Finite Group Databasebeta

← #13245241

Subgroups — #13245241

Order 1792.

1276 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2#11729881C211yesmaximal, characteristic
3#11729880C211yesmaximal, characteristic
4C2×#89333C214yesmaximal, complemented
5C23×C7×Q8C411yescharacteristic, complemented
6C24×C4×C7C2211yescharacteristic
7C2×C7×C22⋊Q8—21no—
8C2×C7×C22⋊Q8—21no—
9C2×C7×C22⋊Q8—28no—
10C2×C7×C22⋊Q8C412yescomplemented
11C2×C7×C22⋊Q8C2212yes—
12C22×C7×C4⋊C4—21no—
13C22×C7×C4⋊C4C411yescharacteristic, complemented
14C22×C7×C22⋊C4—21no—
15C22×C22⋊Dic7—21no—
16C2×C7⋊#95—21no—
17C2×C7⋊#95C2212yes—
18C2×C22⋊(C7⋊C8)—21no—
19C2×C22⋊(C7⋊C8)—21no—
20C2×C22⋊(C7⋊C8)C2212yes—
21#89333—28no—
22#8329—71nomaximal, Sylow
23C25×C7D411yescharacteristic
24C22×C7×Q8—24no—
25C22×C7×Q8—21no—
26C22×C7×Q8—42no—
27C22×C7×Q8—42no—
28C22×C7×Q8C2×C412yes—
29C23×C4×C7—21no—
30C23×C4×C7—21no—
31C23×C4×C7—24no—
32C23×C4×C7—42no—
33C23×C4×C7—42no—
34C23×C4×C7C2×C411yescharacteristic
35C23×C4×C7C2×C411yescharacteristic
36C23×C4×C7D411yescharacteristic
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C23×Dic7—22no—
41C7×C22⋊Q8—28no—
42C7×C22⋊Q8—28no—
43C7×C22⋊Q8—24no—
44C7×C22⋊Q8—24no—
45C7×C22⋊Q8—24no—
46C7×C22⋊Q8D418yes—
47C2×C7×C4⋊C4—28no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C4⋊C4—24no—
52C2×C7×C4⋊C4C2×C412yes—
53C2×C7×C22⋊C4—28no—
54C2×C7×C22⋊C4—22no—
55C2×C7×C22⋊C4—22no—
56C2×C22⋊Dic7—24no—
57C2×C22⋊Dic7—42no—
58C2×C22⋊Dic7—42no—
59C22×C7⋊C8—22no—
60C7⋊#95—24no—
61C7⋊#95—42no—
62C22⋊(C7⋊C8)—24no—
63C22⋊(C7⋊C8)—42no—
64C22⋊(C7⋊C8)—42no—
65C22×C22⋊Q8D711yescharacteristic
66#1779—71no—
67C2×#328—74no—
68C22⋊#95—71no—
69C24×C7—22no—
70C24×C7—22no—
71C24×C7—22no—
72C24×C7—22no—
73C24×C7—41no—
74C24×C7—41no—
75C24×C7C22⋊C411yescharacteristic
76C24×C7C22⋊C411yescharacteristic
77C24×C7SD1612yes—
78C24×C7Q1612yes—
79C24×C7C2×D411yescharacteristic
80C2×C7×Q8—24no—
81C2×C7×Q8—22no—
82C2×C7×Q8—416no—
83C2×C7×Q8—42no—
84C2×C7×Q8—42no—
85C2×C7×Q8—42no—
86C2×C7×Q8—42no—
87C2×C7×Q8C22⋊C414yescomplemented
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—24no—
92C22×C4×C7—24no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—41no—
103C22×C4×C7—41no—
104C22×C4×C7—41no—
105C22×C4×C7—41no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—41no—
109C22×C4×C7—416no—
110C22×C4×C7—42no—
111C22×C4×C7—42no—
112C22×C4×C7—42no—
113C22×C4×C7—42no—
114C22×C4×C7C22⋊C411yescharacteristic
115C22×C4×C7C22⋊C411yescharacteristic
116C22×C4×C7C22×C411yescharacteristic
117C22×C4×C7C2×D411yescharacteristic
118C22×C4×C7C2×D414yes—
119C22×Dic7—24no—
120C22×Dic7—42no—
121C22×Dic7—42no—
122C22×Dic7—42no—
123C22×Dic7—42no—
124C22×Dic7—42no—
125C22×Dic7—42no—
126C7×C4⋊C4—24no—
127C7×C4⋊C4—28no—
128C7×C4⋊C4—22no—
129C7×C4⋊C4—28no—
130C7×C4⋊C4C22⋊C414yescomplemented
131C7×C22⋊C4—28no—
132C7×C22⋊C4—28no—
133C22⋊Dic7—48no—
134C2×C7⋊C8—42no—
135C2×C7⋊C8—42no—
136C2×C7⋊C8—42no—
137C23×Q8Dic711yescharacteristic, complemented
138C24×C4C2×D711yescharacteristic
139C2×C22⋊Q8—21no—
140C2×C22⋊Q8—21no—
141C2×C22⋊Q8—28no—
142C2×C22⋊Q8Dic712yescomplemented
143C2×C22⋊Q8C2×D712yes—
144C22×C4⋊C4—21no—
145C22×C4⋊C4Dic711yescharacteristic, complemented
