Finite Group Databasebeta

← #13322371

Subgroups — #13322371

Order 1792.

950 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2#11736776C211yesmaximal, characteristic
3#11736776C211yesmaximal, characteristic
4C7×#1556C211yesmaximal, characteristic, complemented
5#11729530C211yesmaximal, characteristic, complemented
6#11728577C211yesmaximal, characteristic, complemented
7#11728576C211yesmaximal, characteristic, complemented
8#11726185C214yesmaximal, complemented
9#11726173C214yesmaximal, complemented
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×#6509—21no—
12C2×Q8×Dic7—21no—
13#90419—21no—
14#90419—21no—
15#90419—21no—
16#90419—21no—
17#90419C412yes—
18#90419C412yes—
19#90419C2212yes—
20#90419C2212yes—
21C2×#6460—21no—
22C2×#6455C412yes—
23C2×#6453C412yes—
24C2×#6450—21no—
25C2×C42×D7C2211yescharacteristic
26C2×#6432C2211yescharacteristic
27C7×#432C2211yescharacteristic
28C7×#432C2211yescharacteristic
29#89698C2211yescharacteristic
30#89698C2211yescharacteristic
31#89609C2212yes—
32#89609C2212yes—
33#89605C2212yes—
34#89605C2212yes—
35#89498—21no—
36#89498—21no—
37#89490—21no—
38#89490—21no—
39#89479—21no—
40#89479—21no—
41#89477—21no—
42#89477—21no—
43C7×#332C2214yescomplemented
44#89342C2214yescomplemented
45#89281C2214yescomplemented
46#89277C2214yescomplemented
47#17739—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C22×C4×D7C2×C411yescharacteristic
50C22×C4×D7D411yescharacteristic
51C22×C4×D7D411yescharacteristic
52C22×Dic14—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8C2×C412yes—
56C7×C4⋊Q8C2312yes—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4C2×C412yes—
59C2×C42×C7C2311yescharacteristic
60#6509—22no—
61#6509—22no—
62#6509—22no—
63#6509—22no—
64Q8×Dic7—22no—
65Q8×Dic7—22no—
66Q8×Dic7—22no—
67Q8×Dic7—22no—
68C2×(C2×D7)⋊C4—22no—
69C2×C4⋊Dic7—21no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7C2×C412yes—
72C2×Dic7⋊C4—22no—
73C2×C4×Dic7—22no—
74C2×C4×Dic7C2×C411yescharacteristic
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×#1131—22no—
78C2×#1131—22no—
79#6460—22no—
80#6460—22no—
81#6460—22no—
82#6460—22no—
83#6455—22no—
84#6455—22no—
85#6455C2×C414yes—
86#6455C2×C414yes—
87#6453—22no—
88#6453—22no—
89#6453C2×C414yes—
90#6453C2×C414yes—
91#6450—22no—
92#6450—22no—
93#6450—22no—
94#6450—22no—
95C42×D7C2×C411yescharacteristic
96C42×D7C2×C411yescharacteristic
97C42×D7D411yescharacteristic
98C42×D7D411yescharacteristic
99C42×D7D411yescharacteristic
100C42×D7D411yescharacteristic
101C42×D7C2311yescharacteristic
102C42×D7C2311yescharacteristic
103#6432—21no—
104#6432—21no—
105#6432C2×C412yes—
106#6432C2312yes—
107C7×C4⋊C8C2312yes—
108C7×C4⋊C8C2312yes—
109C7×C22⋊C8—21no—
110C7×C22⋊C8—21no—
111C7×#97—21no—
112C7×#97—21no—
113C22⋊(C7⋊C8)—21no—
114C22⋊(C7⋊C8)—21no—
115(C2×D7)⋊C8—22no—
116(C2×D7)⋊C8—22no—
117Dic7⋊C8—22no—
118Dic7⋊C8—22no—
119C8×Dic7—22no—
120C8×Dic7—22no—
121C4⋊(C7⋊C8)C2312yes—
122C4⋊(C7⋊C8)C2312yes—
123#6377—21no—
124#6377—21no—
125C2×#533—71no—
126#2920—71no—
127#2920—71no—
