Finite Group Databasebeta

← #13360746

Subgroups — #13360746

Order 1792.

868 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2875C211yesmaximal, characteristic, complemented
2#11741250C211yesmaximal, characteristic
3#11741237C212yesmaximal, complemented
4C2×#89489C211yesmaximal, characteristic, complemented
5#11729767C211yesmaximal, characteristic, complemented
6#11729399C212yesmaximal, complemented
7#11727655C211yesmaximal, characteristic, complemented
8#11727652C212yesmaximal, complemented
9#11727651C211yesmaximal, characteristic, complemented
10#11727628C212yesmaximal, complemented
11#11727583C211yesmaximal, characteristic, complemented
12C2×C7×#131C2211yescharacteristic
13C2×C7×#131C2211yescharacteristic
14C4×C7×#41C2211yescharacteristic
15#90513C2212yes—
16C2×#6467C2211yescharacteristic
17C2×#6464C2212yes—
18C7×#405—21no—
19C7×#405—21no—
20C7×#405C2211yescharacteristic
21C7×#405C2211yescharacteristic
22C4×C7×M16—21no—
23C4×C7×M16C2211yescharacteristic
24C2×C7×#97—21no—
25C2×C7×#97C2211yescharacteristic
26#90008C2212yes—
27#89917—24no—
28#89916—24no—
29#89915C412yescomplemented
30#89914C412yescomplemented
31#89913C414yescomplemented
32#89911C2212yes—
33#89904C2212yes—
34#89896C2212yes—
35C2×#6393C2211yescharacteristic
36C2×#6393C2211yescharacteristic
37#89878C2211yescharacteristic
38#89873C2212yes—
39#89778C2212yes—
40C2×#6379C2212yes—
41#89489—21no—
42#89489—21no—
43#89489—21no—
44#89489—21no—
45#89354C2212yes—
46#89328C2212yes—
47#89318C2212yes—
48#21559—71nomaximal, Sylow
49C2×C7×#41C2311yescharacteristic
50C2×#1157C2312yes—
51C7×#131—21no—
52C7×#131—21no—
53C7×#131—22no—
54C7×#131C2×C412yes—
55C7×#131C2×C412yes—
56C7×#131D414yes—
57C2×C7×M16—21no—
58C2×C7×M16—21no—
59C2×C7×M16C2311yescharacteristic
60C2×C7×M16C2311yescharacteristic
61C22×C7×C8—21no—
62C22×C7×C8—21no—
63C22×C7×C8C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C4×C7×Q8—21no—
66C4×C7×D4—21no—
67C4×C7×D4—22no—
68C7×#117—21no—
69C7×#117C2311yescharacteristic
70C2×C42×C7—21no—
71C2×C42×C7C2311yescharacteristic
72#6514C2×C414yes—
73#6513—24no—
74#6512C2×C414yes—
75#6505—22no—
76#6495—22no—
77#6489C2312yes—
78C2×#1136C2312yes—
79C2×#1136C2312yes—
80C2×Dic28C2×C411yescharacteristic, complemented
81#6467—21no—
82#6467—21no—
83#6467—21no—
84#6467—21no—
85C2×D56C2×C411yescharacteristic, complemented
86C2×#1132C2×C412yescomplemented
87#6464—22no—
88#6464—22no—
89#6464—22no—
90#6464—22no—
91C2×#1131—22no—
92C2×C8×D7—22no—
93C7×#97—21no—
94C7×#97—21no—
95C7×#97—21no—
96C7×#97D411yescharacteristic
97C7×#97D411yescharacteristic
98C7×#97D411yescharacteristic
99C7×#97D411yescharacteristic
100C4×C7×C8—21no—
101C4×C7×C8—21no—
102C4×C7×C8—21no—
103#6411—24no—
104#6399—24no—
105#6398C2×C414yescomplemented
106#6397C2×C414yescomplemented
107#6396—24no—
108#6393—21no—
109#6393—21no—
110#6393C2×C412yescomplemented
111#6393C2×C412yescomplemented
112#6392—21no—
113#6391—21no—
114#6389—22no—
115C8×Dic7—22no—
116#6380—24no—
