Finite Group Databasebeta

← #13514273

Subgroups — #13514273

Order 1792.

2094 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×D7×#125C211yesmaximal, characteristic
2C7×#2330C211yesmaximal, characteristic, complemented
3#11740468C211yesmaximal, characteristic, complemented
4#11733646C214yesmaximal
5#11733640C212yesmaximal
6#11733435C214yesmaximal, complemented
7#11733411C212yesmaximal, complemented
8#11732414C214yesmaximal, complemented
9#11732407C214yesmaximal
10#11731767C211yesmaximal, characteristic, complemented
11#11731763C211yesmaximal, characteristic
12#11731195C212yesmaximal
13C4×#6453C212yesmaximal, complemented
14#11730835C211yesmaximal, characteristic, complemented
15#11730827C211yesmaximal, characteristic
16C2×C7×#125C2211yescharacteristic
17D7×#125C4116yescomplemented
18C2×#6455C2212yes—
19C2×D7×C4⋊C4C2212yes—
20C2×D7×C4⋊C4C2214yes—
21C2×#6453C2212yes—
22C2×#6452C2214yes—
23C2×#6435C2211yescharacteristic
24C2×#6435C2211yescharacteristic
25C2×C42×D7C2211yescharacteristic
26C2×#6433C2211yescharacteristic
27C7×#389C2212yes—
28C7×#389C2214yes—
29C7×#387C2214yes—
30C7×#384C2211yescharacteristic
31C4×C7×C4⋊C4C2212yes—
32C7×#376C2211yescharacteristic
33#89759C2214yes—
34#89759C2214yes—
35#89757C2212yes—
36#89757C2214yes—
37#89753C2214yes—
38#89750C2214yes—
39#89750C2214yes—
40#89749C2212yes—
41C4⋊C4×Dic7C2214yes—
42C4⋊C4×Dic7C2212yes—
43C4⋊C4×Dic7C2212yes—
44C4⋊C4×Dic7C2214yes—
45#89745C2212yes—
46#89744C2214yes—
47#89744C2214yes—
48C4×(C2×D7)⋊C4C2214yes—
49#89707C2211yescharacteristic
50C4×C4⋊Dic7C2214yes—
51C4×C4⋊Dic7C2212yes—
52#89704C2211yescharacteristic
53C4×Dic7⋊C4C2214yes—
54C4×Dic7⋊C4C2214yes—
55C42×Dic7C2211yescharacteristic
56C42×Dic7C2211yescharacteristic
57#89700C2212yes—
58#89439C2214yes—
59#89438C2212yes—
60#89438C2212yes—
61#89438C2214yes—
62#89427C2214yes—
63#89427C2214yes—
64#89426C2214yes—
65#89425C2214yes—
66#89421C2212yes—
67#89421C2212yes—
68#89421C2214yes—
69#89421C2214yes—
70#89419C2214yes—
71#89419C2214yes—
72#34217—71nomaximal, Sylow
73C22×C4×D7C2312yes—
74C22×C4×D7C2314yes—
75C22×C4×D7C2311yescharacteristic
76C7×#125C2×C418yes—
77C2×C7×C4⋊C4C2312yes—
78C2×C7×C4⋊C4C2314yes—
79C2×C7×C4⋊C4C2314yes—
80C2×C7×C4⋊C4C2314yes—
81C2×C7×C4⋊C4C2312yes—
82C2×C42×C7C2311yescharacteristic
83C2×C42×C7C2314yes—
84C2×C42×C7C2311yescharacteristic
85C2×C42×C7C2311yescharacteristic
86C2×(C2×D7)⋊C4C2311yescharacteristic
87C2×(C2×D7)⋊C4C2311yescharacteristic
88C2×(C2×D7)⋊C4C2314yes—
89C2×(C2×D7)⋊C4C2312yes—
90C2×C4⋊Dic7C2312yes—
91C2×C4⋊Dic7C2314yes—
92C2×C4⋊Dic7C2314yes—
93C2×C4⋊Dic7C2312yes—
94C2×C4⋊Dic7C2314yes—
95C2×Dic7⋊C4C2311yescharacteristic
96C2×Dic7⋊C4C2311yescharacteristic
97C2×Dic7⋊C4C2314yes—
98C2×Dic7⋊C4C2314yes—
99C2×Dic7⋊C4C2314yes—
100C2×Dic7⋊C4C2314yes—
101C2×Dic7⋊C4C2314yes—
102C2×Dic7⋊C4C2312yes—
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2314yes—
105C2×C4×Dic7C2314yes—
106C2×C4×Dic7C2312yes—
107C2×C4×Dic7C2314yes—
108C2×C4×Dic7C2312yes—
