Finite Group Databasebeta

← #13319848

Subgroups — #13319848

Order 1792.

940 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90225C211yesmaximal, characteristic
2#11736709C211yesmaximal, characteristic, complemented
3#11736708C211yesmaximal, characteristic, complemented
4C7×#1458C211yesmaximal, characteristic, complemented
5#11729325C211yesmaximal, characteristic, complemented
6#11728537C211yesmaximal, characteristic, complemented
7#11728537C211yesmaximal, characteristic, complemented
8#11725883C214yesmaximal, complemented
9#11725858C214yesmaximal, complemented
10C2×C7×#117C2211yescharacteristic
11C2×#6493C411yescharacteristic
12C2×#6487C411yescharacteristic
13#90225—22no—
14#90225—22no—
15#90225—22no—
16#90225—22no—
17C2×#6461C2212yes—
18C2×#6459C412yes—
19C2×#6448C2212yes—
20C2×#6446C412yes—
21C2×#6438C411yescharacteristic
22C2×C4×D28C2211yescharacteristic
23C2×#6432C411yescharacteristic
24C2×C4×Dic14C2211yescharacteristic
25C7×#431C2211yescharacteristic
26C7×#431C2211yescharacteristic
27#89697C2211yescharacteristic
28#89697C2211yescharacteristic
29#89603—22no—
30#89602—22no—
31#89491C2214yes—
32#89490C2214yes—
33#89451—22no—
34C8×Dic14—22no—
35C7×#324C2212yes—
36C7×#323C2212yes—
37#89323C2212yes—
38#89320C2212yes—
39#89263C2214yes—
40#89263C2214yes—
41#17390—71nomaximal, Sylow
42C23×C4×C7C2×C411yescharacteristic
43C22×C22⋊D7C2×C412yes—
44C22×D28C2×C411yescharacteristic
45C22×C4×D7C2×C412yes—
46C22×Dic14C2×C411yescharacteristic
47C22×Dic14C2×C412yes—
48C7×#117—21no—
49C7×#117—21no—
50C7×#117—21no—
51C7×#117—21no—
52C2×C7×C4⋊C4C2×C411yescharacteristic
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C22⋊C4C2×C411yescharacteristic
55C2×C7×C22⋊C4C2311yescharacteristic
56C2×C42×C7C2×C411yescharacteristic
57C2×C42×C7C2311yescharacteristic
58C2×C22⋊Dic7C2×C412yes—
59#6493—22no—
60#6493—22no—
61C2×(C2×D7)⋊C4C2×C412yes—
62C2×(C2×D7)⋊C4C2×C412yes—
63C2×(C2×D7)⋊C4C2×C412yes—
64C2×(C2×D7)⋊C4C2312yes—
65C2×C4⋊Dic7C2×C411yescharacteristic
66C2×C4⋊Dic7C2×C412yes—
67C2×C4⋊Dic7C2×C411yescharacteristic
68#6487—22no—
69#6487—22no—
70C2×Dic7⋊C4C2×C412yes—
71C2×Dic7⋊C4C2×C412yes—
72C2×C4×Dic7C2312yes—
73#6461—24no—
74#6461—24no—
75#6459—24no—
76#6459—24no—
77#6448—24no—
78#6448—24no—
79#6446—24no—
80#6446—24no—
81#6438—22no—
82#6438—22no—
83C4×D28—21no—
84C4×D28—21no—
85C4×D28—21no—
86C4×D28—21no—
87#6432—22no—
88#6432—22no—
89C4×Dic14D412yes—
90C4×Dic14D412yes—
91C4×Dic14D412yes—
92C4×Dic14D412yes—
93C7×C4⋊C8—21no—
94C7×C4⋊C8—21no—
95C7×C22⋊C8C2312yes—
96C7×C22⋊C8C2312yes—
97C4×C7×C8—21no—
98C4×C7×C8—21no—
99C22⋊(C7⋊C8)C2312yes—
100C22⋊(C7⋊C8)C2312yes—
101#6389—24no—
102Dic7⋊C8—24no—
103Dic7⋊C8—24no—
104C8×Dic7—24no—
105C4⋊(C7⋊C8)—21no—
