Finite Group Databasebeta

← #13339282

Subgroups — #13339282

Order 1792.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3035C211yesmaximal, characteristic
2#11740715C211yesmaximal, characteristic, complemented
3#11739978C211yesmaximal, characteristic, complemented
4#11739965C211yesmaximal, characteristic, complemented
5#11729560C211yesmaximal, characteristic, complemented
6#11728807C212yesmaximal, complemented
7#11728791C212yesmaximal, complemented
8#11728777C212yesmaximal, complemented
9#11728761C212yesmaximal, complemented
10C2×#89277C211yesmaximal, characteristic, complemented
11#11728176C211yesmaximal, characteristic, complemented
12C2×C7×#135—21no—
13C7×#532C2211yescharacteristic
14C4×C7×#41C2211yescharacteristic
15C2×#6514—21no—
16C2×#6432C2211yescharacteristic
17C7×#464—21no—
18C7×#463—21no—
19C7×#462C2211yescharacteristic
20C7×#461C2211yescharacteristic
21C7×#460C2211yescharacteristic
22C7×#459C2211yescharacteristic
23C2×C7×C4⋊C8C2211yescharacteristic
24C7×#416—21no—
25#89850C2212yes—
26#89842C2212yes—
27#89805—22no—
28#89803—21no—
29#89803C2212yes—
30Q8×C7⋊C8C2212yes—
31#89791—21no—
32#89791C2212yes—
33D4×C7⋊C8—22no—
34D4×C7⋊C8C2212yes—
35#89779—21no—
36#89779—21no—
37#89776—21no—
38#89697—21no—
39#89697C2211yescharacteristic
40#89696C2211yescharacteristic
41C2×C4⋊(C7⋊C8)—21no—
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89342C2212yes—
44#89339C2212yes—
45#89277C2212yes—
46#89277C2212yes—
47#89274C2212yes—
48#89274C2212yes—
49#19416—71nomaximal, Sylow
50C2×C7×#41—21no—
51C2×C7×#41D411yescharacteristic
52C22×Dic14—21no—
53C7×#135—44no—
54C2×C7×Q16—21no—
55C2×C7×SD16—21no—
56C2×C7×SD16—21no—
57C2×C7×D8—21no—
58C22×C7×C8—21no—
59C7×C4⋊Q8C2311yescharacteristic
60C7×#127C2311yescharacteristic
61C7×#124—21no—
62C7×#121—21no—
63C7×#121—21no—
64C4×C7×Q8D411yescharacteristic
65C4×C7×Q8C2311yescharacteristic
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C4×C7×D4—21no—
69C4×C7×D4—21no—
70C4×C7×D4D411yescharacteristic
71C4×C7×D4C2311yescharacteristic
72C7×#117—21no—
73C7×#117D411yescharacteristic
74C2×C42×C7—21no—
75C2×C42×C7C2311yescharacteristic
76#6514—44no—
77C2×#1144—22no—
78C2×#1142—22no—
79C2×C4⋊Dic7—21no—
80C2×C4⋊Dic7—21no—
81C2×#1136—21no—
82C22×C7⋊C8—21no—
83C22×C7⋊C8—22no—
84#6432—21no—
85#6432—21no—
86#6432—21no—
87#6432C2312yes—
88C7×C4⋊C8—21no—
89C7×C4⋊C8C2311yescharacteristic
90C7×C4⋊C8C2311yescharacteristic
91C7×#103—21no—
92C7×#103—21no—
93C7×#102—21no—
94C7×#102—21no—
95#6406—22no—
96#6406—22no—
97C22⋊(C7⋊C8)—22no—
98C22⋊(C7⋊C8)D412yes—
99#6384—22no—
100#6384—22no—
101#6381—22no—
102#6381—22no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)—22no—
106C4⋊(C7⋊C8)D411yescharacteristic
107C4⋊(C7⋊C8)D411yescharacteristic
108C4⋊(C7⋊C8)C2312yes—
109C4⋊(C7⋊C8)C2312yes—
110C4×C7⋊C8—22no—
111C4×C7⋊C8C2312yes—
112#3226—71no—
113#3039—71no—
114#3035D711yescharacteristic
115#2988—71no—
116#1682—72no—
117#1681—72no—
118#1674—72no—
119#1673—72no—
120#1571—71no—
121#1547—71no—
122C2×#332—71no—
123C7×#41—22no—
124C7×#41—42no—
125C7×#41—42no—
126C7×#41D814yes—
127C2×C7×Q8—21no—
128C2×C7×Q8C2×D411yescharacteristic
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—21no—
133C2×C7×D4—41no—
134C2×C7×D4C2×D411yescharacteristic
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—41no—
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7C2×D411yescharacteristic
141C22×C4×C7#4111yescharacteristic
142C22×Dic7—41no—
143C2×Dic14—22no—
144C2×Dic14—42no—
145C2×Dic14—42no—
146C2×Dic14—41no—
147C7×Q16—42no—
