Finite Group Databasebeta

← #13363302

Subgroups — #13363302

Order 1792.

1030 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90208C211yesmaximal, characteristic
2C2×#89490C211yesmaximal, characteristic
3#11731308C211yesmaximal, characteristic
4C7×#1825C211yesmaximal, characteristic
5#11729329C211yesmaximal, characteristic
6#11727680C211yesmaximal, characteristic
7#11727257C211yesmaximal, characteristic
8#11726558C212yesmaximal, complemented
9#11726558C212yesmaximal, complemented
10#11726554C212yesmaximal, complemented
11#11726553C212yesmaximal, complemented
12C2×C7×#117C2211yescharacteristic
13C2×#6489C2211yescharacteristic
14C22×C4×Dic7C2211yescharacteristic
15#90208—24no—
16#90208C418yescomplemented
17C2×#6453C2211yescharacteristic
18C2×#6453C2211yescharacteristic
19C2×#6451C2211yescharacteristic
20C2×#6451C2211yescharacteristic
21C2×#6441—21no—
22C2×#6441—21no—
23C2×C4×Dic14—21no—
24C2×C4×Dic14—21no—
25C2×C7×C22⋊C8C2211yescharacteristic
26C7×#380C2211yescharacteristic
27#89984C2211yescharacteristic
28C2×C22⋊(C7⋊C8)C2211yescharacteristic
29C2×Dic7⋊C8C2212yes—
30C2×C8×Dic7C2212yes—
31#89749C2212yes—
32C4⋊C4×Dic7C2212yes—
33#89715—21no—
34C22⋊C4×Dic7—21no—
35#89490—24no—
36#89423—21no—
37#89420—21no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21837—71nomaximal, Sylow
47C23×C4×C7C2311yescharacteristic
48C23×Dic7D411yescharacteristic
49C23×Dic7D411yescharacteristic
50C22×Dic14—21no—
51C22×Dic14—21no—
52C22×C7×C8C2312yes—
53C7×#117—22no—
54C7×#117C2×C414yes—
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C4⋊C4C2311yescharacteristic
57C2×C7×C4⋊C4C2311yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C42×C7—21no—
61C2×C22⋊Dic7—21no—
62C2×C22⋊Dic7—21no—
63#6489—22no—
64#6489C2×C414yes—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—21no—
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2311yescharacteristic
69C2×C4⋊Dic7C2311yescharacteristic
70C2×Dic7⋊C4—21no—
71C2×Dic7⋊C4—21no—
72C2×Dic7⋊C4—21no—
73C2×Dic7⋊C4—21no—
74C2×C4×Dic7—21no—
75C2×C4×Dic7—22no—
76C2×C4×Dic7C2×C414yes—
77C2×C4×Dic7D411yescharacteristic
78C2×C4×Dic7D411yescharacteristic
79C2×C4×Dic7D414yes—
80C2×C4×Dic7C2311yescharacteristic
81C2×C4×Dic7C2311yescharacteristic
82C22×C7⋊C8C2312yes—
83#6453—22no—
84#6453—22no—
85#6453C2×C414yes—
86#6453C2×C414yes—
87#6451—22no—
88#6451—22no—
89#6451C2×C414yes—
90#6451C2×C414yes—
91#6441—28no—
92#6441—28no—
93C4×Dic14—28no—
94C4×Dic14—28no—
95C7×C22⋊C8—22no—
96C7×#95—21no—
97C7⋊#95—21no—
98C7⋊#95—22no—
99C7⋊#95—22no—
100C7⋊#95—22no—
101C22⋊(C7⋊C8)—22no—
102Dic7⋊C8—24no—
103C8×Dic7—24no—
104C2×#533—71no—
105C2×#431—71no—
106#2283—71no—
107#1991—72no—
108#1991—72no—
109#1988—72no—
110#1987—72no—
111#1825—71no—
112#1825—71no—
113#1825—71no—
114#1825D711yescharacteristic
115C24×C7C2×D411yescharacteristic
116C22×C4×C7—24no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—22no—
120C22×C4×C7C22×C412yes—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2411yescharacteristic
124C22×Dic7—24no—
125C22×Dic7—21no—
126C22×Dic7—21no—
127C22×Dic7—22no—
128C22×Dic7—41no—
129C22×Dic7—41no—
130C22×Dic7C22⋊C414yes—
131C22×Dic7C22⋊C414yes—
132C22×Dic7C2×D411yescharacteristic
133C22×Dic7C2×D411yescharacteristic
134C22×Dic7C2×D411yescharacteristic
135C22×Dic7C2×D411yescharacteristic
136C2×Dic14—24no—
