Finite Group Databasebeta

← #13408066

Subgroups — #13408066

Order 1792.

1094 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C2×C7×#498C211yesmaximal, characteristic, complemented
3#11740714C211yesmaximal, characteristic
4C2×#89865C211yesmaximal, characteristic, complemented
5#11729541C211yesmaximal, characteristic, complemented
6#11728342C214yesmaximal, complemented
7#11728336C214yesmaximal
8#11727525C211yesmaximal, characteristic, complemented
9C4×(C2×D7)⋊C8C211yesmaximal, characteristic
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C7×C4⋊Q8C2211yescharacteristic
12C2×#6509—21no—
13C2×#6509—21no—
14C2×Q8×Dic7—21no—
15#90419—24no—
16#90419—24no—
17C2×#6460C2212yes—
18C2×#6455—21no—
19C2×#6453—21no—
20C2×#6450C2212yes—
21C2×C42×D7C2211yescharacteristic
22C2×#6432C2211yescharacteristic
23C7×#498—24no—
24C2×C7×#106C2214yes—
25C2×C4×C7×C8C2211yescharacteristic
26#90002—21no—
27C2×(C2×D7)⋊C8C2212yes—
28C2×#6390C2214yes—
29#89865—24no—
30#89759—21no—
31#89759C2212yes—
32#89745—21no—
33#89745C2212yes—
34C2×#6384C2214yes—
35#89707C2211yescharacteristic
36C2×C4×C7⋊C8C2211yescharacteristic
37#89345C2212yes—
38#89324C2214yes—
39#25672—71nomaximal, Sylow
40C22×C7×Q8—21no—
41C22×C7×Q8—21no—
42C22×C4×D7D411yescharacteristic
43C22×C4×D7D411yescharacteristic
44C22×C4×D7C2311yescharacteristic
45C22×Dic14C2312yes—
46C22×C7×C8C2312yes—
47C7×C4⋊Q8—22no—
48C7×C4⋊Q8—22no—
49C7×C4⋊Q8—22no—
50C7×C4⋊Q8D414yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4C2312yes—
54C2×C7×C4⋊C4C2312yes—
55C2×C42×C7C2311yescharacteristic
56#6509—28no—
57#6509—44no—
58Q8×Dic7—28no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4—22no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7—22no—
64C2×C4⋊Dic7—21no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7C2312yes—
67C2×Dic7⋊C4—22no—
68C2×Dic7⋊C4—22no—
69C2×C4×Dic7—22no—
70C2×C4×Dic7D411yescharacteristic
71C2×C4×Dic7D411yescharacteristic
72C2×C4×Dic7C2311yescharacteristic
73C22×C7⋊C8C2312yes—
74#6460—24no—
75#6460—24no—
76#6455—28no—
77#6453—28no—
78#6450—24no—
79#6450—24no—
80C42×D7—22no—
81C42×D7—22no—
82#6432—22no—
83#6432D414yes—
84C7×#106—28no—
85C4×C7×C8—22no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88(C2×D7)⋊C8—24no—
89#6390—28no—
90#6384—28no—
91C4×C7⋊C8—22no—
92C2×#533—71no—
93C2×#498D711yescharacteristic, complemented
94C2×#487—71no—
95#2857—71no—
96#2035—74no—
97#1990—71no—
98#1956—71no—
99#1865—74no—
100C4×C22⋊C8—71no—
101C23×D7C2×D411yescharacteristic
102C2×C7×Q8—24no—
103C2×C7×Q8—42no—
104C2×C7×Q8—42no—
105C2×C7×Q8—42no—
106C2×C7×Q8—42no—
107C2×C7×Q8—42no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2411yescharacteristic
113C22×Dic7—22no—
114C22×Dic7—22no—
115C22×Dic7C2×D411yescharacteristic
116C22×Dic7C2×D411yescharacteristic
117C22×Dic7C2×D411yescharacteristic
118C2×C4×D7—22no—
119C2×C4×D7—22no—
120C2×C4×D7—22no—
121C2×C4×D7—22no—
122C2×C4×D7—22no—
123C2×C4×D7—21no—
124C2×C4×D7—21no—
125C2×C4×D7—22no—
126C2×C4×D7—22no—
127C2×C4×D7SD1614yes—
128C2×Dic14—24no—
129C2×Dic14—24no—
130C2×Dic14—22no—
131C2×Dic14C2×D414yes—
132C2×C7×C8—22no—
133C2×C7×C8—22no—
134C2×C7×C8#4114yes—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—42no—
140C7×C4⋊C4C2×D414yes—
141C42×C7C2×D412yes—
142C42×C7C2×D412yes—
143(C2×D7)⋊C4—28no—
144(C2×D7)⋊C4—44no—
145C4⋊Dic7—28no—
146C4⋊Dic7—24no—
