Finite Group Databasebeta

← #13516675

Subgroups — #13516675

Order 1792.

2238 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×C4⋊Q8C211yesmaximal, characteristic
2C7×#2381C211yesmaximal, characteristic, complemented
3#11740676C211yesmaximal, characteristic, complemented
4#11734059C214yesmaximal
5#11733931C214yesmaximal, complemented
6#11732741C214yesmaximal
7#11732412C214yesmaximal, complemented
8#11732407C214yesmaximal
9#11732393C214yesmaximal, complemented
10#11731710C211yesmaximal, characteristic, complemented
11#11731710C211yesmaximal, characteristic, complemented
12#11731703C211yesmaximal, characteristic
13#11731703C211yesmaximal, characteristic
14C22×Q8×D7C2212yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×#6507C2212yes—
17D7×C4⋊Q8C4116yescomplemented
18C2×D7×C4⋊C4C2212yes—
19C2×D7×C4⋊C4C2212yes—
20C2×#6451C2212yes—
21C2×#6451C2212yes—
22C2×#6450C2214yes—
23C2×C42×D7C2211yescharacteristic
24C2×#6432C2211yescharacteristic
25C2×C4×Dic14C2214yes—
26C7×#391C2214yes—
27C7×#387C2214yes—
28C7×#387C2214yes—
29C7×#383C2211yescharacteristic
30C7×#383C2211yescharacteristic
31#90002C2214yes—
32#89759C2214yes—
33#89753C2214yes—
34#89753C2214yes—
35#89750C2214yes—
36#89744C2214yes—
37C4×(C2×D7)⋊C4C2214yes—
38#89709C2211yescharacteristic
39#89709C2211yescharacteristic
40#89705C2214yes—
41C4×Dic7⋊C4C2214yes—
42#89439C2214yes—
43#89439C2212yes—
44#89439C2212yes—
45#89438C2214yes—
46#89438C2212yes—
47#89438C2212yes—
48#89436C2212yes—
49#89436C2212yes—
50#89427C2214yes—
51#89426C2214yes—
52#89426C2214yes—
53#89425C2214yes—
54#89425C2214yes—
55#89424C2214yes—
56#89422C2212yes—
57#89422C2212yes—
58#89421C2214yes—
59#89421C2212yes—
60#89421C2212yes—
61#89420C2212yes—
62#89420C2212yes—
63#89419C2214yes—
64#89418C2214yes—
65#89418C2214yes—
66#89418C2214yes—
67#89418C2214yes—
68#34371—71nomaximal, Sylow
69C22×C7×Q8C2312yes—
70C2×Q8×D7—44no—
71C2×Q8×D7—44no—
72C2×Q8×D7C2×C4116yes—
73C22×C4×D7C2311yescharacteristic
74C22×C4×D7C2311yescharacteristic
75C22×C4×D7C2311yescharacteristic
76C22×C4×D7C2314yes—
77C22×Dic14C2314yes—
78C22×Dic14C2312yes—
79C7×C4⋊Q8C2×C418yes—
80C2×C7×C4⋊C4C2314yes—
81C2×C7×C4⋊C4C2312yes—
82C2×C7×C4⋊C4C2312yes—
83C2×C42×C7C2314yes—
84C2×C42×C7C2311yescharacteristic
85#6507C2×C4116yes—
86C2×(C2×D7)⋊C4C2314yes—
87C2×(C2×D7)⋊C4C2314yes—
88C2×C4⋊Dic7C2314yes—
89C2×C4⋊Dic7C2312yes—
90C2×C4⋊Dic7C2312yes—
91C2×Dic7⋊C4C2314yes—
92C2×Dic7⋊C4C2314yes—
93C2×Dic7⋊C4C2314yes—
94C2×Dic7⋊C4C2314yes—
95C2×Dic7⋊C4C2314yes—
96C2×Dic7⋊C4C2314yes—
97C2×C4×Dic7C2314yes—
98C2×C4×Dic7C2314yes—
99C2×C4×Dic7C2314yes—
100C2×C4×Dic7C2314yes—
101C2×C4×Dic7C2314yes—
102C2×C4×Dic7C2311yescharacteristic
103C2×C4×Dic7C2311yescharacteristic
