Finite Group Databasebeta

← #13363337

Subgroups — #13363337

Order 1792.

1010 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90379C211yesmaximal, characteristic
2C2×#89607C211yesmaximal, characteristic
3#11731197C211yesmaximal, characteristic
4C7×#1989C211yesmaximal, characteristic, complemented
5#11729492C211yesmaximal, characteristic, complemented
6#11728215C211yesmaximal, characteristic, complemented
7#11727245C211yesmaximal, characteristic, complemented
8#11726627C212yesmaximal
9#11726567C212yesmaximal
10#11726560C212yesmaximal, complemented
11#11726560C212yesmaximal, complemented
12C2×C7×#125C2211yescharacteristic
13#90379—24no—
14#90379C418yescomplemented
15C2×#6459—21no—
16C2×#6459—21no—
17C2×#6455C2211yescharacteristic
18C2×#6455C2211yescharacteristic
19C2×#6451C2211yescharacteristic
20C2×#6451C2211yescharacteristic
21C2×#6450—21no—
22C2×#6450—21no—
23C2×C42×D7C2211yescharacteristic
24C2×#6433C2211yescharacteristic
25C2×C7×C4⋊C8C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C8C2212yes—
28C2×#6389C2212yes—
29#89759C2212yes—
30#89749C2212yes—
31C4×(C2×D7)⋊C4—21no—
32C4×C4⋊Dic7C2211yescharacteristic
33#89704—21no—
34C2×C4⋊(C7⋊C8)C2211yescharacteristic
35#89607—24no—
36#89436—21no—
37#89423—21no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#89324C2212yes—
45#89324C2212yes—
46#21842—71nomaximal, Sylow
47C22×C4×D7D411yescharacteristic
48C22×C4×D7D411yescharacteristic
49C22×C4×D7C2311yescharacteristic
50C22×Dic14—21no—
51C22×Dic14—21no—
52C22×C7×C8C2312yes—
53C7×#125—22no—
54C7×#125C2×C414yes—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C42×C7—21no—
61C2×C42×C7C2311yescharacteristic
62C2×(C2×D7)⋊C4—21no—
63C2×(C2×D7)⋊C4—21no—
64C2×(C2×D7)⋊C4—22no—
65C2×C4⋊Dic7C2312yes—
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2311yescharacteristic
68C2×Dic7⋊C4—21no—
69C2×Dic7⋊C4—21no—
70C2×Dic7⋊C4—21no—
71C2×Dic7⋊C4—21no—
72C2×C4×Dic7—21no—
73C2×C4×Dic7—21no—
74C2×C4×Dic7—21no—
75C2×C4×Dic7D411yescharacteristic
76C2×C4×Dic7D411yescharacteristic
77C2×C4×Dic7C2311yescharacteristic
78C22×C7⋊C8C2312yes—
79#6459—28no—
80#6459—28no—
81#6455—22no—
82#6455—22no—
83#6455C2×C414yes—
84#6455C2×C414yes—
85#6451—22no—
86#6451—22no—
87#6451C2×C414yes—
88#6451C2×C414yes—
89#6450—28no—
90#6450—28no—
91C42×D7C2×C414yes—
92C42×D7D414yes—
93#6433—22no—
94#6433C2×C414yes—
95C7×C4⋊C8—22no—
96C7×#95—21no—
97C7⋊#95—21no—
98C7⋊#95—22no—
99C7⋊#95—22no—
100C7⋊#95—22no—
101(C2×D7)⋊C8—24no—
102#6389—24no—
103C4⋊(C7⋊C8)—22no—
104C2×#533—71no—
105C2×#432—71no—
106#2283—71no—
107#1993—72no—
108#1993—72no—
109#1989—71no—
110#1989—71no—
111#1989—71no—
112#1989D711yescharacteristic, complemented
113#1825—72no—
114#1825—72no—
115C23×D7C2×D411yescharacteristic
116C22×C4×C7—21no—
117C22×C4×C7—21no—
118C22×C4×C7—22no—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D411yescharacteristic
121C22×C4×C7C2411yescharacteristic
122C22×Dic7—21no—
123C22×Dic7—21no—
124C22×Dic7—22no—
125C22×Dic7C2×D411yescharacteristic
126C22×Dic7C2×D411yescharacteristic
127C22×Dic7C2×D411yescharacteristic
128C2×C4×D7—22no—
129C2×C4×D7—22no—
130C2×C4×D7—41no—
131C2×C4×D7—41no—
132C2×C4×D7—42no—
133C2×C4×D7C22⋊C414yes—
134C2×C4×D7C22⋊C414yes—
135C2×C4×D7C22×C412yes—
136C2×C4×D7C2×D412yes—
