Finite Group Databasebeta

← #13363289

Subgroups — #13363289

Order 1792.

1074 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90208C211yesmaximal, characteristic
2C2×#89490C211yesmaximal, characteristic
3#11731309C211yesmaximal, characteristic
4C7×#1825C211yesmaximal, characteristic
5#11729329C211yesmaximal, characteristic
6#11727680C211yesmaximal, characteristic
7#11727257C211yesmaximal, characteristic
8#11726557C212yesmaximal, complemented
9#11726557C212yesmaximal, complemented
10#11726556C212yesmaximal, complemented
11#11726555C212yesmaximal, 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×#6440—21no—
24C2×C4×Dic14—21no—
25C2×C4×Dic14—21no—
26C2×C7×C22⋊C8C2211yescharacteristic
27C7×#380C2211yescharacteristic
28#89984C2211yescharacteristic
29C2×C22⋊(C7⋊C8)C2211yescharacteristic
30C2×Dic7⋊C8C2212yes—
31C2×C8×Dic7C2212yes—
32#89748C2212yes—
33C4⋊C4×Dic7C2212yes—
34#89714—21no—
35#89490—24no—
36#89422—21no—
37#89421—21no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21835—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×Dic7⋊C4—22no—
75C2×Dic7⋊C4—22no—
76C2×C4×Dic7—21no—
77C2×C4×Dic7—22no—
78C2×C4×Dic7C2×C414yes—
79C2×C4×Dic7D411yescharacteristic
80C2×C4×Dic7D411yescharacteristic
81C2×C4×Dic7D414yes—
82C2×C4×Dic7C2311yescharacteristic
83C2×C4×Dic7C2311yescharacteristic
84C22×C7⋊C8C2312yes—
85#6453—22no—
86#6453—22no—
87#6453C2×C414yes—
88#6453C2×C414yes—
89#6451—22no—
90#6451—22no—
91#6451C2×C414yes—
92#6451C2×C414yes—
93#6441—28no—
94#6441—28no—
95#6440—44no—
96C4×Dic14—28no—
97C4×Dic14—28no—
98C7×C22⋊C8—22no—
99C7×#95—21no—
100C7⋊#95—21no—
101C7⋊#95—22no—
102C22⋊(C7⋊C8)—22no—
103Dic7⋊C8—24no—
104C8×Dic7—24no—
105C2×#533—71no—
106C2×#431—71no—
107#2285—71no—
108#1990—72no—
109#1990—72no—
110#1989—72no—
111#1989—72no—
112#1825—71no—
113#1825—71no—
114#1825—71no—
115#1825D711yescharacteristic
116C24×C7C2×D411yescharacteristic
117C22×C4×C7—24no—
118C22×C4×C7—21no—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7C22×C412yes—
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2411yescharacteristic
125C22×Dic7—24no—
126C22×Dic7—22no—
127C22×Dic7—22no—
128C22×Dic7—21no—
129C22×Dic7—21no—
130C22×Dic7—22no—
131C22×Dic7C22⋊C414yes—
132C22×Dic7C22⋊C414yes—
133C22×Dic7C2×D411yescharacteristic
134C22×Dic7C2×D411yescharacteristic
135C22×Dic7C2×D411yescharacteristic
136C22×Dic7C2×D411yescharacteristic
137C2×Dic14—24no—
138C2×Dic14—24no—
139C2×Dic14—44no—
140C2×Dic14—42no—
141C2×Dic14—42no—
142C2×C7×C8—22no—
143C2×C7×C8—24no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—24no—
147C7×C4⋊C4C22×C412yes—
148C7×C4⋊C4C22×C412yes—
149C7×C22⋊C4—24no—
150C7×C22⋊C4—42no—
151C42×C7—24no—
152C22⋊Dic7—24no—
153C22⋊Dic7—42no—
154C4⋊Dic7—21no—
155C4⋊Dic7—21no—
156C4⋊Dic7—24no—
157C4⋊Dic7—24no—
158C4⋊Dic7—24no—
159C4⋊Dic7C22×C412yes—
160C4⋊Dic7C22×C412yes—
161Dic7⋊C4—24no—
162Dic7⋊C4—24no—
163Dic7⋊C4—24no—
164Dic7⋊C4—24no—
165Dic7⋊C4—44no—
166Dic7⋊C4—44no—
167C4×Dic7—24no—
168C4×Dic7—44no—
169C4×Dic7C22⋊C414yes—
170C4×Dic7C22⋊C414yes—
