Finite Group Databasebeta

← #13369945

Subgroups — #13369945

Order 1792.

1132 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90232C211yesmaximal, characteristic
2#11731323C212yesmaximal, complemented
3C7×#1799C211yesmaximal, characteristic
4C4×#6408C211yesmaximal, characteristic
5#11727783C214yesmaximal
6#11727283C212yesmaximal, complemented
7#11726707C212yesmaximal, complemented
8#11726705C212yesmaximal, complemented
9C2×C4×C7×D4C2211yescharacteristic
10C2×#6498C2212yes—
11C2×#6489C411yescharacteristic, complemented
12C22×C4⋊Dic7C2211yescharacteristic
13C2×#6487C411yescharacteristic, complemented
14C2×#6487C2211yescharacteristic
15#90232—24no—
16#90232—22no—
17#90232—22no—
18C2×#6452C412yescomplemented
19C2×#6442C412yescomplemented
20C2×#6440C2212yes—
21C2×#6440C2214yes—
22C2×#6433C411yescharacteristic, complemented
23C2×C4×Dic14C411yescharacteristic, complemented
24C2×C7×#99C2212yes—
25C7×#380C2212yes—
26C2×#6408C2212yes—
27C4×C22⋊Dic7C2212yes—
28C4⋊C4×Dic7—22no—
29#89745—22no—
30#89715C2214yes—
31#89426—24no—
32C7×#342C2212yes—
33#89322C2214yes—
34#89322C2214yes—
35#89321C2211yescharacteristic
36#89321C2211yescharacteristic
37#22478—71nomaximal, Sylow
38C22×C7×D4C2311yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7C2312yes—
42C22×Dic14C2×C411yescharacteristic
43C4×C7×D4—22no—
44C4×C7×D4—22no—
45C2×C7×C4⋊C4—22no—
46C2×C7×C4⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4C2314yes—
49C2×C7×C22⋊C4C2311yescharacteristic
50C2×C42×C7C2×C411yescharacteristic
51C2×C22⋊Dic7C2×C412yes—
52C2×C22⋊Dic7C2×C412yes—
53C2×C22⋊Dic7C2312yes—
54C2×C22⋊Dic7C2312yes—
55C2×C22⋊Dic7C2312yes—
56#6498—24no—
57#6498—24no—
58#6489—22no—
59#6489—21no—
60#6489—21no—
61C2×C4⋊Dic7C2×C411yescharacteristic
62C2×C4⋊Dic7C2×C411yescharacteristic
63C2×C4⋊Dic7C2×C411yescharacteristic
64C2×C4⋊Dic7C2×C412yes—
65C2×C4⋊Dic7D414yes—
66C2×C4⋊Dic7D412yes—
67C2×C4⋊Dic7D412yes—
68#6487—22no—
69#6487—22no—
70#6487—22no—
71#6487—22no—
72C2×Dic7⋊C4—24no—
73C2×Dic7⋊C4C2×C412yes—
74C2×Dic7⋊C4C2×C412yes—
75C2×Dic7⋊C4C2×C412yes—
76C2×Dic7⋊C4C2×C412yes—
77C2×Dic7⋊C4C2×C412yes—
78C2×C4×Dic7—22no—
79C2×C4×Dic7—24no—
80C2×C4×Dic7C2×C412yes—
81C2×C4×Dic7C2×C412yes—
82#6452—24no—
83#6452—24no—
84#6442—24no—
85#6442—24no—
86#6440—24no—
87#6440—24no—
88#6440—28no—
89#6440—44no—
90#6433—22no—
91#6433—22no—
92C4×Dic14—22no—
93C4×Dic14—21no—
94C4×Dic14—21no—
95C7×#99—42no—
96C7×#95—22no—
97#6408—24no—
98C7⋊#95—22no—
99C7⋊#95—24no—
100C7⋊#95—24no—
101C2×#552—71no—
102#2344—72no—
103#1996—72no—
104#1899—74no—
105#1820—72no—
106#1799D711yescharacteristic
107C4×#99—71no—
108#1740—72no—
109C24×C7C22×C411yescharacteristic
110C24×C7C2411yescharacteristic
111C2×C7×D4—22no—
112C2×C7×D4—22no—
113C2×C7×D4—21no—
