4⋊Q8C4×D711yescharacteristic
316#127C4×D711yescharacteristic
317#126—78no—
318#126—78no—
319#125—78no—
320#125—72no—
321#124—78no—
322#124C4×D712yes—
323#123—74no—
324#123—78no—
325#123—78no—
326C22⋊Q8—74no—
327C22⋊Q8—74no—
328C22⋊Q8—78no—
329#121—74no—
330#121—78no—
331#121C4×D714yes—
332#120—74no—
333C4×Q8—74no—
334C4×Q8—74no—
335C4×D4—78no—
336C4×D4—74no—
337C4×D4—78no—
338C4×D4C22×D714yes—
339#117—72no—
340#117—74no—
341#117—72no—
342#117—72no—
343#117—74no—
344C2×C4⋊C4—72no—
345C2×C22⋊C4—72no—
346C2×C42—71no—
347C2×C42—74no—
348C2×C42C22×D711yescharacteristic
349C4⋊C8—78no—
350C4⋊C8—78no—
351C4⋊C8—74no—
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—72no—
362C4×C8—72no—
363C4×C8—74no—
364C4×C8—74no—
365C4×C8—74no—
366C4×C8—72no—
367C4×C8—72no—
368C22×C7—22no—
369C22×C7—24no—
370C22×C7—28no—
371C22×C7C4×#4112yes—
372C22×C7C24×C411yescharacteristic
373C2×D7—22no—
374C2×D7—24no—
375C2×D7—44no—
376C4×C7—24no—
377C4×C7—22no—
378C4×C7—22no—
379C4×C7C4×#4112yes—
380C4×C7C22×#4111yescharacteristic
381C4×C7C22×#4111yescharacteristic
382Dic7—22no—
383Dic7—28no—
384Dic7—24no—
385C24—71no—
386#41—24no—
387#41—24no—
388C2×Q8—74no—
389C2×Q8C2×C4×D712yes—
390C2×D4—78no—
391C2×D4C2×C4×D714yes—
392C2×D4C2×C4×D712yes—
393C2×D4C2×C4×D714yes—
394C22×C4—71no—
395C22×C4—72no—
396C22×C4—78no—
397C22×C4—72no—
398C22×C4—72no—
399C22×C4—142no—
400C22×C4—142no—
401C22×C4C2×C4×D712yes—
402C22×C4C23×D714yes—
403C22×C4C23×D711yescharacteristic
404M16—24no—
405M16—28no—
406M16—148no—
407M16—144no—
408M16—148no—
409C2×C8—22no—
410C2×C8—28no—
411C2×C8—74no—
412C2×C8—71no—
413C2×C8—71no—
414C2×C8—72no—
415C2×C8—74no—
416C2×C8—144no—
417C2×C8—144no—
418C2×C8—148no—
419C2×C8—142no—
420C2×C8#115914yes—
421C2×C8C23×D714yes—
422C2×C8C23×D712yes—
423C2×C8C23×D712yes—
424C4⋊C4—74no—
425C4⋊C4—74no—
426C4⋊C4—78no—
427C4⋊C4—78no—
428C4⋊C4—74no—
429C4⋊C4—78no—
430C4⋊C4—74no—
431C4⋊C4C2×C4×D714yes—
432C22⋊C4—74no—
433C22⋊C4—78no—
434C22⋊C4—78no—
435C22⋊C4—74no—
436C22⋊C4—74no—
437C22⋊C4—78no—
438C22⋊C4C2×C4×D718yes—
439C42—72no—
440C42—78no—
441C42—72no—
442C42—74no—
443C42C2×C4×D711yescharacteristic
444C42C2×C4×D711yescharacteristic
445C42C23×D712yes—
446C2×C7—22no—
447C2×C7—44no—
448C2×C7C2×C4×#4111yescharacteristic
449C2×C7#357312yes—
450D7—42no—
451C23—72no—
452C23—142no—
453C23—282no—
454C23C22×C4×D714yes—
455C23C22×C4×D711yescharacteristic
456Q8—24no—
457Q8—284no—
458D4—28no—
459D4—24no—
460D4—44no—
461D4—288no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—74no—
466C2×C4—78no—
467C2×C4—72no—
468C2×C4—142no—
469C2×C4—144no—
470C2×C4—142no—
471C2×C4—142no—
472C2×C4—282no—
473C2×C4—288no—
474C2×C4C22×C4×D714yes—
475C2×C4C22×C4×D712yes—
476C2×C4C22×C4×D712yes—
477C2×C4C22×C4×D711yescharacteristic
478C2×C4C2×#115912yes—
479C2×C4D7×#4112yes—
480C2×C4C24×D711yescharacteristic
481C8—24no—
482C8—22no—
483C8—72no—
484C8—72no—
485C8—142no—
486C8—144no—
487C8C2×#115914yes—
488C7#6020611yescharacteristic, complemented, Sylow
489C22—22no—
490C22—24no—
491C22—28no—
492C22—142no—
493C22—144no—
494C22—284no—
495C22C4×#115912yes—
496C22C23×C4×D711yescharacteristic
497C4—24no—
498C4—22no—
499C4—22no—
500C4—142no—
501C4—144no—
502C4—148no—
503C4C4×#115912yes—
504C4C22×#115911yescharacteristic
505C4C2×D7×#4111yescharacteristic
506C2—22no—
507C2—44no—
508C2—282no—
509C2C2×C4×#115911yescharacteristic
510C2#1174423412yes—
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2932C2
prime radical#11743304C2
cosocleC2×C4×C7C25
hypercenter#2932D7
hypocenterC7#60206
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2932

Complements

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

NH ≅ G/Ntype
2. C7×#2932C2semidirect
3. #11741240C2semidirect
4. Dic7×#131C2semidirect
5. #11740720C2semidirect
6. #11736900C2semidirect
7. #11736899C2semidirect
8. #11736489C2semidirect
9. #11736484C2semidirect
10. #11736100C2semidirect
11. #11736094C2semidirect
12. C2×#89480C2semidirect
13. #11735925C2semidirect
29. C7×#435C22semidirect
37. #89894C22semidirect
40. M16×Dic7C22semidirect
44. #89873C22semidirect
47. C2×C8×Dic7C22semidirect
48. #89784C22semidirect
49. D4×C7⋊C8C22semidirect
53. #89603C22semidirect
54. #89602C22semidirect
57. #89492C22semidirect
58. #89491C22semidirect
61. #89473C22semidirect
141. #2932D7semidirect
232. #435C2×D7semidirect
488. C7#60206semidirect

Maximal subgroups

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

subgroupindex
#117433042
C7×#29322
#117412402
Dic7×#1312
#117407202
#117369002
#117368992
#117364892
#117364842
#117361002
#117360942
C2×#894802
#117359252
#602067