146C22×C22⋊C4—21no—
147C22×C22⋊C4—141no—
148C2×C22⋊C8—72no—
149C2×C22⋊C8—141no—
150C2×C22⋊C8—141no—
151C2×#95—72no—
152C2×#95—141no—
153#328—148no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7—24no—
163C23×C7—24no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—21no—
168C23×C7—21no—
169C23×C7—21no—
170C23×C7—24no—
171C23×C7—24no—
172C23×C7—42no—
173C23×C7—42no—
174C23×C7—44no—
175C23×C7—44no—
176C23×C7—44no—
177C23×C7—44no—
178C23×C7#9912yes—
179C23×C7#10312yes—
180C23×C7#10312yes—
181C23×C7#10412yes—
182C23×C7C2×C22⋊C411yescharacteristic
183C23×C7#12012yes—
184C23×C7C2×SD1611yescharacteristic
185C23×C7C2×Q1611yescharacteristic
186C23×C7#13712yes—
187C7×Q8—44no—
188C7×Q8—48no—
189C7×Q8—44no—
190C2×C4×C7—28no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—24no—
198C2×C4×C7—24no—
199C2×C4×C7—21no—
200C2×C4×C7—21no—
201C2×C4×C7—21no—
202C2×C4×C7—44no—
203C2×C4×C7—44no—
204C2×C4×C7—44no—
205C2×C4×C7—44no—
206C2×C4×C7—44no—
207C2×C4×C7—48no—
208C2×C4×C7—48no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7#9912yes—
214C2×C4×C7C2×C22⋊C412yes—
215C2×C4×C7#12012yes—
216C2×Dic7—44no—
217C2×Dic7—44no—
218C2×Dic7—44no—
219C2×Dic7—82no—
220C2×Dic7—82no—
221C2×Dic7—82no—
222C2×Dic7—82no—
223C7⋊C8—82no—
224C25C22⋊D711yescharacteristic
225C22×Q8—24no—
226C22×Q8—21no—
227C22×Q8—42no—
228C22×Q8—42no—
229C22×Q8C2×Dic712yes—
230C23×C4—21no—
231C23×C4—21no—
232C23×C4—24no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—142no—
236C23×C4C2×Dic711yescharacteristic
237C23×C4C2×Dic711yescharacteristic
238C23×C4C22⋊D711yescharacteristic
239C23×C4C22⋊D711yescharacteristic
240C23×C4C22⋊D711yescharacteristic
241C23×C4C22×D711yescharacteristic
242C22×C8—142no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—24no—
248C22⋊Q8C22⋊D718yes—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4C2×Dic712yes—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—144no—
259C2×C22⋊C4—282no—
260C2×C22⋊C4—282no—
261C22⋊C8—144no—
262C22⋊C8—282no—
263C22⋊C8—282no—
264#95—144no—
265#95—282no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—22no—
269C22×C7—22no—
270C22×C7—22no—
271C22×C7—22no—
272C22×C7—21no—
273C22×C7—22no—
274C22×C7—21no—
275C22×C7—22no—
276C22×C7—22no—
277C22×C7—22no—
278C22×C7—22no—
279C22×C7—22no—
280C22×C7—22no—
281C22×C7—22no—
282C22×C7—22no—
283C22×C7—48no—
284C22×C7—48no—
285C22×C7—41no—
286C22×C7—41no—
287C22×C7—41no—
288C22×C7—41no—
289C22×C7#32814yescomplemented
290C22×C7#37911yescharacteristic
291C22×C7C2×#9911yescharacteristic
292C22×C7C2×#9911yescharacteristic
293C22×C7C2×#10311yescharacteristic
294C22×C7#41911yescharacteristic
295C22×C7C2×#10411yescharacteristic
296C22×C7#42111yescharacteristic
297C22×C7#44812yes—
298C22×C7#45112yes—
299C4×C7—22no—
300C4×C7—22no—
301C4×C7—22no—
302C4×C7—41no—
303C4×C7—44no—
304C4×C7—44no—
305Dic7—84no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24#114312yes—
313C24#114412yes—
314C24C22⋊Dic711yescharacteristic
315C24C22⋊Dic711yescharacteristic
316C24C2×C22⋊D711yescharacteristic
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—416no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8C22⋊Dic714yescomplemented