171C4×Dic7C22×C412yes—
172C4×Dic7C22×C412yes—
173C4×Dic7C2×D412yes—
174C4×Dic7C2×D412yes—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—24no—
177#533—78no—
178#533—144no—
179C2×C4⋊Q8—71no—
180C2×C4⋊Q8—71no—
181C2×#125—71no—
182C2×#125—71no—
183C2×C22⋊Q8—141no—
184C2×C22⋊Q8—141no—
185C2×C4×Q8—141no—
186C2×C4×Q8—141no—
187C2×#117—71no—
188C2×#117—141no—
189C2×#117C2×D711yescharacteristic
190C22×C42—71no—
191#431—144no—
192C2×C4⋊C8—72no—
193C2×C22⋊C8—71no—
194C2×C22⋊C8C2×D711yescharacteristic
195C2×C4×C8—72no—
196#384—72no—
197#382—141no—
198#380—71no—
199#380—141no—
200#380C2×D711yescharacteristic
201C4×C4⋊C4—72no—
202C4×C4⋊C4—141no—
203#340—72no—
204#340—72no—
205#340—72no—
206#340—72no—
207#340—72no—
208#340—72no—
209#340C2×D712yes—
210#340C2×D712yes—
211C23×C7—22no—
212C23×C7—22no—
213C23×C7—21no—
214C23×C7—21no—
215C23×C7C2×C22⋊C412yes—
216C23×C7C22×D411yescharacteristic
217C2×C4×C7—22no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—42no—
223C2×C4×C7—41no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7C2×C22⋊C412yes—
229C2×C4×C7C23×C411yescharacteristic
230C2×C4×C7C22×D411yescharacteristic
231C2×C4×C72−1+414yes—
232C2×Dic7—22no—
233C2×Dic7—21no—
234C2×Dic7—21no—
235C2×Dic7—28no—
236C2×Dic7—28no—
237C2×Dic7—22no—
238C2×Dic7—22no—
239C2×Dic7—42no—
240C2×Dic7—41no—
241C2×Dic7—41no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7C2×C22⋊C412yes—
246C2×Dic7C2×C22⋊C412yes—
247C2×Dic7C2×C22⋊C412yes—
248C2×Dic7C2×C22⋊C412yes—
249C2×Dic7#13514yes—
250C2×Dic7AGL(1,ℤ/8)12yes—
251C2×Dic7#13712yes—
252Dic14—48no—
253Dic14—48no—
254C7×C8—42no—
255C7⋊C8—42no—
256C22×Q8—141no—
257C22×Q8—141no—
258C23×C4—71no—
259C23×C4—71no—
260C23×C4C22×D711yescharacteristic
261C22×C8—72no—
262C22×C8C22×D712yes—
263C4⋊Q8—74no—
264C4⋊Q8—74no—
265C4⋊Q8—142no—
266C4⋊Q8—142no—
267#125—74no—
268#125—74no—
269#125—142no—
270#125—142no—
271C22⋊Q8—148no—
272C22⋊Q8—148no—
273C4×Q8—148no—
274C4×Q8—148no—
275#117—22no—
276#117—74no—
277#117—142no—
278#117—284no—
279#117C4×D714yes—
280C2×C4⋊C4—71no—
281C2×C4⋊C4—71no—
282C2×C4⋊C4—72no—
283C2×C4⋊C4—141no—
284C2×C4⋊C4—141no—
285C2×C4⋊C4—141no—
286C2×C4⋊C4—141no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4—141no—
289C2×C4⋊C4—142no—
290C2×C4⋊C4—142no—
291C2×C4⋊C4C22×D712yes—
292C2×C4⋊C4C22×D711yescharacteristic
293C2×C4⋊C4C22×D711yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—141no—
297C2×C22⋊C4—141no—
298C2×C42—21no—
299C2×C42—74no—
300C2×C42—71no—
301C2×C42—71no—
302C2×C42—74no—
303C2×C42—71no—
304C2×C42—71no—
305C2×C42—141no—
306C2×C42—142no—
307C4⋊C8—144no—
308C22⋊C8—22no—
309C22⋊C8—142no—
310C4×C8—144no—
311#95—21no—
312#95—142no—
313#95—141no—
314C22×C7—22no—
315C22×C7—22no—
316C22×C7—22no—
317C22×C7—42no—
318C22×C7—42no—
319C22×C7C22×C22⋊C411yescharacteristic
320C22×C7#53612yes—
321C22×C7C2×#13512yes—
322C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
323C22×C7C2×#13711yescharacteristic
324C4×C7—22no—
325C4×C7—42no—
326C4×C7—42no—
327C4×C7—82no—
328C4×C7#51912yes—
329C4×C7#53612yes—
