Finite Group Databasebeta

← #13516712

Subgroups — #13516712

Order 1792.

1898 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90419C211yesmaximal, characteristic
2C7×#2381C211yesmaximal, characteristic, complemented
3#11740676C211yesmaximal, characteristic, complemented
4#11733934C214yesmaximal, complemented
5#11733932C214yesmaximal
6#11732416C214yesmaximal, complemented
7#11732408C214yesmaximal, complemented
8#11732396C214yesmaximal
9#11732389C214yesmaximal
10#11731710C211yesmaximal, characteristic, complemented
11#11731710C211yesmaximal, characteristic, complemented
12#11731703C211yesmaximal, characteristic
13#11731703C211yesmaximal, characteristic
14C2×C7×C4⋊Q8C2211yescharacteristic
15C2×#6509C2212yes—
16C2×Q8×Dic7C2212yes—
17#90419C4116yescomplemented
18C2×#6460C2212yes—
19C2×#6455C2212yes—
20C2×#6453C2212yes—
21C2×#6450C2212yes—
22C2×C42×D7C2211yescharacteristic
23C2×#6432C2211yescharacteristic
24C7×#391C2214yes—
25C7×#387C2214yes—
26C7×#387C2214yes—
27C7×#383C2211yescharacteristic
28C7×#383C2211yescharacteristic
29#90002C2214yes—
30#89762C2214yes—
31#89753C2214yes—
32#89753C2214yes—
33#89750C2214yes—
34#89748C2214yes—
35C4⋊C4×Dic7C2214yes—
36#89712C2214yes—
37#89709C2211yescharacteristic
38#89709C2211yescharacteristic
39C4×C4⋊Dic7C2214yes—
40C4×Dic7⋊C4C2214yes—
41C4×Dic7⋊C4C2214yes—
42#89442C2214yes—
43#89441C2214yes—
44#89439C2212yes—
45#89439C2212yes—
46#89438C2212yes—
47#89438C2212yes—
48#89436C2212yes—
49#89436C2212yes—
50#89427C2214yes—
51#89426C2214yes—
52#89426C2214yes—
53#89426C2214yes—
54#89425C2214yes—
55#89425C2214yes—
56#89425C2214yes—
57#89422C2214yes—
58#89422C2212yes—
59#89422C2212yes—
60#89421C2214yes—
61#89421C2214yes—
62#89421C2212yes—
63#89421C2212yes—
64#89420C2212yes—
65#89420C2212yes—
66#89418C2214yes—
67#89418C2214yes—
68#34375—71nomaximal, Sylow
69C22×C7×Q8C2312yes—
70C22×C4×D7C2311yescharacteristic
71C22×C4×D7C2311yescharacteristic
72C22×C4×D7C2311yescharacteristic
73C22×Dic14C2312yes—
74C7×C4⋊Q8C2×C418yes—
75C2×C7×C4⋊C4C2314yes—
76C2×C7×C4⋊C4C2312yes—
77C2×C7×C4⋊C4C2312yes—
78C2×C42×C7C2314yes—
79C2×C42×C7C2311yescharacteristic
80#6509C2×C4116yes—
81Q8×Dic7C2×C4116yes—
82C2×(C2×D7)⋊C4C2314yes—
83C2×(C2×D7)⋊C4C2314yes—
84C2×(C2×D7)⋊C4C2314yes—
85C2×C4⋊Dic7C2314yes—
86C2×C4⋊Dic7C2312yes—
87C2×C4⋊Dic7C2312yes—
88C2×C4⋊Dic7C2314yes—
89C2×Dic7⋊C4C2314yes—
90C2×Dic7⋊C4C2314yes—
91C2×Dic7⋊C4C2314yes—
92C2×Dic7⋊C4C2314yes—
93C2×Dic7⋊C4C2314yes—
94C2×C4×Dic7C2314yes—
95C2×C4×Dic7C2314yes—
96C2×C4×Dic7C2314yes—
97C2×C4×Dic7C2314yes—
98C2×C4×Dic7C2314yes—
99C2×C4×Dic7C2311yescharacteristic
