Finite Group Databasebeta

← #13364321

Subgroups — #13364321

Order 1792.

834 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89749C211yesmaximal, characteristic
2C2×#89490C211yesmaximal, characteristic
3#11731308C211yesmaximal, characteristic
4C7×#1853C211yesmaximal, characteristic
5#11729309C211yesmaximal, characteristic
6#11728318C212yesmaximal, complemented
7#11728309C212yesmaximal, complemented
8#11727682C211yesmaximal, characteristic
9#11727258C211yesmaximal, characteristic
10#11726602C212yesmaximal, complemented
11#11726551C212yesmaximal, complemented
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C4⋊Dic7C2211yescharacteristic
14C22×C4×Dic7C2211yescharacteristic
15C2×C7×#106C2212yes—
16C2×C7×C22⋊C8C2211yescharacteristic
17C7×#380C2211yescharacteristic
18#89984C2211yescharacteristic
19C2×C7⋊#95—21no—
20C2×C7⋊#95—21no—
21C2×C22⋊(C7⋊C8)C2211yescharacteristic
22C2×#6391C2212yes—
23C2×Dic7⋊C8C2212yes—
24C2×C8×Dic7C2212yes—
25#89749—21no—
26#89749—21no—
27#89749C2211yescharacteristic
28#89749C2212yes—
29#89749C2211yescharacteristic
30#89749C2212yes—
31C4⋊C4×Dic7C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33C2×#6381C2212yes—
34C2×#6381C2212yes—
35#89715—21no—
36C22⋊C4×Dic7—21no—
37#89490C418yescomplemented
38#89423—21no—
39#89420—21no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21983—71nomaximal, Sylow
45C23×C4×C7—21no—
46C23×C4×C7C2311yescharacteristic
47C23×Dic7—21no—
48C23×Dic7D411yescharacteristic
49C23×Dic7D411yescharacteristic
50C22×C7×C8C2312yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—42no—
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C4⋊C4C2312yes—
55C2×C7×C4⋊C4C2311yescharacteristic
56C2×C7×C22⋊C4—21no—
57C2×C22⋊Dic7—21no—
58C2×C4⋊Dic7—21no—
59C2×C4⋊Dic7—42no—
60C2×C4⋊Dic7C2311yescharacteristic
61C2×C4⋊Dic7C2312yes—
62C2×C4⋊Dic7C2311yescharacteristic
63C2×C4×Dic7—21no—
64C2×C4×Dic7—21no—
65C2×C4×Dic7C2×C414yes—
66C2×C4×Dic7D411yescharacteristic
67C2×C4×Dic7D411yescharacteristic
68C2×C4×Dic7Q814yes—
69C2×C4×Dic7C2311yescharacteristic
70C2×C4×Dic7C2311yescharacteristic
71C22×C7⋊C8C2312yes—
72C7×#106—42no—
73C7×C22⋊C8C2×C414yes—
74C7×#95—21no—
75C7⋊#95—21no—
76C7⋊#95—21no—
77C7⋊#95—21no—
78C7⋊#95—21no—
79C7⋊#95—21no—
80C7⋊#95—21no—
81C7⋊#95—21no—
82C7⋊#95—22no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95—22no—
86C22⋊(C7⋊C8)C2×C414yes—
87#6391—42no—
88Dic7⋊C8C2×C418yescomplemented
89C8×Dic7C2×C418yescomplemented
90#6381—42no—
91#6381—42no—
92C2×#431—71no—
93#2283—71no—
94C2×#382—71no—
95#1949—72no—
96#1949—72no—
97#1853—71no—
98#1853D711yescharacteristic
99#1852—71no—
100#1852—71no—
101#1846—72no—
102#1835—72no—
103C24×C7C2×D411yescharacteristic
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—42no—
110C22×C4×C7—42no—
111C22×C4×C7—42no—
112C22×C4×C7C22×C412yes—
113C22×C4×C7C2×D411yescharacteristic
114C22×C4×C7C2×Q812yes—
115C22×C4×C7C2411yescharacteristic
116C22×Dic7—22no—
117C22×Dic7—22no—
118C22×Dic7—21no—
119C22×Dic7—21no—
120C22×Dic7—22no—
121C22×Dic7—41no—
122C22×Dic7—41no—
123C22×Dic7—42no—
124C22×Dic7—42no—
125C22×Dic7—42no—
126C22×Dic7C4⋊C414yes—
127C22×Dic7C4⋊C414yes—
128C22×Dic7C2×D411yescharacteristic
129C22×Dic7C2×D411yescharacteristic
130C22×Dic7C2×D411yescharacteristic
131C22×Dic7C2×D411yescharacteristic
132C2×C7×C8—22no—
133C2×C7×C8—22no—