104C2×C4×Dic7C2311yescharacteristic
105D7×C4⋊C4C2×C4116yes—
106D7×C4⋊C4C2×C4116yes—
107#6451C2×C4116yes—
108#6451C2×C4116yes—
109#6450—48no—
110C42×D7C2×C418yes—
111#6432C2×C418yes—
112C4×Dic14—48no—
113C7×#95C2312yes—
114C7×#95C2312yes—
115C7×#95C2314yes—
116C7×#95C2314yes—
117C7×#95C2312yes—
118C7×#95C2312yes—
119C7×#95C2312yes—
120C7×#95C2312yes—
121C7⋊#95C2314yes—
122C7⋊#95C2312yes—
123C7⋊#95C2314yes—
124C7⋊#95C2312yes—
125C7⋊#95C2314yes—
126C7⋊#95C2314yes—
127C7⋊#95C2314yes—
128C7⋊#95C2314yes—
129C7⋊#95C2314yes—
130C7⋊#95C2314yes—
131C7⋊#95C2312yes—
132C7⋊#95C2312yes—
133C7⋊#95C2312yes—
134C7⋊#95C2312yes—
135C22×C4⋊Q8—71no—
136#2381—71no—
137#2381—71no—
138#2381—71no—
139#2381D711yescharacteristic, complemented
140#2365—74no—
141#2345—74no—
142Q8×C22⋊C4—74no—
143#2331—74no—
144#2327—74no—
145#2313—74no—
146#2283—71no—
147#2283—71no—
148C23×D7C2411yescharacteristic
149C2×C7×Q8—42no—
150C2×C7×Q8—42no—
151C2×C7×Q8C22×C418yes—
152C22×C4×C7C2414yes—
153C22×C4×C7C2414yes—
154C22×C4×C7C2411yescharacteristic
155C22×C4×C7C2411yescharacteristic
156C22×C4×C7C2411yescharacteristic
157C22×C4×C7C2414yes—
158C22×Dic7C2414yes—
159C22×Dic7C2414yes—
160C22×Dic7C2411yescharacteristic
161C22×Dic7C2414yes—
162C22×Dic7C2411yescharacteristic
163C22×Dic7C2411yescharacteristic
164Q8×D7—432no—
165C2×C4×D7—24no—
166C2×C4×D7—24no—
167C2×C4×D7—24no—
168C2×C4×D7—44no—
169C2×C4×D7—44no—
170C2×C4×D7C22×C414yes—
171C2×C4×D7C22×C414yes—
172C2×C4×D7C22×C414yes—
173C2×C4×D7C22×C4116yes—
174C2×Dic14—28no—
175C2×Dic14—28no—
176C2×Dic14—42no—
177C2×Dic14—42no—
178C2×Dic14C22×C4116yes—
179C2×Dic14C22×C418yes—
180C7×C4⋊C4—44no—
181C7×C4⋊C4C22×C418yes—
182C7×C4⋊C4C22×C418yes—
183C42×C7—44no—
184C42×C7C22×C414yes—
185(C2×D7)⋊C4—28no—
186(C2×D7)⋊C4—28no—
187C4⋊Dic7—44no—
188C4⋊Dic7C22×C418yes—
189C4⋊Dic7C22×C418yes—
190Dic7⋊C4—28no—
191Dic7⋊C4—28no—
192Dic7⋊C4—44no—
193Dic7⋊C4—44no—
194Dic7⋊C4C22×C4116yes—
195Dic7⋊C4C22×C4116yes—
196C4×Dic7—28no—
197C4×Dic7—28no—
198C4×Dic7—44no—
199C4×Dic7—44no—
200C4×Dic7—44no—
201C4×Dic7C22×C414yes—
202C4×Dic7C22×C414yes—
203C4×Dic7C22×C414yes—
204C23×Q8—72no—
205C2×C4⋊Q8—72no—
206C2×C4⋊Q8—716no—
207C2×C4⋊Q8—72no—
208C2×C4⋊Q8—72no—
209C2×C4⋊Q8—71no—
210C2×C4⋊Q8C2×D711yescharacteristic
211C2×C4×Q8—74no—
212C2×C4×Q8—74no—
213C22×C4⋊C4—72no—
214C22×C4⋊C4—72no—
215C22×C42—71no—
216#391—74no—
217#391—74no—
218#391—74no—
219#391—74no—
220#391—74no—
221#391C2×D714yes—
222#389—74no—
223#389—74no—
224#387—74no—
225#387—74no—
226#387—74no—
227#387—74no—
228#387—74no—
229#387—74no—
230#387—74no—
231#387—74no—
232#387—74no—
233#387—74no—
234#387C2×D714yes—
235#387C2×D714yes—
236#383—71no—
237#383—71no—
238#383C2×D711yescharacteristic
