Finite Group Databasebeta

← #13508482

Subgroups — #13508482

Order 1792.

556 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739363C211yesmaximal, characteristic, complemented
2#11737033C211yesmaximal, characteristic
3C7×#2228C211yesmaximal, characteristic, complemented
4#11729110C211yesmaximal, characteristic, complemented
5#11728641C211yesmaximal, characteristic, complemented
6#11727107C211yesmaximal, characteristic, complemented
7#11727100C211yesmaximal, characteristic, complemented
8#11727088C211yesmaximal, characteristic, complemented
9#11727079C211yesmaximal, characteristic, complemented
10#11727043C211yesmaximal, characteristic, complemented
11#11727039C211yesmaximal, characteristic, complemented
12#11726944C211yesmaximal, characteristic
13D7×#357C211yesmaximal, characteristic
14#11726934C211yesmaximal, characteristic, complemented
15#11726932C211yesmaximal, characteristic, complemented
16C2×D7×D8C2211yescharacteristic
17#90380—21no—
18#90236—21no—
19C7×#499C2211yescharacteristic, complemented
20C4×C7×D8C2211yescharacteristic
21D8×Dic7C2211yescharacteristic
22#89837—21no—
23#89786—21no—
24C2×#6372C2211yescharacteristic
25C2×D7×C16C2211yescharacteristic
26#89652C2211yescharacteristic
27#89652C2211yescharacteristic
28D7×#107C2211yescharacteristic
29#89650C2211yescharacteristic, complemented
30#89646C2211yescharacteristic, complemented
31#89644C2211yescharacteristic
32#89633—21no—
33#89628—21no—
34#89557—21no—
35#89542—21no—
36C4×D56C2211yescharacteristic
37#89461C2211yescharacteristic
38#89454C2211yescharacteristic, complemented
39C7×#367C2211yescharacteristic, complemented
40C7×#366C2211yescharacteristic, complemented
41C7×C4⋊C16C2211yescharacteristic, complemented
42C7×#357C2211yescharacteristic
43C7×#357C2211yescharacteristic
44#89357C2211yescharacteristic
45#89357C2211yescharacteristic
46#89304C2211yescharacteristic
47#89304C2211yescharacteristic
48#89299C2211yescharacteristic, complemented
49#89299C2211yescharacteristic, complemented
50Dic7⋊C16C2211yescharacteristic, complemented
51Dic7⋊C16C2211yescharacteristic, complemented
52#89286C2211yescharacteristic
53#89286C2211yescharacteristic
54#89285C2211yescharacteristic, complemented
55#89285C2211yescharacteristic, complemented
56#89285C2211yescharacteristic, complemented
57#89284C2211yescharacteristic, complemented
58C4⋊(C7⋊C16)C2211yescharacteristic, complemented
59#33631—71nomaximal, Sylow
60C2×D4×D7—21no—
61C2×C7×D8C2311yescharacteristic
62C7×#125—21no—
63C4×C7×D4—21no—
64D4×Dic7—21no—
65C2×#1141—21no—
66C4×C22⋊D7—41no—
67C2×(C2×D7)⋊C4—41no—
68D7×D8—22no—
69D7×D8Q814yescomplemented
70C2×D56C2311yescharacteristic
71C2×#1131D411yescharacteristic
72C2×C8×D7C2311yescharacteristic
73#6460—41no—
74#6459—41no—
75#6456—21no—
76#6455—21no—
77#6455—21no—
78D7×C4⋊C4—21no—
79#6453—21no—
80#6452—41no—
81#6451—21no—
82#6451—41no—
83(C22⋊D7)⋊C4—41no—
84D7×C22⋊C4—41no—
85C4×D28—21no—
86#6435D411yescharacteristic
87#6432—21no—
88C2×C7×C16C2311yescharacteristic
89C2×C7×C16C2311yescharacteristic
90C7×#107C2311yescharacteristic
91C7×#107C2311yescharacteristic
92C7×#107C2311yescharacteristic
93C7×#106—21no—
94C7×#102—21no—
95C4×C7×C8C2311yescharacteristic
96#6406—21no—
97#6395—21no—
98#6392—21no—
99#6392C2311yescharacteristic
100#6392C2311yescharacteristic
101#6392C2311yescharacteristic
102#6389D411yescharacteristic
103C8×Dic7C2311yescharacteristic
104C2×C7⋊C16C2311yescharacteristic
105C2×C7⋊C16C2311yescharacteristic
106#6383—21no—
107#6382—21no—
108#6382—21no—
109#6382—21no—