134C2×C7×C8C22×C414yes—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—42no—
140C7×C22⋊C4—42no—
141C22⋊Dic7—42no—
142C4⋊Dic7—22no—
143C4⋊Dic7—22no—
144C4⋊Dic7—22no—
145C4⋊Dic7—22no—
146C4⋊Dic7—42no—
147C4×Dic7—42no—
148C4×Dic7—42no—
149C4×Dic7C4⋊C414yes—
150C4×Dic7C4⋊C414yes—
151C4×Dic7C22×C412yes—
152C4×Dic7C22×C412yes—
153C4×Dic7C2×Q812yes—
154C4×Dic7C2×Q812yes—
155C2×C7⋊C8—22no—
156C2×C7⋊C8—22no—
157C2×C7⋊C8C22×C414yes—
158C22×C4⋊C4—71no—
159C22×C4⋊C4C2×D711yescharacteristic
160C22×C42—71no—
161#431—78no—
162C2×#107—72no—
163C2×#107—72no—
164C2×#106—72no—
165C2×#106C2×D712yes—
166C2×C4⋊C8—72no—
167C2×C22⋊C8—71no—
168C2×C22⋊C8C2×D711yescharacteristic
169C2×C4×C8—72no—
170#383—141no—
171#382—71no—
172#382—72no—
173#382—71no—
174#382—72no—
175#382—141no—
176#382—141no—
177#381—141no—
178#380—71no—
179#380C2×D711yescharacteristic
180C4×C4⋊C4—71no—
181C4×C4⋊C4—71no—
182C4×C22⋊C4—141no—
183#376—141no—
184C2×#95—141no—
185C2×#95—141no—
186#340—72no—
187#340—72no—
188#340—72no—
189#340C2×D712yes—
190C23×C7—21no—
191C23×C7—21no—
192C23×C7—21no—
193C23×C7—21no—
194C23×C7C2×C4⋊C412yes—
195C23×C7#13512yes—
196C23×C7#13512yes—
197C23×C7C22×D411yescharacteristic
198C2×C4×C7—21no—
199C2×C4×C7—21no—
200C2×C4×C7—24no—
201C2×C4×C7—24no—
202C2×C4×C7—42no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—42no—
208C2×C4×C7—41no—
209C2×C4×C7—41no—
210C2×C4×C7—42no—
211C2×C4×C7—84no—
212C2×C4×C7C2×C4⋊C412yes—
213C2×C4×C7C23×C411yescharacteristic
214C2×C4×C7C22×Q811yescharacteristic
215C2×C4×C7H(5,𝔽2)12yes—
216C2×C4×C72−1+412yes—
217C2×Dic7—21no—
218C2×Dic7—21no—
219C2×Dic7—21no—
220C2×Dic7—21no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7—21no—
224C2×Dic7—21no—
225C2×Dic7—28no—
226C2×Dic7—44no—
227C2×Dic7—42no—
228C2×Dic7—41no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7—41no—
232C2×Dic7—42no—
233C2×Dic7—41no—
234C2×Dic7—41no—
235C2×Dic7—42no—
236C2×Dic7—84no—
237C2×Dic7C2×C4⋊C412yes—
238C2×Dic7C2×C4⋊C412yes—
239C2×Dic7C2×C4⋊C412yes—
240C2×Dic7C2×C4⋊C412yes—
241C7×C8—24no—
242C7⋊C8—24no—
243C23×C4—21no—
244C23×C4—71no—
245C23×C4—71no—
246C23×C4—141no—
247C23×C4C22×D711yescharacteristic
248C22×C8—72no—
249C22×C8C22×D712yes—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—72no—
253C2×C4⋊C4—71no—
254C2×C4⋊C4—71no—
255C2×C4⋊C4—141no—
256C2×C4⋊C4—282no—
257C2×C4⋊C4C22×D712yes—
258C2×C4⋊C4C22×D711yescharacteristic
259C2×C4⋊C4C22×D711yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—141no—
262C2×C42—74no—
263C2×C42—71no—
264C2×C42—71no—
265C2×C42—74no—
266C2×C42—71no—
267C2×C42—71no—
268C2×C42—141no—
269C2×C42—141no—
270#107—282no—
271#107—282no—
272#106—42no—
273#106—282no—
274C4⋊C8—78no—
275C22⋊C8—74no—
276C22⋊C8C4×D714yes—
277C4×C8—78no—
278#95—21no—
279#95—142no—
280#95—142no—
281#95—142no—
282#95—142no—
283#95—141no—
284#95—141no—
285#95—141no—
286#95—141no—
287#95—141no—
288#95—141no—
289#95—141no—
290C22×C7—22no—
291C22×C7—22no—
292C22×C7—22no—
293C22×C7—22no—
294C22×C7—22no—
295C22×C7—22no—
296C22×C7#42714yes—
297C22×C7C22×C4⋊C411yescharacteristic
298C22×C7#53511yescharacteristic
299C22×C7#53611yescharacteristic
300C22×C7C2×#13511yescharacteristic
301C22×C7C2×#13511yescharacteristic
302C22×C7#57511yescharacteristic
303C22×C7#57511yescharacteristic
304C4×C7—22no—
305C4×C7—81no—
306C4×C7—82no—
307C4×C7—81no—
308C4×C7#52012yes—
309C4×C7#54412yes—
310Dic7—22no—