117#6379—22no—
118#6379—22no—
119#3067—71no—
120#2955—72no—
121C2×#444—71no—
122#2875D711yescharacteristic, complemented
123#1976—71no—
124#1953—71no—
125#1932—72no—
126#1906—72no—
127#1814—71no—
128#1773—71no—
129#1757—72no—
130C7×#41—21no—
131C7×#41—21no—
132C7×#41C22×C412yes—
133C7×#41C2×D412yes—
134C2×C7×Q8#4111yescharacteristic
135C2×C7×D4—21no—
136C2×C7×D4#4111yescharacteristic
137C22×C4×C7—21no—
138C22×C4×C7—22no—
139C22×C4×C7—21no—
140C22×C4×C7C2411yescharacteristic
141C2×C22⋊D7—22no—
142#1157—22no—
143#1157—22no—
144#1157—22no—
145#1157—22no—
146C2×D28C22×C412yes—
147C2×C4×D7—22no—
148C2×Dic14C22×C412yes—
149C7×M16—21no—
150C7×M16—21no—
151C7×M16—22no—
152C7×M16—22no—
153C7×M16—24no—
154C7×M16C22×C412yes—
155C7×M16C2×D412yes—
156C2×C7×C8—21no—
157C2×C7×C8—21no—
158C2×C7×C8—22no—
159C2×C7×C8—22no—
160C2×C7×C8—24no—
161C2×C7×C8C22×C411yescharacteristic
162C2×C7×C8C22×C411yescharacteristic
163C2×C7×C8C2×D411yescharacteristic
164C2×C7×C8C2×D411yescharacteristic
165C2×C7×C8C2×D412yes—
166C2×C7×C8C2×D411yescharacteristic
167C2×C7×C8C2×D411yescharacteristic
168C2×C7×C8#4111yescharacteristic
169C2×C7×C8#4111yescharacteristic
170C2×C7×C8#4112yes—
171C7×C4⋊C4—22no—
172C7×C4⋊C4—21no—
173C7×C22⋊C4—22no—
174C7×C22⋊C4#4112yes—
175C42×C7—22no—
176C42×C7—21no—
177C42×C7—21no—
178C42×C7C2×D411yescharacteristic
179C42×C7C2×D411yescharacteristic
180C22⋊Dic7—22no—
181#1144—24no—
182#1143—24no—
183#1142—24no—
184#1141—24no—
185C4⋊Dic7—22no—
186C4×Dic7—22no—
187#1136—22no—
188#1136—22no—
189#1136—22no—
190#1136C22×C414yes—
191C2×C7⋊C8—22no—
192C2×C7⋊C8—22no—
193Dic28—21no—
194Dic28—21no—
195D56—21no—
196D56—21no—
197#1132—22no—
198#1132—22no—
199#1131—42no—
200#1131—42no—
201C8×D7—42no—
202C8×D7—42no—
203#575—72no—
204C2×#135—71no—
205C2×#131—72no—
206C2×#131C2×D711yescharacteristic
207C2×#131C2×D711yescharacteristic
208C4×#41C2×D711yescharacteristic
209#473—72no—
210#472—72no—
211#471—74no—
212#444—144no—
213#444—141no—
214#444—141no—
215#444—141no—
216#444—141no—
217#443—144no—
218#430—72no—
219C2×#108—71no—
220C2×#108—71no—
221#427—71no—
222#421—72no—
223#421—72no—
224C2×#104—72no—
225#413—72no—
226#410—72no—
227#405—21no—
228#405—21no—
229#405—72no—
230#405C2×D711yescharacteristic
231#405C2×D711yescharacteristic
232C4×M16—21no—
233C4×M16C2×D711yescharacteristic
234C2×#97—21no—
235C2×#97C2×D711yescharacteristic
236#344—72no—
237#344—72no—
238#339—72no—
239C23×C7—21no—
240C23×C7C2×#4111yescharacteristic
241C22×D7—22no—
242C7×Q8—21no—
243C7×Q8C4×D412yes—
244C7×D4—22no—
245C7×D4—22no—
246C7×D4—21no—
247C7×D4C4×D412yes—
248C2×C4×C7—21no—
249C2×C4×C7—22no—
250C2×C4×C7—22no—
251C2×C4×C7—21no—
252C2×C4×C7—22no—
253C2×C4×C7—21no—
254C2×C4×C7—81no—
255C2×C4×C7—82no—
256C2×C4×C7C23×C411yescharacteristic
257C2×C4×C7C22×D411yescharacteristic
258C2×C4×C7C22×D411yescharacteristic
259C2×C4×C7C2×#4111yescharacteristic
260C2×C4×C7C2×#4111yescharacteristic
261C2×C4×C7C2×#4111yescharacteristic
262C22⋊D7—42no—
263C22⋊D7—42no—