330Dic7—22no—
331Dic7—22no—
332Dic7—42no—
333Dic7—42no—
334Dic7—82no—
335Dic7#41614yes—
336Dic7#41714yes—
337C24D4×D711yescharacteristic
338C2×Q8—144no—
339C2×Q8—144no—
340C2×Q8—284no—
341C2×Q8—282no—
342C2×Q8—282no—
343C22×C4—24no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—71no—
348C22×C4—71no—
349C22×C4—74no—
350C22×C4—74no—
351C22×C4—71no—
352C22×C4—71no—
353C22×C4—142no—
354C22×C4—144no—
355C22×C4—142no—
356C22×C4—141no—
357C22×C4—141no—
358C22×C4—142no—
359C22×C4C2×C4×D712yes—
360C22×C4D4×D711yescharacteristic
361C22×C4D4×D712yes—
362C22×C4C23×D711yescharacteristic
363C2×C8—22no—
364C2×C8—24no—
365C2×C8—142no—
366C2×C8—144no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—24no—
370C4⋊C4—72no—
371C4⋊C4—72no—
372C4⋊C4—144no—
373C4⋊C4—144no—
374C4⋊C4—144no—
375C4⋊C4—144no—
376C4⋊C4—144no—
377C4⋊C4—144no—
378C4⋊C4—141no—
379C4⋊C4—141no—
380C4⋊C4—144no—
381C4⋊C4—284no—
382C4⋊C4—284no—
383C4⋊C4C2×C4×D712yes—
384C4⋊C4C2×C4×D712yes—
385C22⋊C4—24no—
386C22⋊C4—42no—
387C22⋊C4—144no—
388C22⋊C4—282no—
389C42—24no—
390C42—74no—
391C42—74no—
392C42—72no—
393C42—72no—
394C42—72no—
395C42—72no—
396C42—144no—
397C42—284no—
398C2×C7—42no—
399C2×C7#231011yescharacteristic
400C2×C7C2×#41611yescharacteristic
401C2×C7C2×#41711yescharacteristic
402C2×C7#322812yes—
403C2×C7#323112yes—
404C23—22no—
405C23—21no—
406C23—21no—
407C23—22no—
408C23D7×C22⋊C412yes—
409C23C2×D4×D711yescharacteristic
410Q8—288no—
411Q8—288no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—72no—
424C2×C4—72no—
425C2×C4—74no—
426C2×C4—72no—
427C2×C4—72no—
428C2×C4—72no—
429C2×C4—72no—
430C2×C4—148no—
431C2×C4—148no—
432C2×C4—142no—
433C2×C4—141no—
434C2×C4—141no—
435C2×C4—142no—
436C2×C4—142no—
437C2×C4—282no—
438C2×C4—281no—
439C2×C4—281no—
440C2×C4—282no—
441C2×C4—282no—
442C2×C4—282no—
443C2×C4D7×C22⋊C412yes—
444C2×C4C22×C4×D711yescharacteristic
445C2×C4C2×D4×D711yescharacteristic
446C2×C4#655414yes—
447C8—42no—
448C8—282no—
449C7#2183511yescharacteristic, complemented, Sylow
450C22—22no—
451C22—22no—
452C22—22no—
453C22—42no—
454C22—42no—
455C22C2×D7×C22⋊C411yescharacteristic
456C22#9033912yes—
457C22D7×#13512yes—
458C22D7×AGL(1,ℤ/8)11yescharacteristic
459C22D7×#13711yescharacteristic
460C4—22no—
461C4—42no—
462C4—42no—
463C4—74no—
464C4—74no—
465C4—82no—
466C4—142no—
467C4—142no—
468C4—282no—
469C4—282no—
470C4—562no—
471C4#9020712yes—
472C4#9033912yes—
473C2—42no—
474C2#1173149811yescharacteristic
475C2D7×#41611yescharacteristic
476C2D7×#41711yescharacteristic
477C2#1173880112yes—
478C2#1173894512yes—
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#21835
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. #11726557C2semidirect
9. #11726557C2semidirect
10. #11726556C2semidirect
11. #11726555C2semidirect
16. #90208C4semidirect
449. C7#21835semidirect

Maximal subgroups

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

subgroupindex
C2×#902082
C2×#894902
#117313092
C7×#18252
#117293292
#117276802
#117272572
#117265572
#117265562
#117265552
#218357