Finite Group Databasebeta

← #13247183

Subgroups — #13247183

Order 1792.

868 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2280C211yesmaximal, characteristic
2#11729881C211yesmaximal, characteristic
3#11729880C211yesmaximal, characteristic
4#11725430C212yesmaximal, complemented
5#11725430C212yesmaximal, complemented
6C24×C4×C7C2211yescharacteristic
7C2×C7×#117C411yescharacteristic, complemented
8C2×C7×#117C411yescharacteristic, complemented
9C22×C22⋊Dic7—21no—
10C7×#381—21no—
11C7×#380C411yescharacteristic, complemented
12C7×#380C411yescharacteristic, complemented
13C7×#380C2211yescharacteristic
14C7×#380C2211yescharacteristic
15C7×#379—21no—
16C4×C7×C22⋊C4—21no—
17C4×C7×C22⋊C4—21no—
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2212yes—
20C2×C22⋊(C7⋊C8)—21no—
21C2×C22⋊(C7⋊C8)—21no—
22C2×C22⋊(C7⋊C8)C2212yes—
23#8715—71nomaximal, Sylow
24C25×C7D411yescharacteristic
25C23×C4×C7—24no—
26C23×C4×C7—42no—
27C23×C4×C7—42no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7C2311yescharacteristic
34C23×Dic7—22no—
35C7×#117—42no—
36C7×#117—42no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C42×C7—21no—
48C2×C42×C7—21no—
49C2×C22⋊Dic7—24no—
50C2×C22⋊Dic7—42no—
51C2×C22⋊Dic7—42no—
52C22×C7⋊C8—22no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7⋊#95—24no—
58C7⋊#95—42no—
59C22⋊(C7⋊C8)—24no—
60C22⋊(C7⋊C8)—42no—
61C22⋊(C7⋊C8)—42no—
62#2280D711yescharacteristic
63#1779—71no—
64C22⋊#95—71no—
65#1294—72no—
66#1294—72no—
67C24×C7—22no—
68C24×C7—22no—
69C24×C7—21no—
70C24×C7—21no—
71C24×C7—22no—
72C24×C7—22no—
73C24×C7—41no—
74C24×C7—41no—
75C24×C7C22⋊C411yescharacteristic
76C24×C7C22⋊C411yescharacteristic
77C24×C7C2×D411yescharacteristic
78C22×C4×C7—22no—
79C22×C4×C7—22no—
80C22×C4×C7—22no—
81C22×C4×C7—22no—
82C22×C4×C7—22no—
83C22×C4×C7—22no—
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7—41no—
97C22×C4×C7—41no—
98C22×C4×C7—41no—
99C22×C4×C7—416no—
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22×C411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7#4114yes—
105C22×Dic7—24no—
106C22×Dic7—42no—
107C22×Dic7—42no—
108C22×Dic7—42no—
109C22×Dic7—42no—
110C22×Dic7—42no—
111C22×Dic7—42no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—42no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C42×C7—42no—
124C42×C7—42no—
125C22⋊Dic7—48no—
126C2×C7⋊C8—42no—
127C2×C7⋊C8—42no—
128C2×C7⋊C8—42no—
129C24×C4C2×D711yescharacteristic
130C2×#117Dic711yescharacteristic, complemented
131C2×#117Dic711yescharacteristic, complemented
132C22×C22⋊C4—141no—
133C2×C22⋊C8—72no—
134C2×C22⋊C8—141no—
135C2×C22⋊C8—141no—
136#381—21no—
137#380Dic711yescharacteristic, complemented
138#380Dic711yescharacteristic, complemented
139#380C2×D711yescharacteristic
140#380C2×D711yescharacteristic
141#379—21no—
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C2×#95—72no—
145C2×#95—141no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—24no—
151C23×C7—24no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7—21no—
163C23×C7—21no—
164C23×C7—44no—
165C23×C7—44no—
166C23×C7—44no—
167C23×C7—44no—
168C23×C7—42no—
169C23×C7—42no—
170C23×C7—42no—
171C23×C7—42no—
172C23×C7#9912yes—
173C23×C7#10412yes—
174C23×C7C2×C22⋊C411yescharacteristic
175C23×C7#12312yes—
176C23×C7#13512yes—
177C23×C7AGL(1,ℤ/8)11yescharacteristic
178C23×C7#13711yescharacteristic
179C2×C4×C7—28no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—21no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—44no—
190C2×C4×C7—44no—
191C2×C4×C7—44no—
192C2×C4×C7—44no—
193C2×C4×C7—44no—
194C2×C4×C7—41no—
195C2×C4×C7—41no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7#9912yes—
207C2×C4×C7#11712yes—
208C2×C4×C7#12312yes—
209C2×Dic7—44no—
210C2×Dic7—44no—
211C2×Dic7—44no—
212C2×Dic7—82no—
213C2×Dic7—82no—
214C2×Dic7—82no—
215C2×Dic7—82no—
216C7⋊C8—82no—
217C25C22⋊D711yescharacteristic
218C23×C4—24no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—142no—
222C23×C4C2×Dic711yescharacteristic
223C23×C4C2×Dic711yescharacteristic
224C23×C4C22⋊D711yescharacteristic
225C23×C4C22⋊D711yescharacteristic
226C23×C4C22⋊D711yescharacteristic
227C23×C4C22×D711yescharacteristic
228C22×C8—142no—
229#117—42no—
230#117—42no—