109C2×C4×Dic7C2312yes—
110C2×C4×Dic7C2312yes—
111C2×C4×Dic7C2314yes—
112C2×C4×Dic7C2312yes—
113C2×C4×Dic7C2311yescharacteristic
114C2×C4×Dic7C2311yescharacteristic
115#6455—28no—
116D7×C4⋊C4C2×C4132yes—
117D7×C4⋊C4C2×C4116yes—
118#6453C2×C4116yes—
119#6452C2×C4132yes—
120#6435—24no—
121#6435Q818yes—
122C42×D7C2×C418yes—
123#6433C2×C418yes—
124C7×#95C2312yes—
125C7×#95C2314yes—
126C7×#95C2314yes—
127C7×#95C2312yes—
128C7⋊#95C2314yes—
129C7⋊#95C2314yes—
130C7⋊#95C2314yes—
131C7⋊#95C2314yes—
132C7⋊#95C2312yes—
133C7⋊#95C2312yes—
134C7⋊#95C2314yes—
135C7⋊#95C2314yes—
136C7⋊#95C2314yes—
137C7⋊#95C2314yes—
138C22×#125—71no—
139#2370—74no—
140#2370—74no—
141#2345—74no—
142#2345—74no—
143#2330—71no—
144#2330—71no—
145#2330—71no—
146#2330D711yescharacteristic, complemented
147#2316—72no—
148C4×#125—72no—
149C4×#125—72no—
150#2285—71no—
151#2274—72no—
152C4×#117—71no—
153C23×D7C2411yescharacteristic
154C22×C4×C7C2414yes—
155C22×C4×C7C2411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×C4×C7C2414yes—
158C22×C4×C7C2412yes—
159C22×C4×C7C2412yes—
160C22×C4×C7C2411yescharacteristic
161C22×Dic7C2411yescharacteristic
162C22×Dic7C2411yescharacteristic
163C22×Dic7C2414yes—
164C22×Dic7C2412yes—
165C22×Dic7C2414yes—
166C22×Dic7C2412yes—
167C22×Dic7C2411yescharacteristic
168C2×C4×D7—42no—
169C2×C4×D7—42no—
170C2×C4×D7—44no—
171C2×C4×D7—44no—
172C2×C4×D7C22×C418yes—
173C2×C4×D7C22×C4116yes—
174C2×C4×D7C22×C414yes—
175C2×C4×D7C2×Q814yes—
176C2×C4×D7#4114yes—
177C7×C4⋊C4—24no—
178C7×C4⋊C4—44no—
179C7×C4⋊C4—44no—
180C7×C4⋊C4C22×C4116yes—
181C7×C4⋊C4C22×C418yes—
182C42×C7—22no—
183C42×C7—44no—
184C42×C7C22×C414yes—
185C42×C7C2×Q814yes—
186(C2×D7)⋊C4—22no—
187(C2×D7)⋊C4—22no—
188(C2×D7)⋊C4—28no—
189(C2×D7)⋊C4C2×Q818yes—
190C4⋊Dic7—24no—
191C4⋊Dic7—44no—
192C4⋊Dic7—44no—
193C4⋊Dic7C22×C418yes—
194C4⋊Dic7C22×C4116yes—
195Dic7⋊C4—22no—
196Dic7⋊C4—22no—
197Dic7⋊C4—44no—
198Dic7⋊C4—44no—
199Dic7⋊C4C22×C4116yes—
200Dic7⋊C4C22×C4116yes—
201Dic7⋊C4C22×C4116yes—
202Dic7⋊C4C2×Q818yes—
203C4×Dic7—22no—
204C4×Dic7—28no—
205C4×Dic7—42no—
206C4×Dic7—42no—
207C4×Dic7—44no—
208C4×Dic7—44no—
209C4×Dic7—42no—
210C4×Dic7—42no—
211C4×Dic7—44no—
212C4×Dic7C22×C418yes—
213C4×Dic7C22×C414yes—
214C4×Dic7C2×Q814yes—
215C2×#125—716no—
216C2×#125—72no—
217C2×#125—74no—
218C2×#125—71no—
219C2×#125C2×D711yescharacteristic
220C2×#117—72no—
221C2×#117—71no—
222C2×#117—71no—
223C22×C4⋊C4—72no—
224C22×C4⋊C4—74no—
225C22×C42—71no—
226#389—72no—
227#389—72no—
228#389—74no—
229#389—74no—
230#389—74no—
231#389—72no—
232#389—74no—
233#389—74no—
234#389C2×D712yes—
235#389C2×D714yes—
236#387—74no—
237#387—74no—
238#387—74no—
239#387—74no—
240#387—74no—
241#387—74no—
242#387—74no—
243#387C2×D714yes—
244#384—71no—
245#384C2×D711yescharacteristic
246#382—72no—
247#380—74no—
248#380—74no—
249#380—74no—
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—74no—
257C4×C4⋊C4—74no—
258C4×C4⋊C4—74no—
259C4×C4⋊C4—72no—
260C4×C4⋊C4—72no—
