Finite Group Databasebeta

← #13407454

Subgroups — #13407454

Order 1792.

1094 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×#90209C211yesmaximal, characteristic, complemented
2C7×#2620C211yesmaximal, characteristic
3C2×#89832C211yesmaximal, characteristic, complemented
4C2×#89828C211yesmaximal, characteristic, complemented
5#11729454C212yesmaximal, complemented
6#11729439C212yesmaximal
7#11729336C211yesmaximal, characteristic, complemented
8#11727444C212yesmaximal, complemented
9#11727414C212yesmaximal, complemented
10#11727337C211yesmaximal, characteristic, complemented
11#11727333C211yesmaximal, characteristic, complemented
12C2×C7×#124C2211yescharacteristic
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×C7×#121C2211yescharacteristic
15C2×C7×#117C2211yescharacteristic
16C22×#1144C2212yes—
17C22×#1142C2212yes—
18C2×#6487C2212yes—
19#90209—24no—
20#90209—24no—
21C2×#6452—22no—
22C2×#6442—22no—
23C2×#6433C2211yescharacteristic
24C2×#6432C2211yescharacteristic
25C2×C4×Dic14C2212yes—
26C7×#396—22no—
27C7×#394—22no—
28C7×#391C2211yescharacteristic
29C7×#390C2211yescharacteristic
30C7×#384C2211yescharacteristic
31C2×#6409C2212yes—
32C2×#6406C2212yes—
33C2×C22⋊(C7⋊C8)C2212yes—
34#89832—24no—
35#89828—24no—
36C2×#6384C2212yes—
37C2×#6384C2212yes—
38C2×#6377C2211yescharacteristic
39C2×C4×C7⋊C8C2211yescharacteristic
40#89324C2212yes—
41#89324C2212yes—
42#89319C2212yes—
43#25612—71nomaximal, Sylow
44C22×C7×Q8C2311yescharacteristic
45C22×C7×D4—21no—
46C22×C7×D4C2311yescharacteristic
47C23×C4×C7C2311yescharacteristic
48C22×Dic14C2312yes—
49C7×#124—22no—
50C7×#124—22no—
51C7×C22⋊Q8—42no—
52C7×#121—42no—
53C7×#117—22no—
54C7×#117—22no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4D412yes—
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C4⋊C4C2311yescharacteristic
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—22no—
63C2×C7×C22⋊C4—22no—
64C2×C7×C22⋊C4D412yes—
65C2×C42×C7C2311yescharacteristic
66C2×C42×C7C2311yescharacteristic
67C2×C22⋊Dic7—22no—
68C2×#1144—24no—
69C2×#1144—24no—
70C2×#1144—42no—
71C2×#1142—24no—
72C2×#1142—24no—
73C2×#1142—42no—
74C2×C4⋊Dic7—22no—
75C2×C4⋊Dic7C2312yes—
76#6487—24no—
77#6487—24no—
78C2×Dic7⋊C4—22no—
79C2×Dic7⋊C4—22no—
80C2×Dic7⋊C4—22no—
81C2×C4×Dic7—22no—
82C22×C7⋊C8C2312yes—
83C22×C7⋊C8C2312yes—
84#6452—216no—
85#6442—216no—
86#6433—22no—
87#6433D414yescomplemented
88#6432—22no—
89#6432D414yescomplemented
90C4×Dic14—24no—
91C4×Dic14—24no—
92C7×#95—22no—
93#6409—24no—
94#6406—24no—
95C22⋊(C7⋊C8)—24no—
96#6384—42no—
97#6384—42no—
98#6377—22no—
99C4×C7⋊C8—22no—
100C2×#541—71no—
101C2×#497—71no—
102C2×#488—71no—
103#2620D711yescharacteristic
104#2001—72no—
105#2000—72no—
106#1992—71no—
107#1956—71no—
108#1880—72no—
109#1878—72no—
110#1750—71no—
111C24×C7—21no—
112C24×C7C2×D411yescharacteristic
113C2×C7×Q8—21no—
114C2×C7×Q8—21no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×D4—21no—
118C2×C7×D4—21no—
119C2×C7×D4—22no—
120C2×C7×D4—22no—
121C2×C7×D4—42no—
122C2×C7×D4—82no—
123C22×C4×C7—21no—
124C22×C4×C7—21no—
125C22×C4×C7—22no—
126C22×C4×C7—22no—
127C22×C4×C7—21no—
128C22×C4×C7—21no—
129C22×C4×C7—21no—
130C22×C4×C7C2×D412yes—
131C22×C4×C7C2×D411yescharacteristic
132C22×C4×C7C2×D412yes—
133C22×C4×C7C2411yescharacteristic
134C22×Dic7—22no—
135C22×Dic7—22no—
136C2×Dic14—22no—
137C2×Dic14—22no—