264C2×Dic7—22no—
265C2×Dic7—42no—
266D28—22no—
267D28—22no—
268C4×D7—42no—
269C4×D7—42no—
270Dic14—22no—
271Dic14—22no—
272C7×C8—22no—
273C7×C8—22no—
274C7×C8C4×D411yescharacteristic
275C7×C8C4×D411yescharacteristic
276C7×C8C4×D411yescharacteristic
277C7×C8C4×D411yescharacteristic
278C7×C8#12112yes—
279C7×C8#12112yes—
280C7⋊C8—24no—
281C7⋊C8—42no—
282C2×#41—72no—
283C2×#41C22×D711yescharacteristic
284#137—74no—
285#137—142no—
286AGL(1,ℤ/8)—74no—
287AGL(1,ℤ/8)—142no—
288#135—144no—
289#135—141no—
290#135—141no—
291#135—141no—
292#135—141no—
293C2×Q16—71no—
294C2×SD16—72no—
295C2×D8—71no—
296#131—21no—
297#131—21no—
298#131—22no—
299#131—142no—
300#131—142no—
301#131—142no—
302#131—142no—
303#131C4×D712yes—
304#131C4×D712yes—
305#131D2814yes—
306C2×M16—21no—
307C2×M16—21no—
308C2×M16—72no—
309C2×M16—72no—
310C2×M16—142no—
311C2×M16C22×D711yescharacteristic
312C2×M16C22×D711yescharacteristic
313C22×C8—21no—
314C22×C8—21no—
315C22×C8—142no—
316C22×C8C22×D711yescharacteristic
317C22×C8C22×D711yescharacteristic
318C4×Q8—21no—
319C4×D4—21no—
320C4×D4—22no—
321#117—21no—
322#117—72no—
323#117C22×D711yescharacteristic
324C2×C42—21no—
325C2×C42C22×D711yescharacteristic
326#108—72no—
327#108—72no—
328#108—144no—
329#108—141no—
330#108—141no—
331#107—141no—
332#106—141no—
333#104—144no—
334#104—144no—
335#104—142no—
336#104—142no—
337#101—74no—
338#100—74no—
339#99—144no—
340#97—21no—
341#97—21no—
342#97—21no—
343#97—142no—
344#97C22⋊D711yescharacteristic
345#97C22⋊D711yescharacteristic
346#97C22⋊D711yescharacteristic
347#97C22⋊D711yescharacteristic
348C4×C8—21no—
349C4×C8—21no—
350C4×C8—21no—
351C4×C8—142no—
352C22×C7—22no—
353C22×C7—41no—
354C22×C7—42no—
355C22×C7C4×#4111yescharacteristic
356C22×C7#52511yescharacteristic
357C22×C7#53211yescharacteristic
358C2×D7—42no—
359C2×D7—82no—
360C4×C7—22no—
361C4×C7—42no—
362C4×C7—41no—
363C4×C7—41no—
364C4×C7C2×C4×D411yescharacteristic
365C4×C7C2×C4×D411yescharacteristic
366C4×C7C2×#12111yescharacteristic
367C4×C7#52911yescharacteristic
368Dic7—42no—
369Dic7—82no—
370#41—21no—
371#41—21no—
372#41—142no—
373#41—142no—
374#41—142no—
375#41—142no—
376#41C2×C4×D712yes—
377#41C2×D2812yes—
378C2×Q8—72no—
379C2×Q8#115711yescharacteristic
380C2×D4—21no—
381C2×D4—72no—
382C2×D4—142no—
383C2×D4#115711yescharacteristic
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—142no—
388C22×C4C23×D711yescharacteristic
389Q16—144no—
390Q16—141no—
391Q16—141no—
392SD16—144no—
393SD16—144no—
394SD16—142no—
395SD16—142no—
396D8—144no—
397D8—141no—
398D8—141no—
399M16—21no—
400M16—21no—
401M16—22no—
402M16—22no—
403M16—24no—
404M16—74no—
405M16—142no—
406M16—142no—
407M16—142no—
408M16—282no—
409M16—282no—
410M16C2×C4×D712yes—
411M16C2×D2812yes—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—142no—
418C2×C8—142no—
419C2×C8—282no—
420C2×C8—282no—
421C2×C8C2×C4×D711yescharacteristic
422C2×C8C2×C4×D711yescharacteristic
423C2×C8C2×D2812yes—
424C2×C8#115711yescharacteristic
