Finite Group Databasebeta

← #13319706

Subgroups — #13319706

Order 1792.

1210 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90214C211yesmaximal, characteristic
2#11736780C212yesmaximal, complemented
3C7×#1464C211yesmaximal, characteristic, complemented
4#11729326C211yesmaximal, characteristic, complemented
5#11728584C212yesmaximal, complemented
6#11725898C212yesmaximal, complemented
7D7×#324C212yesmaximal, complemented
8#11725875C212yesmaximal, complemented
9#11725873C212yesmaximal, complemented
10C22×#1157C411yescharacteristic
11C2×C7×#117C2211yescharacteristic
12C22×C4⋊Dic7C411yescharacteristic
13#90214—24no—
14#90214—24no—
15C2×#6455C411yescharacteristic
16C2×#6455C411yescharacteristic
17C2×D7×C4⋊C4C2211yescharacteristic
18C2×D7×C4⋊C4C2211yescharacteristic
19C2×(C22⋊D7)⋊C4C411yescharacteristic
20C2×(C22⋊D7)⋊C4C411yescharacteristic
21C2×(C22⋊D7)⋊C4C2211yescharacteristic
22C2×(C22⋊D7)⋊C4C2211yescharacteristic
23C2×C4×D28C411yescharacteristic
24C2×C4×D28C2211yescharacteristic
25C2×C4×Dic14C411yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C7×#432C2212yes—
28#89698C2212yes—
29#89608—24no—
30#89498C2214yes—
31D7×C22⋊C8C2214yes—
32#89478—24no—
33C7×#324C2212yes—
34C7×#323C2212yes—
35#89323C2212yes—
36#89320C2212yes—
37#89264C2214yes—
38#89264C2214yes—
39#17368—71nomaximal, Sylow
40C23×C4×C7C2×C411yescharacteristic
41C22×C22⋊D7C2×C411yescharacteristic
42C22×C22⋊D7C2×C411yescharacteristic
43C23×Dic7C2×C411yescharacteristic
44C23×Dic7C2×C411yescharacteristic
45C2×#1157—24no—
46C2×#1157—22no—
47C2×#1157—22no—
48C2×#1157—24no—
49C22×D28C2×C411yescharacteristic
50C22×C4×D7C2×C411yescharacteristic
51C22×C4×D7C2×C411yescharacteristic
52C22×C4×D7C2×C411yescharacteristic
53C22×C4×D7C2×C411yescharacteristic
54C22×C4×D7C2311yescharacteristic
55C22×C4×D7C2311yescharacteristic
56C22×Dic14C2×C411yescharacteristic
57C7×#117—22no—
58C7×#117—22no—
59C2×C7×C4⋊C4C2×C411yescharacteristic
60C2×C7×C4⋊C4C2311yescharacteristic
61C2×C7×C22⋊C4C2×C411yescharacteristic
62C2×C7×C22⋊C4C2311yescharacteristic
63C2×C42×C7C2×C411yescharacteristic
64C2×C42×C7C2311yescharacteristic
65C2×(C2×D7)⋊C4C2×C411yescharacteristic
66C2×(C2×D7)⋊C4C2×C411yescharacteristic
67C2×(C2×D7)⋊C4C2×C411yescharacteristic
68C2×(C2×D7)⋊C4C2×C411yescharacteristic
69C2×C4⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C2×C4⋊Dic7C2×C411yescharacteristic
72C2×C4⋊Dic7C2×C411yescharacteristic
73C2×C4⋊Dic7C2×C411yescharacteristic
74C2×C4⋊Dic7C2×C411yescharacteristic
75C2×Dic7⋊C4C2×C411yescharacteristic
76C2×Dic7⋊C4C2×C411yescharacteristic
77C2×Dic7⋊C4C2311yescharacteristic
78C2×Dic7⋊C4C2311yescharacteristic
79C2×C4×Dic7C2×C411yescharacteristic
80C2×C4×Dic7C2×C411yescharacteristic
81C2×C4×Dic7C2×C411yescharacteristic
82C2×C4×Dic7C2×C411yescharacteristic
83C2×#1131—24no—
84C2×C8×D7—24no—
85#6455—24no—
86#6455—24no—
87D7×C4⋊C4—24no—
88D7×C4⋊C4—24no—
89(C22⋊D7)⋊C4—24no—
90(C22⋊D7)⋊C4—24no—
91(C22⋊D7)⋊C4—24no—
92(C22⋊D7)⋊C4—24no—
93C4×D28—22no—
94C4×D28—22no—
95C4×D28D414yes—
96C4×D28D414yes—
97C4×Dic14—22no—
98C4×Dic14—22no—
99C4×Dic14—22no—
100C4×Dic14—22no—
101C7×C4⋊C8—22no—
102C7×C22⋊C8C2314yes—
103C7×#97—22no—