114C2×C7×D4—21no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—24no—
118C22×C4×C7—42no—
119C22×C4×C7—42no—
120C22×C4×C7C22×C411yescharacteristic
121C22×C4×C7C22×C411yescharacteristic
122C22×C4×C7C22×C411yescharacteristic
123C22×C4×C7C22×C411yescharacteristic
124C22×C4×C7C22×C411yescharacteristic
125C22×C4×C7C2×D412yes—
126C22×C4×C7C2×D412yes—
127C22×Dic7—24no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7C22×C412yes—
133C22×Dic7C22×C412yes—
134C22×Dic7C22×C412yes—
135C22×Dic7C22×C412yes—
136C22×Dic7C2×D414yes—
137C22×Dic7C2×D414yes—
138C2×Dic14—21no—
139C2×Dic14—21no—
140C2×Dic14—42no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—44no—
143C7×C22⋊C4—22no—
144C7×C22⋊C4—22no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—24no—
147C42×C7—21no—
148C42×C7—21no—
149C22⋊Dic7—24no—
150C22⋊Dic7—22no—
151C22⋊Dic7—24no—
152C22⋊Dic7—22no—
153C22⋊Dic7—22no—
154C22⋊Dic7—22no—
155C22⋊Dic7—22no—
156C22⋊Dic7#4114yes—
157C4⋊Dic7—21no—
158C4⋊Dic7—21no—
159C4⋊Dic7C22⋊C412yes—
160C4⋊Dic7C22⋊C411yescharacteristic
161C4⋊Dic7C22⋊C411yescharacteristic
162C4⋊Dic7C22⋊C412yes—
163C4⋊Dic7C22⋊C412yes—
164C4⋊Dic7C22⋊C414yes—
165C4⋊Dic7C22⋊C414yes—
166Dic7⋊C4—24no—
167Dic7⋊C4—22no—
168Dic7⋊C4—22no—
169Dic7⋊C4—24no—
170Dic7⋊C4—24no—
171Dic7⋊C4—24no—
172Dic7⋊C4—48no—
173C4×Dic7—22no—
174C4×Dic7—22no—
175C4×Dic7—22no—
176C4×Dic7—22no—
177C4×Dic7—48no—
178C4×Dic7—44no—
179#552—142no—
180#552—144no—
181#552—142no—
182C2×#126—72no—
183C2×#125—72no—
184C2×#125—71no—
185C2×#123—72no—
186C2×C22⋊Q8—71no—
187C2×C22⋊Q8—71no—
188C2×C4×Q8—71no—
189C2×C4×D4C2×D711yescharacteristic
190C2×#117—71no—
191C2×#117—72no—
192C2×#117—74no—
193C22×C4⋊C4—71no—
194C2×#99—72no—
195C2×#99C2×D712yes—
196#389—142no—
197#387—144no—
198#381—74no—
199#380C2×D712yes—
200C4×C4⋊C4—142no—
201C4×C22⋊C4—72no—
202#342—71no—
203#342—74no—
204#342—74no—
205#342—71no—
206#342C2×D712yes—
207C23×C7—21no—
208C23×C7—21no—
209C23×C7—21no—
210C23×C7—41no—
211C23×C7—41no—
212C23×C7—41no—
213C23×C7—41no—
214C23×C7C23×C411yescharacteristic
215C23×C7C22×D412yes—
216C23×C7C2×#4112yes—
217C23×C7H(5,𝔽2)11yescharacteristic
218C23×C72−1+411yescharacteristic
219C7×D4—42no—
220C7×D4—42no—
221C2×C4×C7—24no—
222C2×C4×C7—22no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—41no—
229C2×C4×C7—42no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—44no—
235C2×C4×C7—44no—
236C2×C4×C7—44no—
237C2×C4×C7C2×C22⋊C412yes—
238C2×C4×C7C2×C22⋊C411yescharacteristic
239C2×C4×C7C2×C22⋊C411yescharacteristic
240C2×Dic7—22no—
241C2×Dic7—22no—
242C2×Dic7—22no—
243C2×Dic7—22no—
244C2×Dic7—24no—
245C2×Dic7—24no—
246C2×Dic7—24no—
247C2×Dic7—24no—
248C2×Dic7—22no—
249C2×Dic7—22no—
250C2×Dic7—24no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7—42no—
