Finite Group Databasebeta

← #13515184

Subgroups — #13515184

Order 1792.

2070 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90422C211yesmaximal, characteristic
2C4×C7×C4⋊Q8C211yesmaximal, characteristic, complemented
3#11740672C211yesmaximal, characteristic, complemented
4#11739933C212yesmaximal
5#11739895C212yesmaximal, complemented
6#11733924C212yesmaximal
7#11733922C212yesmaximal, complemented
8#11733436C214yesmaximal
9#11733435C214yesmaximal, complemented
10#11731803C211yesmaximal, characteristic
11#11731767C211yesmaximal, characteristic, complemented
12#11731111C214yesmaximal
13#11731108C214yesmaximal, complemented
14C4×#6433C211yesmaximal, characteristic, complemented
15C4×#6435C211yesmaximal, characteristic
16C2×C7×C4⋊Q8C2211yescharacteristic
17C2×C4×C7×Q8C2212yes—
18C2×#6509C2212yes—
19C2×Q8×Dic7C2212yes—
20C2×Q8×Dic7C2212yes—
21#90422C4116yescomplemented
22C2×#6461C2214yes—
23C2×#6452C2214yes—
24C2×#6435C2211yescharacteristic
25C2×#6435C2211yescharacteristic
26C2×C42×D7C2211yescharacteristic
27C2×#6433C2211yescharacteristic
28C7×#391C2212yes—
29C7×#389C2214yes—
30C7×#384C2211yescharacteristic
31C4×C7×C4⋊C4C2214yes—
32C43×C7C2211yescharacteristic
33#90002C2212yes—
34#89762C2214yes—
35#89762C2214yes—
36#89759C2212yes—
37#89753C2214yes—
38#89753C2214yes—
39#89750C2214yes—
40#89750C2214yes—
41C4⋊C4×Dic7C2212yes—
42C4⋊C4×Dic7C2212yes—
43C4⋊C4×Dic7C2214yes—
44C4⋊C4×Dic7C2214yes—
45#89745C2212yes—
46#89744C2214yes—
47#89744C2214yes—
48#89712C2214yes—
49C4×(C2×D7)⋊C4C2212yes—
50C4×(C2×D7)⋊C4C2212yes—
51#89708C2211yescharacteristic
52C4×C4⋊Dic7C2212yes—
53C4×C4⋊Dic7C2214yes—
54#89705C2214yes—
55#89705C2214yes—
56#89704C2211yescharacteristic
57#89704C2211yescharacteristic
58#89704C2214yes—
59C4×Dic7⋊C4C2212yes—
60C4×Dic7⋊C4C2212yes—
61C4×Dic7⋊C4C2214yes—
62C42×Dic7C2211yescharacteristic
63#89700C2212yes—
64#89440C2214yes—
65#89438C2214yes—
66#89427C2214yes—
67#89426C2214yes—
68#89425C2214yes—
69#89421C2214yes—
70#89421C2214yes—
71#89420C2214yes—
72#34281—71nomaximal, Sylow
73C22×C7×Q8C2312yes—
74C22×C4×D7C2312yes—
75C22×C4×D7C2311yescharacteristic
76C7×C4⋊Q8C2×C418yescomplemented
77C4×C7×Q8—28no—
78C2×C7×C4⋊C4C2314yes—
79C2×C7×C4⋊C4C2312yes—
80C2×C7×C4⋊C4C2314yes—
81C2×C7×C4⋊C4C2314yes—
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2312yes—
84C2×C42×C7C2314yes—
85C2×C42×C7C2312yes—
86C2×C42×C7C2311yescharacteristic
87C2×C42×C7C2311yescharacteristic