106C4⋊(C7⋊C8)—21no—
107C4×C7⋊C8—21no—
108C4×C7⋊C8—21no—
109C2×#542—71no—
110#2983—71no—
111#2982—71no—
112#1473—74no—
113#1469—74no—
114#1460—71no—
115#1460—71no—
116#1458—71no—
117#1458D711yescharacteristic, complemented
118C24×C7C22×C411yescharacteristic
119C23×D7C22×C412yes—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7C22×C411yescharacteristic
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C22×C411yescharacteristic
129C22×C4×C7C2411yescharacteristic
130C2×C22⋊D7—24no—
131C2×C22⋊D7—82no—
132C22×Dic7C22×C412yes—
133C22×Dic7C22×C412yes—
134C22×Dic7C22×C412yes—
135C2×D28—21no—
136C2×D28—21no—
137C2×D28—22no—
138C2×D28—22no—
139C2×C4×D7—22no—
140C2×C4×D7—22no—
141C2×C4×D7—82no—
142C2×Dic14—22no—
143C2×Dic14—22no—
144C2×Dic14—24no—
145C2×Dic14—82no—
146C2×Dic14C22⋊C412yes—
147C2×Dic14C22⋊C412yes—
148C2×C7×C8—22no—
149C2×C7×C8—22no—
150C7×C4⋊C4—21no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—21no—
154C7×C22⋊C4—21no—
155C7×C22⋊C4—21no—
156C7×C22⋊C4—21no—
157C7×C22⋊C4—21no—
158C42×C7—21no—
159C42×C7—21no—
160C42×C7C2×D411yescharacteristic
161C42×C7C2×D411yescharacteristic
162C42×C7C2×D411yescharacteristic
163C42×C7C2×D411yescharacteristic
164C22⋊Dic7—24no—
165(C2×D7)⋊C4—22no—
166(C2×D7)⋊C4—22no—
167(C2×D7)⋊C4—24no—
168(C2×D7)⋊C4—24no—
169(C2×D7)⋊C4—24no—
170C4⋊Dic7—21no—
171C4⋊Dic7—21no—
172C4⋊Dic7—24no—
173C4⋊Dic7C22⋊C412yes—
174C4⋊Dic7C22⋊C412yes—
175Dic7⋊C4—24no—
176Dic7⋊C4C22⋊C414yes—
177Dic7⋊C4C22⋊C414yes—
178C4×Dic7C2×D414yes—
179C4×Dic7C2×D414yes—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—22no—
182#542—142no—
183#542—142no—
184#542—142no—
185#542—142no—
186C2×C4⋊Q8—71no—
187C2×#126—72no—
188C2×#124—72no—
189C2×#124—71no—
190C2×#123—72no—
191C2×C22⋊Q8—71no—
192C2×C22⋊Q8—72no—
193C2×#121—71no—
194C2×C4×Q8—71no—
195C2×C4×D4—71no—
196C2×#117C2×D711yescharacteristic
197#446—142no—
198#446—142no—
199C8×Q8—142no—
200C8×Q8—142no—
201#432—74no—
202#431—71no—
203#431—71no—
204#431—74no—
205#431C2×D711yescharacteristic
206#431C2×D711yescharacteristic
207#324—72no—
208#324C2×D712yes—
209#323—72no—
210#323—74no—
211#323—74no—
212#323C2×D712yes—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7—21no—
216C23×C7—41no—
217C23×C7—41no—
218C23×C7C23×C411yescharacteristic
219C22×D7—22no—
220C22×D7—22no—
221C22×D7—22no—
222C22×D7—42no—
223C22×D7—42no—
224C2×C4×C7—21no—
225C2×C4×C7—21no—
226C2×C4×C7—21no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—41no—
235C2×C4×C7C2×C22⋊C411yescharacteristic
236C2×C4×C7C2×C22⋊C411yescharacteristic
237C2×C4×C7C2×C22⋊C411yescharacteristic
238C2×C4×C7C2×C22⋊C411yescharacteristic
239C2×C4×C7#13112yes—
240C2×C4×C7#13112yes—
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C22×D411yescharacteristic
