Finite Group Databasebeta

← #14218200

Subgroups — #14218200

Order 1792.

1810 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11743342C211yesmaximal, characteristic
2C7×#2960C211yesmaximal, characteristic, complemented
3#11740774C211yesmaximal, characteristic, complemented
4#11737406C218yesmaximal, complemented
5#11737363C218yesmaximal, complemented
6D7×#424C212yesmaximal
7#11736521C212yesmaximal, complemented
8#11736493C211yesmaximal, characteristic
9#11736490C211yesmaximal, characteristic, complemented
10#11735993C212yesmaximal
11#11735988C212yesmaximal, complemented
12C2×#89461C211yesmaximal, characteristic
13#11735890C211yesmaximal, characteristic, complemented
14C7×#533C2211yescharacteristic
15C2×D7×M16C2212yes—
16C22×#1131C2211yescharacteristic
17C22×C8×D7C2211yescharacteristic
18#90420C414yes—
19#90380C414yes—
20#90325C414yes—
21#90314C414yes—
22#90269C2218yes—
23D7×#117C2212yes—
24#90208C2212yes—
25C2×#6435C2211yescharacteristic
26#90161C2211yescharacteristic
27C7×#446C2214yescomplemented
28C7×C8×Q8C2214yescomplemented
29C7×#424C2212yes—
30C7×#423C2211yescharacteristic
31C7×#405C2212yes—
32C2×C4×C7×C8C2211yescharacteristic
33#89894C2212yes—
34#89891C2212yes—
35#89890C2212yes—
36#89874C2212yes—
37#89873C2212yes—
38C2×#6389C2211yescharacteristic
39C2×C8×Dic7C2211yescharacteristic
40#89796C2218yescomplemented
41#89771C2212yes—
42#89769C2212yes—
43#89696C2211yescharacteristic
44C2×#6377C2211yescharacteristic
45#89611C2218yes—
46#89611C2218yes—
47D7×C4⋊C8C2218yes—
48#89602C2218yescomplemented
49#89600C2218yescomplemented
50#89600C2218yescomplemented
51#89480C2214yes—
52#89479C2214yes—
53#89473C2218yescomplemented
54#89461C2214yes—
55#89461C2214yes—
56#89461C2214yes—
57C4×#1131C2214yes—
58#89451C2214yescomplemented
59#89451C2218yescomplemented
60C8×Dic14C2214yescomplemented
61#60193—71nomaximal, Sylow
62C22×C4×D7C2311yescharacteristic
63C2×C7×M16C2312yes—
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C7×C4⋊Q8C2×C412yes—
67C7×#125C2×C412yes—
68C7×C22⋊Q8C2×C414yes—
69C4×C7×Q8C2314yes—
70C7×#117C2312yes—
71C2×C42×C7C2311yescharacteristic
72#6509C2×C418yes—
73#6507C2×C418yes—
74C2×(C2×D7)⋊C4C2×C412yes—
75#6489C2312yes—
76#6487C2×C414yes—
77C2×Dic7⋊C4C2×C412yes—
78C2×C4×Dic7C2311yescharacteristic
79C2×#1136C2312yes—
80C22×C7⋊C8C2311yescharacteristic
81C22×C7⋊C8C2311yescharacteristic
82D7×M16—28no—
83C2×#1131Q818yescomplemented
84C2×#1131C2314yes—
85C2×#1131C2312yes—
86C2×#1131C2312yes—
87C2×C8×D7—24no—
88C2×C8×D7C2314yes—
89C2×C8×D7C2314yes—
90#6460C2×C418yes—
91#6459C2×C418yes—
92#6459C2×C418yes—
93#6455C2×C414yes—
94D7×C4⋊C4C2×C414yes—
95#6453C2×C414yes—
96#6452C2×C418yes—
97#6452C2×C418yes—
98#6451C2×C414yes—
99#6451C2×C418yes—
100D7×C22⋊C4C2×C414yes—
101#6441C2×C418yes—
102#6440C2×C414yes—
