Finite Group Databasebeta

← #13388730

Subgroups — #13388730

Order 1792.

1068 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic
2C2×#89862C211yesmaximal, characteristic, complemented
3#11739925C211yesmaximal, characteristic, complemented
4C2×#89790C211yesmaximal, characteristic
5#11729540C211yesmaximal, characteristic, complemented
6#11729483C212yesmaximal
7#11729476C212yesmaximal
8#11729389C211yesmaximal, characteristic, complemented
9#11729386C212yesmaximal, complemented
10#11729382C212yesmaximal, complemented
11#11729359C211yesmaximal, characteristic
12C7×#556—24no—
13C7×#556—24no—
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×C7×#126—22no—
16C2×C7×#124C2212yes—
17C2×C7×C22⋊Q8—21no—
18C2×C4×C7×Q8C2212yes—
19C2×C4×C7×Q8C2211yescharacteristic
20C2×C4×C7×D4C2211yescharacteristic
21C2×C7×#117—21no—
22C22×#1144C2212yes—
23C22×#1142C2211yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25#90001—21no—
26C2×#6409C2212yes—
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6406C2211yescharacteristic
30C2×#6406C2212yes—
31#89862—24no—
32#89790—24no—
33#89754—21no—
34#89750—22no—
35C2×#6384C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89708C2211yescharacteristic
38#89701C2212yes—
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24201—71nomaximal, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2312yes—
48C22×C7×D4C2311yescharacteristic
49C23×C4×C7—21no—
50C22×Dic14C2311yescharacteristic
51C7×C4⋊Q8—22no—
52C7×C4⋊Q8D414yes—
53C7×#126—216no—
54C7×#124—24no—
55C7×#124—24no—
56C7×C22⋊Q8—28no—
57C4×C7×Q8—24no—
58C4×C7×Q8—24no—
59C4×C7×Q8—22no—
60C4×C7×Q8—22no—
61C4×C7×D4—22no—
62C4×C7×D4D414yes—
63C7×#117—28no—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4D412yes—
68C2×C7×C4⋊C4D411yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—22no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4—22no—
73C2×C42×C7—22no—
74C2×C42×C7C2312yes—
75C2×C42×C7C2311yescharacteristic
76C2×#1144—24no—
77C2×#1144—24no—
78C2×#1144—42no—
79C2×#1142—22no—
80C2×#1142—22no—
81C2×#1142—22no—
82C2×C4⋊Dic7C2311yescharacteristic
83C2×C4⋊Dic7C2312yes—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—22no—
86C2×C4×Dic7—21no—
87C22×C7⋊C8C2311yescharacteristic
88C22×C7⋊C8C2312yes—
89C22×C7⋊C8C2311yescharacteristic
90C4×Dic14—22no—
91C4×Dic14—22no—
92#6409—24no—
93#6409—24no—
94#6409—42no—
95#6406—22no—
96#6406—42no—
97C7⋊#95—22no—
98C7⋊#95—22no—
99C7⋊#95—22no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102#6384—22no—
103#6381—22no—
104C4⋊(C7⋊C8)—22no—
105#6377—22no—
106C2×#556D711yescharacteristic
107C2×#462—71no—
108C2×#440—71no—
109#2581—71no—
110#2042—72no—
111#2015—72no—
112#1985—71no—
113#1960—71no—
114#1878—72no—
115#1863—72no—
116#1804—71no—
117C24×C7—21no—
118C2×C7×Q8—22no—
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—42no—
125C2×C7×Q8C2×D414yes—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4C2×D412yes—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—24no—
135C22×C4×C7—44no—
136C22×C4×C7C2×D412yes—
137C22×C4×C7C2×D412yes—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2411yescharacteristic
141C22×Dic7—21no—
142C22×Dic7—22no—
143C22×Dic7—21no—
144C2×Dic14—21no—
145C2×Dic14—21no—
146C2×Dic14—21no—
147C2×Dic14—21no—
148C2×Dic14—21no—
149C2×Dic14#4112yes—
150C7×C4⋊C4—28no—
151C7×C4⋊C4—24no—
152C7×C4⋊C4—24no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4C2×D412yes—