261C4×C4⋊C4—74no—
262C4×C4⋊C4C2×D712yes—
263C4×C22⋊C4—74no—
264C4×C22⋊C4—72no—
265C4×C22⋊C4—72no—
266C4×C22⋊C4—74no—
267#376—71no—
268#376C2×D711yescharacteristic
269C43—71no—
270C43—71no—
271C23×C7C2511yescharacteristic
272C22×D7—22no—
273C22×D7—22no—
274C22×D7C23×C412yes—
275C22×D7C22×Q812yes—
276C22×D7C2×#4112yes—
277C2×C4×C7—22no—
278C2×C4×C7—24no—
279C2×C4×C7—22no—
280C2×C4×C7—22no—
281C2×C4×C7—24no—
282C2×C4×C7—22no—
283C2×C4×C7—24no—
284C2×C4×C7—21no—
285C2×C4×C7—21no—
286C2×C4×C7—22no—
287C2×C4×C7—24no—
288C2×C4×C7—22no—
289C2×C4×C7—24no—
290C2×C4×C7C23×C414yes—
291C2×C4×C7C23×C418yes—
292C2×C4×C7C23×C412yes—
293C2×C4×C7C22×Q814yes—
294C2×C4×C7C22×Q812yes—
295C2×C4×C7C2×#4112yes—
296C2×Dic7—22no—
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—22no—
300C2×Dic7—24no—
301C2×Dic7—24no—
302C2×Dic7—21no—
303C2×Dic7—21no—
304C2×Dic7—24no—
305C2×Dic7—24no—
306C2×Dic7—22no—
307C2×Dic7—22no—
308C2×Dic7—24no—
309C2×Dic7C23×C414yes—
310C2×Dic7C23×C418yes—
311C2×Dic7C23×C412yes—
312C2×Dic7C22×Q814yes—
313C2×Dic7C22×Q812yes—
314C2×Dic7C2×#4112yes—
315C4×D7—48no—
316C4×D7—416no—
317C4×D7C4×Q8116yes—
318C23×C4—71no—
319C23×C4—74no—
320C23×C4—72no—
321#125—732no—
322#125—716no—
323#125—78no—
324#125C4×D718yes—
325#117—78no—
326#117—144no—
327#117—148no—
328C2×C4⋊C4—74no—
329C2×C4⋊C4—74no—
330C2×C4⋊C4—74no—
331C2×C4⋊C4—74no—
332C2×C4⋊C4—72no—
333C2×C4⋊C4—72no—
334C2×C4⋊C4—716no—
335C2×C4⋊C4—732no—
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⋊C4—74no—
343C2×C4⋊C4C22×D712yes—
344C2×C4⋊C4C22×D714yes—
345C2×C4⋊C4C22×D714yes—
346C2×C4⋊C4C22×D712yes—
347C2×C4⋊C4C22×D714yes—
348C2×C22⋊C4—72no—
349C2×C22⋊C4—74no—
350C2×C22⋊C4—71no—
351C2×C22⋊C4—71no—
352C2×C42—71no—
353C2×C42—71no—
354C2×C42—72no—
355C2×C42—78no—
356C2×C42—71no—
357C2×C42—74no—
358C2×C42—72no—
359C2×C42—72no—
360C2×C42—72no—
361C2×C42—72no—
362C2×C42—74no—
363C2×C42—74no—
364C2×C42—74no—
365C2×C42C22×D711yescharacteristic
366C2×C42C22×D714yes—
367C2×C42C22×D711yescharacteristic
368C2×C42C22×D711yescharacteristic
369#95—72no—
370#95—72no—
371#95—74no—
372#95—74no—
373#95—74no—
374#95—74no—
375#95—74no—
376#95—74no—
377#95—74no—
378#95—74no—
379#95C22×D712yes—
380#95C22×D712yes—
381#95C22×D714yes—
382#95C22×D714yes—
383C22×C7C24×C411yescharacteristic
384C22×C7C23×Q811yescharacteristic
385C22×C7C22×#4111yescharacteristic
386C22×C7#58512yes—
387C22×C7#58512yes—
388C2×D7—24no—
389C2×D7—24no—
390C2×D7—22no—
391C2×D7—22no—
392C2×D7C2×C4×Q814yes—
393C4×C7—28no—
394C4×C7—24no—
395C4×C7—24no—
396C4×C7—44no—
397C4×C7—41no—
398C4×C7—41no—
399C4×C7C2×C4×Q814yes—
400Dic7—24no—
401Dic7—28no—
402Dic7—24no—
403Dic7—41no—
404Dic7—41no—
405Dic7—44no—
406Dic7C2×C4×Q814yes—
407C24—71no—
408C22×C4—71no—
409C22×C4—74no—
410C22×C4—74no—
411C22×C4—74no—
412C22×C4—72no—
413C22×C4—72no—
414C22×C4—74no—
415C22×C4—78no—
416C22×C4—716no—
417C22×C4—71no—
418C22×C4—74no—
419C22×C4—71no—
420C22×C4—284no—
421C22×C4—282no—
422C22×C4—284no—
423C22×C4—282no—