103#6435C2×C412yes—
104#6435C2×C412yes—
105#6435C2314yes—
106#6435C2314yes—
107C42×D7C2314yes—
108#6433C2×C412yes—
109#6432C2×C412yes—
110C4×Dic14C2314yes—
111C4×Dic14C2318yes—
112C7×C4⋊C8C2314yes—
113C7×C4⋊C8C2318yes—
114C7×#97C2314yes—
115C4×C7×C8C2312yes—
116C4×C7×C8C2314yes—
117C4×C7×C8C2312yes—
118#6389C2314yes—
119#6389C2312yes—
120#6389C2312yes—
121#6389C2314yes—
122Dic7⋊C8C2318yes—
123Dic7⋊C8C2318yes—
124Dic7⋊C8C2318yes—
125C8×Dic7C2314yes—
126C8×Dic7C2314yes—
127C8×Dic7C2314yes—
128C4⋊(C7⋊C8)C2318yes—
129C4⋊(C7⋊C8)C2314yes—
130#6377C2314yes—
131#6377C2314yes—
132C4×C7⋊C8C2314yes—
133#3458—71no—
134#2962—78no—
135#2962—78no—
136#2961—71no—
137#2961—72no—
138#2961—71no—
139#2960—72no—
140#2960—71no—
141#2960D711yescharacteristic, complemented
142#2905—71no—
143C2×#424—72no—
144#2873—72no—
145C2×#405—71no—
146C23×D7C22×C411yescharacteristic
147C2×C7×Q8C22×C414yes—
148C22×C4×C7C22×C412yes—
149C22×C4×C7C2411yescharacteristic
150C22×Dic7C22×C412yes—
151C22×Dic7C22×C411yescharacteristic
152C2×C4×D7C22×C418yes—
153C2×C4×D7C22×C412yes—
154C2×C4×D7C2×Q814yes—
155C2×C4×D7#4114yes—
156C2×C4×D7C2412yes—
157C2×Dic14C22×C414yes—
158C2×Dic14C22×C418yes—
159C7×M16—24no—
160C2×C7×C8—22no—
161C2×C7×C8C2×Q814yes—
162C2×C7×C8C2412yes—
163C2×C7×C8C2414yes—
164C2×C7×C8C2411yescharacteristic
165C2×C7×C8C2411yescharacteristic
166C7×C4⋊C4C22×C418yes—
167C7×C4⋊C4C22×C414yes—
168C7×C4⋊C4C22×C414yes—
169C7×C22⋊C4C22×C414yes—
170C42×C7C22×C411yescharacteristic
171C42×C7C22×C411yescharacteristic
172C42×C7C2412yes—
173C42×C7C2414yes—
174C22⋊Dic7C22×C414yes—
175(C2×D7)⋊C4C22×C414yes—
176(C2×D7)⋊C4C22×C414yes—
177(C2×D7)⋊C4C22×C418yes—
178C4⋊Dic7C22×C414yes—
179C4⋊Dic7C22×C418yes—
180C4⋊Dic7C22×C414yes—
181Dic7⋊C4C22×C418yes—
182Dic7⋊C4C22×C418yes—
183Dic7⋊C4C22×C418yes—
184Dic7⋊C4C22×C414yes—
185Dic7⋊C4C22×C414yes—
186Dic7⋊C4C22×C418yes—
187C4×Dic7C22×C418yes—
188C4×Dic7C22×C412yes—
189C4×Dic7C2414yes—
190C4×Dic7C2412yes—
191#1136—24no—
192C2×C7⋊C8—22no—
193C2×C7⋊C8C2×Q814yes—
194C2×C7⋊C8C2412yes—
195C2×C7⋊C8C2414yes—
196C2×C7⋊C8C2412yes—
197#1131—28no—
198#1131C2×Q818yes—
199#1131C2×Q818yes—
200C8×D7—28no—
201C8×D7—28no—
202C22×M16—72no—
203C22×M16—71no—
204C23×C8—71no—
205#544—74no—
206#544—74no—
207#544—74no—
208#543—74no—
209#533—78no—
210#533—72no—
211#533—71no—
212#533C2×D711yescharacteristic
213C2×#117—72no—
214C2×#117—71no—
215#446—78no—
216#446—78no—
217#446—78no—
218#446—78no—
219#446—74no—
220#446—78no—
221#446C2×D714yescomplemented
222C8×Q8—78no—
223C8×Q8—74no—
224C8×Q8C2×D714yescomplemented
225#424—72no—
226#424—72no—
227#424—72no—
228#424—78no—
229#424—78no—
230#424C2×D712yes—
231#423—72no—
232#423—71no—
233#423C2×D711yescharacteristic
234C2×C4⋊C8—78no—
235#405—72no—
236#405—72no—
237#405—72no—
238#405—74no—
239#405—74no—