104C22⋊(C7⋊C8)C2314yes—
105(C2×D7)⋊C8—24no—
106(C2×D7)⋊C8—24no—
107C4⋊(C7⋊C8)—22no—
108#6377—22no—
109C2×#520—71no—
110#2973—72no—
111#1467—72no—
112#1465—72no—
113#1464—71no—
114#1464D711yescharacteristic, complemented
115#1463—72no—
116#1459—72no—
117C2×#324—72no—
118C24×C7C22×C411yescharacteristic
119C23×D7C22×C411yescharacteristic
120C23×D7C22×C411yescharacteristic
121C22×C4×C7—22no—
122C22×C4×C7—21no—
123C22×C4×C7—21no—
124C22×C4×C7—22no—
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C22×C411yescharacteristic
129C22×C4×C7C22×C411yescharacteristic
130C22×C4×C7C2411yescharacteristic
131C2×C22⋊D7—22no—
132C2×C22⋊D7—22no—
133C2×C22⋊D7—22no—
134C2×C22⋊D7—22no—
135C2×C22⋊D7—22no—
136C2×C22⋊D7M1614yescomplemented
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—81no—
140C22×Dic7—81no—
141C22×Dic7C22×C411yescharacteristic
142C22×Dic7C22×C411yescharacteristic
143C22×Dic7C22×C411yescharacteristic
144C22×Dic7C22×C411yescharacteristic
145C22×Dic7C22×C411yescharacteristic
146C22×Dic7C22×C411yescharacteristic
147#1157—216no—
148#1157—216no—
149C2×D28—22no—
150C2×D28—22no—
151C2×D28C22⋊C412yes—
152C2×D28C22⋊C412yes—
153C2×C4×D7—22no—
154C2×C4×D7—22no—
155C2×C4×D7—22no—
156C2×C4×D7—22no—
157C2×C4×D7—22no—
158C2×C4×D7—22no—
159C2×C4×D7—22no—
160C2×C4×D7—81no—
161C2×C4×D7—81no—
162C2×C4×D7—81no—
163C2×C4×D7—81no—
164C2×C4×D7M1614yescomplemented
165C2×C4×D7C2×D414yes—
166C2×C4×D7C2×D414yes—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—22no—
170C2×Dic14—22no—
171C2×C7×C8—24no—
172C7×C4⋊C4—22no—
173C7×C4⋊C4—22no—
174C7×C22⋊C4—22no—
175C7×C22⋊C4—22no—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7C2×D412yes—
179C42×C7C2×D412yes—
180(C2×D7)⋊C4—22no—
181(C2×D7)⋊C4—22no—
182(C2×D7)⋊C4C22⋊C414yes—
183(C2×D7)⋊C4C22⋊C414yes—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—22no—
187C4⋊Dic7—22no—
188C4⋊Dic7C22⋊C412yes—
189C4⋊Dic7C22⋊C412yes—
190Dic7⋊C4—22no—
191Dic7⋊C4—22no—
192Dic7⋊C4—22no—
193Dic7⋊C4—22no—
194C4×Dic7—22no—
195C4×Dic7—22no—
196C4×Dic7—22no—
197C4×Dic7—22no—
198C2×C7⋊C8—24no—
199#1131—84no—
200C8×D7—84no—
201C22×#41—71no—
202#520—144no—
203#520—144no—
204C2×C4×Q8—71no—
205C2×C4×Q8—71no—
206C2×C4×D4—71no—
207C2×C4×D4—71no—
208C2×C4×D4—71no—
209C2×C4×D4—71no—
210C2×C4×D4—71no—
211C2×C4×D4—71no—
212C2×#117—71no—
213C2×#117—71no—
214C2×#117C2×D711yescharacteristic
215C22×C4⋊C4—71no—
216C22×C4⋊C4—71no—
217C22×C4⋊C4—71no—
218#435—144no—
219#435—144no—
220#432—72no—
221#432C2×D712yes—
222#407—74no—
223C2×C22⋊C8—74no—
224#324—72no—
225#324—74no—
226#324—74no—
227#324C2×D712yes—
228#323—72no—
229#323C2×D712yes—
230C23×C7—21no—
231C23×C7—21no—
232C23×C7—41no—
233C23×C7—41no—
234C23×C7C2×M1612yes—
235C23×C7C23×C411yescharacteristic
236C22×D7—21no—
237C22×D7—21no—
238C22×D7—41no—
239C22×D7—41no—
240C22×D7—41no—
241C22×D7—41no—
242C22×D7C2×C22⋊C412yes—
243C22×D7C2×C22⋊C412yes—
244C22×D7C2×M1612yes—
245C22×D7C2×M1612yes—
246C2×C4×C7—24no—
247C2×C4×C7—24no—
248C2×C4×C7—21no—
249C2×C4×C7—21no—
250C2×C4×C7—21no—
251C2×C4×C7—21no—
252C2×C4×C7—21no—
253C2×C4×C7—21no—
254C2×C4×C7—41no—
255C2×C4×C7—41no—
256C2×C4×C7—41no—
257C2×C4×C7—41no—
258C2×C4×C7C2×C22⋊C412yes—