110#6381—21no—
111#6381—21no—
112#6377D411yescharacteristic
113#6372—22no—
114D7×C16—22no—
115#3164—71no—
116#2948—71no—
117#2228—71no—
118#2228—71no—
119#2228—71no—
120#2228D711yescharacteristic, complemented
121#2226—71no—
122#2226—71no—
123#2224—71no—
124#2224—71no—
125#2159—71no—
126#2158—71no—
127#2151—71no—
128#2140—71no—
129C2×#357—71no—
130C23×D7—41no—
131C2×C7×D4—21no—
132C22×C4×C7—41no—
133C2×C22⋊D7—41no—
134C22×Dic7—41no—
135D4×D7—24no—
136D4×D7—42no—
137C2×D28—21no—
138C2×C4×D7—41no—
139C2×C4×D7—41no—
140C2×C4×D7—41no—
141C2×C4×D7C2×D411yescharacteristic
142C2×Dic14—41no—
143C2×Dic14—41no—
144C7×D8—21no—
145C7×D8C2×Q812yes—
146C2×C7×C8C2×D411yescharacteristic
147C2×C7×C8C2411yescharacteristic
148C7×C4⋊C4—21no—
149C7×C4⋊C4—21no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—41no—
152C7×C4⋊C4—41no—
153C7×C22⋊C4—41no—
154C42×C7C2×D411yescharacteristic
155C22⋊Dic7—41no—
156#1141—24no—
157(C2×D7)⋊C4—41no—
158(C2×D7)⋊C4—41no—
159(C2×D7)⋊C4—41no—
160(C2×D7)⋊C4—41no—
161(C2×D7)⋊C4D811yescharacteristic
162(C2×D7)⋊C4D811yescharacteristic
163C4⋊Dic7—21no—
164C4⋊Dic7—21no—
165C4⋊Dic7—21no—
166C4⋊Dic7—41no—
167C4⋊Dic7—41no—
168Dic7⋊C4—41no—
169Dic7⋊C4—41no—
170Dic7⋊C4—41no—
171Dic7⋊C4D811yescharacteristic
172Dic7⋊C4D811yescharacteristic
173C4×Dic7—41no—
174C4×Dic7—41no—
175C4×Dic7C2×D411yescharacteristic
176C2×C7⋊C8C2×D411yescharacteristic
177C2×C7⋊C8C2×D411yescharacteristic
178D56—21no—
179D56C2×Q812yes—
180#1131—22no—
181C8×D7C2×Q812yes—
182C8×D7#4112yes—
183C7×C16—21no—
184C7×C16—21no—
185C7⋊C16—21no—
186C7⋊C16—21no—
187C22×D8—71no—
188#544—141no—
189#518—141no—
190C2×M32—71no—
191C22×C16—71no—
192#501—141no—
193#501—141no—
194#500—71no—
195#500—71no—
196#499—71no—
197#499C2×D711yescharacteristic, complemented
198#442—141no—
199#442—141no—
200C4×D8—71no—
201C4×D8—71no—
202C4×D8—71no—
203C4×D8C2×D711yescharacteristic
204#428—141no—
205#427—71no—
206#427—71no—
207C2×#107—71no—
208#418—141no—
209#405—71no—
210#367—71no—
211#367—71no—
212#367—71no—
213#367C2×D711yescharacteristic, complemented
214#366—71no—
215#366—71no—
216#366—71no—
217#366C2×D711yescharacteristic, complemented
218C4⋊C16—71no—
219C4⋊C16—71no—
220C4⋊C16—71no—
221C4⋊C16C2×D711yescharacteristic, complemented
222#357—71no—
223#357—71no—
224#357—71no—
225#357—71no—
226#357—71no—
227#357—71no—
228#357C2×D711yescharacteristic
229#357C2×D711yescharacteristic
230C23×C7—41no—
231C22×D7—41no—
232C22×D7—44no—
233C22×D7—161no—
234C22×D7—161no—
235C22×D7C2×D811yescharacteristic
236C7×D4—22no—
237C7×D4—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—161no—
242C2×C4×C7C2×D811yescharacteristic
243C2×C4×C7C2×D811yescharacteristic
244C2×C4×C7C22×D411yescharacteristic
245C22⋊D7—44no—
246C2×Dic7—41no—
247C2×Dic7—41no—
248C2×Dic7—41no—
249C2×Dic7—161no—
250C2×Dic7C2×D811yescharacteristic
251C2×Dic7C2×D811yescharacteristic
252C2×Dic7C2×D811yescharacteristic
253D28—22no—
254D28—41no—
255C4×D7—161no—
256C4×D7—161no—
257C4×D7—161no—
258C4×D7C22⋊Q812yes—
259C4×D7#13712yes—
260Dic14—161no—
261Dic14—161no—
262C7×C8—21no—
263C7×C8C22×Q811yescharacteristic
264C7×C8C2×#4111yescharacteristic
265C7⋊C8—21no—
266C7⋊C8C22⋊Q812yes—
267C22×D4—141no—
268C2×D8—74no—
269C2×D8—71no—
270C2×D8—142no—
271C2×D8—141no—
272C2×D8C22×D711yescharacteristic
273C2×M16—71no—
274C22×C8—71no—
275C4⋊Q8—141no—
276C4⋊Q8—141no—
277C4⋊Q8—281no—
278#125—21no—
279#125—141no—
280#125—281no—
281C22⋊Q8—281no—
282C22⋊Q8—281no—
283C4×D4—21no—
284C4×D4—141no—
285C4×D4—141no—
286C4×D4—141no—
287C4×D4—281no—
288C4×D4—281no—