128#1561—74no—
129#1556—71no—
130#1556—71no—
131#1556—71no—
132#1556D711yescharacteristic, complemented
133#1544—74no—
134C23×D7C22⋊C411yescharacteristic
135C2×C7×Q8—22no—
136C2×C7×Q8—22no—
137C2×C7×Q8—44no—
138C22×C4×C7—22no—
139C22×C4×C7C22×C411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7C2×D411yescharacteristic
142C22×Dic7—22no—
143C22×Dic7C22⋊C411yescharacteristic
144C22×Dic7C22⋊C411yescharacteristic
145C22×Dic7C22⋊C411yescharacteristic
146C2×C4×D7—24no—
147C2×C4×D7—42no—
148C2×C4×D7—42no—
149C2×C4×D7C22⋊C411yescharacteristic
150C2×C4×D7C22⋊C411yescharacteristic
151C2×C4×D7C22⋊C411yescharacteristic
152C2×C4×D7C22⋊C411yescharacteristic
153C2×C4×D7C22⋊C411yescharacteristic
154C2×C4×D7C22⋊C411yescharacteristic
155C2×C4×D7C22×C411yescharacteristic
156C2×C4×D7C22×C411yescharacteristic
157C2×C4×D7C2×D411yescharacteristic
158C2×C4×D7C2×D411yescharacteristic
159C2×C4×D7C2×D411yescharacteristic
160C2×C4×D7C2×D411yescharacteristic
161C2×Dic14—22no—
162C2×Dic14—22no—
163C2×Dic14—44no—
164C2×C7×C8—22no—
165C2×C7×C8—22no—
166C7×C4⋊C4—22no—
167C7×C4⋊C4—22no—
168C7×C4⋊C4—22no—
169C7×C4⋊C4C22×C414yes—
170C42×C7C22×C411yescharacteristic
171C42×C7C2×D411yescharacteristic
172C42×C7C2×D411yescharacteristic
173C42×C7C2411yescharacteristic
174(C2×D7)⋊C4—24no—
175(C2×D7)⋊C4—24no—
176C4⋊Dic7—22no—
177C4⋊Dic7—22no—
178C4⋊Dic7—22no—
179C4⋊Dic7—24no—
180C4⋊Dic7—24no—
181C4⋊Dic7C22×C414yes—
182Dic7⋊C4—24no—
183Dic7⋊C4—24no—
184C4×Dic7—24no—
185C4×Dic7—24no—
186C4×Dic7C22⋊C411yescharacteristic
187C4×Dic7C22⋊C411yescharacteristic
188C4×Dic7C22⋊C411yescharacteristic
189C4×Dic7C22⋊C411yescharacteristic
190C4×Dic7C22⋊C411yescharacteristic
191C4×Dic7C22⋊C411yescharacteristic
192C4×Dic7C22×C411yescharacteristic
193C4×Dic7C22×C411yescharacteristic
194C4×Dic7C2×D411yescharacteristic
195C4×Dic7C2×D411yescharacteristic
196C4×Dic7C2×D411yescharacteristic
197C4×Dic7C2×D411yescharacteristic
198C2×C7⋊C8—22no—
199C2×C7⋊C8—22no—
200#1131—44no—
201#1131—44no—
202#533—72no—
203#533—72no—
204#533—72no—
205#533—72no—
206#533—141no—
207#533—141no—
208#533—141no—
209#533—141no—
210C2×C4⋊Q8—71no—
211C2×C4⋊Q8C2×D711yescharacteristic
212C2×#125—72no—
213C2×C22⋊Q8—141no—
214C2×C22⋊Q8—141no—
215C2×C4×Q8—141no—
216C2×C4×Q8—141no—
217C2×#117—72no—
218C22×C42—71no—
219#432—71no—
220#432—71no—
221#432—141no—
222#432—141no—
223#432C2×D711yescharacteristic
224#432C2×D711yescharacteristic
225#431—72no—
226#431—72no—
227#431—141no—
228#431—141no—
229#423—72no—
230#423—72no—
231#407—141no—
232#407—141no—
233C4×M16—141no—
234C4×M16—141no—
235#332—74no—
236#332—74no—
237#332—74no—
238#332C2×D714yescomplemented
239C23×C7C2×C22⋊C411yescharacteristic
240C22×D7—22no—
241C22×D7—41no—
242C22×D7—41no—
243C22×D7C2×C22⋊C411yescharacteristic
244C22×D7C2×C22⋊C411yescharacteristic
245C7×Q8—44no—
246C7×Q8—44no—
247C2×C4×C7—22no—
248C2×C4×C7—22no—
249C2×C4×C7—22no—
250C2×C4×C7—24no—
251C2×C4×C7—44no—
252C2×C4×C7C2×C22⋊C411yescharacteristic
253C2×C4×C7C2×C22⋊C411yescharacteristic
254C2×C4×C7C2×C22⋊C411yescharacteristic