259C2×C4×C7C2×C22⋊C411yescharacteristic
260C2×C4×C7C2×C22⋊C411yescharacteristic
261C2×C4×C7C2×M1612yes—
262C2×C4×C7C22×D412yes—
263C22⋊D7—28no—
264C22⋊D7—28no—
265C2×Dic7—21no—
266C2×Dic7—21no—
267C2×Dic7—21no—
268C2×Dic7—21no—
269C2×Dic7—21no—
270C2×Dic7—21no—
271C2×Dic7—41no—
272C2×Dic7—41no—
273C2×Dic7—41no—
274C2×Dic7—41no—
275C2×Dic7—82no—
276C2×Dic7—82no—
277C2×Dic7C2×C22⋊C412yes—
278C2×Dic7C2×C22⋊C412yes—
279C2×Dic7C2×M1612yes—
280C2×Dic7C2×M1612yes—
281D28—24no—
282D28—24no—
283C4×D7—28no—
284C4×D7—28no—
285C4×D7—44no—
286C4×D7—44no—
287C4×D7—82no—
288C4×D7—82no—
289Dic14—24no—
290Dic14—24no—
291C7×C8—44no—
292C7⋊C8—44no—
293C2×#41—144no—
294C2×#41—142no—
295C2×#41—142no—
296C2×#41—144no—
297C22×Q8—71no—
298C22×D4—71no—
299C22×D4—71no—
300C22×D4—71no—
301C23×C4—71no—
302C23×C4—71no—
303C23×C4—71no—
304C23×C4—71no—
305C23×C4—71no—
306C23×C4—71no—
307C23×C4—71no—
308C23×C4—71no—
309C23×C4C4×D711yescharacteristic
310C2×M16—144no—
311C22×C8—144no—
312C4×Q8—142no—
313C4×Q8—142no—
314C4×Q8—142no—
315C4×Q8—142no—
316C4×D4—74no—
317C4×D4—74no—
318C4×D4—142no—
319C4×D4—142no—
320C4×D4—144no—
321C4×D4—144no—
322C4×D4—144no—
323C4×D4—144no—
324#117—22no—
325#117—22no—
326#117—144no—
327#117—144no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—144no—
337C2×C4⋊C4—144no—
338C2×C4⋊C4—142no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4C4×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C22⋊C4—71no—
343C2×C22⋊C4—71no—
344C2×C22⋊C4—71no—
345C2×C22⋊C4—71no—
346C2×C22⋊C4C4×D711yescharacteristic
347C2×C22⋊C4C22×D711yescharacteristic
348C2×C42—71no—
349C2×C42—71no—
350C2×C42—71no—
351C2×C42—71no—
352C2×C42C4×D711yescharacteristic
353C2×C42C22×D711yescharacteristic
354C4⋊C8—22no—
355C4⋊C8—142no—
356C22⋊C8—74no—
357C22⋊C8—144no—
358C22⋊C8—144no—
359C22⋊C8C22×D714yes—
360#97—22no—
361#97—142no—
362C22×C7—21no—
363C22×C7—21no—
364C22×C7—22no—
365C22×C7—42no—
366C22×C7—42no—
367C22×C7—41no—
368C22×C7—41no—
369C22×C7C22×C22⋊C411yescharacteristic
370C22×C7C22×M1611yescharacteristic
371C22×C7C22×M1611yescharacteristic
372C2×D7—22no—
373C2×D7—22no—
374C2×D7—22no—
375C2×D7—22no—
376C2×D7—22no—
377C2×D7—42no—
378C2×D7—42no—
379C2×D7—42no—
380C2×D7—42no—
381C2×D7#40714yes—
382C4×C7—21no—
383C4×C7—21no—
384C4×C7—22no—
385C4×C7—22no—
386C4×C7—22no—
387C4×C7—41no—
388C4×C7—41no—
389Dic7—22no—
390Dic7—22no—
391Dic7—22no—
392Dic7—22no—
393Dic7—41no—
394Dic7—41no—
395C24—71no—
396C24—71no—
397C24C2×C4×D711yescharacteristic
398#41—1416no—
399#41—1416no—
400C2×Q8—142no—
401C2×Q8—142no—
402C2×Q8—141no—
403C2×Q8—141no—
404C2×D4—74no—
405C2×D4—72no—
406C2×D4—72no—
407C2×D4—142no—
408C2×D4—142no—
409C2×D4—142no—
410C2×D4—142no—
411C2×D4—142no—
412C2×D4—142no—
413C2×D4—142no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—71no—
419C22×C4—71no—
420C22×C4—71no—
421C22×C4—71no—
422C22×C4—74no—
423C22×C4—74no—
424C22×C4—74no—
425C22×C4—71no—
426C22×C4—71no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—561no—
437C22×C4—561no—
438C22×C4—561no—
439C22×C4—561no—
440C22×C4—561no—
441C22×C4—561no—
442C22×C4C2×C4×D711yescharacteristic
443C22×C4C2×C4×D711yescharacteristic