88#6509C2×C4116yes—
89Q8×Dic7—28no—
90Q8×Dic7C2×C4116yescomplemented
91C2×(C2×D7)⋊C4C2311yescharacteristic
92C2×(C2×D7)⋊C4C2311yescharacteristic
93C2×(C2×D7)⋊C4C2314yes—
94C2×(C2×D7)⋊C4C2312yes—
95C2×(C2×D7)⋊C4C2314yes—
96C2×C4⋊Dic7C2314yes—
97C2×C4⋊Dic7C2312yes—
98C2×C4⋊Dic7C2312yes—
99C2×C4⋊Dic7C2314yes—
100C2×Dic7⋊C4C2311yescharacteristic
101C2×Dic7⋊C4C2311yescharacteristic
102C2×Dic7⋊C4C2314yes—
103C2×Dic7⋊C4C2314yes—
104C2×Dic7⋊C4C2314yes—
105C2×Dic7⋊C4C2312yes—
106C2×Dic7⋊C4C2314yes—
107C2×C4×Dic7C2312yes—
108C2×C4×Dic7C2311yescharacteristic
109C2×C4×Dic7C2312yes—
110C2×C4×Dic7C2314yes—
111C2×C4×Dic7C2314yes—
112C2×C4×Dic7C2312yes—
113C2×C4×Dic7C2314yes—
114C2×C4×Dic7C2314yes—
115C2×C4×Dic7C2311yescharacteristic
116C2×C4×Dic7C2314yes—
117C2×C4×Dic7C2311yescharacteristic
118#6461C2×C4132yes—
119#6452C2×C4132yescomplemented
120#6435—24no—
121#6435C2×C418yes—
122C42×D7—24no—
123#6433C2×C418yescomplemented
124C7×#95C2314yes—
125C7×#95C2314yes—
126C7⋊#95C2314yes—
127C7⋊#95C2314yes—
128C7⋊#95C2314yes—
129C7⋊#95C2314yes—
130C7⋊#95C2314yes—
131C7⋊#95C2314yes—
132C7⋊#95C2314yes—
133C7⋊#95C2312yes—
134C7⋊#95C2312yes—
135C7⋊#95C2312yes—
136C7⋊#95C2312yes—
137C7⋊#95C2314yes—
138C7⋊#95C2314yes—
139C2×#541—71no—
140#2373—74no—
141#2373—74no—
142#2371—74no—
143#2370—74no—
144#2330—71no—
145#2327—71no—
146#2323—72no—
147#2314—71no—
148C4×C4⋊Q8D711yescharacteristic, complemented
149C4×#125—71no—
150C4×C22⋊Q8—72no—
151#2295—72no—
152#2276—72no—
153C4×#117—71no—
154C23×D7C2411yescharacteristic
155C2×C7×Q8—24no—
156C2×C7×Q8—24no—
157C2×C7×Q8C22×C418yes—
158C22×C4×C7C2414yes—
159C22×C4×C7C2412yes—
160C22×C4×C7C2414yes—
161C22×C4×C7C2411yescharacteristic
162C22×C4×C7C2411yescharacteristic
163C22×C4×C7C2412yes—
164C22×C4×C7C2411yescharacteristic
165C22×Dic7C2411yescharacteristic
166C22×Dic7C2414yes—
167C22×Dic7C2411yescharacteristic
168C22×Dic7C2412yes—
169C22×Dic7C2414yes—
170C22×Dic7C2411yescharacteristic
171C22×Dic7C2412yes—
172C2×C4×D7—22no—
173C2×C4×D7—22no—
174C2×C4×D7—24no—
175C2×C4×D7—24no—
176C2×C4×D7—42no—
177C2×C4×D7C22×C414yes—
178C7×C4⋊C4—28no—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—44no—
181C7×C4⋊C4C22×C4116yes—
182C42×C7—22no—
183C42×C7—28no—
184C42×C7—24no—
185C42×C7—42no—
186C42×C7C22×C414yes—
187C42×C7#4114yes—
188(C2×D7)⋊C4—22no—
189(C2×D7)⋊C4—22no—
190(C2×D7)⋊C4—44no—
191(C2×D7)⋊C4C22×C418yes—
192(C2×D7)⋊C4C22×C4116yes—
193C4⋊Dic7—24no—
194C4⋊Dic7—28no—