239#383C2×D711yescharacteristic
240#382—72no—
241#382—72no—
242#381—72no—
243#381—72no—
244#380—74no—
245#380—74no—
246#380—72no—
247#380—72no—
248C4×C4⋊C4—74no—
249C4×C4⋊C4—74no—
250C4×C4⋊C4—72no—
251C4×C4⋊C4—72no—
252C4×C22⋊C4—74no—
253C4×C22⋊C4—74no—
254C4×C22⋊C4—72no—
255C4×C22⋊C4—72no—
256#376—72no—
257#376—72no—
258C23×C7C2511yescharacteristic
259C22×D7—22no—
260C22×D7—22no—
261C22×D7C23×C412yes—
262C22×D72−1+414yes—
263C7×Q8—48no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—24no—
270C2×C4×C7—24no—
271C2×C4×C7—24no—
272C2×C4×C7—24no—
273C2×C4×C7—24no—
274C2×C4×C7—24no—
275C2×C4×C7C23×C412yes—
276C2×C4×C7C23×C412yes—
277C2×C4×C7C23×C412yes—
278C2×C4×C7C23×C418yes—
279C2×Dic7—24no—
280C2×Dic7—24no—
281C2×Dic7—22no—
282C2×Dic7—24no—
283C2×Dic7—24no—
284C2×Dic7—24no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7C23×C412yes—
289C2×Dic7C23×C418yes—
290C2×Dic7C23×C412yes—
291C2×Dic7C23×C412yes—
292C2×Dic7H(5,𝔽2)14yes—
293C2×Dic72−1+414yes—
294C4×D7—28no—
295C4×D7—28no—
296C4×D7—28no—
297C4×D7—416no—
298Dic14—232no—
299Dic14—48no—
300C22×Q8—716no—
301C22×Q8—74no—
302C22×Q8—72no—
303C22×Q8—284no—
304C22×Q8—284no—
305C22×Q8C22×D712yes—
306C23×C4—71no—
307C23×C4—71no—
308C23×C4—71no—
309C23×C4—74no—
310C4⋊Q8—716no—
311C4⋊Q8—716no—
312C4⋊Q8—716no—
313C4⋊Q8—78no—
314C4⋊Q8C4×D718yes—
315C4×Q8—288no—
316C4×Q8—288no—
317C2×C4⋊C4—74no—
318C2×C4⋊C4—74no—
319C2×C4⋊C4—72no—
320C2×C4⋊C4—72no—
321C2×C4⋊C4—716no—
322C2×C4⋊C4—716no—
323C2×C4⋊C4—74no—
324C2×C4⋊C4—74no—
325C2×C4⋊C4—74no—
326C2×C4⋊C4—74no—
327C2×C4⋊C4—74no—
328C2×C4⋊C4C22×D714yes—
329C2×C4⋊C4C22×D712yes—
330C2×C4⋊C4C22×D712yes—
331C2×C22⋊C4—74no—
332C2×C22⋊C4—74no—
333C2×C42—71no—
334C2×C42—71no—
335C2×C42—71no—
336C2×C42—78no—
337C2×C42—74no—
338C2×C42—74no—
339C2×C42—74no—
340C2×C42—74no—
341C2×C42—74no—
342C2×C42C22×D714yes—
343C2×C42C22×D711yescharacteristic
344#95—72no—
345#95—72no—
346#95—74no—
347#95—74no—
348#95—72no—
349#95—72no—
350#95—74no—
351#95—74no—
352#95—74no—
353#95—74no—
354#95—74no—
355#95—74no—
356#95—72no—
357#95—72no—
358#95C22×D712yes—
359#95C22×D712yes—
360#95C22×D712yes—
361#95C22×D712yes—
362#95C22×D714yes—
363#95C22×D714yes—
364#95C22×D712yes—
365#95C22×D712yes—
366C22×C7C24×C411yescharacteristic
367C22×C7C2×H(5,𝔽2)12yes—
368C22×C7C2×2−1+412yes—
369C22×C7C2×2−1+412yes—
370C2×D7—22no—
371C2×D7—22no—
372C2×D7—28no—
373C2×D7#51914yes—
374C4×C7—28no—
375C4×C7—22no—
376C4×C7—22no—
377C4×C7—22no—
378C4×C7—44no—
379C4×C7—44no—
380Dic7—28no—
381Dic7—22no—
382Dic7—44no—
383Dic7—44no—
384Dic7#51814yes—
385Dic7#51914yes—
386C24—71no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—716no—
390C2×Q8—78no—
391C2×Q8—148no—
392C2×Q8—148no—
393C2×Q8—282no—
394C2×Q8—282no—