243C22⋊D7—84no—
244C2×Dic7—22no—
245C2×Dic7—22no—
246C2×Dic7—22no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7C2×C22⋊C414yes—
250C2×Dic7C2×C22⋊C414yes—
251D28—42no—
252D28—42no—
253C4×D7—82no—
254C4×D7—82no—
255Dic14—24no—
256Dic14—24no—
257Dic14—84no—
258C7×C8—42no—
259C7×C8—42no—
260C7⋊C8—42no—
261C7⋊C8—42no—
262C22×Q8—71no—
263C22×Q8—72no—
264C22×D4—72no—
265C22×D4—71no—
266C23×C4—72no—
267C23×C4C4×D711yescharacteristic
268C4⋊Q8—142no—
269C4⋊Q8—142no—
270#126—144no—
271#126—144no—
272#124—142no—
273#124—142no—
274#124—144no—
275#124—144no—
276#123—144no—
277#123—144no—
278C22⋊Q8—144no—
279C22⋊Q8—144no—
280C22⋊Q8—142no—
281C22⋊Q8—142no—
282#121—142no—
283#121—142no—
284C4×Q8—72no—
285C4×Q8—72no—
286C4×Q8—72no—
287C4×Q8—72no—
288C4×D4—141no—
289C4×D4—141no—
290C4×D4—141no—
291C4×D4—141no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296C2×C4⋊C4—71no—
297C2×C4⋊C4—72no—
298C2×C4⋊C4—71no—
299C2×C4⋊C4—72no—
300C2×C4⋊C4—72no—
301C2×C4⋊C4C4×D711yescharacteristic
302C2×C4⋊C4C22×D711yescharacteristic
303C2×C22⋊C4—72no—
304C2×C22⋊C4—72no—
305C2×C22⋊C4—72no—
306C2×C22⋊C4—72no—
307C2×C22⋊C4—72no—
308C2×C22⋊C4C4×D711yescharacteristic
309C2×C22⋊C4C22×D711yescharacteristic
310C2×C42—72no—
311C2×C42C4×D711yescharacteristic
312C2×C42C22×D711yescharacteristic
313C4⋊C8—21no—
314C4⋊C8—21no—
315C4⋊C8—144no—
316C4⋊C8—144no—
317C4⋊C8—141no—
318C4⋊C8—141no—
319C22⋊C8—72no—
320C22⋊C8—72no—
321C22⋊C8C22×D712yes—
322C22⋊C8C22×D712yes—
323#97—144no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—144no—
327C4×C8—141no—
328C4×C8—141no—
329C22×C7—21no—
330C22×C7—21no—
331C22×C7—41no—
332C22×C7—41no—
333C22×C7—41no—
334C22×C7—81no—
335C22×C7—81no—
336C22×C7C22×C22⋊C411yescharacteristic
337C22×C7C2×#13111yescharacteristic
338C22×C7C2×#13111yescharacteristic
339C2×D7—42no—
340C2×D7—42no—
341C2×D7—42no—
342C2×D7—82no—
343C2×D7—82no—
344C4×C7—21no—
345C4×C7—21no—
346C4×C7—21no—
347C4×C7—21no—
348C4×C7—41no—
349C4×C7—41no—
350C4×C7—41no—
351C4×C7#40812yes—
352C4×C7#40812yes—
353Dic7—24no—
354Dic7—24no—
355Dic7—42no—
356C24—72no—
357C24C2×C4×D711yescharacteristic
358C2×Q8—72no—
359C2×Q8—72no—
360C2×Q8—144no—
361C2×Q8—142no—
362C2×Q8—142no—
363C2×Q8—562no—
364C2×D4—144no—
365C2×D4—141no—
366C2×D4—141no—
367C2×D4—142no—
368C2×D4—142no—
369C2×D4—562no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—72no—
375C22×C4—72no—
376C22×C4—72no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—562no—
380C22×C4C2×C4×D711yescharacteristic
381C22×C4C2×C4×D711yescharacteristic
382C22×C4C2×C4×D711yescharacteristic
383C22×C4C2×C4×D711yescharacteristic
384C22×C4C2×C4×D711yescharacteristic
385C22×C4C23×D711yescharacteristic