240#405—74no—
241#405—74no—
242#405C2×D712yes—
243C4×M16—74no—
244C4×M16—74no—
245C2×#97—71no—
246C2×#97—71no—
247C2×C4×C8—71no—
248C2×C4×C8C2×D711yescharacteristic
249C23×C7C23×C411yescharacteristic
250C22×D7—42no—
251C22×D7C4×Q814yes—
252C22×D7C23×C412yes—
253C7×Q8—44no—
254C2×C4×C7—42no—
255C2×C4×C7C23×C418yes—
256C2×C4×C7C23×C412yes—
257C2×C4×C7C23×C412yes—
258C2×C4×C7C23×C411yescharacteristic
259C2×C4×C7C22×Q812yes—
260C2×C4×C7C2×#4112yes—
261C2×C4×C7C2511yescharacteristic
262C2×Dic7—42no—
263C2×Dic7C4×Q814yes—
264C2×Dic7C23×C414yes—
265C2×Dic7C23×C418yes—
266C2×Dic7C23×C412yes—
267C4×D7—48no—
268C4×D7C4×Q818yes—
269C4×D7C22×Q814yes—
270C4×D7C2×#4114yes—
271Dic14—44no—
272Dic14—48no—
273C7×C8—22no—
274C7×C8—24no—
275C7×C8C22×Q812yes—
276C7×C8C22×Q812yes—
277C7⋊C8—22no—
278C7⋊C8—24no—
279C7⋊C8C22×Q814yes—
280C23×C4—71no—
281C2×M16—72no—
282C2×M16—78no—
283C2×M16—72no—
284C2×M16—72no—
285C2×M16—74no—
286C2×M16—148no—
287C2×M16C22×D712yes—
288C22×C8—71no—
289C22×C8—71no—
290C22×C8—74no—
291C22×C8—74no—
292C22×C8—144no—
293C22×C8C22×D711yescharacteristic
294C22×C8C22×D711yescharacteristic
295C4⋊Q8—78no—
296C4⋊Q8—78no—
297C4⋊Q8—74no—
298C4⋊Q8—72no—
299C4⋊Q8C4×D712yes—
300#125—78no—
301#125—78no—
302#125—74no—
303#125—72no—
304#125C4×D712yes—
305C22⋊Q8—78no—
306C22⋊Q8—74no—
307C22⋊Q8—78no—
308C22⋊Q8—78no—
309C22⋊Q8—78no—
310C22⋊Q8—78no—
311C22⋊Q8C4×D714yes—
312C4×Q8—78no—
313C4×Q8—74no—
314C4×Q8C22×D714yes—
315#117—72no—
316#117—74no—
317#117—74no—
318#117—72no—
319#117—72no—
320#117—74no—
321#117—74no—
322#117C22×D712yes—
323C2×C4⋊C4—72no—
324C2×C4⋊C4—74no—
325C2×C22⋊C4—72no—
326C2×C22⋊C4—74no—
327C2×C42—71no—
328C2×C42—74no—
329C2×C42C22×D711yescharacteristic
330C4⋊C8—78no—
331C4⋊C8—78no—
332C4⋊C8—78no—
333C4⋊C8—78no—
334C4⋊C8—74no—
335C4⋊C8C22×D718yes—
336C4⋊C8C22×D714yes—
337#97—72no—
338#97—72no—
339#97—74no—
340#97—74no—
341#97—74no—
342#97—74no—
343#97C22×D714yes—
344C4×C8—74no—
345C4×C8—74no—
346C4×C8—74no—
347C4×C8—74no—
348C4×C8C22×D712yes—
349C4×C8C22×D712yes—
350C4×C8C22×D714yes—
351C22×C7—22no—
352C22×C7C2×C4×Q812yes—
353C22×C7C24×C411yescharacteristic
354C2×D7—24no—
355C2×D7—44no—
356C2×D7C2×C4×Q814yes—
357C4×C7—22no—
358C4×C7—28no—
359C4×C7—22no—
360C4×C7C2×C4×Q812yes—
361C4×C7C23×Q811yescharacteristic
362C4×C7C22×#4111yescharacteristic
363Dic7—28no—
364Dic7—24no—
365Dic7C2×C4×Q814yes—
366C24—71no—
367C2×Q8—78no—
368C2×Q8—74no—
369C2×Q8C2×C4×D714yes—
370C22×C4—71no—
371C22×C4—72no—
372C22×C4—74no—
373C22×C4—74no—
374C22×C4—72no—
375C22×C4—72no—
376C22×C4—78no—
377C22×C4C2×C4×D712yes—
378C22×C4C23×D711yescharacteristic
379M16—24no—
380M16—78no—
381M16—78no—
382M16—148no—
383M16—144no—
384C2×C8—22no—
385C2×C8—74no—
386C2×C8—72no—
387C2×C8—74no—
388C2×C8—72no—
389C2×C8—148no—
390C2×C8—148no—