195C4⋊Dic7C22×C418yes—
196C4⋊Dic7C22×C4116yes—
197Dic7⋊C4—22no—
198Dic7⋊C4—22no—
199Dic7⋊C4—28no—
200Dic7⋊C4—44no—
201Dic7⋊C4C22×C4116yes—
202Dic7⋊C4C22×C418yes—
203Dic7⋊C4C22×C4116yes—
204C4×Dic7—22no—
205C4×Dic7—24no—
206C4×Dic7—24no—
207C4×Dic7—28no—
208C4×Dic7—44no—
209C4×Dic7—42no—
210C4×Dic7—44no—
211C4×Dic7—44no—
212C4×Dic7C22×C414yes—
213C4×Dic7C22×C4116yes—
214C4×Dic7#4114yes—
215#541—716no—
216C2×C4⋊Q8C2×D711yescharacteristic
217C2×#126—74no—
218C2×#125—74no—
219C2×#125—71no—
220C2×C22⋊Q8—72no—
221C2×C4×Q8—72no—
222C2×C4×Q8—72no—
223C2×C4×Q8C2×D712yes—
224C2×#117—71no—
225C2×#117—71no—
226C22×C42—71no—
227#391—72no—
228#391C2×D712yes—
229#389—72no—
230#389—74no—
231#389—74no—
232#389—72no—
233#389C2×D714yes—
234#388—74no—
235#388—74no—
236#388—74no—
237#387—74no—
238#387—74no—
239#387—74no—
240#387—74no—
241#387—74no—
242#387—74no—
243#387—74no—
244#387—74no—
245#387—74no—
246#385—74no—
247#384C2×D711yescharacteristic
248#382—71no—
249#380—72no—
250C4×C4⋊C4—72no—
251C4×C4⋊C4—72no—
252C4×C4⋊C4—74no—
253C4×C4⋊C4—74no—
254C4×C4⋊C4—72no—
255C4×C4⋊C4—74no—
256C4×C4⋊C4—72no—
257C4×C4⋊C4—72no—
258C4×C4⋊C4—74no—
259C4×C4⋊C4—74no—
260C4×C4⋊C4—74no—
261C4×C4⋊C4C2×D714yes—
262C4×C22⋊C4—72no—
263C4×C22⋊C4—72no—
264C4×C22⋊C4—74no—
265#376—71no—
266#376—74no—
267#376—71no—
268#376—74no—
269C43—71no—
270C43C2×D711yescharacteristic
271C23×C7C2511yescharacteristic
272C22×D7—21no—
273C22×D7—21no—
274C22×D7—22no—
275C22×D7—22no—
276C22×D7C23×C412yes—
277C7×Q8—216no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—24no—
281C2×C4×C7—22no—
282C2×C4×C7—22no—
283C2×C4×C7—22no—
284C2×C4×C7—22no—
285C2×C4×C7—22no—
286C2×C4×C7—24no—
287C2×C4×C7—24no—
288C2×C4×C7—24no—
289C2×C4×C7C23×C418yes—
290C2×C4×C7C23×C414yes—
291C2×C4×C7C23×C412yes—
292C2×C4×C7C2×#4112yes—
293C2×C4×C7C2×#4112yes—
294C2×C4×C7C2×#4112yes—
295C2×C4×C7C2×#4112yes—
296C2×C4×C72−1+414yes—
297C2×Dic7—21no—
298C2×Dic7—21no—
299C2×Dic7—22no—
300C2×Dic7—24no—
301C2×Dic7—22no—
302C2×Dic7—22no—
303C2×Dic7—24no—
304C2×Dic7—22no—
305C2×Dic7—21no—
306C2×Dic7—21no—
307C2×Dic7—24no—
308C2×Dic7—22no—
309C2×Dic7—24no—
310C2×Dic7—24no—
311C2×Dic7—22no—
312C2×Dic7C23×C414yes—
313C2×Dic7C23×C418yes—
314C2×Dic7C23×C412yes—
315C2×Dic7C2×#4114yes—
316C4×D7—28no—
317C4×D7—48no—
318C22×Q8C22×D712yes—
319C23×C4—71no—
320C23×C4—72no—
321C4⋊Q8C4×D718yescomplemented
322#126—732no—
323#125—732no—
324#125—78no—
325C22⋊Q8—716no—
326C4×Q8—28no—