147C4⋊Dic7—24no—
148C4⋊Dic7—24no—
149C4⋊Dic7—42no—
150C4⋊Dic7—42no—
151Dic7⋊C4—28no—
152Dic7⋊C4—44no—
153C4×Dic7—28no—
154C4×Dic7—22no—
155C4×Dic7—22no—
156C4×Dic7—21no—
157C4×Dic7—21no—
158C2×C7⋊C8—22no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—22no—
161#533—144no—
162#533—144no—
163C2×C4⋊Q8—71no—
164C2×C4⋊Q8C2×D711yescharacteristic
165C2×C4⋊Q8C2×D711yescharacteristic
166C2×#125—141no—
167C2×C22⋊Q8—72no—
168C2×C22⋊Q8—141no—
169C2×C22⋊Q8—141no—
170C2×C4×Q8—72no—
171C2×C4×Q8—141no—
172C2×#117—141no—
173C22×C42—71no—
174#498—24no—
175#487—144no—
176C2×#106C2×D714yes—
177C2×#103—74no—
178C2×#103—74no—
179C2×C22⋊C8—72no—
180C2×C4×C8—71no—
181C2×C4×C8C2×D711yescharacteristic
182#391—141no—
183#389—72no—
184#389—141no—
185#384—71no—
186#380—72no—
187#380—141no—
188#340—74no—
189#336—72no—
190C23×C7C22×D411yescharacteristic
191C22×D7—21no—
192C22×D7—21no—
193C22×D7—21no—
194C22×D7—21no—
195C22×D7—21no—
196C22×D7—21no—
197C22×D7C2×SD1612yes—
198C7×Q8—48no—
199C7×Q8—84no—
200C2×C4×C7—24no—
201C2×C4×C7—42no—
202C2×C4×C7—42no—
203C2×C4×C7—42no—
204C2×C4×C7—42no—
205C2×C4×C7—42no—
206C2×C4×C7#12012yes—
207C2×C4×C7#12012yes—
208C2×C4×C7C2×SD1612yes—
209C2×C4×C7#13512yes—
210C2×C4×C7#13712yes—
211C2×C4×C7#13712yes—
212C2×C4×C7C22×D411yescharacteristic
213C2×C4×C7C22×D411yescharacteristic
214C2×C4×C7C2×#4112yes—
215C2×C4×C72−1+412yes—
216C2×Dic7—21no—
217C2×Dic7—21no—
218C2×Dic7—21no—
219C2×Dic7—21no—
220C2×Dic7—24no—
221C2×Dic7—21no—
222C2×Dic7—21no—
223C2×Dic7—21no—
224C2×Dic7—21no—
225C2×Dic7—42no—
226C2×Dic7—42no—
227C2×Dic7—42no—
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7C2×SD1612yes—
231C4×D7—28no—
232C4×D7—42no—
233C4×D7—42no—
234C4×D7—44no—
235Dic14—28no—
236Dic14—44no—
237C7×C8—24no—
238C7⋊C8—42no—
239C22×Q8—21no—
240C22×Q8—21no—
241C22×Q8—72no—
242C23×C4—71no—
243C23×C4—71no—
244C23×C4—71no—
245C22×C8—72no—
246C22×C8C22×D712yes—
247C4⋊Q8—22no—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—74no—
251C4⋊Q8—142no—
252C4⋊Q8C22⋊D714yes—
253#125—148no—
254C22⋊Q8—144no—
255C22⋊Q8—148no—
256C22⋊Q8—144no—
257C22⋊Q8—284no—
258C4×Q8—144no—
259C4×Q8—148no—
260C4×Q8—144no—
261#117—148no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—72no—
265C2×C4⋊C4—142no—
266C2×C4⋊C4—142no—
267C2×C4⋊C4—141no—
268C2×C4⋊C4—141no—
269C2×C4⋊C4—142no—
270C2×C4⋊C4—141no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4C22×D712yes—
273C2×C4⋊C4C22×D712yes—
274C2×C22⋊C4—142no—
275C2×C22⋊C4—142no—
276C2×C42—71no—
277C2×C42—71no—
278C2×C42—71no—
279C2×C42—142no—
280C2×C42—142no—
281C2×C42—142no—
282C2×C42C22×D711yescharacteristic
283#106—28no—
284#103—148no—
285#103—148no—
286C22⋊C8—144no—
287C4×C8—22no—
288C4×C8—142no—
289#95—142no—
290#95—142no—
291C22×C7C2×#12011yescharacteristic
292C22×C7#53211yescharacteristic
293C22×C7#53611yescharacteristic
294C22×C7C22×SD1611yescharacteristic
295C22×C7C2×#13511yescharacteristic
296C22×C7C2×#13711yescharacteristic
297C22×C7C2×#13711yescharacteristic
298C2×D7—22no—
299C2×D7—22no—
300C2×D7—22no—
301C2×D7—41no—
302C2×D7—41no—
303C2×D7—41no—
304C2×D7—41no—
305C4×C7—44no—
306C4×C7—82no—
307C4×C7#44814yes—
308C4×C7#45214yes—
309C4×C7#54712yes—
310C4×C7#54812yes—
311Dic7—22no—
312Dic7—44no—
313Dic7—41no—
314Dic7—41no—
315Dic7—82no—
316C24—71no—
317C2×Q8—24no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—74no—