391C2×C8—142no—
392C2×C8Q8×D714yes—
393C2×C8C23×D712yes—
394C2×C8C23×D714yes—
395C2×C8C23×D711yescharacteristic
396C2×C8C23×D711yescharacteristic
397C4⋊C4—78no—
398C4⋊C4—74no—
399C4⋊C4—74no—
400C4⋊C4—78no—
401C4⋊C4—78no—
402C4⋊C4—78no—
403C4⋊C4—74no—
404C4⋊C4—78no—
405C4⋊C4—74no—
406C4⋊C4C2×C4×D714yes—
407C4⋊C4C2×C4×D714yes—
408C4⋊C4C2×C4×D718yes—
409C22⋊C4—74no—
410C22⋊C4—78no—
411C22⋊C4—74no—
412C22⋊C4—74no—
413C22⋊C4C2×C4×D714yes—
414C42—78no—
415C42—72no—
416C42—72no—
417C42—74no—
418C42C2×C4×D711yescharacteristic
419C42C2×C4×D711yescharacteristic
420C42C23×D714yes—
421C42C23×D712yes—
422C2×C7—22no—
423C2×C7C22×C4×Q811yescharacteristic
424C2×C7#357312yes—
425D7—24no—
426C23—72no—
427C23—74no—
428C23—282no—
429C23C22×C4×D711yescharacteristic
430Q8—44no—
431Q8—284no—
432Q8—288no—
433C2×C4—42no—
434C2×C4—74no—
435C2×C4—72no—
436C2×C4—78no—
437C2×C4—74no—
438C2×C4—74no—
439C2×C4—78no—
440C2×C4—74no—
441C2×C4—288no—
442C2×C4—282no—
443C2×C4C22×C4×D718yes—
444C2×C4C22×C4×D712yes—
445C2×C4C22×C4×D712yes—
446C2×C4C22×C4×D711yescharacteristic
447C2×C4C2×Q8×D712yes—
448C2×C4D7×#4112yes—
449C2×C4C24×D711yescharacteristic
450C8—22no—
451C8—24no—
452C8—74no—
453C8—142no—
454C8—144no—
455C8C2×Q8×D712yes—
456C8C2×Q8×D712yes—
457C7#6019311yescharacteristic, complemented, Sylow
458C22—22no—
459C22—74no—
460C22—144no—
461C22—284no—
462C22C4×Q8×D712yes—
463C22C23×C4×D711yescharacteristic
464C4—22no—
465C4—22no—
466C4—28no—
467C4—74no—
468C4—148no—
469C4—144no—
470C4C4×Q8×D712yes—
471C4C22×Q8×D711yescharacteristic
472C4C2×D7×#4111yescharacteristic
473C2—22no—
474C2—144no—
475C2C2×C4×Q8×D711yescharacteristic
476C2#1174422812yes—
subgroupG/H
center, FrattiniC2×C4C24×D7
commutator, socleC22×C7C24×C4
FittingC7×#2960C2
prime radicalC23×D7C22×C4
cosocleC2×C4×C7C25
hypercenter#2960D7
hypocenterC7#60193
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #2960

Complements

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

NH ≅ G/Ntype
2. C7×#2960C2semidirect
3. #11740774C2semidirect
4. #11737406C2semidirect
5. #11737363C2semidirect
7. #11736521C2semidirect
9. #11736490C2semidirect
11. #11735988C2semidirect
13. #11735890C2semidirect
27. C7×#446C22semidirect
28. C7×C8×Q8C22semidirect
40. #89796C22semidirect
48. #89602C22semidirect
49. #89600C22semidirect
50. #89600C22semidirect
53. #89473C22semidirect
58. #89451C22semidirect
59. #89451C22semidirect
60. C8×Dic14C22semidirect
83. C2×#1131Q8semidirect
141. #2960D7semidirect
221. #446C2×D7semidirect
224. C8×Q8C2×D7semidirect
457. C7#60193semidirect

Maximal subgroups

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

subgroupindex
#117433422
C7×#29602
#117407742
#117374062
#117373632
D7×#4242
#117365212
#117364932
#117364902
#117359932
#117359882
C2×#894612
#117358902
#601937