Finite Group Databasebeta

← #14218330

Subgroups — #14218330

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
1C7×#3455C211yesmaximal, characteristic
2#11740774C211yesmaximal, characteristic, complemented
3#11740773C211yesmaximal, characteristic, complemented
4#11740721C211yesmaximal, characteristic, complemented
5#11740720C211yesmaximal, characteristic, complemented
6#11740572C218yesmaximal, complemented
7#11740571C218yesmaximal, complemented
8#11740362C212yesmaximal, complemented
9#11740361C212yesmaximal, complemented
10#11739959C212yesmaximal, complemented
11C4×#6511C212yesmaximal, complemented
12#11739654C211yesmaximal, characteristic, complemented
13C2×#89769C211yesmaximal, characteristic, complemented
14C7×#542C414yes—
15C7×#541C412yes—
16C7×#540C412yes—
17C7×#539C414yes—
18C7×#538C414yes—
19C7×#533C2211yescharacteristic
20C7×#532C2211yescharacteristic
21C7×#529C2218yes—
22C7×#525C2212yes—
23C4×C7×#41C2212yes—
24C2×C7×#117C2211yescharacteristic
25C2×#6511C2212yes—
26C2×#6511C2212yes—
27C22×#1136C2211yescharacteristic
28C23×C7⋊C8C2211yescharacteristic
29#90006C2212yes—
30#90006C2212yes—
31#89979C2212yes—
32#89796C2214yescomplemented
33#89796C2218yescomplemented
34Q8×C7⋊C8C2214yescomplemented
35#89784C2218yes—
36#89784C2214yescomplemented
37#89783C2218yes—
38#89783C2218yes—
39#89783C2218yescomplemented
40D4×C7⋊C8C2218yes—
41D4×C7⋊C8C2214yescomplemented
42#89772C2218yes—
43#89772C2218yescomplemented
44#89771C2212yes—
45#89771C2212yes—
46#89769C2212yes—
47#89769C2212yes—
48#89769C2214yes—
49#89769C2214yes—
50#89769C2214yescomplemented
51#89769C2214yescomplemented
52#89698C2218yescomplemented
53#89696C2211yescharacteristic
54#89696C2211yescharacteristic
55C4×#1136C2212yes—
56C4×#1136C2214yescomplemented
57C2×#6377C2211yescharacteristic
58C2×#6377C2211yescharacteristic
59C2×C4×C7⋊C8C2211yescharacteristic
60C2×C4×C7⋊C8C2211yescharacteristic
61C2×C4×C7⋊C8C2214yescomplemented
62#60206—71nomaximal, Sylow
63C2×C7×#41C2312yes—
64C23×C4×C7C2311yescharacteristic
65C7×C4⋊Q8C2×C411yescharacteristic
66C7×C4⋊Q8C2×C412yes—
67C7×#127C2×C411yescharacteristic
68C7×#126C2×C418yes—
69C7×#126C2×C418yes—
70C7×#125C2×C418yes—
71C7×#125C2×C412yes—
72C7×#124C2×C418yes—
73C7×#124C2×C412yes—
74C7×#123C2×C414yes—
75C7×#123C2×C418yes—
76C7×#123C2×C418yes—
77C7×C22⋊Q8C2×C414yes—
78C7×C22⋊Q8C2×C418yes—
79C7×C22⋊Q8C2×C414yes—
80C7×#121C2×C414yes—
81C7×#121C2×C418yes—
82C7×#121C2×C414yes—
83C7×#120C2×C414yes—
84C4×C7×Q8C2×C414yes—
85C4×C7×Q8C2314yes—
86C4×C7×D4C2×C418yes—
87C4×C7×D4C2×C414yes—
88C4×C7×D4C2318yes—
89C4×C7×D4C2314yes—
90C7×#117C2×C412yes—
91C7×#117C2×C412yes—
92C7×#117C2314yes—
93C7×#117C2312yes—
94C7×#117C2314yes—
95C2×C7×C4⋊C4C2×C412yes—
96C2×C7×C22⋊C4C2×C412yes—
97C2×C42×C7C2311yescharacteristic
98C2×C42×C7C2311yescharacteristic
99C2×C42×C7C2314yes—
100#6511—28no—
101#6511—28no—
102C2×#1136—24no—
103C2×#1136C2312yes—
104C2×#1136C2312yes—
105C2×#1136C2314yes—