138C2×Dic14—22no—
139C2×Dic14—22no—
140C2×Dic14—22no—
141C2×Dic14C2×D414yes—
142C7×C4⋊C4—24no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—42no—
146C7×C4⋊C4—82no—
147C7×C22⋊C4—24no—
148C7×C22⋊C4—42no—
149C7×C22⋊C4—42no—
150C7×C22⋊C4—44no—
151C7×C22⋊C4—82no—
152C42×C7—21no—
153C42×C7—21no—
154C42×C7C2×D412yes—
155C42×C7C2×D412yes—
156C22⋊Dic7—28no—
157#1144—48no—
158#1142—48no—
159C4⋊Dic7—28no—
160C4⋊Dic7—22no—
161C4⋊Dic7C2×D414yes—
162Dic7⋊C4—28no—
163Dic7⋊C4—28no—
164Dic7⋊C4—28no—
165C4×Dic7—28no—
166C2×C7⋊C8—22no—
167C2×C7⋊C8—22no—
168C2×C7⋊C8—22no—
169C2×C7⋊C8—22no—
170C2×C7⋊C8—22no—
171C2×C7⋊C8#4114yes—
172C22×Q16—72no—
173C22×SD16—72no—
174#541—144no—
175#541—144no—
176C2×C4⋊Q8—71no—
177C2×#126—142no—
178C2×#125—71no—
179C2×#125—142no—
180C2×#124C2×D711yescharacteristic
181C2×C22⋊Q8—72no—
182C2×C22⋊Q8C2×D711yescharacteristic
183C2×#121C2×D711yescharacteristic
184C2×C4×Q8—72no—
185C2×#117C2×D711yescharacteristic
186#497—144no—
187#488—144no—
188C2×#103—72no—
189C2×#103—72no—
190C2×#103—72no—
191C2×#102—72no—
192C2×C22⋊C8—72no—
193C2×#97—71no—
194C2×C4×C8—71no—
195#396—22no—
196#394—22no—
197#391C2×D711yescharacteristic
198#390C2×D711yescharacteristic
199#384C2×D711yescharacteristic
200#340—72no—
201#340—72no—
202#336—72no—
203C23×C7—21no—
204C23×C7—21no—
205C23×C7—21no—
206C23×C7—22no—
207C23×C7—22no—
208C23×C7—41no—
209C23×C7—41no—
210C23×C7—41no—
211C23×C7—42no—
212C23×C7—42no—
213C23×C7C22×D411yescharacteristic
214C7×Q8—42no—
215C7×Q8—42no—
216C7×D4—42no—
217C7×D4—42no—
218C7×D4—84no—
219C2×C4×C7—22no—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—21no—
223C2×C4×C7—21no—
224C2×C4×C7—21no—
225C2×C4×C7—22no—
226C2×C4×C7—22no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—41no—
230C2×C4×C7—41no—
231C2×C4×C7—41no—
232C2×C4×C7—41no—
233C2×C4×C7—41no—
234C2×C4×C7—42no—
235C2×C4×C7—42no—
236C2×C4×C7#12014yes—
237C2×C4×C7C22×D411yescharacteristic
238C2×C4×C7C22×D411yescharacteristic
239C2×C4×C7C2×#4112yes—
240C2×C4×C7H(5,𝔽2)12yes—
241C2×Dic7—24no—
242C2×Dic7—24no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247Dic14—24no—
248Dic14—24no—
249Dic14—42no—
250Dic14—42no—
251C7⋊C8—24no—
252C7⋊C8—42no—
253C22×Q8—72no—
254C22×Q8C22×D711yescharacteristic
255C22×D4—21no—
256C22×D4C22×D711yescharacteristic
257C23×C4C22×D711yescharacteristic
258C2×Q16—144no—
259C2×Q16—144no—
260C2×Q16—282no—
261C2×SD16—144no—
262C2×SD16—144no—
263C2×SD16—282no—
264C22×C8—72no—
265C22×C8—72no—
266C4⋊Q8—74no—
267C4⋊Q8—142no—
268#126—1416no—
269#125—74no—
270#125—1416no—
271#125—142no—
272#124—22no—
273#124—22no—
274C22⋊Q8—42no—
275C22⋊Q8—144no—
276C22⋊Q8—144no—
277#121—42no—
278C4×Q8—144no—
279C4×Q8—144no—
280#117—22no—
281#117—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—72no—
285C2×C4⋊C4—142no—
286C2×C4⋊C4—142no—
287C2×C4⋊C4—142no—
288C2×C4⋊C4—142no—
289C2×C4⋊C4C22⋊D712yes—
290C2×C4⋊C4C22×D711yescharacteristic
291C2×C4⋊C4C22×D711yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—142no—
297C2×C22⋊C4C22⋊D712yes—
298C2×C42—142no—
299C2×C42C22×D711yescharacteristic
300C2×C42C22×D711yescharacteristic
301#103—144no—
302#103—282no—
303#103—282no—
304#102—144no—
305C22⋊C8—144no—
306#97—142no—
307C4×C8—142no—
308#95—22no—
309C22×C7—42no—
310C22×C7—41no—
311C22×C7—41no—
312C22×C7—41no—
313C22×C7—42no—
314C22×C7—82no—
315C22×C7—81no—
316C22×C7—81no—
317C22×C7—81no—