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—416no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—144no—
352C22×C4—282no—
353C22×C4—282no—
354C22×C4—282no—
355C22×C4—282no—
356C22×C4—282no—
357C22×C4—282no—
358C22×C4C22⋊Dic711yescharacteristic
359C22×C4C22⋊Dic711yescharacteristic
360C22×C4C22×Dic711yescharacteristic
361C22×C4C2×C22⋊D711yescharacteristic
362C22×C4C2×C22⋊D714yes—
363C2×C8—282no—
364C2×C8—282no—
365C2×C8—282no—
366C4⋊C4—24no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—28no—
370C4⋊C4C22⋊Dic714yescomplemented
371C22⋊C4—28no—
372C22⋊C4—28no—
373C22⋊C4—288no—
374C2×C7—22no—
375C2×C7—24no—
376C2×C7—22no—
377C2×C7—42no—
378C2×C7C2×#32811yescharacteristic
379C2×C7#150811yescharacteristic
380C2×C7#151111yescharacteristic
381C2×C7#152011yescharacteristic
382C2×C7#178111yescharacteristic
383C2×C7#178711yescharacteristic
384C2×C7#178911yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—24no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—24no—
402C23—24no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23#640812yes—
410C23#640912yes—
411C23#640912yes—
412C23#641112yes—
413C23C2×#114311yescharacteristic
414C23#650512yes—
415C23C2×#114411yescharacteristic
416C23C2×C22⋊Dic711yescharacteristic
417C23#651612yes—
418Q8—44no—
419Q8—48no—
420Q8—44no—
421C2×C4—28no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—21no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—284no—
445C2×C4—284no—
446C2×C4—284no—
447C2×C4—562no—
448C2×C4—562no—
449C2×C4—562no—
450C2×C4—562no—
451C2×C4#640812yes—
452C2×C4C2×C22⋊Dic712yes—
453C2×C4#651612yes—
454C8—562no—
455C7#832911yescharacteristic, complemented, Sylow
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—48no—
474C22—48no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#8933314yescomplemented
480C22C2×#640811yescharacteristic
481C22C2×#640811yescharacteristic
482C22C2×#640911yescharacteristic
483C22#8999711yescharacteristic
484C22#8999912yes—
485C22#9000012yes—
486C22C2×#641111yescharacteristic
487C22#9000811yescharacteristic
488C22#9001911yescharacteristic
489C4—22no—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—44no—
494C4—44no—
495C4—564no—
496C2—22no—
497C2—22no—
498C2—24no—
499C2—42no—
500C2C2×#8933311yescharacteristic
501C2#1172944211yescharacteristic
502C2#1172944311yescharacteristic
503C2#1172944411yescharacteristic
504C2#1172988411yescharacteristic
505C2#1172988611yescharacteristic
506C2#1172989211yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC22×C7×C22⋊Q8C2
socleC23×C7C2×C22⋊C4
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenterC22×C22⋊Q8D7
hypocenterC7#8329
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
4. C2×#89333C2semidirect
5. C23×C7×Q8C4semidirect
10. C2×C7×C22⋊Q8C4semidirect
13. C22×C7×C4⋊C4C4semidirect
87. C2×C7×Q8C22⋊C4semidirect
130. C7×C4⋊C4C22⋊C4semidirect
137. C23×Q8Dic7semidirect
142. C2×C22⋊Q8Dic7semidirect
145. C22×C4⋊C4Dic7semidirect
289. C22×C7#328semidirect
324. C2×Q8C22⋊Dic7semidirect
370. C4⋊C4C22⋊Dic7semidirect
455. C7#8329semidirect
479. C22#89333semidirect

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
#117298812
#117298802
C2×#893332
#83297