106C2×#1136C2314yes—
107C2×#1136C2314yes—
108C22×C7⋊C8—24no—
109C22×C7⋊C8C2311yescharacteristic
110C22×C7⋊C8C2314yes—
111C22×C7⋊C8C2314yes—
112C22×C7⋊C8C2311yescharacteristic
113C22×C7⋊C8C2312yes—
114C22×C7⋊C8C2312yes—
115C22×C7⋊C8C2311yescharacteristic
116C22×C7⋊C8C2311yescharacteristic
117C22⋊(C7⋊C8)C2318yes—
118C22⋊(C7⋊C8)C2318yes—
119C22⋊(C7⋊C8)C2318yes—
120C4⋊(C7⋊C8)C2314yes—
121C4⋊(C7⋊C8)C2314yes—
122C4⋊(C7⋊C8)C2318yes—
123C4⋊(C7⋊C8)C2318yes—
124#6377C2314yes—
125#6377C2314yes—
126#6377C2314yes—
127#6377C2314yes—
128#6377C2314yes—
129C4×C7⋊C8C2312yes—
130C4×C7⋊C8C2312yes—
131C4×C7⋊C8C2312yes—
132C4×C7⋊C8C2312yes—
133C4×C7⋊C8C2314yes—
134C4×C7⋊C8C2314yes—
135C4×C7⋊C8C2314yes—
136#3455D711yescharacteristic
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×#405—71no—
149C24×C7C22×C411yescharacteristic
150C7×#41—24no—
151C7×#41—24no—
152C2×C7×Q8C22×C412yes—
153C2×C7×Q8C22×C414yes—
154C2×C7×D4C22×C414yes—
155C2×C7×D4C22×C412yes—
156C2×C7×D4C22×C418yes—
157C2×C7×D4C22×C414yes—
158C22×C4×C7—22no—
159C22×C4×C7—22no—
160C22×C4×C7C22×C411yescharacteristic
161C22×C4×C7C22×C412yes—
162C22×C4×C7C22×C412yes—
163C22×C4×C7C22×C412yes—
164C22×C4×C7C22×C418yes—
165C22×C4×C7C2414yes—
166C22×C4×C7C2412yes—
167C22×C4×C7C2411yescharacteristic
168C7×C4⋊C4C22×C414yes—
169C7×C4⋊C4C22×C414yes—
170C7×C4⋊C4C22×C414yes—
171C7×C4⋊C4C22×C418yes—
172C7×C4⋊C4C22×C414yes—
173C7×C4⋊C4C22×C418yes—
174C7×C4⋊C4C22×C418yes—
175C7×C4⋊C4C22×C414yes—
176C7×C22⋊C4C22×C414yes—
177C7×C22⋊C4C22×C414yes—
178C7×C22⋊C4C22×C414yes—
179C7×C22⋊C4C22×C418yes—
180C7×C22⋊C4C22×C418yes—
181C7×C22⋊C4C22×C418yes—
182C7×C22⋊C4C22×C418yes—
183C42×C7C22×C412yes—
184C42×C7C22×C411yescharacteristic
185C42×C7C22×C411yescharacteristic
186C42×C7C22×C418yes—
187C42×C7C2412yes—
188C42×C7C2412yes—
189C42×C7C2414yes—
190#1136—24no—
191#1136—24no—
192#1136—28no—
193#1136—28no—
194#1136—28no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197C2×C7⋊C8—24no—
198C2×C7⋊C8—24no—
199C2×C7⋊C8—28no—
200C2×C7⋊C8—28no—
201C2×C7⋊C8#4114yes—
202C2×C7⋊C8#4114yes—
203C2×C7⋊C8C2412yes—
204C2×C7⋊C8C2414yes—
205C2×C7⋊C8C2412yes—
206C2×C7⋊C8C2414yes—
207C2×C7⋊C8C2411yescharacteristic
208C2×C7⋊C8C2412yes—
209C2×C7⋊C8C2411yescharacteristic
210C2×#131—72no—
211C2×#131—72no—
212C22×M16—71no—
213C23×C8—71no—
214#542Dic714yes—
215#541Dic712yes—
216#540Dic712yes—
217#539Dic714yes—
218#538Dic714yes—
219#533C2×D711yescharacteristic
220#532C2×D711yescharacteristic
221#529C2×D718yes—
222#525C2×D712yes—
223C4×#41C2×D712yes—
224C2×#117C2×D711yescharacteristic
225#446—74no—
226#446—78no—
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—74no—
246#405—74no—
247#405—72no—
248#405—72no—
249#405—74no—
250#405—74no—
251C4×M16—72no—
252C4×M16—74no—
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×C414yes—
262C23×C7C23×C411yescharacteristic
263C7×Q8—24no—
264C7×Q8—44no—
265C7×D4—28no—
266C7×D4—24no—
267C7×D4—44no—
268C7×D4—48no—