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—142no—
389C2×C8—142no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—74no—
395C4⋊C4—74no—
396C4⋊C4—72no—
397C4⋊C4—72no—
398C4⋊C4—144no—
399C4⋊C4—144no—
400C4⋊C4—141no—
401C4⋊C4—141no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—144no—
407C22⋊C4—144no—
408C22⋊C4—144no—
409C22⋊C4—144no—
410C22⋊C4—142no—
411C22⋊C4—142no—
412C42—21no—
413C42—21no—
414C42—74no—
415C42—74no—
416C42D4×D711yescharacteristic
417C42D4×D711yescharacteristic
418C42D4×D711yescharacteristic
419C42D4×D711yescharacteristic
420C2×C7—21no—
421C2×C7—21no—
422C2×C7—81no—
423C2×C7C2×#40811yescharacteristic
424C2×C7#288411yescharacteristic
425C2×C7#288811yescharacteristic
426D7—82no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—41no—
431C23—41no—
432C23—142no—
433C23—142no—
434C23—142no—
435C23—282no—
436C23—282no—
437C23C22×C4×D711yescharacteristic
438Q8—144no—
439Q8—144no—
440Q8—564no—
441D4—282no—
442D4—282no—
443D4—564no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—74no—
456C2×C4—74no—
457C2×C4—142no—
458C2×C4—142no—
459C2×C4—142no—
460C2×C4—282no—
461C2×C4—282no—
462C2×C4—562no—
463C2×C4—562no—
464C2×C4D7×C22⋊C411yescharacteristic
465C2×C4D7×C22⋊C411yescharacteristic
466C2×C4D7×C22⋊C411yescharacteristic
467C2×C4D7×C22⋊C411yescharacteristic
468C2×C4#646412yes—
469C2×C4#647012yes—
470C2×C4C2×D4×D711yescharacteristic
471C2×C4C2×D4×D711yescharacteristic
472C8—42no—
473C8—42no—
474C8—282no—
475C8—282no—
476C7#1739011yescharacteristic, complemented, Sylow
477C22—21no—
478C22—21no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—81no—
483C22—81no—
484C22—282no—
485C22—282no—
486C22—282no—
487C22—562no—
488C22—562no—
489C22C2×D7×C22⋊C411yescharacteristic
490C22C2×#646411yescharacteristic
491C22C2×#647011yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—144no—
500C4—144no—
501C4—282no—
502C4#8949912yes—
503C4#8949912yes—
504C2—21no—
505C2—21no—
506C2—81no—
507C2—562no—
508C2C2×#8949911yescharacteristic
509C2#1173613411yescharacteristic
510C2#1173615911yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1458C2
socleC22×C7C22×C22⋊C4
prime radicalC2×#6493C4
cosocleC22×C4×C7C24
hypercenter#1458D7
hypocenterC7#17390
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #11736709C2semidirect
3. #11736708C2semidirect
4. C7×#1458C2semidirect
5. #11729325C2semidirect
6. #11728537C2semidirect
7. #11728537C2semidirect
8. #11725883C2semidirect
9. #11725858C2semidirect
117. #1458D7semidirect
476. C7#17390semidirect

Maximal subgroups

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

subgroupindex
C2×#902252
#117367092
#117367082
C7×#14582
#117293252
#117285372
#117258832
#117258582
#173907