137C2×Dic14—24no—
138C2×Dic14—42no—
139C2×Dic14—42no—
140C2×Dic14—44no—
141C2×C7×C8—22no—
142C2×C7×C8—24no—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—24no—
146C7×C4⋊C4C22×C412yes—
147C7×C4⋊C4C22×C412yes—
148C7×C22⋊C4—24no—
149C7×C22⋊C4—42no—
150C42×C7—24no—
151C22⋊Dic7—24no—
152C22⋊Dic7—42no—
153C4⋊Dic7—21no—
154C4⋊Dic7—21no—
155C4⋊Dic7—24no—
156C4⋊Dic7—24no—
157C4⋊Dic7—24no—
158C4⋊Dic7C22×C412yes—
159C4⋊Dic7C22×C412yes—
160Dic7⋊C4—24no—
161Dic7⋊C4—24no—
162Dic7⋊C4—24no—
163Dic7⋊C4—24no—
164C4×Dic7—24no—
165C4×Dic7—44no—
166C4×Dic7C22⋊C414yes—
167C4×Dic7C22⋊C414yes—
168C4×Dic7C22×C412yes—
169C4×Dic7C22×C412yes—
170C4×Dic7C2×D412yes—
171C4×Dic7C2×D412yes—
172C2×C7⋊C8—22no—
173C2×C7⋊C8—24no—
174#533—78no—
175#533—144no—
176C2×C4⋊Q8—71no—
177C2×C4⋊Q8—71no—
178C2×#125—71no—
179C2×#125—71no—
180C2×C22⋊Q8—141no—
181C2×C22⋊Q8—141no—
182C2×C4×Q8—141no—
183C2×C4×Q8—141no—
184C2×#117—71no—
185C2×#117C2×D711yescharacteristic
186C22×C42—71no—
187#431—144no—
188C2×C4⋊C8—72no—
189C2×C22⋊C8—71no—
190C2×C22⋊C8C2×D711yescharacteristic
191C2×C4×C8—72no—
192#383—141no—
193#382—72no—
194#381—141no—
195#380—71no—
196#380C2×D711yescharacteristic
197C4×C4⋊C4—72no—
198C4×C22⋊C4—141no—
199#376—141no—
200#340—72no—
201#340—72no—
202#340—72no—
203#340—72no—
204#340—72no—
205#340—72no—
206#340C2×D712yes—
207#340C2×D712yes—
208C23×C7—22no—
209C23×C7—22no—
210C23×C7—21no—
211C23×C7—21no—
212C23×C7C2×C22⋊C412yes—
213C23×C7C22×D411yescharacteristic
214C2×C4×C7—22no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—22no—
218C2×C4×C7—22no—
219C2×C4×C7—42no—
220C2×C4×C7—41no—
221C2×C4×C7—41no—
222C2×C4×C7—42no—
223C2×C4×C7—42no—
224C2×C4×C7—42no—
225C2×C4×C7C2×C22⋊C412yes—
226C2×C4×C7C23×C411yescharacteristic
227C2×C4×C7C22×D411yescharacteristic
228C2×C4×C72−1+414yes—
229C2×Dic7—22no—
230C2×Dic7—21no—
231C2×Dic7—21no—
232C2×Dic7—21no—
233C2×Dic7—21no—
234C2×Dic7—28no—
235C2×Dic7—22no—
236C2×Dic7—22no—
237C2×Dic7—44no—
238C2×Dic7—42no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C2×Dic7C2×C22⋊C412yes—
245C2×Dic7C2×C22⋊C412yes—
246C2×Dic7C2×C22⋊C412yes—
247C2×Dic7C2×C22⋊C412yes—
248C2×Dic7#13514yes—
249Dic14—48no—
250Dic14—48no—
251C7×C8—42no—
252C7⋊C8—42no—
253C22×Q8—141no—
254C22×Q8—141no—
255C23×C4—71no—
256C23×C4—71no—
257C23×C4C22×D711yescharacteristic
258C22×C8—72no—
259C22×C8C22×D712yes—
260C4⋊Q8—74no—
261C4⋊Q8—74no—
262C4⋊Q8—142no—
263C4⋊Q8—142no—
264#125—74no—
265#125—74no—
266#125—142no—
267#125—142no—
268C22⋊Q8—148no—
269C22⋊Q8—148no—
270C4×Q8—148no—
271C4×Q8—148no—
272#117—22no—
273#117—74no—
274#117—142no—
275#117C4×D714yes—
276C2×C4⋊C4—71no—
277C2×C4⋊C4—71no—
278C2×C4⋊C4—72no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4—141no—
281C2×C4⋊C4—141no—
282C2×C4⋊C4—141no—
283C2×C4⋊C4—141no—
284C2×C4⋊C4—141no—
285C2×C4⋊C4C22×D712yes—
286C2×C4⋊C4C22×D711yescharacteristic
287C2×C4⋊C4C22×D711yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—141no—
291C2×C22⋊C4—141no—
292C2×C42—21no—
293C2×C42—74no—
294C2×C42—71no—
295C2×C42—71no—
296C2×C42—74no—
297C2×C42—71no—
298C2×C42—71no—
299C2×C42—141no—
300C2×C42—142no—
301C4⋊C8—144no—
302C22⋊C8—22no—
303C22⋊C8—142no—
304C4×C8—144no—
305#95—21no—
306#95—142no—
307#95—142no—
308#95—142no—
309#95—141no—
310C22×C7—22no—
311C22×C7—22no—
312C22×C7—22no—
313C22×C7—42no—
314C22×C7—42no—