269C2×C4×C7—22no—
270C2×C4×C7—24no—
271C2×C4×C7—22no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—22no—
275C2×C4×C7—42no—
276C2×C4×C7—48no—
277C2×C4×C7—42no—
278C2×C4×C7C23×C414yes—
279C2×C4×C7C23×C414yes—
280C2×C4×C7C23×C412yes—
281C2×C4×C7C23×C418yes—
282C2×C4×C7C23×C412yes—
283C2×C4×C7C23×C412yes—
284C2×C4×C7C23×C411yescharacteristic
285C2×C4×C7C2×#4112yes—
286C2×C4×C7C2×#4112yes—
287C2×C4×C7C2511yescharacteristic
288C7⋊C8—22no—
289C7⋊C8—22no—
290C7⋊C8—24no—
291C7⋊C8—24no—
292C7⋊C8C2×#4112yes—
293C7⋊C8C2×#4112yes—
294C7⋊C8C2×#4114yes—
295C2×#41C22×D712yes—
296C23×C4C22×D711yescharacteristic
297#131—148no—
298#131—148no—
299C2×M16—74no—
300C2×M16—72no—
301C2×M16—72no—
302C2×M16—74no—
303C2×M16—74no—
304C2×M16—144no—
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⋊Q8C2×Dic711yescharacteristic
315C4⋊Q8C2×Dic712yes—
316#127C2×Dic711yescharacteristic
317#126C2×Dic718yes—
318#126C2×Dic718yes—
319#125C2×Dic718yes—
320#125C2×Dic712yes—
321#124C2×Dic718yes—
322#124C2×Dic712yes—
323#123C2×Dic714yes—
324#123C2×Dic718yes—
325#123C2×Dic718yes—
326C22⋊Q8C2×Dic714yes—
327C22⋊Q8C2×Dic718yes—
328C22⋊Q8C2×Dic714yes—
329#121C2×Dic714yes—
330#121C2×Dic718yes—
331#121C2×Dic714yes—
332#120C2×Dic714yes—
333C4×Q8C2×Dic714yes—
334C4×Q8C22×D714yes—
335C4×D4C2×Dic718yes—
336C4×D4C2×Dic714yes—
337C4×D4C22×D718yes—
338C4×D4C22×D714yes—
339#117C2×Dic712yes—
340#117C2×Dic712yes—
341#117C22×D714yes—
342#117C22×D712yes—
343#117C22×D714yes—
344C2×C4⋊C4C2×Dic712yes—
345C2×C22⋊C4C2×Dic712yes—
346C2×C42C22×D711yescharacteristic
347C2×C42C22×D711yescharacteristic
348C2×C42C22×D714yes—
349C4⋊C8—74no—
350C4⋊C8—74no—
351C4⋊C8—78no—
352C4⋊C8—78no—
353C22⋊C8—78no—
354C22⋊C8—78no—
355C22⋊C8—78no—
356#97—74no—
357#97—74no—
358#97—74no—
359#97—74no—
360#97—74no—
361C4×C8—72no—
362C4×C8—72no—
363C4×C8—74no—
364C4×C8—72no—
365C4×C8—74no—
366C4×C8—72no—
367C4×C8—74no—
368C22×C7—22no—
369C22×C7—22no—
370C22×C7—24no—
371C22×C7—28no—
372C22×C7—24no—
373C22×C7—44no—
374C22×C7C4×#4112yes—
375C22×C7C24×C411yescharacteristic
376C4×C7—22no—
377C4×C7—24no—
378C4×C7—24no—
379C4×C7—22no—
380C4×C7—28no—
381C4×C7—22no—
382C4×C7C4×#4112yes—
383C4×C7C22×#4111yescharacteristic
384C4×C7C22×#4111yescharacteristic
385C24C22×Dic711yescharacteristic
386#41—24no—
387#41—24no—
388C2×Q8C22×Dic712yes—
389C2×Q8C22×Dic714yes—
390C2×D4C22×Dic714yes—
391C2×D4C22×Dic712yes—
392C2×D4C22×Dic718yes—
393C2×D4C22×Dic714yes—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4C22×Dic711yescharacteristic
397C22×C4C22×Dic712yes—
398C22×C4C22×Dic712yes—
399C22×C4C22×Dic712yes—
400C22×C4C22×Dic718yes—
401C22×C4C23×D714yes—
402C22×C4C23×D712yes—
403C22×C4C23×D711yescharacteristic
404M16—148no—
405M16—148no—
406M16—148no—
407M16—144no—
408M16—144no—
409C2×C8—74no—
410C2×C8—74no—
411C2×C8—71no—
412C2×C8—72no—
413C2×C8—71no—
414C2×C8—74no—
415C2×C8—74no—
416C2×C8—72no—
417C2×C8—72no—
418C2×C8—144no—
419C2×C8—144no—
420C2×C8—148no—