255C2×C4×C7C23×C411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C22×D411yescharacteristic
258C2×Dic7—24no—
259C2×Dic7—41no—
260C2×Dic7—41no—
261C2×Dic7—44no—
262C2×Dic7#10114yes—
263C2×Dic7C2×C22⋊C411yescharacteristic
264C2×Dic7C2×C22⋊C411yescharacteristic
265C2×Dic7C2×C22⋊C411yescharacteristic
266C2×Dic7C2×C22⋊C411yescharacteristic
267C2×Dic7C2×C22⋊C411yescharacteristic
268C2×Dic7C2×C22⋊C411yescharacteristic
269C4×D7—22no—
270C4×D7—22no—
271C4×D7—22no—
272C4×D7—22no—
273C4×D7—22no—
274C4×D7—22no—
275C4×D7—22no—
276C4×D7—22no—
277C4×D7—41no—
278C4×D7—41no—
279C4×D7—41no—
280C4×D7—41no—
281Dic14—44no—
282Dic14—44no—
283C7×C8—42no—
284C7×C8—42no—
285C7⋊C8—42no—
286C7⋊C8—42no—
287C22×Q8—21no—
288C22×Q8—141no—
289C23×C4—71no—
290C23×C4—71no—
291C23×C4—71no—
292C2×M16—142no—
293C2×M16—142no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8—72no—
297C4⋊Q8—72no—
298C4⋊Q8—141no—
299C4⋊Q8—141no—
300C4⋊Q8C4×D712yes—
301C4⋊Q8C22×D712yes—
302#125—74no—
303#125—74no—
304#125—142no—
305#125—142no—
306C22⋊Q8—142no—
307C22⋊Q8—142no—
308C22⋊Q8—142no—
309C22⋊Q8—142no—
310C22⋊Q8—142no—
311C22⋊Q8—142no—
312C22⋊Q8—142no—
313C22⋊Q8—142no—
314C4×Q8—142no—
315C4×Q8—142no—
316C4×Q8—142no—
317C4×Q8—142no—
318C4×Q8—142no—
319C4×Q8—142no—
320C4×Q8—142no—
321C4×Q8—142no—
322#117—74no—
323#117—74no—
324#117—142no—
325#117—142no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—72no—
328C2×C4⋊C4—142no—
329C2×C4⋊C4—142no—
330C2×C4⋊C4—141no—
331C2×C4⋊C4C4×D712yes—
332C2×C22⋊C4—142no—
333C2×C42—71no—
334C2×C42—71no—
335C2×C42—71no—
336C2×C42—71no—
337C2×C42—71no—
338C2×C42—71no—
339C2×C42—71no—
340C2×C42—71no—
341C2×C42—71no—
342C2×C42—71no—
343C2×C42—71no—
344C2×C42—142no—
345C2×C42C22×D711yescharacteristic
346C4⋊C8—72no—
347C4⋊C8—72no—
348C4⋊C8—142no—
349C4⋊C8—142no—
350C4⋊C8C22×D712yes—
351C4⋊C8C22×D712yes—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—142no—
355C22⋊C8—142no—
356C22⋊C8—141no—
357C22⋊C8—141no—
358#97—21no—
359#97—21no—
360#97—141no—
361#97—141no—
362C4×C8—142no—
363C4×C8—142no—
364C22×C7—21no—
365C22×C7—21no—
366C22×C7C2×#10112yes—
367C22×C7C22×C22⋊C411yescharacteristic
368C2×D7—21no—
369C2×D7—21no—
370C2×D7—21no—
371C2×D7—21no—
372C2×D7—21no—
373C2×D7—21no—
374C2×D7—44no—
375C4×C7—21no—
376C4×C7—21no—
377C4×C7—21no—
378C4×C7—21no—
379C4×C7—44no—
380C4×C7#57512yes—
381C4×C7#57512yes—
382Dic7—21no—
383Dic7—21no—
384Dic7—21no—
385Dic7—21no—
386Dic7—44no—
387Dic7C2×#10112yes—
388Dic7C2×#10112yes—
389C24—71no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—44no—
393C2×Q8—142no—
394C2×Q8—142no—
395C2×Q8—284no—
396C22×C4—22no—
397C22×C4—71no—
398C22×C4—71no—
399C22×C4—71no—
400C22×C4—71no—
401C22×C4—71no—
402C22×C4—71no—
403C22×C4—71no—
404C22×C4—71no—
405C22×C4—71no—
406C22×C4—71no—
407C22×C4—71no—
408C22×C4—71no—
409C22×C4—71no—
410C22×C4—71no—
411C22×C4—71no—
412C22×C4—144no—
413C22×C4—142no—
414C22×C4—282no—
415C22×C4—282no—
416C22×C4C2×C4×D711yescharacteristic
417C22×C4D4×D711yescharacteristic