157C7×C22⋊C4—28no—
158C7×C22⋊C4—28no—
159C7×C22⋊C4—24no—
160C42×C7—28no—
161C42×C7—22no—
162C42×C7—22no—
163C42×C7C2×D412yes—
164C42×C7C2×D412yes—
165#1144—48no—
166#1142—28no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—22no—
170C4⋊Dic7—22no—
171Dic7⋊C4—44no—
172Dic7⋊C4—42no—
173C4×Dic7—42no—
174C2×C7⋊C8—21no—
175C2×C7⋊C8—21no—
176C2×C7⋊C8—21no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—21no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8#4112yes—
183C22×Q16—72no—
184C22×SD16—71no—
185#556—24no—
186#556—24no—
187C2×C4⋊Q8C2×D711yescharacteristic
188C2×#126—22no—
189C2×#124C2×D712yes—
190C2×C22⋊Q8—21no—
191C2×C4×Q8—71no—
192C2×C4×Q8C2×D712yes—
193C2×C4×Q8C2×D711yescharacteristic
194C2×C4×D4C2×D711yescharacteristic
195C2×#117—21no—
196#462—144no—
197#440—144no—
198C2×#107—71no—
199C2×C4⋊C8—71no—
200C2×#103—72no—
201C2×#103—72no—
202C2×#103—71no—
203C2×#103—72no—
204C2×#102—71no—
205C2×#102—72no—
206C2×#97—71no—
207#400—141no—
208#395—141no—
209#391—72no—
210#389—72no—
211#387—142no—
212#382—71no—
213#340—72no—
214#336—72no—
215#336—72no—
216C23×C7—22no—
217C23×C7—41no—
218C23×C7—41no—
219C23×C7—42no—
220C23×C7—42no—
221C23×C7C22×D411yescharacteristic
222C7×Q8—24no—
223C7×Q8—24no—
224C7×Q8—44no—
225C7×Q8—42no—
226C7×Q8—42no—
227C7×D4—22no—
228C7×D4—22no—
229C7×D4—41no—
230C7×D4—41no—
231C2×C4×C7—22no—
232C2×C4×C7—22no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—24no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—24no—
240C2×C4×C7—48no—
241C2×C4×C7—42no—
242C2×C4×C7—42no—
243C2×C4×C7—42no—
244C2×C4×C7#12012yes—
245C2×C4×C7#12012yes—
246C2×C4×C7C22×D411yescharacteristic
247C2×C4×C7C22×D411yescharacteristic
248C2×C4×C7C2×#4111yescharacteristic
249C2×C4×C7C2×#4111yescharacteristic
250C2×C4×C7H(5,𝔽2)11yescharacteristic
251C2×C4×C72−1+411yescharacteristic
252C2×Dic7—22no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7—42no—
257C2×Dic7—42no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260Dic14—22no—
261Dic14—22no—
262Dic14—41no—
263Dic14—41no—
264C7⋊C8—22no—
265C7⋊C8—41no—
266C7⋊C8—42no—
267C22×Q8—71no—
268C22×Q8C22⋊D711yescharacteristic
269C22×Q8C22×D712yes—
270C22×D4C22×D711yescharacteristic
271C23×C4—21no—
272C2×Q16—144no—
273C2×Q16—144no—
274C2×Q16—282no—
275C2×SD16—142no—
276C2×SD16—142no—
277C2×SD16—142no—
278C22×C8—71no—
279C22×C8—72no—
280C22×C8—71no—
281C4⋊Q8—22no—
282C4⋊Q8C22⋊D714yes—
283#126—216no—
284#124—24no—
285#124—24no—
286C22⋊Q8—28no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—142no—
292C4×Q8—142no—
293C4×D4—22no—
294C4×D4C22⋊D714yes—
295#117—28no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—71no—
300C2×C4⋊C4—72no—
301C2×C4⋊C4—141no—
302C2×C4⋊C4—142no—
303C2×C4⋊C4C22⋊D711yescharacteristic
304C2×C4⋊C4C22⋊D712yes—
305C2×C4⋊C4C22×D711yescharacteristic
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—22no—
309C2×C42—22no—
310C2×C42—141no—
311C2×C42C22×D712yes—
312C2×C42C22×D711yescharacteristic
313#107—142no—
314C4⋊C8—142no—
315#103—144no—
316#103—144no—
317#103—142no—
318#103—282no—
319#102—142no—
320#102—282no—
321#97—142no—
322#95—141no—
323#95—141no—
324#95—142no—
325#95—142no—
326#95—142no—
327C22×C7—42no—
328C22×C7—42no—
329C22×C7—42no—
330C22×C7—82no—
331C22×C7—82no—
332C22×C7C2×#12011yescharacteristic
333C22×C7#54811yescharacteristic
334C22×C7#55011yescharacteristic
335C22×C7#57512yes—
336C22×C7#57812yes—
337C4×C7—22no—
