Finite Group Databasebeta

← #13339260

Subgroups — #13339260

Order 1792.

662 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3035C211yesmaximal, characteristic
2#11740716C211yesmaximal, characteristic, complemented
3#11739965C211yesmaximal, characteristic, complemented
4#11739953C211yesmaximal, characteristic, complemented
5#11729560C211yesmaximal, characteristic, complemented
6#11728806C212yesmaximal, complemented
7#11728790C212yesmaximal, complemented
8#11728776C212yesmaximal, complemented
9#11728760C212yesmaximal, complemented
10C2×#89276C211yesmaximal, characteristic, complemented
11#11728175C211yesmaximal, characteristic, complemented
12C2×C7×#135—21no—
13C7×#532C2211yescharacteristic
14C4×C7×#41C2211yescharacteristic
15C2×#6432C2211yescharacteristic
16C7×#464—21no—
17C7×#463—21no—
18C7×#462C2211yescharacteristic
19C7×#461C2211yescharacteristic
20C7×#460C2211yescharacteristic
21C7×#459C2211yescharacteristic
22C2×C7×C4⋊C8C2211yescharacteristic
23C7×#416—21no—
24#90004—21no—
25#89851C2212yes—
26#89842C2212yes—
27#89806—22no—
28Q8×C7⋊C8C2212yes—
29#89793C2212yes—
30#89792—21no—
31D4×C7⋊C8—22no—
32D4×C7⋊C8C2212yes—
33#89780—21no—
34#89780—21no—
35#89780C2212yes—
36#89779—21no—
37#89767—21no—
38#89697—21no—
39#89697C2211yescharacteristic
40#89696C2211yescharacteristic
41C2×C4⋊(C7⋊C8)—21no—
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89342C2212yes—
44#89339C2212yes—
45#89276C2212yes—
46#89276C2212yes—
47#89275C2212yes—
48#89275C2212yes—
49#19414—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
76C2×C4⋊Dic7—21no—
77C2×C4⋊Dic7—21no—
78C2×#1136—21no—
79C22×C7⋊C8—21no—
80C22×C7⋊C8—22no—
81#6432—21no—
82#6432—21no—
83#6432—21no—
84#6432C2312yes—
85C7×C4⋊C8—21no—
86C7×C4⋊C8C2311yescharacteristic
87C7×C4⋊C8C2311yescharacteristic
88C7×#103—21no—
89C7×#103—21no—
90C7×#102—21no—
91C7×#102—21no—
92#6409—22no—
93#6406—22no—
94#6406—22no—
95#6406—22no—
96C22⋊(C7⋊C8)—22no—
97C22⋊(C7⋊C8)D412yes—
98#6382—22no—
99#6382—22no—
100#6382—22no—
101#6381—22no—
102C4⋊(C7⋊C8)—21no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—22no—
105C4⋊(C7⋊C8)D411yescharacteristic
106C4⋊(C7⋊C8)D411yescharacteristic
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2312yes—
109C4×C7⋊C8—22no—
110C4×C7⋊C8C2312yes—
111#3227—71no—
112#3079—71no—
113#3035D711yescharacteristic
114#2988—71no—
115#1678—72no—
116#1677—72no—
117#1672—72no—
118#1671—72no—
119#1570—71no—
120#1547—71no—
121C2×#333—71no—
122C7×#41—22no—
123C7×#41—42no—
124C7×#41—42no—
125C7×#41SD1614yes—
126C2×C7×Q8—21no—
127C2×C7×Q8C2×D411yescharacteristic
128C2×C7×D4—21no—
129C2×C7×D4—21no—
130C2×C7×D4—21no—
131C2×C7×D4—21no—
132C2×C7×D4—41no—
133C2×C7×D4C2×D411yescharacteristic
134C22×C4×C7—21no—
135C22×C4×C7—21no—
136C22×C4×C7—21no—
137C22×C4×C7—41no—
138C22×C4×C7C2×D411yescharacteristic
139C22×C4×C7C2×D411yescharacteristic
140C22×C4×C7#4111yescharacteristic
141C22×Dic7—41no—
142C2×Dic14—22no—
143C2×Dic14—41no—
144C2×Dic14—82no—
145C7×Q16—42no—
146C7×SD16—42no—
147C7×SD16—42no—
148C7×D8—42no—
149C2×C7×C8—21no—
150C2×C7×C8—21no—
151C2×C7×C8—42no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C4⋊C4—21no—
155C7×C4⋊C4—21no—
156C7×C4⋊C4C2×D411yescharacteristic
157C7×C4⋊C4C2×D411yescharacteristic
158C7×C22⋊C4—21no—
159C7×C22⋊C4—21no—
160C7×C22⋊C4—21no—
161C7×C22⋊C4—41no—
162C7×C22⋊C4—41no—
163C42×C7—21no—
164C42×C7—21no—