311Dic7—22no—
312Dic7—22no—
313Dic7—22no—
314Dic7—81no—
315Dic7—82no—
316Dic7—81no—
317C24D4×D711yescharacteristic
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—74no—
327C22×C4—71no—
328C22×C4—71no—
329C22×C4—71no—
330C22×C4—71no—
331C22×C4—74no—
332C22×C4—142no—
333C22×C4—142no—
334C22×C4—141no—
335C22×C4—141no—
336C22×C4—142no—
337C22×C4—281no—
338C22×C4—281no—
339C22×C4—282no—
340C22×C4—282no—
341C22×C4—282no—
342C22×C4C2×C4×D712yes—
343C22×C4D4×D711yescharacteristic
344C22×C4Q8×D712yes—
345C22×C4C23×D711yescharacteristic
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—74no—
349C2×C8—142no—
350C2×C8—142no—
351C2×C8C2×C4×D714yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—42no—
357C4⋊C4—142no—
358C4⋊C4—142no—
359C4⋊C4—142no—
360C4⋊C4—142no—
361C4⋊C4—282no—
362C22⋊C4—42no—
363C22⋊C4—282no—
364C42—74no—
365C42—74no—
366C42—72no—
367C42—72no—
368C42—72no—
369C42—72no—
370C42—282no—
371C42—282no—
372C2×C7—24no—
373C2×C7#231711yescharacteristic
374C2×C7C2×#42711yescharacteristic
375C2×C7#291111yescharacteristic
376C2×C7#321111yescharacteristic
377C2×C7#321111yescharacteristic
378C2×C7#323211yescharacteristic
379C2×C7#323211yescharacteristic
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23D7×C4⋊C412yes—
385C23#647912yes—
386C23#647912yes—
387C23C2×D4×D711yescharacteristic
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—72no—
402C2×C4—72no—
403C2×C4—72no—
404C2×C4—72no—
405C2×C4—84no—
406C2×C4—148no—
407C2×C4—141no—
408C2×C4—141no—
409C2×C4—141no—
410C2×C4—141no—
411C2×C4—141no—
412C2×C4—141no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—284no—
416C2×C4—281no—
417C2×C4—281no—
418C2×C4—282no—
419C2×C4—281no—
420C2×C4—281no—
421C2×C4—282no—
422C2×C4—282no—
423C2×C4—281no—
424C2×C4—281no—
425C2×C4—564no—
426C2×C4D7×C4⋊C412yes—
427C2×C4C22×C4×D711yescharacteristic
428C2×C4#654812yes—
429C2×C4C2×Q8×D711yescharacteristic
430C2×C4#655412yes—
431C8—24no—
432C8—144no—
433C7#2198311yescharacteristic, complemented, Sylow
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22#8963214yes—
441C22C2×D7×C4⋊C411yescharacteristic
442C22#9028311yescharacteristic
443C22#9033911yescharacteristic
444C22C2×#647911yescharacteristic
445C22C2×#647911yescharacteristic
446C22#9046411yescharacteristic
447C22#9046811yescharacteristic
448C4—22no—
449C4—81no—
450C4—81no—
451C4—82no—
452C4—142no—
453C4—142no—
454C4—142no—
455C4—142no—
456C4—562no—
457C4—561no—
458C4—561no—
459C4#9023712yes—
460C4#9031412yes—
461C2—24no—
462C2#1173182711yescharacteristic
463C2C2×#8963211yescharacteristic
464C2#1173665311yescharacteristic
465C2#1173750211yescharacteristic
466C2#1173758411yescharacteristic
467C2#1173876311yescharacteristic
468C2#1173886411yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1853C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1853D7
hypocenterC7#21983
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
6. #11728318C2semidirect
7. #11728309C2semidirect
10. #11726602C2semidirect
11. #11726551C2semidirect
37. #89490C4semidirect
88. Dic7⋊C8C2×C4semidirect
89. C8×Dic7C2×C4semidirect
433. C7#21983semidirect

Maximal subgroups

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

subgroupindex
C2×#897492
C2×#894902
#117313082
C7×#18532
#117293092
#117283182
#117283092
#117276822
#117272582
#117266022
#117265512
#219837