425C2×C8#115711yescharacteristic
426C2×C8#115712yes—
427C2×C8C2×C22⋊D711yescharacteristic
428C2×C8C2×C22⋊D711yescharacteristic
429C2×C8C2×C22⋊D711yescharacteristic
430C2×C8C2×C22⋊D711yescharacteristic
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—142no—
434C22⋊C4—22no—
435C22⋊C4—142no—
436C22⋊C4#115712yes—
437C42—22no—
438C42—21no—
439C42—21no—
440C42—142no—
441C42C2×C22⋊D711yescharacteristic
442C42C2×C22⋊D711yescharacteristic
443C2×C7—21no—
444C2×C7—21no—
445C2×C7—21no—
446C2×C7—42no—
447C2×C7C4×#12111yescharacteristic
448D7—82no—
449C23—21no—
450C23—142no—
451C23C2×#115711yescharacteristic
452Q8—21no—
453Q8—142no—
454Q8—142no—
455Q8C4×D2812yes—
456D4—22no—
457D4—22no—
458D4—21no—
459D4—142no—
460D4—142no—
461D4—282no—
462D4—282no—
463D4C4×D2812yes—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—142no—
473C2×C4—282no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4C22×C4×D711yescharacteristic
477C2×C4C22×D2811yescharacteristic
478C2×C4C2×#115711yescharacteristic
479C2×C4C2×#115711yescharacteristic
480C2×C4C2×#115711yescharacteristic
481C2×C4C22×C22⋊D711yescharacteristic
482C8—22no—
483C8—22no—
484C8—144no—
485C8—282no—
486C8C4×C22⋊D711yescharacteristic
487C8C4×C22⋊D711yescharacteristic
488C8C4×C22⋊D711yescharacteristic
489C8C4×C22⋊D711yescharacteristic
490C8#649312yes—
491C8#649312yes—
492C7#2155911yescharacteristic, complemented, Sylow
493C22—22no—
494C22—41no—
495C22—42no—
496C22—282no—
497C22—562no—
498C22C4×#115711yescharacteristic
499C22#9016811yescharacteristic
500C22#9048211yescharacteristic
501C4—22no—
502C4—41no—
503C4—42no—
504C4—41no—
505C4—282no—
506C4—562no—
507C4C2×C4×D2811yescharacteristic
508C4#9017011yescharacteristic
509C4C2×C4×C22⋊D711yescharacteristic
510C4C2×#649311yescharacteristic
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—562no—
516C2C4×#649311yescharacteristic
subgroupG/H
centerC4C2×#6493
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2875C2
socleC2×C7C4×#121
prime radical#11741250C2
cosocleC22×C4×C7C24
hypercenter#2875D7
hypocenterC7#21559
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2875

Complements

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

NH ≅ G/Ntype
1. C7×#2875C2semidirect
3. #11741237C2semidirect
4. C2×#89489C2semidirect
5. #11729767C2semidirect
6. #11729399C2semidirect
7. #11727655C2semidirect
8. #11727652C2semidirect
9. #11727651C2semidirect
10. #11727628C2semidirect
11. #11727583C2semidirect
29. #89915C4semidirect
30. #89914C4semidirect
31. #89913C4semidirect
80. C2×Dic28C2×C4semidirect
85. C2×D56C2×C4semidirect
86. C2×#1132C2×C4semidirect
105. #6398C2×C4semidirect
106. #6397C2×C4semidirect
110. #6393C2×C4semidirect
111. #6393C2×C4semidirect
122. #2875D7semidirect
492. C7#21559semidirect

Maximal subgroups

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

subgroupindex
C7×#28752
#117412502
#117412372
C2×#894892
#117297672
#117293992
#117276552
#117276522
#117276512
#117276282
#117275832
#215597