100C2×C4×Dic7C2311yescharacteristic
101C2×C4×Dic7C2311yescharacteristic
102C2×C4×Dic7C2314yes—
103#6460C2×C4116yes—
104#6455C2×C4116yes—
105#6453C2×C4116yes—
106#6450C2×C4116yes—
107C42×D7C2×C418yes—
108#6432C2×C418yes—
109C7×#95C2312yes—
110C7×#95C2312yes—
111C7×#95C2312yes—
112C7×#95C2312yes—
113C7×#95C2314yes—
114C7×#95C2312yes—
115C7×#95C2314yes—
116C7×#95C2312yes—
117C7⋊#95C2312yes—
118C7⋊#95C2312yes—
119C7⋊#95C2314yes—
120C7⋊#95C2314yes—
121C7⋊#95C2314yes—
122C7⋊#95C2314yes—
123C7⋊#95C2314yes—
124C7⋊#95C2314yes—
125C7⋊#95C2312yes—
126C7⋊#95C2312yes—
127C7⋊#95C2312yes—
128C7⋊#95C2314yes—
129C7⋊#95C2312yes—
130C7⋊#95C2314yes—
131C2×#533—71no—
132#2381—71no—
133#2381—71no—
134#2381—71no—
135#2381D711yescharacteristic, complemented
136#2371—74no—
137#2355—74no—
138#2355—74no—
139#2332—74no—
140#2332—74no—
141#2314—74no—
142#2283—71no—
143#2283—71no—
144C23×D7C2411yescharacteristic
145C2×C7×Q8—42no—
146C2×C7×Q8—42no—
147C2×C7×Q8C22×C418yes—
148C22×C4×C7C2414yes—
149C22×C4×C7C2414yes—
150C22×C4×C7C2411yescharacteristic
151C22×C4×C7C2411yescharacteristic
152C22×C4×C7C2411yescharacteristic
153C22×C4×C7C2414yes—
154C22×Dic7C2414yes—
155C22×Dic7C2414yes—
156C22×Dic7C2411yescharacteristic
157C22×Dic7C2411yescharacteristic
158C22×Dic7C2414yes—
159C22×Dic7C2411yescharacteristic
160C2×C4×D7—24no—
161C2×C4×D7—42no—
162C2×C4×D7—42no—
163C2×C4×D7C22×C414yes—
164C2×C4×D7C22×C414yes—
165C2×C4×D7C22×C414yes—
166C2×Dic14—42no—
167C2×Dic14—42no—
168C2×Dic14C22×C418yes—
169C7×C4⋊C4—44no—
170C7×C4⋊C4C22×C418yes—
171C7×C4⋊C4C22×C418yes—
172C42×C7—44no—
173C42×C7C22×C414yes—
174(C2×D7)⋊C4—44no—
175(C2×D7)⋊C4—44no—
176(C2×D7)⋊C4C22×C4116yes—
177C4⋊Dic7—44no—
178C4⋊Dic7C22×C418yes—
179C4⋊Dic7C22×C418yes—
180C4⋊Dic7C22×C4116yes—
181Dic7⋊C4—28no—
182Dic7⋊C4—28no—
183Dic7⋊C4—44no—
184Dic7⋊C4—44no—
185Dic7⋊C4C22×C4116yes—
186C4×Dic7—44no—
187C4×Dic7—44no—
188C4×Dic7—44no—
189C4×Dic7—44no—
190C4×Dic7—44no—
191C4×Dic7C22×C414yes—
192C4×Dic7C22×C414yes—
193C4×Dic7C22×C414yes—
194C4×Dic7C22×C4116yes—
195#533—716no—
196C2×C4⋊Q8—71no—
197C2×C4⋊Q8C2×D711yescharacteristic
198C2×#125—72no—
199C2×C22⋊Q8—72no—
200C2×C22⋊Q8—72no—
201C2×C4×Q8—72no—
202C2×C4×Q8—72no—
203C2×#117—72no—
204C22×C42—71no—
205#391—74no—
206#391—74no—
207#391—74no—
208#391C2×D714yes—
209#388—74no—
210#388—74no—
211#388—74no—
212#388—74no—
213#387—74no—
214#387—74no—
215#387—74no—
216#387—74no—
217#387—74no—
218#387—74no—
219#387—74no—
220#387—74no—
221#387—74no—
222#387—74no—
223#387C2×D714yes—
224#387C2×D714yes—
225#384—74no—
226#383—71no—
227#383—71no—
228#383C2×D711yescharacteristic