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—24no—
145C7×C4⋊C4—24no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4C22×C412yes—
151C7×C4⋊C4C22×C412yes—
152C42×C7—42no—
153C42×C7C22×C412yes—
154C42×C7C2×D412yes—
155(C2×D7)⋊C4—24no—
156(C2×D7)⋊C4—24no—
157(C2×D7)⋊C4—44no—
158C4⋊Dic7—21no—
159C4⋊Dic7—21no—
160C4⋊Dic7—22no—
161C4⋊Dic7—22no—
162C4⋊Dic7C22×C412yes—
163C4⋊Dic7C22×C412yes—
164Dic7⋊C4—24no—
165Dic7⋊C4—24no—
166Dic7⋊C4—24no—
167Dic7⋊C4—24no—
168C4×Dic7—24no—
169C4×Dic7—24no—
170C4×Dic7—42no—
171C4×Dic7C22⋊C414yes—
172C4×Dic7C22⋊C414yes—
173C4×Dic7C22×C412yes—
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×#125C2×D711yescharacteristic
183C2×C22⋊Q8—141no—
184C2×C22⋊Q8—141no—
185C2×C4×Q8—141no—
186C2×C4×Q8—141no—
187C2×#117—71no—
188C2×#117—71no—
189C22×C42—71no—
190#432—144no—
191C2×C4⋊C8—71no—
192C2×C4⋊C8C2×D711yescharacteristic
193C2×C22⋊C8—72no—
194C2×#97—72no—
195#383—141no—
196#382—72no—
197#381—141no—
198#380—72no—
199C4×C4⋊C4—71no—
200C4×C4⋊C4C2×D711yescharacteristic
201C4×C22⋊C4—141no—
202#376—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×C7C22×D411yescharacteristic
212C22×D7—22no—
213C22×D7—22no—
214C22×D7—21no—
215C22×D7—21no—
216C22×D7C2×C22⋊C412yes—
217C2×C4×C7—21no—
218C2×C4×C7—21no—
219C2×C4×C7—22no—
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×C7C2×C22⋊C412yes—
230C2×C4×C7C23×C411yescharacteristic
231C2×C4×C7C22×D411yescharacteristic
232C2×C4×C72−1+412yes—
233C2×C4×C72−1+412yes—
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—21no—
237C2×Dic7—21no—
238C2×Dic7—22no—
239C2×Dic7—22no—
240C2×Dic7—42no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—42no—
246C2×Dic7C2×C22⋊C412yes—
247C2×Dic7C2×C22⋊C412yes—
248C2×Dic7C2×C22⋊C412yes—
249C4×D7—24no—
250C4×D7—22no—
251C4×D7—22no—
252C4×D7—44no—
253C4×D7—44no—
254Dic14—48no—
255Dic14—48no—
256C7×C8—42no—
257C7⋊C8—42no—
258C22×Q8—141no—
259C22×Q8—141no—
260C23×C4—71no—
261C23×C4—71no—
262C23×C4—71no—
263C22×C8—72no—
264C22×C8C22×D712yes—
265C4⋊Q8—74no—
266C4⋊Q8—74no—
267C4⋊Q8—142no—
268C4⋊Q8—142no—
269#125—22no—
270#125—74no—
271#125—142no—
272#125C4×D714yes—
273C22⋊Q8—148no—
274C22⋊Q8—148no—
275C4×Q8—148no—
276C4×Q8—148no—
277#117—74no—
278#117—74no—
279#117—142no—
280#117—142no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—71no—
284C2×C4⋊C4—71no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—141no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4—141no—
289C2×C4⋊C4—141no—
290C2×C4⋊C4C22×D712yes—
291C2×C4⋊C4C22×D711yescharacteristic
292C2×C4⋊C4C22×D711yescharacteristic
293C2×C22⋊C4—141no—
294C2×C22⋊C4—141no—
295C2×C22⋊C4—142no—
296C2×C42—21no—
297C2×C42—71no—
298C2×C42—71no—
299C2×C42—71no—
300C2×C42—74no—
301C2×C42—74no—
302C2×C42—141no—
303C2×C42—141no—
304C2×C42—141no—
305C2×C42C22×D711yescharacteristic
306C4⋊C8—22no—
307C4⋊C8—142no—
308C22⋊C8—144no—
309#97—144no—
310#95—21no—
311#95—142no—
312#95—142no—
313#95—142no—
314#95—141no—
315C22×C7C22×C22⋊C411yescharacteristic
316C22×C7#53612yes—
317C22×C7#57512yes—
318C22×C7#57511yescharacteristic
319C22×C7#57511yescharacteristic
320C2×D7—22no—
321C2×D7—22no—
322C2×D7—22no—
323C2×D7—42no—
324C2×D7—42no—
325C4×C7—22no—
326C4×C7—22no—
327C4×C7—42no—
328C4×C7—42no—
329C4×C7—82no—
330C4×C7#51912yes—