421C2×C8—148no—
422C2×C8—142no—
423C2×C8—142no—
424C4⋊C4C22×Dic714yes—
425C4⋊C4C22×Dic714yes—
426C4⋊C4C22×Dic714yes—
427C4⋊C4C22×Dic718yes—
428C4⋊C4C22×Dic714yes—
429C4⋊C4C22×Dic718yes—
430C4⋊C4C22×Dic718yes—
431C4⋊C4C22×Dic714yes—
432C22⋊C4C22×Dic714yes—
433C22⋊C4C22×Dic714yes—
434C22⋊C4C22×Dic714yes—
435C22⋊C4C22×Dic718yes—
436C22⋊C4C22×Dic718yes—
437C22⋊C4C22×Dic718yes—
438C22⋊C4C22×Dic718yes—
439C42C22×Dic712yes—
440C42C22×Dic711yescharacteristic
441C42C22×Dic711yescharacteristic
442C42C22×Dic718yes—
443C42C23×D712yes—
444C42C23×D712yes—
445C42C23×D714yes—
446C2×C7—22no—
447C2×C7—44no—
448C2×C7—42no—
449C2×C7C2×C4×#4111yescharacteristic
450C2×C7#357312yes—
451C23—22no—
452C23—42no—
453C23C23×Dic712yes—
454C23C23×Dic714yes—
455C23C23×Dic711yescharacteristic
456Q8—24no—
457Q8—44no—
458D4—28no—
459D4—24no—
460D4—44no—
461D4—48no—
462C2×C4—22no—
463C2×C4—24no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—42no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4D7×#4112yes—
472C2×C4D7×#4112yes—
473C2×C4C23×Dic714yes—
474C2×C4C23×Dic714yes—
475C2×C4C23×Dic712yes—
476C2×C4C23×Dic718yes—
477C2×C4C23×Dic712yes—
478C2×C4C23×Dic712yes—
479C2×C4C23×Dic711yescharacteristic
480C2×C4C24×D711yescharacteristic
481C8—72no—
482C8—72no—
483C8—74no—
484C8—142no—
485C8—144no—
486C8—144no—
487C8—142no—
488C7#6020611yescharacteristic, complemented, Sylow
489C22—22no—
490C22—22no—
491C22—24no—
492C22—28no—
493C22—24no—
494C22—44no—
495C22#41×Dic712yes—
496C22C24×Dic711yescharacteristic
497C4—22no—
498C4—24no—
499C4—24no—
500C4—22no—
501C4—28no—
502C4—22no—
503C4#41×Dic712yes—
504C4C2×D7×#4111yescharacteristic
505C4C2×D7×#4111yescharacteristic
506C2—22no—
507C2—42no—
508C2—44no—
509C2C2×#41×Dic711yescharacteristic
510C2#1174436412yes—
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
Fitting, prime radicalC7×#3455C2
cosocleC2×C4×C7C25
hypercenter#3455D7
hypocenterC7#60206
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #3455

Complements

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

NH ≅ G/Ntype
2. #11740774C2semidirect
3. #11740773C2semidirect
4. #11740721C2semidirect
5. #11740720C2semidirect
6. #11740572C2semidirect
7. #11740571C2semidirect
8. #11740362C2semidirect
9. #11740361C2semidirect
10. #11739959C2semidirect
11. C4×#6511C2semidirect
12. #11739654C2semidirect
13. C2×#89769C2semidirect
32. #89796C22semidirect
33. #89796C22semidirect
34. Q8×C7⋊C8C22semidirect
36. #89784C22semidirect
39. #89783C22semidirect
41. D4×C7⋊C8C22semidirect
43. #89772C22semidirect
50. #89769C22semidirect
51. #89769C22semidirect
52. #89698C22semidirect
56. C4×#1136C22semidirect
61. C2×C4×C7⋊C8C22semidirect
488. C7#60206semidirect

Maximal subgroups

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

subgroupindex
C7×#34552
#117407742
#117407732
#117407212
#117407202
#117405722
#117405712
#117403622
#117403612
#117399592
C4×#65112
#117396542
C2×#897692
#602067