444C22×C4C2×C4×D711yescharacteristic
445C22×C4C2×C4×D711yescharacteristic
446C22×C4C2×C4×D711yescharacteristic
447C22×C4C23×D711yescharacteristic
448M16—564no—
449C2×C8—24no—
450C2×C8—144no—
451C2×C8—564no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—72no—
455C4⋊C4—72no—
456C4⋊C4—142no—
457C4⋊C4—142no—
458C4⋊C4—142no—
459C4⋊C4—142no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—142no—
463C4⋊C4—142no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—74no—
467C22⋊C4—74no—
468C22⋊C4—142no—
469C22⋊C4—142no—
470C42—21no—
471C42—21no—
472C42—142no—
473C42—142no—
474C42—142no—
475C42—142no—
476C42D4×D712yes—
477C42D4×D712yes—
478C2×C7—21no—
479C2×C7—21no—
480C2×C7—42no—
481C2×C7C2×#40711yescharacteristic
482C2×C7#288311yescharacteristic
483C2×C7#288911yescharacteristic
484D7—24no—
485D7—42no—
486C23—21no—
487C23—21no—
488C23—41no—
489C23—41no—
490C23—72no—
491C23—72no—
492C23—72no—
493C23—72no—
494C23—141no—
495C23—141no—
496C23—281no—
497C23—281no—
498C23—281no—
499C23—281no—
500C23D7×M1612yes—
501C23C22×C4×D711yescharacteristic
502Q8—144no—
503Q8—144no—
504D4—144no—
505D4—144no—
506D4—148no—
507D4—148no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—72no—
521C2×C4—72no—
522C2×C4—72no—
523C2×C4—72no—
524C2×C4—148no—
525C2×C4—148no—
526C2×C4—141no—
527C2×C4—141no—
528C2×C4—141no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—141no—
532C2×C4—284no—
533C2×C4—284no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—562no—
539C2×C4—562no—
540C2×C4—562no—
541C2×C4—562no—
542C2×C4D7×C22⋊C411yescharacteristic
543C2×C4D7×C22⋊C411yescharacteristic
544C2×C4D7×C22⋊C412yes—
545C2×C4D7×M1612yes—
546C2×C4C2×D4×D712yes—
547C8—44no—
548C8—284no—
549C7#1736811yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—22no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—74no—
558C22—142no—
559C22—142no—
560C22—142no—
561C22—142no—
562C22—142no—
563C22—282no—
564C22—282no—
565C22—282no—
566C22—282no—
567C22C2×D7×C22⋊C411yescharacteristic
568C22C2×D7×M1611yescharacteristic
569C22C2×D7×M1611yescharacteristic
570C4—22no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—142no—
578C4—142no—
579C4—142no—
580C4—142no—
581C4—281no—
582C4—281no—
583C2—21no—
584C2—21no—
585C2—42no—
586C2—144no—
587C2—282no—
588C2D7×#40711yescharacteristic
589C2#1173613911yescharacteristic
590C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1464C2
socleC22×C7C22×C22⋊C4
prime radicalC22×#1157C4
cosocleC22×C4×C7C24
hypercenter#1464D7
hypocenterC7#17368
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1464

Complements

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

NH ≅ G/Ntype
2. #11736780C2semidirect
3. C7×#1464C2semidirect
4. #11729326C2semidirect
5. #11728584C2semidirect
6. #11725898C2semidirect
7. D7×#324C2semidirect
8. #11725875C2semidirect
9. #11725873C2semidirect
114. #1464D7semidirect
136. C2×C22⋊D7M16semidirect
164. C2×C4×D7M16semidirect
549. C7#17368semidirect

Maximal subgroups

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

subgroupindex
C2×#902142
#117367802
C7×#14642
#117293262
#117285842
#117258982
D7×#3242
#117258752
#117258732
#173687