289#117—71no—
290#117—141no—
291#117—141no—
292C2×C4⋊C4—141no—
293C2×C22⋊C4—281no—
294C2×C22⋊C4—281no—
295M32—142no—
296C2×C16—71no—
297C2×C16—71no—
298C2×C16—142no—
299C2×C16C22×D711yescharacteristic
300C2×C16C22×D711yescharacteristic
301#107—71no—
302#107—71no—
303#107—71no—
304#107—141no—
305#107—141no—
306#107—141no—
307#107C22×D711yescharacteristic
308#107C22×D711yescharacteristic
309#107C22×D711yescharacteristic
310#106—21no—
311#106—141no—
312#106—141no—
313#106—141no—
314#102—21no—
315#102—141no—
316#102—141no—
317#102—141no—
318#97—71no—
319#97—71no—
320C4×C8—71no—
321C4×C8C22×D711yescharacteristic
322C22×C7—42no—
323C22×C7—81no—
324C22×C7—81no—
325C22×C7C22×D811yescharacteristic
326C2×D7—21no—
327C2×D7—21no—
328C2×D7—42no—
329C2×D7—81no—
330C2×D7—81no—
331C2×D7—164no—
332C2×D7#47412yes—
333C4×C7—21no—
334C4×C7—21no—
335C4×C7—81no—
336C4×C7—81no—
337C4×C7—81no—
338C4×C7C2×C22⋊Q811yescharacteristic
339C4×C7C2×#13711yescharacteristic
340Dic7—21no—
341Dic7—21no—
342Dic7—81no—
343Dic7—81no—
344Dic7—81no—
345Dic7#47412yes—
346C24—281no—
347C2×Q8—281no—
348C2×Q8—281no—
349C2×D4—21no—
350C2×D4—144no—
351C2×D4—141no—
352C2×D4—282no—
353C2×D4—281no—
354C22×C4—41no—
355C22×C4—71no—
356C22×C4—281no—
357C22×C4—281no—
358C22×C4—281no—
359C22×C4—281no—
360D8—21no—
361D8—72no—
362D8—144no—
363D8—141no—
364D8Q8×D712yes—
365M16—142no—
366C2×C8—71no—
367C2×C8—72no—
368C2×C8—72no—
369C2×C8—71no—
370C2×C8D4×D711yescharacteristic
371C2×C8C23×D711yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—71no—
378C4⋊C4—71no—
379C4⋊C4—141no—
380C4⋊C4—141no—
381C4⋊C4—141no—
382C4⋊C4—281no—
383C4⋊C4—281no—
384C4⋊C4—281no—
385C4⋊C4—281no—
386C4⋊C4—281no—
387C22⋊C4—41no—
388C22⋊C4—71no—
389C22⋊C4—71no—
390C22⋊C4—281no—
391C22⋊C4—281no—
392C22⋊C4—281no—
393C22⋊C4—281no—
394C22⋊C4—281no—
395C42—71no—
396C42—281no—
397C42—281no—
398C42D4×D711yescharacteristic
399C16—21no—
400C16—21no—
401C16—141no—
402C16—141no—
403C2×C7—82no—
404C2×C7C2×#47411yescharacteristic
405C2×C7#341511yescharacteristic
406C2×C7#341611yescharacteristic
407D7—22no—
408D7—82no—
409C23—41no—
410C23—71no—
411C23—284no—
412C23—281no—
413C23—1121no—
414C23—1121no—
415Q8—1121no—
416Q8—1121no—
417D4—22no—
418D4—41no—
419D4—142no—
420D4—284no—
421D4—281no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—71no—
426C2×C4—72no—
427C2×C4—72no—
428C2×C4—71no—
429C2×C4—71no—
430C2×C4—161no—
431C2×C4—281no—
432C2×C4—281no—
433C2×C4—281no—
434C2×C4—1121no—
435C2×C4—1121no—
436C2×C4—1121no—
437C2×C4—1121no—
438C2×C4D7×D811yescharacteristic
439C2×C4D7×D811yescharacteristic
440C2×C4C2×D4×D711yescharacteristic
441C8—21no—
442C8—72no—
443C8—141no—
444C8C2×Q8×D711yescharacteristic
445C8D7×#4111yescharacteristic
446C7#3363111yescharacteristic, complemented, Sylow
447C22—42no—
448C22—72no—
449C22—81no—
450C22—81no—
451C22—141no—
452C22—141no—
453C22—282no—
454C22—561no—
455C22—561no—
456C22—1124no—
457C22C2×D7×D811yescharacteristic
458C4—21no—
459C4—21no—
460C4—72no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—141no—
465C4—141no—
466C4—561no—
467C4—561no—
468C4—561no—
469C4D7×C22⋊Q811yescharacteristic
470C4D7×#13711yescharacteristic
471C2—82no—
472C2—142no—
473C2—562no—
474C2D7×#47411yescharacteristic
475C2#1173947611yescharacteristic
476C2#1173948211yescharacteristic
subgroupG/H
centerC22C2×D7×D8
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC7×#2228C2
socleC22×C7C22×D8
prime radicalC2×D7×D8C22
hypercenter#2228D7
hypocenterC7#33631
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2228