254C2×Dic7—42no—
255C2×Dic7—42no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7C2×C22⋊C412yes—
259C2×Dic7C2×C22⋊C412yes—
260C2×Dic7C2×C22⋊C414yes—
261Dic14—42no—
262Dic14—42no—
263C22×Q8—71no—
264C22×D4C22×D711yescharacteristic
265C23×C4—72no—
266C23×C4C4×D711yescharacteristic
267C23×C4C22×D711yescharacteristic
268#126—144no—
269#126—144no—
270#125—142no—
271#125—142no—
272#125—144no—
273#125—144no—
274#123—144no—
275#123—144no—
276C22⋊Q8—142no—
277C22⋊Q8—142no—
278C22⋊Q8—142no—
279C22⋊Q8—142no—
280C4×Q8—142no—
281C4×Q8—141no—
282C4×Q8—141no—
283C4×D4—22no—
284C4×D4—22no—
285#117—144no—
286#117—144no—
287#117—142no—
288#117—141no—
289#117—141no—
290#117—148no—
291#117—284no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—71no—
294C2×C4⋊C4—71no—
295C2×C4⋊C4—72no—
296C2×C4⋊C4—72no—
297C2×C4⋊C4—74no—
298C2×C4⋊C4—72no—
299C2×C4⋊C4—72no—
300C2×C4⋊C4—71no—
301C2×C4⋊C4—72no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—72no—
304C2×C4⋊C4—72no—
305C2×C4⋊C4—144no—
306C2×C4⋊C4C4×D711yescharacteristic
307C2×C22⋊C4—72no—
308C2×C22⋊C4—72no—
309C2×C22⋊C4—72no—
310C2×C22⋊C4—72no—
311C2×C22⋊C4—72no—
312C2×C22⋊C4C4×D711yescharacteristic
313C2×C22⋊C4C22×D714yes—
314C2×C22⋊C4C22×D711yescharacteristic
315C2×C42—72no—
316C2×C42—72no—
317C2×C42—144no—
318C2×C42—142no—
319C2×C42C4×D711yescharacteristic
320#99—42no—
321#99—144no—
322#95—22no—
323#95—144no—
324#95—144no—
325#95—142no—
326C22×C7—22no—
327C22×C7—21no—
328C22×C7—22no—
329C22×C7—21no—
330C22×C7—42no—
331C22×C7—41no—
332C22×C7—41no—
333C22×C7—41no—
334C22×C7—41no—
335C22×C7—41no—
336C22×C7—81no—
337C22×C7—81no—
338C22×C7C22×C22⋊C411yescharacteristic
339C22×C7C4×#4111yescharacteristic
340C22×C7#52011yescharacteristic
341C22×C7#54612yes—
342C22×C7#54712yes—
343C4×C7—21no—
344C4×C7—22no—
345C4×C7—41no—
346C4×C7—41no—
347C4×C7—41no—
348C4×C7—84no—
349C4×C7#41012yes—
350Dic7—22no—
351Dic7—24no—
352Dic7—22no—
353Dic7—44no—
354Dic7—42no—
355Dic7—42no—
356Dic7—82no—
357C24C2×C4×D711yescharacteristic
358C24C23×D711yescharacteristic
359C2×Q8—141no—
360C2×Q8—141no—
361C2×Q8—282no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—72no—
372C22×C4—72no—
373C22×C4—74no—
374C22×C4—74no—
375C22×C4—72no—
376C22×C4—72no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—144no—
380C22×C4—142no—
381C22×C4—142no—
382C22×C4C2×C4×D711yescharacteristic
383C22×C4C2×C4×D711yescharacteristic
384C22×C4C2×C4×D711yescharacteristic
385C22×C4C2×C4×D711yescharacteristic
386C22×C4C2×C4×D711yescharacteristic
387C22×C4D4×D712yes—
388C22×C4D4×D712yes—
389C4⋊C4—22no—
390C4⋊C4—44no—
391C4⋊C4—72no—
392C4⋊C4—71no—
393C4⋊C4—71no—
394C4⋊C4—72no—
395C4⋊C4—72no—
396C4⋊C4—74no—
397C4⋊C4—74no—
398C4⋊C4—144no—
399C4⋊C4—144no—