229#383C2×D711yescharacteristic
230#382—74no—
231#382—72no—
232#382—72no—
233#381—72no—
234#381—72no—
235#380—72no—
236#380—72no—
237C4×C4⋊C4—74no—
238C4×C4⋊C4—74no—
239C4×C4⋊C4—74no—
240C4×C4⋊C4—74no—
241C4×C4⋊C4—74no—
242C4×C4⋊C4—74no—
243C4×C4⋊C4—72no—
244C4×C4⋊C4—72no—
245C4×C22⋊C4—72no—
246C4×C22⋊C4—72no—
247#376—72no—
248#376—72no—
249C23×C7C2511yescharacteristic
250C22×D7—22no—
251C22×D7—22no—
252C22×D7—22no—
253C22×D7C23×C412yes—
254C7×Q8—48no—
255C2×C4×C7—22no—
256C2×C4×C7—24no—
257C2×C4×C7—24no—
258C2×C4×C7—22no—
259C2×C4×C7—22no—
260C2×C4×C7—24no—
261C2×C4×C7—24no—
262C2×C4×C7—24no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7C23×C418yes—
267C2×C4×C7C23×C412yes—
268C2×C4×C7C23×C412yes—
269C2×C4×C7C23×C412yes—
270C2×Dic7—24no—
271C2×Dic7—22no—
272C2×Dic7—22no—
273C2×Dic7—24no—
274C2×Dic7—24no—
275C2×Dic7—24no—
276C2×Dic7—24no—
277C2×Dic7—24no—
278C2×Dic7—24no—
279C2×Dic7—24no—
280C2×Dic7C23×C412yes—
281C2×Dic7C23×C412yes—
282C2×Dic7C23×C418yes—
283C2×Dic7C23×C412yes—
284C2×Dic72−1+414yes—
285C4×D7—28no—
286C4×D7—44no—
287C4×D7—44no—
288Dic14—48no—
289C22×Q8—72no—
290C22×Q8C22×D712yes—
291C23×C4—71no—
292C23×C4—71no—
293C23×C4—71no—
294C4⋊Q8—78no—
295C4⋊Q8C4×D718yes—
296#125—716no—
297C22⋊Q8—716no—
298C22⋊Q8—716no—
299C4×Q8—716no—
300C4×Q8—716no—
301#117—716no—
302C2×C4⋊C4—72no—
303C2×C4⋊C4—74no—
304C2×C4⋊C4—74no—
305C2×C4⋊C4—72no—
306C2×C4⋊C4—74no—
307C2×C4⋊C4—74no—
308C2×C4⋊C4—74no—
309C2×C4⋊C4—74no—
310C2×C4⋊C4—74no—
311C2×C4⋊C4C22×D714yes—
312C2×C4⋊C4C22×D712yes—
313C2×C4⋊C4C22×D712yes—
314C2×C22⋊C4—74no—
315C2×C22⋊C4—74no—
316C2×C22⋊C4—74no—
317C2×C42—74no—
318C2×C42—71no—
319C2×C42—71no—
320C2×C42—71no—
321C2×C42—78no—
322C2×C42—74no—
323C2×C42—74no—
324C2×C42—74no—
325C2×C42—74no—
326C2×C42—74no—
327C2×C42C22×D714yes—
328C2×C42C22×D711yescharacteristic
329#95—74no—
330#95—74no—
331#95—72no—
332#95—72no—
333#95—72no—
334#95—72no—
335#95—72no—
336#95—72no—
337#95—74no—
338#95—74no—
339#95—74no—
340#95—74no—
341#95—74no—
342#95—74no—
343#95C22×D712yes—
344#95C22×D712yes—
345#95C22×D712yes—
346#95C22×D712yes—
347#95C22×D712yes—
348#95C22×D712yes—
349#95C22×D714yes—
350#95C22×D714yes—
351C22×C7C24×C411yescharacteristic
352C22×C7C2×2−1+412yes—
353C22×C7#58512yes—
354C22×C7#58512yes—
355C2×D7—22no—
356C2×D7—22no—
357C2×D7—22no—
358C2×D7—44no—
359C4×C7—22no—
360C4×C7—22no—
361C4×C7—22no—
362C4×C7—28no—
363C4×C7—44no—
364C4×C7—44no—
365Dic7—28no—
366Dic7—22no—
367Dic7—22no—
368Dic7—44no—
369Dic7—44no—
370Dic7#51914yes—
371C24—71no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8—78no—
375C2×Q8—282no—
376C2×Q8—282no—