338C4×C7—22no—
339C4×C7—44no—
340C4×C7—42no—
341C4×C7—42no—
342C4×C7—42no—
343C4×C7#52511yescharacteristic
344C4×C7#53711yescharacteristic
345C4×C7#54611yescharacteristic
346C4×C7#54711yescharacteristic
347Dic7—42no—
348Dic7—82no—
349Dic7—81no—
350C24—21no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—42no—
358C2×Q8—72no—
359C2×Q8—141no—
360C2×Q8—141no—
361C2×Q8—141no—
362C2×Q8—141no—
363C2×Q8—141no—
364C2×Q8C2×C22⋊D714yes—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4C2×C22⋊D712yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—44no—
375C22×C4—141no—
376C22×C4—142no—
377C22×C4—141no—
378C22×C4C2×C22⋊D712yes—
379C22×C4C2×C22⋊D712yes—
380C22×C4C2×C22⋊D711yescharacteristic
381C22×C4C2×C22⋊D711yescharacteristic
382C22×C4C23×D711yescharacteristic
383Q16—288no—
384SD16—148no—
385C2×C8—72no—
386C2×C8—141no—
387C2×C8—141no—
388C2×C8—142no—
389C2×C8—142no—
390C2×C8—141no—
391C2×C8—141no—
392C2×C8—141no—
393C2×C8—142no—
394C4⋊C4—28no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—142no—
401C4⋊C4—142no—
402C4⋊C4—141no—
403C4⋊C4—141no—
404C4⋊C4—284no—
405C4⋊C4—282no—
406C4⋊C4C2×C22⋊D712yes—
407C22⋊C4—28no—
408C22⋊C4—28no—
409C22⋊C4—24no—
410C42—28no—
411C42—22no—
412C42—22no—
413C42—282no—
414C42C2×C22⋊D712yes—
415C42C2×C22⋊D712yes—
416C2×C7—82no—
417C2×C7#240711yescharacteristic
418C2×C7#300511yescharacteristic
419C2×C7#301111yescharacteristic
420C2×C7#337712yes—
421C2×C7#340212yes—
422C23—22no—
423C23—42no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23C22×C22⋊D711yescharacteristic
428Q8—24no—
429Q8—24no—
430Q8—44no—
431Q8—42no—
432Q8—42no—
433Q8—142no—
434Q8—142no—
435Q8—281no—
436Q8—281no—
437D4—22no—
438D4—22no—
439D4—41no—
440D4—41no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—48no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—142no—
455C2×C4—281no—
456C2×C4—281no—
457C2×C4—282no—
458C2×C4—282no—
459C2×C4—282no—
460C2×C4—282no—
461C2×C4—281no—
462C2×C4#651612yes—
463C2×C4#651612yes—
464C2×C4D7×#4111yescharacteristic
465C2×C4D7×#4111yescharacteristic
466C2×C4#655311yescharacteristic
467C2×C4#655411yescharacteristic
468C2×C4C22×C22⋊D711yescharacteristic
469C2×C4C22×C22⋊D711yescharacteristic
470C8—142no—
471C8—282no—
472C8—281no—
473C7#2420111yescharacteristic, complemented, Sylow
474C22—42no—
475C22—42no—
476C22—42no—
477C22—82no—
478C22—82no—
479C22#9051312yes—
480C22#9052211yescharacteristic
481C22#9052511yescharacteristic
482C22#9052912yes—
483C22C2×#651611yescharacteristic
484C4—22no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—44no—
489C4—42no—
490C4—282no—
491C4—561no—
492C4—562no—
493C4#9051511yescharacteristic
494C4#9051911yescharacteristic
495C4#9052011yescharacteristic
496C4#9052411yescharacteristic
497C2—82no—
498C2#1174112312yes—
499C2#1174112612yes—
500C2#1174152611yescharacteristic
501C2#1174154211yescharacteristic
502C2#1174156211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#556C2
socleC23×C7C22×D4
prime radicalC22×C7×D4C23
hypercenterC2×#556D7
hypocenterC7#24201
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. C2×#89862C2semidirect
3. #11739925C2semidirect
5. #11729540C2semidirect
8. #11729389C2semidirect
9. #11729386C2semidirect
10. #11729382C2semidirect
473. C7#24201semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5562
C2×#898622
#117399252
C2×#897902
#117295402
#117294832
#117294762
#117293892
#117293862
#117293822
#117293592
#242017