395C2×Q8—2832no—
396C2×Q8C2×C4×D718yes—
397C22×C4—71no—
398C22×C4—71no—
399C22×C4—71no—
400C22×C4—74no—
401C22×C4—74no—
402C22×C4—74no—
403C22×C4—74no—
404C22×C4—716no—
405C22×C4—74no—
406C22×C4—74no—
407C22×C4—144no—
408C22×C4—144no—
409C22×C4—144no—
410C22×C4—284no—
411C22×C4—284no—
412C22×C4C23×D714yes—
413C22×C4C23×D714yes—
414C22×C4C23×D711yescharacteristic
415C22×C4C23×D711yescharacteristic
416C22×C4C23×D711yescharacteristic
417C22×C4C23×D714yes—
418C4⋊C4—44no—
419C4⋊C4—716no—
420C4⋊C4—716no—
421C4⋊C4—78no—
422C4⋊C4—78no—
423C4⋊C4—148no—
424C4⋊C4—148no—
425C4⋊C4—284no—
426C4⋊C4—284no—
427C4⋊C4—284no—
428C4⋊C4C2×C4×D718yes—
429C4⋊C4C2×C4×D718yes—
430C22⋊C4—148no—
431C22⋊C4—148no—
432C42—44no—
433C42—74no—
434C42—74no—
435C42—74no—
436C42—148no—
437C42—148no—
438C42—284no—
439C42—284no—
440C42—284no—
441C42C2×C4×D714yes—
442C2×C7C2×#51811yescharacteristic
443C2×C7C2×#51911yescharacteristic
444C2×C7C2×#51911yescharacteristic
445C2×C7#355214yes—
446D7—24no—
447C23—72no—
448C23—74no—
449C23—142no—
450C23—142no—
451C23C24×D711yescharacteristic
452Q8—48no—
453Q8—1432no—
454Q8—288no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—72no—
467C2×C4—72no—
468C2×C4—74no—
469C2×C4—74no—
470C2×C4—72no—
471C2×C4—78no—
472C2×C4—148no—
473C2×C4—148no—
474C2×C4—148no—
475C2×C4—144no—
476C2×C4—144no—
477C2×C4—142no—
478C2×C4—144no—
479C2×C4—144no—
480C2×C4—144no—
481C2×C4—144no—
482C2×C4—144no—
483C2×C4—144no—
484C2×C4—2816no—
485C2×C4C22×C4×D712yes—
486C2×C4C22×C4×D712yes—
487C2×C4C22×C4×D712yes—
488C2×C4C22×C4×D718yes—
489C7#3437111yescharacteristic, complemented, Sylow
490C22—74no—
491C22—148no—
492C22—142no—
493C22—142no—
494C22C23×C4×D711yescharacteristic
495C22D7×H(5,𝔽2)12yes—
496C22D7×2−1+412yes—
497C22D7×2−1+412yes—
498C4—22no—
499C4—28no—
500C4—22no—
501C4—22no—
502C4—44no—
503C4—44no—
504C4—74no—
505C4—74no—
506C4—148no—
507C4—142no—
508C4—284no—
509C4—284no—
510C2—144no—
511C2D7×#51811yescharacteristic
512C2D7×#51911yescharacteristic
513C2D7×#51911yescharacteristic
514C2#1174420014yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2381C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenter#2381D7
hypocenterC7#34371
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. C7×#2381C2semidirect
3. #11740676C2semidirect
5. #11733931C2semidirect
7. #11732412C2semidirect
9. #11732393C2semidirect
10. #11731710C2semidirect
11. #11731710C2semidirect
17. D7×C4⋊Q8C4semidirect
139. #2381D7semidirect
489. C7#34371semidirect

Maximal subgroups

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

subgroupindex
C2×D7×C4⋊Q82
C7×#23812
#117406762
#117340592
#117339312
#117327412
#117324122
#117324072
#117323932
#117317102
#117317032
#343717