Complements

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

NH ≅ G/Ntype
1. #11739363C2semidirect
3. C7×#2228C2semidirect
4. #11729110C2semidirect
5. #11728641C2semidirect
6. #11727107C2semidirect
7. #11727100C2semidirect
8. #11727088C2semidirect
9. #11727079C2semidirect
10. #11727043C2semidirect
11. #11727039C2semidirect
14. #11726934C2semidirect
15. #11726932C2semidirect
19. C7×#499C22semidirect
29. #89650C22semidirect
30. #89646C22semidirect
38. #89454C22semidirect
39. C7×#367C22semidirect
40. C7×#366C22semidirect
41. C7×C4⋊C16C22semidirect
48. #89299C22semidirect
49. #89299C22semidirect
50. Dic7⋊C16C22semidirect
51. Dic7⋊C16C22semidirect
54. #89285C22semidirect
55. #89285C22semidirect
56. #89285C22semidirect
57. #89284C22semidirect
58. C4⋊(C7⋊C16)C22semidirect
69. D7×D8Q8semidirect
120. #2228D7semidirect
197. #499C2×D7semidirect
213. #367C2×D7semidirect
217. #366C2×D7semidirect
221. C4⋊C16C2×D7semidirect
446. C7#33631semidirect

Maximal subgroups

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

subgroupindex
#117393632
#117370332
C7×#22282
#117291102
#117286412
#117271072
#117271002
#117270882
#117270792
#117270432
#117270392
#117269442
D7×#3572
#117269342
#117269322
#336317