148C7×SD16—42no—
149C7×SD16—42no—
150C7×D8—42no—
151C2×C7×C8—21no—
152C2×C7×C8—21no—
153C2×C7×C8—42no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4—21no—
157C7×C4⋊C4—21no—
158C7×C4⋊C4C2×D411yescharacteristic
159C7×C4⋊C4C2×D411yescharacteristic
160C7×C22⋊C4—21no—
161C7×C22⋊C4—21no—
162C7×C22⋊C4—21no—
163C7×C22⋊C4—41no—
164C7×C22⋊C4—41no—
165C42×C7—21no—
166C42×C7—21no—
167C42×C7—21no—
168C42×C7C2×D411yescharacteristic
169C42×C7C2×D411yescharacteristic
170C42×C7#4111yescharacteristic
171C42×C7C2411yescharacteristic
172#1144—44no—
173#1142—44no—
174C4⋊Dic7—22no—
175C4⋊Dic7—22no—
176C4⋊Dic7—41no—
177C4⋊Dic7—41no—
178#1136—42no—
179C2×C7⋊C8—22no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—82no—
182C2×C7⋊C8—81no—
183C2×C7⋊C8C2×D412yes—
184C2×C7⋊C8C2×D412yes—
185C2×#137—141no—
186C2×#135—21no—
187#532C2×D711yescharacteristic
188C2×C4⋊Q8—71no—
189C4×#41C2×D711yescharacteristic
190#500—72no—
191#486—72no—
192#480—142no—
193#474—141no—
194#474—141no—
195#464—21no—
196#463—21no—
197#462—72no—
198#462—141no—
199#462C2×D711yescharacteristic
200#461—72no—
201#461—141no—
202#461C2×D711yescharacteristic
203#460C2×D711yescharacteristic
204#459C2×D711yescharacteristic
205C8×Q8—72no—
206C8×D4—72no—
207C8×D4—142no—
208#431—71no—
209#431—141no—
210#423—71no—
211C2×C4⋊C8—71no—
212C2×C4⋊C8—141no—
213C2×C4⋊C8C2×D711yescharacteristic
214#419—141no—
215#416—21no—
216#335—72no—
217#335—72no—
218#332—72no—
219#332—72no—
220#332—72no—
221#331—72no—
222C23×C7—21no—
223C23×C7—41no—
224C23×C7#12111yescharacteristic
225C7×Q8—21no—
226C7×Q8—41no—
227C7×Q8—41no—
228C7×Q8C2×D812yes—
229C7×D4—24no—
230C7×D4—21no—
231C7×D4—41no—
232C7×D4—41no—
233C7×D4—41no—
234C7×D4—41no—
235C7×D4—81no—
236C7×D4—82no—
237C7×D4C2×D812yes—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—21no—
241C2×C4×C7—24no—
242C2×C4×C7—21no—
243C2×C4×C7—21no—
244C2×C4×C7—41no—
245C2×C4×C7—41no—
246C2×C4×C7—41no—
247C2×C4×C7—41no—
248C2×C4×C7—81no—
249C2×C4×C7—82no—
250C2×C4×C7#12111yescharacteristic
251C2×C4×C7#12111yescharacteristic
252C2×C4×C7#12111yescharacteristic
253C2×C4×C7C2×D812yes—
254C2×C4×C7#13512yes—
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C2×#4111yescharacteristic
258C2×Dic7—42no—
259C2×Dic7—82no—
260Dic14—42no—
261Dic14—42no—
262Dic14—82no—
263C7×C8—41no—
264C7×C8—41no—
265C7⋊C8—22no—
266C7⋊C8—22no—
267C7⋊C8—42no—
268C7⋊C8—42no—
269C2×#41—21no—
270C2×#41C22⋊D711yescharacteristic
271C22×Q8—141no—
272#137—284no—
273#135—44no—
274C2×Q16—21no—
275C2×Q16—142no—
276C2×SD16—21no—
277C2×SD16—21no—
278C2×SD16—142no—
279C2×D8—21no—
280C2×M16—141no—
281C22×C8—21no—
282C22×C8—142no—
283C22×C8—141no—
284C4⋊Q8—72no—
285C4⋊Q8—141no—
286C4⋊Q8—141no—
287C4⋊Q8—141no—
288C4⋊Q8C22×D711yescharacteristic
289#127C22×D711yescharacteristic
290#124—21no—
291#121—21no—
292#121—21no—
293C4×Q8C22⋊D711yescharacteristic
294C4×Q8C22×D711yescharacteristic
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4C22⋊D711yescharacteristic
300C4×D4C22×D711yescharacteristic
301#117—21no—
302#117C22⋊D711yescharacteristic
303C2×C4⋊C4—141no—
304C2×C4⋊C4—141no—
305C2×C42—21no—
306C2×C42C22×D711yescharacteristic
307#107—142no—
308#107—142no—
309C4⋊C8—21no—
310C4⋊C8—71no—
311C4⋊C8—71no—
312C4⋊C8—72no—
313C4⋊C8—72no—
314C4⋊C8—142no—
315C4⋊C8—141no—
316C4⋊C8—141no—
317C4⋊C8C22×D711yescharacteristic
318C4⋊C8C22×D711yescharacteristic
319#103—21no—
320#103—21no—
321#103—142no—
322#103—142no—
323#102—21no—
324#102—21no—
325#102—142no—
326#102—142no—
327C22⋊C8—72no—
328C22⋊C8—142no—