318C22×C7—82no—
319C22×C7C2×#12011yescharacteristic
320C22×C7#53211yescharacteristic
321C22×C7#53511yescharacteristic
322C22×C7#57712yes—
323C22×C7#57812yes—
324C4×C7—24no—
325C4×C7—41no—
326C4×C7—42no—
327C4×C7—81no—
328C4×C7—81no—
329C4×C7—81no—
330C4×C7#54612yes—
331C4×C7#55012yes—
332Dic7—44no—
333Dic7—44no—
334C24—21no—
335C24C2×C22⋊D711yescharacteristic
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—74no—
341C2×Q8—142no—
342C2×Q8—142no—
343C2×Q8—142no—
344C2×Q8—142no—
345C2×Q8—142no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—42no—
351C2×D4—82no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—142no—
360C22×C4—142no—
361C22×C4C2×C22⋊D712yes—
362C22×C4C2×C22⋊D711yescharacteristic
363C22×C4C2×C22⋊D712yes—
364C22×C4C23×D711yescharacteristic
365Q16—288no—
366SD16—288no—
367C2×C8—74no—
368C2×C8—142no—
369C2×C8—142no—
370C2×C8—142no—
371C2×C8—142no—
372C2×C8—142no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—42no—
377C4⋊C4—74no—
378C4⋊C4—82no—
379C4⋊C4—148no—
380C4⋊C4—148no—
381C4⋊C4—148no—
382C4⋊C4—148no—
383C4⋊C4—142no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—82no—
389C22⋊C4—148no—
390C42—21no—
391C42—21no—
392C42—148no—
393C42D4×D712yes—
394C42D4×D712yes—
395C2×C7—81no—
396C2×C7—161no—
397C2×C7#243111yescharacteristic
398C2×C7#301312yes—
399C2×C7#317812yes—
400C2×C7#322211yescharacteristic
401C2×C7#322311yescharacteristic
402C23—21no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23—42no—
412C23C22×C22⋊D711yescharacteristic
413Q8—42no—
414Q8—42no—
415Q8—144no—
416Q8—144no—
417Q8—282no—
418Q8—282no—
419D4—42no—
420D4—42no—
421D4—84no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—144no—
440C2×C4—144no—
441C2×C4—282no—
442C2×C4—282no—
443C2×C4—282no—
444C2×C4—282no—
445C2×C4#650014yes—
446C2×C4C2×D4×D711yescharacteristic
447C2×C4C2×D4×D711yescharacteristic
448C2×C4#654812yes—
449C2×C4D7×#4112yes—
450C8—144no—
451C8—282no—
452C7#2561211yescharacteristic, complemented, Sylow
453C22—41no—
454C22—41no—
455C22—41no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—82no—
462C22—82no—
463C22C2×#650011yescharacteristic
464C22#9049611yescharacteristic
465C22#9052311yescharacteristic
466C22#9052712yes—
467C22#9052912yes—
468C4—24no—
469C4—41no—
470C4—42no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—284no—
475C4—284no—
476C4#9029012yes—
477C4#9030312yes—
478C2—81no—
479C2—161no—
480C2#1174009711yescharacteristic
481C2#1174083811yescharacteristic
482C2#1174085911yescharacteristic
483C2#1174087212yes—
484C2#1174090312yes—
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2620C2
socleC23×C7C22×D4
prime radicalC2×C7×#121C22
hypercenter#2620D7
hypocenterC7#25612
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2620

Complements

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

NH ≅ G/Ntype
1. C2×#90209C2semidirect
3. C2×#89832C2semidirect
4. C2×#89828C2semidirect
5. #11729454C2semidirect
7. #11729336C2semidirect
8. #11727444C2semidirect
9. #11727414C2semidirect
10. #11727337C2semidirect
11. #11727333C2semidirect
87. #6433D4semidirect
89. #6432D4semidirect
452. C7#25612semidirect

Maximal subgroups

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

subgroupindex
C2×#902092
C7×#26202
C2×#898322
C2×#898282
#117294542
#117294392
#117293362
#117274442
#117274142
#117273372
#117273332
#256127