324C2×Q8—142no—
325C2×Q8—144no—
326C2×Q8—144no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—71no—
330C22×C4—74no—
331C22×C4—71no—
332C22×C4—71no—
333C22×C4—142no—
334C22×C4—142no—
335C22×C4—142no—
336C22×C4—142no—
337C22×C4—142no—
338C22×C4—142no—
339C22×C4—141no—
340C22×C4—142no—
341C22×C4—142no—
342C22×C4—141no—
343C22×C4—142no—
344C22×C4D4×D711yescharacteristic
345C22×C4D4×D711yescharacteristic
346C22×C4C23×D711yescharacteristic
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—142no—
350C2×C8—142no—
351C2×C8—142no—
352C2×C8#116114yes—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—42no—
358C4⋊C4—148no—
359C4⋊C4—148no—
360C4⋊C4—144no—
361C4⋊C4—144no—
362C4⋊C4—144no—
363C4⋊C4—284no—
364C4⋊C4—282no—
365C4⋊C4—282no—
366C4⋊C4C2×C22⋊D714yes—
367C22⋊C4—148no—
368C22⋊C4—284no—
369C42—148no—
370C42—142no—
371C42—142no—
372C42—141no—
373C42—141no—
374C42D4×D712yes—
375C42C2×C22⋊D712yes—
376C2×C7#243211yescharacteristic
377C2×C7C2×#44811yescharacteristic
378C2×C7C2×#45211yescharacteristic
379C2×C7#316711yescharacteristic
380C2×C7#318511yescharacteristic
381C2×C7#322911yescharacteristic
382C2×C7#323111yescharacteristic
383D7—42no—
384C23—72no—
385C23—141no—
386C23—141no—
387C23—141no—
388C23—141no—
389C23—141no—
390C23—141no—
391C23C2×D4×D711yescharacteristic
392Q8—48no—
393Q8—84no—
394Q8—148no—
395Q8—284no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—72no—
403C2×C4—141no—
404C2×C4—141no—
405C2×C4—148no—
406C2×C4—141no—
407C2×C4—141no—
408C2×C4—141no—
409C2×C4—141no—
410C2×C4—141no—
411C2×C4—141no—
412C2×C4—144no—
413C2×C4—282no—
414C2×C4—282no—
415C2×C4—282no—
416C2×C4—284no—
417C2×C4—282no—
418C2×C4—282no—
419C2×C4—282no—
420C2×C4—282no—
421C2×C4D7×SD1612yes—
422C2×C4#647812yes—
423C2×C4#647812yes—
424C2×C4#647912yes—
425C2×C4#650012yes—
426C2×C4#650012yes—
427C2×C4C2×D4×D711yescharacteristic
428C2×C4C2×#116112yes—
429C2×C4#655112yes—
430C2×C4C22×C22⋊D711yescharacteristic
431C8—24no—
432C8—282no—
433C7#2567211yescharacteristic, complemented, Sylow
434C22—142no—
435C22—142no—
436C22—142no—
437C22—281no—
438C22—281no—
439C22—281no—
440C22—281no—
441C22#9041511yescharacteristic
442C22#9041611yescharacteristic
443C22C2×D7×SD1611yescharacteristic
444C22C2×#647811yescharacteristic
445C22C2×#647811yescharacteristic
446C22C2×#647911yescharacteristic
447C22C2×#650011yescharacteristic
448C4—44no—
449C4—82no—
450C4—142no—
451C4—281no—
452C4—284no—
453C4—281no—
454C4—562no—
455C4#8994214yes—
456C4#8994514yes—
457C4#9041712yes—
458C4#9050812yes—
459C2—282no—
460C2#1173933111yescharacteristic
461C2#1173933411yescharacteristic
462C2#1173934111yescharacteristic
463C2#1173934211yescharacteristic
464C2#1174070011yescharacteristic
465C2C2×#8994211yescharacteristic
466C2C2×#8994511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#498C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
hypercenterC2×#498D7
hypocenterC7#25672
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
2. C2×C7×#498C2semidirect
4. C2×#89865C2semidirect
5. #11729541C2semidirect
6. #11728342C2semidirect
8. #11727525C2semidirect
93. C2×#498D7semidirect
433. C7#25672semidirect

Maximal subgroups

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

subgroupindex
C2×#904192
C2×C7×#4982
#117407142
C2×#898652
#117295412
#117283422
#117283362
#117275252
C4×(C2×D7)⋊C82
#256727