418C22×C4D4×D711yescharacteristic
419M16—284no—
420M16—284no—
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—142no—
424C2×C8—142no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—74no—
429C4⋊C4—144no—
430C4⋊C4—144no—
431C4⋊C4—144no—
432C4⋊C4—144no—
433C4⋊C4—142no—
434C4⋊C4—142no—
435C4⋊C4—142no—
436C4⋊C4C2×C4×D714yes—
437C22⋊C4—144no—
438C22⋊C4—144no—
439C42—71no—
440C42—71no—
441C42—71no—
442C42—71no—
443C42—71no—
444C42—71no—
445C42—71no—
446C42—71no—
447C42—71no—
448C42—71no—
449C42—71no—
450C42—71no—
451C42—144no—
452C42—144no—
453C42C2×C4×D711yescharacteristic
454C42D4×D711yescharacteristic
455C42D4×D711yescharacteristic
456C42C23×D711yescharacteristic
457C2×C7—22no—
458C2×C7C22×#10111yescharacteristic
459C2×C7#289611yescharacteristic
460C2×C7#289611yescharacteristic
461D7—41no—
462D7—41no—
463C23—71no—
464C23—71no—
465C23—142no—
466C23—281no—
467C23—281no—
468C23D7×C22⋊C411yescharacteristic
469Q8—44no—
470Q8—44no—
471Q8—284no—
472Q8—284no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—44no—
478C2×C4—71no—
479C2×C4—71no—
480C2×C4—74no—
481C2×C4—71no—
482C2×C4—71no—
483C2×C4—71no—
484C2×C4—71no—
485C2×C4—142no—
486C2×C4—142no—
487C2×C4—142no—
488C2×C4—142no—
489C2×C4—142no—
490C2×C4—142no—
491C2×C4—142no—
492C2×C4—142no—
493C2×C4—144no—
494C2×C4—281no—
495C2×C4—281no—
496C2×C4—281no—
497C2×C4—281no—
498C2×C4—281no—
499C2×C4—281no—
500C2×C4—284no—
501C2×C4D7×C22⋊C411yescharacteristic
502C2×C4D7×C22⋊C411yescharacteristic
503C2×C4D7×C22⋊C411yescharacteristic
504C2×C4C22×C4×D711yescharacteristic
505C2×C4C2×D4×D711yescharacteristic
506C2×C4C2×D4×D711yescharacteristic
507C8—42no—
508C8—42no—
509C8—282no—
510C8—282no—
511C7#1773911yescharacteristic, complemented, Sylow
512C22—21no—
513C22—21no—
514C22—141no—
515C22—141no—
516C22—141no—
517C22—141no—
518C22—141no—
519C22—141no—
520C22—284no—
521C22D7×#10112yes—
522C22C2×D7×C22⋊C411yescharacteristic
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—44no—
528C4—72no—
529C4—72no—
530C4—141no—
531C4—141no—
532C4—141no—
533C4—141no—
534C4—284no—
535C4#9046412yes—
536C4#9046812yes—
537C2—22no—
538C2—281no—
539C2—281no—
540C2C2×D7×#10111yescharacteristic
541C2#1173636911yescharacteristic
542C2#1173640111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1556C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC42×C7C24
hypercenter#1556D7
hypocenterC7#17739
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1556

Complements

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

NH ≅ G/Ntype
4. C7×#1556C2semidirect
5. #11729530C2semidirect
6. #11728577C2semidirect
7. #11728576C2semidirect
8. #11726185C2semidirect
9. #11726173C2semidirect
43. C7×#332C22semidirect
44. #89342C22semidirect
45. #89281C22semidirect
46. #89277C22semidirect
132. #1556D7semidirect
238. #332C2×D7semidirect
511. C7#17739semidirect

Maximal subgroups

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

subgroupindex
C2×#904192
#117367762
C7×#15562
#117295302
#117285772
#117285762
#117261852
#117261732
#177397