331C4×C7#53611yescharacteristic
332C4×C7#53611yescharacteristic
333Dic7—22no—
334Dic7—22no—
335Dic7—22no—
336Dic7—42no—
337Dic7—42no—
338Dic7—82no—
339C24—71no—
340C2×Q8—144no—
341C2×Q8—144no—
342C2×Q8—284no—
343C2×Q8—282no—
344C2×Q8—282no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—71no—
349C22×C4—71no—
350C22×C4—71no—
351C22×C4—72no—
352C22×C4—72no—
353C22×C4—74no—
354C22×C4—74no—
355C22×C4—142no—
356C22×C4—142no—
357C22×C4—141no—
358C22×C4—141no—
359C22×C4—142no—
360C22×C4—282no—
361C22×C4—281no—
362C22×C4—281no—
363C22×C4D4×D711yescharacteristic
364C22×C4D4×D711yescharacteristic
365C22×C4C23×D711yescharacteristic
366C2×C8—22no—
367C2×C8—24no—
368C2×C8—142no—
369C2×C8—144no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—72no—
377C4⋊C4—72no—
378C4⋊C4—144no—
379C4⋊C4—144no—
380C4⋊C4—144no—
381C4⋊C4—144no—
382C4⋊C4—141no—
383C4⋊C4—141no—
384C4⋊C4—142no—
385C4⋊C4—142no—
386C4⋊C4C2×C4×D712yes—
387C4⋊C4C2×C4×D712yes—
388C22⋊C4—144no—
389C22⋊C4—144no—
390C22⋊C4—284no—
391C42—42no—
392C42—74no—
393C42—74no—
394C42—72no—
395C42—72no—
396C42—144no—
397C42—144no—
398C42—282no—
399C42C2×C4×D712yes—
400C42D4×D712yes—
401C2×C7#231011yescharacteristic
402C2×C7#289611yescharacteristic
403C2×C7#289611yescharacteristic
404C2×C7#324012yes—
405C2×C7#324012yes—
406D7—42no—
407C23—72no—
408C23—142no—
409C23—141no—
410C23—141no—
411C23—142no—
412C23C2×D4×D711yescharacteristic
413Q8—288no—
414Q8—288no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—72no—
427C2×C4—72no—
428C2×C4—72no—
429C2×C4—142no—
430C2×C4—141no—
431C2×C4—141no—
432C2×C4—142no—
433C2×C4—142no—
434C2×C4—144no—
435C2×C4—142no—
436C2×C4—142no—
437C2×C4—142no—
438C2×C4—284no—
439C2×C4—284no—
440C2×C4—282no—
441C2×C4—281no—
442C2×C4—281no—
443C2×C4—282no—
444C2×C4—282no—
445C2×C4—282no—
446C2×C4D7×C22⋊C412yes—
447C2×C4D7×C22⋊C412yes—
448C2×C4C22×C4×D711yescharacteristic
449C2×C4C2×D4×D711yescharacteristic
450C2×C4#655112yes—
451C2×C4#655412yes—
452C8—42no—
453C8—282no—
454C7#2184211yescharacteristic, complemented, Sylow
455C22—142no—
456C22—142no—
457C22—142no—
458C22—282no—
459C22—282no—
460C22C2×D7×C22⋊C411yescharacteristic
461C22#9031912yes—
462C22#9045912yes—
463C22#9046411yescharacteristic
464C22#9046811yescharacteristic
465C4—22no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—82no—
470C4—142no—
471C4—142no—
472C4—142no—
473C4—282no—
474C4—282no—
475C4—562no—
476C4#9026312yes—
477C4#9036611yescharacteristic
478C4#9041511yescharacteristic
479C2—282no—
480C2#1173405511yescharacteristic
481C2#1173627511yescharacteristic
482C2#1173632611yescharacteristic
483C2#1173862012yes—
484C2#1173868412yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1989C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1989D7
hypocenterC7#21842
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. C7×#1989C2semidirect
5. #11729492C2semidirect
6. #11728215C2semidirect
7. #11727245C2semidirect
10. #11726560C2semidirect
11. #11726560C2semidirect
14. #90379C4semidirect
112. #1989D7semidirect
454. C7#21842semidirect

Maximal subgroups

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

subgroupindex
C2×#903792
C2×#896072
#117311972
C7×#19892
#117294922
#117282152
#117272452
#117266272
#117265672
#117265602
#218427