231C2×C4⋊C4C2×Dic711yescharacteristic
232C2×C4⋊C4C2×Dic711yescharacteristic
233C2×C4⋊C4C2×Dic711yescharacteristic
234C2×C4⋊C4C2×Dic711yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—144no—
242C2×C22⋊C4—282no—
243C2×C22⋊C4—282no—
244C2×C42—21no—
245C2×C42—21no—
246C22⋊C8—144no—
247C22⋊C8—282no—
248C22⋊C8—282no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—144no—
254#95—282no—
255C22×C7—22no—
256C22×C7—22no—
257C22×C7—22no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—22no—
263C22×C7—22no—
264C22×C7—22no—
265C22×C7—22no—
266C22×C7—22no—
267C22×C7—22no—
268C22×C7—22no—
269C22×C7—48no—
270C22×C7—48no—
271C22×C7—41no—
272C22×C7—41no—
273C22×C7—41no—
274C22×C7—41no—
275C22×C7—41no—
276C22×C7—41no—
277C22×C7—41no—
278C22×C7—41no—
279C22×C7#38111yescharacteristic
280C22×C7C2×#9911yescharacteristic
281C22×C7C2×#9911yescharacteristic
282C22×C7#41611yescharacteristic
283C22×C7#41711yescharacteristic
284C22×C7C2×#10411yescharacteristic
285C22×C7#42111yescharacteristic
286C22×C7#48212yes—
287C22×C7#48512yes—
288C4×C7—22no—
289C4×C7—22no—
290C4×C7—22no—
291C4×C7—41no—
292C4×C7—81no—
293C4×C7—81no—
294C4×C7—81no—
295C4×C7—81no—
296Dic7—84no—
297C24—22no—
298C24—22no—
299C24—21no—
300C24—21no—
301C24—22no—
302C24—22no—
303C24—41no—
304C24—41no—
305C24C22⋊Dic711yescharacteristic
306C24C22⋊Dic711yescharacteristic
307C24C2×C22⋊D711yescharacteristic
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—416no—
330C22×C4—144no—
331C22×C4—282no—
332C22×C4—282no—
333C22×C4—282no—
334C22×C4—282no—
335C22×C4—282no—
336C22×C4—282no—
337C22×C4C22⋊Dic711yescharacteristic
338C22×C4C22⋊Dic711yescharacteristic
339C22×C4#115714yes—
340C22×C4C22×Dic711yescharacteristic
341C22×C4C2×C22⋊D711yescharacteristic
342C2×C8—282no—
343C2×C8—282no—
344C2×C8—282no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—41no—
354C22⋊C4—42no—
355C22⋊C4—41no—
356C22⋊C4—288no—
357C42—42no—
358C42—42no—
359C2×C7—22no—
360C2×C7—22no—
361C2×C7—42no—
362C2×C7—42no—
363C2×C7#150111yescharacteristic
364C2×C7#150411yescharacteristic
365C2×C7#151311yescharacteristic
366C2×C7#151611yescharacteristic
367C2×C7#181911yescharacteristic
368C2×C7#182511yescharacteristic
369C2×C7#182611yescharacteristic
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—24no—
375C23—24no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23#640812yes—
397C23#641112yes—
398C23#649212yes—
399C23#651211yescharacteristic
400C23#651312yes—
401C23#651411yescharacteristic
402C23C2×C22⋊Dic711yescharacteristic
403C2×C4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—284no—
431C2×C4—284no—
432C2×C4—284no—
433C2×C4—562no—
434C2×C4—562no—
435C2×C4—562no—
436C2×C4—562no—
437C2×C4#640812yes—
438C2×C4#648912yes—
439C2×C4#649212yes—
440C8—562no—
441C7#871511yescharacteristic, complemented, Sylow
442C22—22no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—48no—
457C22—48no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22#8976812yes—
467C22#8977512yes—
468C22#8998311yescharacteristic
469C22C2×#640811yescharacteristic
470C22C2×#640811yescharacteristic
471C22#9000411yescharacteristic
472C22#9000511yescharacteristic
473C22C2×#641111yescharacteristic
474C22#9000811yescharacteristic
475C4—22no—
476C4—22no—
477C4—22no—
478C4—41no—
479C4—81no—
480C4—81no—
481C4—81no—
482C4—81no—
483C4—564no—
484C2—22no—
485C2—22no—
486C2—42no—
487C2—42no—
488C2#1172929611yescharacteristic
489C2#1172932911yescharacteristic
490C2#1172934411yescharacteristic
491C2#1172946111yescharacteristic
492C2#1172946211yescharacteristic
493C2#1172946311yescharacteristic
494C2#1172946411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2280C2
socleC23×C7C2×C22⋊C4
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2280D7
hypocenterC7#8715
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
4. #11725430C2semidirect
5. #11725430C2semidirect
7. C2×C7×#117C4semidirect
8. C2×C7×#117C4semidirect
11. C7×#380C4semidirect
12. C7×#380C4semidirect
130. C2×#117Dic7semidirect
131. C2×#117Dic7semidirect
137. #380Dic7semidirect
138. #380Dic7semidirect
441. C7#8715semidirect

Maximal subgroups

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

subgroupindex
C7×#22802
#117298812
#117298802
#117254302
#87157