165C42×C7—21no—
166C42×C7C2×D411yescharacteristic
167C42×C7C2×D411yescharacteristic
168C42×C7#4111yescharacteristic
169C42×C7C2411yescharacteristic
170C4⋊Dic7—22no—
171C4⋊Dic7—22no—
172C4⋊Dic7—41no—
173C4⋊Dic7—41no—
174#1136—42no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177C2×C7⋊C8—82no—
178C2×C7⋊C8—81no—
179C2×C7⋊C8C2×D412yes—
180C2×C7⋊C8C2×D412yes—
181C2×#135—21no—
182#532C2×D711yescharacteristic
183C2×C4⋊Q8—71no—
184C4×#41C2×D711yescharacteristic
185#498—72no—
186#486—72no—
187#481—142no—
188#477—72no—
189#476—72no—
190#476—141no—
191#476—141no—
192#475—141no—
193#474—141no—
194#464—21no—
195#463—21no—
196#462C2×D711yescharacteristic
197#461C2×D711yescharacteristic
198#460C2×D711yescharacteristic
199#459C2×D711yescharacteristic
200C8×Q8—72no—
201C8×D4—72no—
202C8×D4—142no—
203#431—71no—
204#431—141no—
205#428—141no—
206#423—71no—
207C2×C4⋊C8—71no—
208C2×C4⋊C8—141no—
209C2×C4⋊C8C2×D711yescharacteristic
210#417—141no—
211#416—21no—
212#334—72no—
213#334—72no—
214#333—72no—
215#333—72no—
216#332—72no—
217#331—72no—
218C23×C7—21no—
219C23×C7—41no—
220C23×C7#12111yescharacteristic
221C7×Q8—21no—
222C7×Q8—41no—
223C7×Q8—41no—
224C7×Q8C2×SD1612yes—
225C7×D4—24no—
226C7×D4—21no—
227C7×D4—41no—
228C7×D4—41no—
229C7×D4—41no—
230C7×D4—41no—
231C7×D4—81no—
232C7×D4—82no—
233C7×D4C2×SD1612yes—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—21no—
237C2×C4×C7—24no—
238C2×C4×C7—21no—
239C2×C4×C7—21no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—41no—
243C2×C4×C7—41no—
244C2×C4×C7—81no—
245C2×C4×C7—82no—
246C2×C4×C7#12111yescharacteristic
247C2×C4×C7#12111yescharacteristic
248C2×C4×C7#12111yescharacteristic
249C2×C4×C7C2×SD1612yes—
250C2×C4×C7#13512yes—
251C2×C4×C7C22×D411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×Dic7—42no—
255C2×Dic7—82no—
256Dic14—82no—
257Dic14—82no—
258C7×C8—41no—
259C7×C8—41no—
260C7⋊C8—22no—
261C7⋊C8—22no—
262C7⋊C8—42no—
263C7⋊C8—42no—
264C2×#41—21no—
265C2×#41C22⋊D711yescharacteristic
266C22×Q8—141no—
267#135—44no—
268C2×Q16—21no—
269C2×SD16—21no—
270C2×SD16—21no—
271C2×D8—21no—
272C2×M16—141no—
273C22×C8—21no—
274C22×C8—142no—
275C22×C8—141no—
276C4⋊Q8—72no—
277C4⋊Q8—141no—
278C4⋊Q8—141no—
279C4⋊Q8—141no—
280C4⋊Q8C22×D711yescharacteristic
281#127C22×D711yescharacteristic
282#124—21no—
283#121—21no—
284#121—21no—
285C4×Q8C22⋊D711yescharacteristic
286C4×Q8C22×D711yescharacteristic
287C4×D4—21no—
288C4×D4—21no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4C22⋊D711yescharacteristic
292C4×D4C22×D711yescharacteristic
293#117—21no—
294#117C22⋊D711yescharacteristic
295C2×C4⋊C4—141no—
296C2×C4⋊C4—141no—
297C2×C42—21no—
298C2×C42C22×D711yescharacteristic
299#107—142no—
300#106—142no—
301#106—142no—
302#106—142no—
303C4⋊C8—21no—
304C4⋊C8—71no—
305C4⋊C8—71no—
306C4⋊C8—72no—
307C4⋊C8—72no—
308C4⋊C8—142no—
309C4⋊C8—141no—
310C4⋊C8—141no—
311C4⋊C8C22×D711yescharacteristic
312C4⋊C8C22×D711yescharacteristic
313#103—21no—
314#103—21no—
315#103—142no—
316#102—21no—
317#102—21no—
318#102—142no—
319#102—142no—
320#102—142no—
321C22⋊C8—72no—
322C22⋊C8—142no—
323C4×C8—72no—
324C4×C8—142no—
325C22×C7—21no—
326C22×C7—21no—
327C22×C7—22no—
328C22×C7—41no—
329C22×C7—41no—
330C22×C7—81no—
331C22×C7—81no—
332C22×C7—81no—
333C22×C7#46512yes—
334C22×C7C2×#12111yescharacteristic
335C4×C7—21no—
336C4×C7—22no—
337C4×C7—21no—
338C4×C7—21no—
339C4×C7—41no—