327C4×Q8—716no—
328C4×Q8—148no—
329#117—78no—
330#117—144no—
331C2×C4⋊C4—72no—
332C2×C4⋊C4—72no—
333C2×C4⋊C4—74no—
334C2×C4⋊C4—74no—
335C2×C4⋊C4—74no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—72no—
339C2×C4⋊C4—74no—
340C2×C4⋊C4—74no—
341C2×C4⋊C4—74no—
342C2×C4⋊C4C22×D712yes—
343C2×C4⋊C4C22×D714yes—
344C2×C4⋊C4C22×D714yes—
345C2×C4⋊C4C22×D714yes—
346C2×C22⋊C4—72no—
347C2×C22⋊C4—74no—
348C2×C22⋊C4—71no—
349C2×C22⋊C4—71no—
350C2×C22⋊C4—74no—
351C2×C42—71no—
352C2×C42—71no—
353C2×C42—74no—
354C2×C42—71no—
355C2×C42—74no—
356C2×C42—72no—
357C2×C42—72no—
358C2×C42—72no—
359C2×C42—74no—
360C2×C42—74no—
361C2×C42—74no—
362C2×C42—144no—
363C2×C42C22×D714yes—
364C2×C42C22×D711yescharacteristic
365C2×C42C22×D712yes—
366C2×C42C22×D712yes—
367C2×C42C22×D711yescharacteristic
368C2×C42C22×D711yescharacteristic
369#95—72no—
370#95—72no—
371#95—72no—
372#95—74no—
373#95—72no—
374#95—74no—
375#95—74no—
376#95—74no—
377#95—74no—
378#95—74no—
379#95—74no—
380#95—74no—
381#95—74no—
382#95C22×D714yes—
383#95C22×D714yes—
384C22×C7C24×C411yescharacteristic
385C22×C7C22×#4111yescharacteristic
386C22×C7C22×#4111yescharacteristic
387C22×C7C2×2−1+412yes—
388C22×C7#58512yes—
389C2×D7—22no—
390C2×D7—24no—
391C2×D7—42no—
392C2×D7—41no—
393C2×D7—41no—
394C4×C7—28no—
395C4×C7—24no—
396C4×C7—24no—
397C4×C7—22no—
398C4×C7—22no—
399C4×C7—44no—
400C4×C7C4×#4114yes—
401Dic7—22no—
402Dic7—24no—
403Dic7—28no—
404Dic7—24no—
405Dic7—44no—
406Dic7—41no—
407Dic7—41no—
408C24—71no—
409C2×Q8—24no—
410C2×Q8—24no—
411C2×Q8C2×C4×D718yes—
412C22×C4—72no—
413C22×C4—72no—
414C22×C4—74no—
415C22×C4—71no—
416C22×C4—74no—
417C22×C4—71no—
418C22×C4—74no—
419C22×C4—71no—
420C22×C4—144no—
421C22×C4—142no—
422C22×C4—142no—
423C22×C4—144no—
424C22×C4—282no—
425C22×C4C23×D714yes—
426C22×C4C23×D712yes—
427C22×C4C23×D714yes—
428C22×C4C23×D711yescharacteristic
429C22×C4C23×D711yescharacteristic
430C22×C4C23×D712yes—
431C22×C4C23×D711yescharacteristic
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—44no—
435C4⋊C4—78no—
436C4⋊C4—716no—
437C4⋊C4—716no—
438C4⋊C4—78no—
439C4⋊C4—716no—
440C4⋊C4—148no—
441C4⋊C4—142no—
442C4⋊C4—142no—
443C4⋊C4—148no—
444C4⋊C4—144no—
445C4⋊C4—284no—
446C4⋊C4C2×C4×D7116yes—
447C22⋊C4—78no—
448C22⋊C4—716no—
449C22⋊C4—142no—
450C22⋊C4—142no—
451C22⋊C4—284no—
452C42—28no—
453C42—24no—
454C42—22no—
455C42—42no—
456C42—74no—
457C42—716no—
458C42—74no—
459C42—144no—
460C42—148no—
461C42—142no—