377C2×Q8C2×C4×D718yes—
378C22×C4—71no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—74no—
382C22×C4—74no—
383C22×C4—74no—
384C22×C4—74no—
385C22×C4—74no—
386C22×C4—74no—
387C22×C4—144no—
388C22×C4—282no—
389C22×C4—282no—
390C22×C4C23×D714yes—
391C22×C4C23×D714yes—
392C22×C4C23×D711yescharacteristic
393C22×C4C23×D711yescharacteristic
394C22×C4C23×D711yescharacteristic
395C22×C4C23×D714yes—
396C4⋊C4—44no—
397C4⋊C4—716no—
398C4⋊C4—78no—
399C4⋊C4—716no—
400C4⋊C4—78no—
401C4⋊C4—148no—
402C4⋊C4—148no—
403C4⋊C4—284no—
404C4⋊C4—284no—
405C4⋊C4—284no—
406C4⋊C4C2×C4×D718yes—
407C4⋊C4C2×C4×D718yes—
408C22⋊C4—716no—
409C22⋊C4—284no—
410C22⋊C4—284no—
411C42—44no—
412C42—74no—
413C42—74no—
414C42—74no—
415C42—716no—
416C42—284no—
417C42—284no—
418C42—284no—
419C42—284no—
420C42—284no—
421C42C2×C4×D714yes—
422C2×C7C2×#51911yescharacteristic
423C2×C7#342811yescharacteristic
424C2×C7#342811yescharacteristic
425C2×C7#355714yes—
426D7—42no—
427C23—72no—
428C23—142no—
429C23—142no—
430C23—142no—
431C23C24×D711yescharacteristic
432Q8—48no—
433Q8—288no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—72no—
446C2×C4—74no—
447C2×C4—72no—
448C2×C4—72no—
449C2×C4—78no—
450C2×C4—148no—
451C2×C4—142no—
452C2×C4—142no—
453C2×C4—144no—
454C2×C4—144no—
455C2×C4—144no—
456C2×C4—144no—
457C2×C4—144no—
458C2×C4—144no—
459C2×C4—144no—
460C2×C4—144no—
461C2×C4—284no—
462C2×C4—284no—
463C2×C4C22×C4×D712yes—
464C2×C4C22×C4×D712yes—
465C2×C4C22×C4×D712yes—
466C2×C4C22×C4×D718yes—
467C7#3437511yescharacteristic, complemented, Sylow
468C22—142no—
469C22—142no—
470C22—142no—
471C22—284no—
472C22C23×C4×D711yescharacteristic
473C22#9061812yes—
474C22D7×2−1+412yes—
475C22#9062012yes—
476C4—22no—
477C4—28no—
478C4—22no—
479C4—22no—
480C4—44no—
481C4—44no—
482C4—74no—
483C4—142no—
484C4—142no—
485C4—148no—
486C4—284no—
487C4—284no—
488C2—282no—
489C2#1174292411yescharacteristic
490C2#1174294011yescharacteristic
491C2D7×#51911yescharacteristic
492C2#1174419014yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2381C2
socle, cosocleC23×C7C25
prime radicalC23×D7C24
hypercenter#2381D7
hypocenterC7#34375
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
4. #11733934C2semidirect
6. #11732416C2semidirect
7. #11732408C2semidirect
10. #11731710C2semidirect
11. #11731710C2semidirect
17. #90419C4semidirect
135. #2381D7semidirect
467. C7#34375semidirect

Maximal subgroups

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

subgroupindex
C2×#904192
C7×#23812
#117406762
#117339342
#117339322
#117324162
#117324082
#117323962
#117323892
#117317102
#117317032
#343757