315C22×C7C22×C22⋊C411yescharacteristic
316C22×C7#53612yes—
317C22×C7C2×#13512yes—
318C22×C7#57511yescharacteristic
319C22×C7#57511yescharacteristic
320C4×C7—22no—
321C4×C7—42no—
322C4×C7—42no—
323C4×C7—82no—
324C4×C7#51912yes—
325C4×C7#53612yes—
326Dic7—22no—
327Dic7—22no—
328Dic7—22no—
329Dic7—42no—
330Dic7—42no—
331Dic7—82no—
332Dic7#41614yes—
333C24D4×D711yescharacteristic
334C2×Q8—144no—
335C2×Q8—144no—
336C2×Q8—284no—
337C2×Q8—282no—
338C2×Q8—282no—
339C22×C4—24no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—71no—
344C22×C4—74no—
345C22×C4—71no—
346C22×C4—71no—
347C22×C4—71no—
348C22×C4—74no—
349C22×C4—144no—
350C22×C4—141no—
351C22×C4—141no—
352C22×C4—142no—
353C22×C4—281no—
354C22×C4—281no—
355C22×C4C2×C4×D712yes—
356C22×C4D4×D711yescharacteristic
357C22×C4D4×D712yes—
358C22×C4C23×D711yescharacteristic
359C2×C8—22no—
360C2×C8—24no—
361C2×C8—142no—
362C2×C8—144no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—24no—
366C4⋊C4—72no—
367C4⋊C4—72no—
368C4⋊C4—144no—
369C4⋊C4—144no—
370C4⋊C4—144no—
371C4⋊C4—144no—
372C4⋊C4—144no—
373C4⋊C4—144no—
374C4⋊C4—141no—
375C4⋊C4—141no—
376C4⋊C4—144no—
377C4⋊C4C2×C4×D712yes—
378C4⋊C4C2×C4×D712yes—
379C22⋊C4—24no—
380C22⋊C4—42no—
381C22⋊C4—144no—
382C22⋊C4—282no—
383C42—24no—
384C42—74no—
385C42—74no—
386C42—72no—
387C42—72no—
388C42—72no—
389C42—72no—
390C42—144no—
391C42—284no—
392C2×C7—42no—
393C2×C7#231011yescharacteristic
394C2×C7C2×#41611yescharacteristic
395C2×C7#289611yescharacteristic
396C2×C7#323312yes—
397C2×C7#323312yes—
398C23—22no—
399C23—21no—
400C23—21no—
401C23—22no—
402C23D7×C22⋊C412yes—
403C23C2×D4×D711yescharacteristic
404Q8—288no—
405Q8—288no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—72no—
418C2×C4—74no—
419C2×C4—72no—
420C2×C4—72no—
421C2×C4—72no—
422C2×C4—148no—
423C2×C4—142no—
424C2×C4—141no—
425C2×C4—141no—
426C2×C4—141no—
427C2×C4—141no—
428C2×C4—142no—
429C2×C4—142no—
430C2×C4—284no—
431C2×C4—282no—
432C2×C4—281no—
433C2×C4—281no—
434C2×C4—282no—
435C2×C4—282no—
436C2×C4—282no—
437C2×C4D7×C22⋊C412yes—
438C2×C4C22×C4×D711yescharacteristic
439C2×C4C2×D4×D711yescharacteristic
440C2×C4#655414yes—
441C8—42no—
442C8—282no—
443C7#2183711yescharacteristic, complemented, Sylow
444C22—22no—
445C22—22no—
446C22—22no—
447C22—42no—
448C22—42no—
449C22C2×D7×C22⋊C411yescharacteristic
450C22#9033912yes—
451C22D7×#13512yes—
452C22#9046411yescharacteristic
453C22#9046811yescharacteristic
454C4—22no—
455C4—42no—
456C4—42no—
457C4—74no—
458C4—82no—
459C4—142no—
460C4—142no—
461C4—142no—
462C4—282no—
463C4—282no—
464C4—562no—
465C4#9020712yes—
466C4#9033912yes—
467C2—42no—
468C2#1173149811yescharacteristic
469C2D7×#41611yescharacteristic
470C2#1173632611yescharacteristic
471C2#1173880212yes—
472C2#1173894612yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1825C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#1825D7
hypocenterC7#21837
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
8. #11726558C2semidirect
9. #11726558C2semidirect
10. #11726554C2semidirect
11. #11726553C2semidirect
16. #90208C4semidirect
443. C7#21837semidirect

Maximal subgroups

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

subgroupindex
C2×#902082
C2×#894902
#117313082
C7×#18252
#117293292
#117276802
#117272572
#117265582
#117265542
#117265532
#218377