400C4⋊C4—142no—
401C4⋊C4—142no—
402C4⋊C4—144no—
403C4⋊C4—144no—
404C4⋊C4—141no—
405C4⋊C4—141no—
406C4⋊C4—288no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—74no—
412C22⋊C4—142no—
413C22⋊C4—142no—
414C22⋊C4—144no—
415C22⋊C4—144no—
416C22⋊C4—142no—
417C22⋊C4—142no—
418C22⋊C4—142no—
419C42—21no—
420C42—21no—
421C42—142no—
422C42—142no—
423C42—142no—
424C42—142no—
425C42—284no—
426C42—288no—
427C2×C7—22no—
428C2×C7—22no—
429C2×C7—21no—
430C2×C7—21no—
431C2×C7—81no—
432C2×C7#233911yescharacteristic
433C2×C7C2×#41011yescharacteristic
434C2×C7#288411yescharacteristic
435C23—21no—
436C23—21no—
437C23—21no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23C22×C4×D711yescharacteristic
443C23C2×D4×D712yes—
444C23#654811yescharacteristic
445C23#655111yescharacteristic
446C23D7×#4112yes—
447Q8—282no—
448Q8—282no—
449D4—42no—
450D4—42no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—72no—
468C2×C4—72no—
469C2×C4—74no—
470C2×C4—142no—
471C2×C4—142no—
472C2×C4—142no—
473C2×C4—142no—
474C2×C4—144no—
475C2×C4—144no—
476C2×C4—144no—
477C2×C4—144no—
478C2×C4—142no—
479C2×C4—142no—
480C2×C4—144no—
481C2×C4—282no—
482C2×C4—282no—
483C2×C4—282no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4—282no—
487C2×C4—282no—
488C2×C4D7×C22⋊C411yescharacteristic
489C2×C4D7×C22⋊C412yes—
490C2×C4D7×C22⋊C411yescharacteristic
491C7#2247811yescharacteristic, complemented, Sylow
492C22—22no—
493C22—21no—
494C22—22no—
495C22—21no—
496C22—41no—
497C22—41no—
498C22—42no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—81no—
504C22C2×D7×C22⋊C411yescharacteristic
505C22#9019511yescharacteristic
506C22#9021311yescharacteristic
507C22#9030312yes—
508C22#9032212yes—
509C4—21no—
510C4—22no—
511C4—41no—
512C4—41no—
513C4—41no—
514C4—84no—
515C4—142no—
516C4—142no—
517C4—144no—
518C4—284no—
519C4—282no—
520C4—282no—
521C4—562no—
522C4#8951212yes—
523C2—22no—
524C2—22no—
525C2—21no—
526C2—21no—
527C2—81no—
528C2#1173145311yescharacteristic
529C2C2×#8951211yescharacteristic
530C2#1173613411yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC24C23×D7
FittingC7×#1799C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4C23
cosocleC24×C7C24
hypercenter#1799D7
hypocenterC7#22478
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. #11731323C2semidirect
6. #11727283C2semidirect
7. #11726707C2semidirect
8. #11726705C2semidirect
11. C2×#6489C4semidirect
13. C2×#6487C4semidirect
18. C2×#6452C4semidirect
19. C2×#6442C4semidirect
22. C2×#6433C4semidirect
23. C2×C4×Dic14C4semidirect
491. C7#22478semidirect

Maximal subgroups

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

subgroupindex
C2×#902322
#117313232
C7×#17992
C4×#64082
#117277832
#117272832
#117267072
#117267052
#224787