340C4×C7—41no—
341C4×C7—81no—
342C4×C7#46512yes—
343C4×C7C22×SD1611yescharacteristic
344C4×C7C2×#13511yescharacteristic
345C4×C7#57711yescharacteristic
346C4×C7#57811yescharacteristic
347Dic7—82no—
348#41—22no—
349#41—42no—
350#41—42no—
351#41#114214yes—
352C2×Q8—21no—
353C2×Q8—142no—
354C2×Q8—281no—
355C2×Q8—562no—
356C2×Q8C2×C22⋊D711yescharacteristic
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4C2×C22⋊D711yescharacteristic
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—41no—
367C22×C4—281no—
368C22×C4D4×D711yescharacteristic
369C22×C4#115911yescharacteristic
370C22×C4C2×C22⋊D711yescharacteristic
371Q16—42no—
372SD16—42no—
373SD16—42no—
374D8—42no—
375M16—282no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—42no—
379C2×C8—72no—
380C2×C8—72no—
381C2×C8—142no—
382C2×C8—142no—
383C2×C8—562no—
384C2×C8—561no—
385C4⋊C4—21no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—142no—
390C4⋊C4—142no—
391C4⋊C4—281no—
392C4⋊C4—281no—
393C4⋊C4C2×C22⋊D711yescharacteristic
394C4⋊C4C2×C22⋊D711yescharacteristic
395C22⋊C4—21no—
396C22⋊C4—21no—
397C22⋊C4—21no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42D4×D711yescharacteristic
404C42#115911yescharacteristic
405C42C2×C22⋊D711yescharacteristic
406C42C23×D711yescharacteristic
407C2×C7—22no—
408C2×C7—42no—
409C2×C7—161no—
410C2×C7#304611yescharacteristic
411C2×C7C2×#46511yescharacteristic
412C2×C7#306811yescharacteristic
413C23—21no—
414C23—41no—
415C23#650211yescharacteristic
416Q8—21no—
417Q8—41no—
418Q8—41no—
419Q8—562no—
420Q8—562no—
421Q8C2×#114212yes—
422D4—24no—
423D4—21no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—41no—
428D4—81no—
429D4—82no—
430D4C2×#114212yes—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—282no—
444C2×C4—562no—
445C2×C4C2×#114212yes—
446C2×C4#650211yescharacteristic
447C2×C4#650211yescharacteristic
448C2×C4#650211yescharacteristic
449C2×C4#651312yes—
450C2×C4C2×D4×D711yescharacteristic
451C2×C4C2×#115911yescharacteristic
452C2×C4C22×C22⋊D711yescharacteristic
453C8—41no—
454C8—41no—
455C8—142no—
456C8—142no—
457C8—282no—
458C8—282no—
459C7#1941411yescharacteristic, complemented, Sylow
460C22—21no—
461C22—21no—
462C22—22no—
463C22—41no—
464C22—41no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22#8981312yes—
469C22C2×#650211yescharacteristic
470C4—21no—
471C4—21no—
472C4—22no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—81no—
477C4—562no—
478C4#8981312yes—
479C4#9046111yescharacteristic
480C4#9046211yescharacteristic
481C4C22×#114211yescharacteristic
482C4C2×#651311yescharacteristic
483C2—22no—
484C2—42no—
485C2—161no—
486C2C2×#8981311yescharacteristic
487C2#1174134711yescharacteristic
488C2#1174144711yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3035C2
socleC22×C7C2×#121
hypercenter#3035D7
hypocenterC7#19414
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. #11740716C2semidirect
3. #11739965C2semidirect
4. #11739953C2semidirect
5. #11729560C2semidirect
6. #11728806C2semidirect
7. #11728790C2semidirect
8. #11728776C2semidirect
9. #11728760C2semidirect
10. C2×#89276C2semidirect
11. #11728175C2semidirect
459. C7#19414semidirect

Maximal subgroups

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

subgroupindex
C7×#30352
#117407162
#117399652
#117399532
#117295602
#117288062
#117287902
#117287762
#117287602
C2×#892762
#117281752
#194147