329C4×C8—72no—
330C4×C8—142no—
331C22×C7—21no—
332C22×C7—21no—
333C22×C7—22no—
334C22×C7—41no—
335C22×C7—41no—
336C22×C7—81no—
337C22×C7—81no—
338C22×C7—81no—
339C22×C7#46612yes—
340C22×C7C2×#12111yescharacteristic
341C4×C7—21no—
342C4×C7—22no—
343C4×C7—21no—
344C4×C7—21no—
345C4×C7—41no—
346C4×C7—41no—
347C4×C7—81no—
348C4×C7#46612yes—
349C4×C7C22×D811yescharacteristic
350C4×C7C2×#13511yescharacteristic
351C4×C7#57711yescharacteristic
352C4×C7#57811yescharacteristic
353Dic7—82no—
354#41—22no—
355#41—42no—
356#41—42no—
357#41#114114yes—
358C2×Q8—21no—
359C2×Q8—142no—
360C2×Q8—281no—
361C2×Q8—282no—
362C2×Q8—282no—
363C2×Q8C2×C22⋊D711yescharacteristic
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4C2×C22⋊D711yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—281no—
375C22×C4D4×D711yescharacteristic
376C22×C4#115911yescharacteristic
377C22×C4C2×C22⋊D711yescharacteristic
378Q16—42no—
379Q16—284no—
380SD16—42no—
381SD16—42no—
382SD16—284no—
383D8—42no—
384M16—282no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—42no—
388C2×C8—72no—
389C2×C8—72no—
390C2×C8—142no—
391C2×C8—142no—
392C2×C8—562no—
393C2×C8—561no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—142no—
399C4⋊C4—142no—
400C4⋊C4—281no—
401C4⋊C4—281no—
402C4⋊C4C2×C22⋊D711yescharacteristic
403C4⋊C4C2×C22⋊D711yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42D4×D711yescharacteristic
413C42#115911yescharacteristic
414C42C2×C22⋊D711yescharacteristic
415C42C23×D711yescharacteristic
416C2×C7—22no—
417C2×C7—42no—
418C2×C7—161no—
419C2×C7#304611yescharacteristic
420C2×C7C2×#46611yescharacteristic
421C2×C7#306911yescharacteristic
422C23—21no—
423C23—41no—
424C23#650211yescharacteristic
425Q8—21no—
426Q8—41no—
427Q8—41no—
428Q8—282no—
429Q8—282no—
430Q8—562no—
431Q8C2×#114112yes—
432D4—24no—
433D4—21no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—41no—
438D4—81no—
439D4—82no—
440D4C2×#114112yes—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—282no—
454C2×C4—562no—
455C2×C4C2×#114112yes—
456C2×C4#650211yescharacteristic
457C2×C4#650211yescharacteristic
458C2×C4#650211yescharacteristic
459C2×C4#651312yes—
460C2×C4C2×D4×D711yescharacteristic
461C2×C4C2×#115911yescharacteristic
462C2×C4C22×C22⋊D711yescharacteristic
463C8—41no—
464C8—41no—
465C8—142no—
466C8—142no—
467C8—282no—
468C8—282no—
469C7#1941611yescharacteristic, complemented, Sylow
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22#8981012yes—
479C22C2×#650211yescharacteristic
480C4—21no—
481C4—21no—
482C4—22no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—562no—
488C4#8981012yes—
489C4#9046111yescharacteristic
490C4#9046211yescharacteristic
491C4C22×#114111yescharacteristic
492C4C2×#651311yescharacteristic
493C2—22no—
494C2—42no—
495C2—161no—
496C2C2×#8981011yescharacteristic
497C2#1174134711yescharacteristic
498C2#1174144811yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3035C2
socleC22×C7C2×#121
hypercenter#3035D7
hypocenterC7#19416
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
2. #11740715C2semidirect
3. #11739978C2semidirect
4. #11739965C2semidirect
5. #11729560C2semidirect
6. #11728807C2semidirect
7. #11728791C2semidirect
8. #11728777C2semidirect
9. #11728761C2semidirect
10. C2×#89277C2semidirect
11. #11728176C2semidirect
469. C7#19416semidirect

Maximal subgroups

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

subgroupindex
C7×#30352
#117407152
#117399782
#117399652
#117295602
#117288072
#117287912
#117287772
#117287612
C2×#892772
#117281762
#194167