424C22×C4C23×D711yescharacteristic
425C22×C4C23×D714yes—
426C22×C4C23×D711yescharacteristic
427C22×C4C23×D714yes—
428C22×C4C23×D712yes—
429C22×C4C23×D712yes—
430C22×C4C23×D711yescharacteristic
431C4⋊C4—24no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—716no—
435C4⋊C4—716no—
436C4⋊C4—716no—
437C4⋊C4—716no—
438C4⋊C4—78no—
439C4⋊C4—78no—
440C4⋊C4—142no—
441C4⋊C4—142no—
442C4⋊C4—144no—
443C4⋊C4—284no—
444C4⋊C4—284no—
445C4⋊C4—284no—
446C4⋊C4—284no—
447C4⋊C4C2×C4×D7116yes—
448C4⋊C4C2×C4×D718yes—
449C22⋊C4—78no—
450C22⋊C4—142no—
451C22⋊C4—142no—
452C22⋊C4—148no—
453C42—22no—
454C42—44no—
455C42—78no—
456C42—74no—
457C42—74no—
458C42—148no—
459C42—142no—
460C42—282no—
461C42—282no—
462C42—284no—
463C42—284no—
464C42—284no—
465C42—282no—
466C42—282no—
467C42C2×C4×D714yes—
468C42Q8×D714yes—
469C2×C7C22×C4×Q811yescharacteristic
470C2×C7#342812yes—
471C2×C7#350412yes—
472C2×C7#351211yescharacteristic
473C2×C7#351211yescharacteristic
474D7—24no—
475C23—72no—
476C23—72no—
477C23—72no—
478C23—142no—
479C23—142no—
480C23C24×D711yescharacteristic
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—22no—
494C2×C4—72no—
495C2×C4—72no—
496C2×C4—72no—
497C2×C4—716no—
498C2×C4—74no—
499C2×C4—78no—
500C2×C4—74no—
501C2×C4—142no—
502C2×C4—142no—
503C2×C4—144no—
504C2×C4—142no—
505C2×C4—142no—
506C2×C4—144no—
507C2×C4—141no—
508C2×C4—144no—
509C2×C4—141no—
510C2×C4—142no—
511C2×C4—144no—
512C2×C4—142no—
513C2×C4—144no—
514C2×C4—288no—
515C2×C4—2816no—
516C2×C4C22×C4×D714yes—
517C2×C4C22×C4×D718yes—
518C2×C4C22×C4×D712yes—
519C2×C4C2×Q8×D714yes—
520C2×C4C2×Q8×D712yes—
521C2×C4D7×#4112yes—
522C7#3421711yescharacteristic, complemented, Sylow
523C22—74no—
524C22—142no—
525C22—142no—
526C22—144no—
527C22—144no—
528C22C23×C4×D711yescharacteristic
529C22C22×Q8×D711yescharacteristic
530C22C2×D7×#4111yescharacteristic
531C22#9061812yes—
532C22#9062012yes—
533C4—24no—
534C4—28no—
535C4—24no—
536C4—41no—
537C4—44no—
538C4—41no—
539C4—74no—
540C4—144no—
541C4—144no—
542C4—148no—
543C4—281no—
544C4—284no—
545C4—281no—
546C4C4×Q8×D714yes—
547C2—144no—
548C2C2×C4×Q8×D711yescharacteristic
549C2#1174295312yes—
550C2#1174353711yescharacteristic
551C2#1174356511yescharacteristic
552C2#1174365512yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2330C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenter#2330D7
hypocenterC7#34217
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
2. C7×#2330C2semidirect
3. #11740468C2semidirect
6. #11733435C2semidirect
7. #11733411C2semidirect
8. #11732414C2semidirect
10. #11731767C2semidirect
13. C4×#6453C2semidirect
14. #11730835C2semidirect
17. D7×#125C4semidirect
146. #2330D7semidirect
522. C7#34217semidirect

Maximal subgroups

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

subgroupindex
C2×D7×#1252
C7×#23302
#117404682
#117336462
#117336402
#117334352
#117334112
#117324142
#117324072
#117317672
#117317632
#117311952
C4×#64532
#117308352
#117308272
#342177