462C42—144no—
463C42—284no—
464C42—282no—
465C42—284no—
466C42—284no—
467C42C2×C4×D714yes—
468C42#115914yes—
469C2×C7C2×C4×#4111yescharacteristic
470C2×C7C4×2−1+412yes—
471C2×C7#349512yes—
472C2×C7#350911yescharacteristic
473C2×C7#351211yescharacteristic
474D7—42no—
475C23—72no—
476C23—141no—
477C23—142no—
478C23—141no—
479C23—142no—
480C23C24×D711yescharacteristic
481Q8—216no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—74no—
494C2×C4—78no—
495C2×C4—72no—
496C2×C4—74no—
497C2×C4—141no—
498C2×C4—142no—
499C2×C4—144no—
500C2×C4—142no—
501C2×C4—141no—
502C2×C4—142no—
503C2×C4—144no—
504C2×C4—142no—
505C2×C4—148no—
506C2×C4—141no—
507C2×C4—141no—
508C2×C4—142no—
509C2×C4—144no—
510C2×C4—144no—
511C2×C4—144no—
512C2×C4—142no—
513C2×C4—288no—
514C2×C4C22×C4×D718yes—
515C2×C4C22×C4×D714yes—
516C2×C4C22×C4×D712yes—
517C2×C4C2×#115912yes—
518C2×C4C2×#115912yes—
519C2×C4C2×#115912yes—
520C2×C4#655114yes—
521C2×C4D7×#4112yes—
522C7#3428111yescharacteristic, complemented, Sylow
523C22—142no—
524C22—144no—
525C22—281no—
526C22—281no—
527C22—282no—
528C22C23×C4×D711yescharacteristic
529C22C22×#115911yescharacteristic
530C22C2×#655112yes—
531C22C2×D7×#4111yescharacteristic
532C22#9061612yes—
533C4—22no—
534C4—22no—
535C4—28no—
536C4—24no—
537C4—24no—
538C4—44no—
539C4—142no—
540C4—144no—
541C4—144no—
542C4—148no—
543C4—281no—
544C4—281no—
545C4—284no—
546C4C4×#115914yes—
547C2—282no—
548C2C2×C4×#115911yescharacteristic
549C2C4×#655112yes—
550C2#1174327211yescharacteristic
551C2#1174329511yescharacteristic
552C2#1174334712yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC4×C7×C4⋊Q8C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenterC4×C4⋊Q8D7
hypocenterC7#34281
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C4×C7×C4⋊Q8C2semidirect
3. #11740672C2semidirect
5. #11739895C2semidirect
7. #11733922C2semidirect
9. #11733435C2semidirect
11. #11731767C2semidirect
13. #11731108C2semidirect
14. C4×#6433C2semidirect
21. #90422C4semidirect
76. C7×C4⋊Q8C2×C4semidirect
90. Q8×Dic7C2×C4semidirect
119. #6452C2×C4semidirect
123. #6433C2×C4semidirect
148. C4×C4⋊Q8D7semidirect
321. C4⋊Q8C4×D7semidirect
522. C7#34281semidirect

Maximal subgroups

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

subgroupindex
C2×#904222
C4×C7×C4⋊Q82
#117406722
#117399332
#117398952
#117339242
#117339222
#117334362
#117334352
#117318032
#117317672
#117311112
#117311082
C4×#64332
C4×#64352
#342817