Finite Group Databasebeta

← #13319727

Subgroups — #13319727

Order 1792.

1098 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90263C211yesmaximal, characteristic
2#11736759C212yesmaximal, complemented
3C7×#1467C211yesmaximal, characteristic, complemented
4#11729379C211yesmaximal, characteristic, complemented
5#11728580C212yesmaximal
6#11725881C212yesmaximal, complemented
7#11725880C212yesmaximal
8#11725879C212yesmaximal, complemented
9#11725878C212yesmaximal
10C22×Q8×D7C411yescharacteristic
11C2×C4×C7×Q8C2211yescharacteristic
12C2×Q8×Dic7C411yescharacteristic
13#90263—24no—
14#90263—24no—
15C2×#6455C411yescharacteristic
16C2×#6455C2211yescharacteristic
17C2×#6455C2211yescharacteristic
18C2×#6450C411yescharacteristic
19C2×#6450C2211yescharacteristic
20C2×#6450C2211yescharacteristic
21C2×#6435C411yescharacteristic
22C2×#6435C411yescharacteristic
23C2×#6435C2211yescharacteristic
24C2×C4×Dic14C411yescharacteristic
25C2×C4×Dic14C411yescharacteristic
26C2×C4×Dic14C2211yescharacteristic
27C7×#432C2212yes—
28#89698C2212yes—
29#89600—24no—
30#89491C2214yes—
31#89490C2214yes—
32#89473—24no—
33C7×#324C2212yes—
34C7×#324C2212yes—
35#89323C2212yes—
36#89323C2212yes—
37#89264C2214yes—
38#89263C2214yes—
39#17371—71nomaximal, Sylow
40C22×C7×Q8C2×C411yescharacteristic
41C2×Q8×D7—24no—
42C2×Q8×D7—24no—
43C2×Q8×D7—24no—
44C22×C4×D7C2×C411yescharacteristic
45C22×C4×D7C2×C411yescharacteristic
46C22×C4×D7C2×C411yescharacteristic
47C22×Dic14C2×C411yescharacteristic
48C22×Dic14C2×C411yescharacteristic
49C22×Dic14C2×C411yescharacteristic
50C4×C7×Q8—22no—
51C4×C7×Q8—22no—
52C2×C7×C4⋊C4C2×C411yescharacteristic
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C42×C7C2×C411yescharacteristic
56C2×C42×C7C2×C411yescharacteristic
57C2×C42×C7C2311yescharacteristic
58Q8×Dic7—24no—
59C2×(C2×D7)⋊C4C2×C411yescharacteristic
60C2×(C2×D7)⋊C4C2×C411yescharacteristic
61C2×(C2×D7)⋊C4C2×C411yescharacteristic
62C2×(C2×D7)⋊C4C2311yescharacteristic
63C2×C4⋊Dic7C2×C411yescharacteristic
64C2×C4⋊Dic7C2×C411yescharacteristic
65C2×C4⋊Dic7C2×C411yescharacteristic
66C2×Dic7⋊C4C2×C411yescharacteristic
67C2×Dic7⋊C4C2×C411yescharacteristic
68C2×Dic7⋊C4C2×C411yescharacteristic
69C2×Dic7⋊C4C2×C411yescharacteristic
70C2×Dic7⋊C4C2×C411yescharacteristic
71C2×Dic7⋊C4C2311yescharacteristic
72C2×C4×Dic7C2×C411yescharacteristic
73C2×C4×Dic7C2×C411yescharacteristic
74C2×C4×Dic7C2×C411yescharacteristic
75C2×C4×Dic7C2×C411yescharacteristic
76C2×C4×Dic7C2×C411yescharacteristic
77C2×C4×Dic7C2×C411yescharacteristic
78C2×C4×Dic7C2×C411yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C2×C4×Dic7C2311yescharacteristic
81#6455—24no—
82#6455—24no—
83#6455—24no—
84#6450—24no—
85#6450—24no—
86#6450—24no—
87#6435—22no—
88#6435—22no—
89#6435—22no—
90#6435—22no—
91#6435—22no—
92#6435—22no—
93C4×Dic14—22no—
94C4×Dic14—22no—
95C4×Dic14—22no—
96C4×Dic14—22no—
97C4×Dic14D414yes—
98C4×Dic14D414yes—
99C7×C4⋊C8—22no—
100C7×C22⋊C8C2314yes—
101C7×#97—22no—
102C22⋊(C7⋊C8)C2314yes—
103#6389—24no—
104Dic7⋊C8—24no—
105Dic7⋊C8—24no—
106C8×Dic7—24no—
107C4⋊(C7⋊C8)—22no—
108#6377—22no—
109C2×#519—71no—
110#2983—72no—
111#1467—72no—
112#1467—71no—
113#1467D711yescharacteristic, complemented
114#1464—72no—
115#1464—72no—
116#1461—72no—
117#1458—72no—
118C23×D7C22×C411yescharacteristic
119C2×C7×Q8—22no—
120C2×C7×Q8—22no—
121C2×C7×Q8—22no—
122C22×C4×C7C22×C411yescharacteristic
123C22×C4×C7C22×C411yescharacteristic
124C22×C4×C7C22×C411yescharacteristic
125C22×C4×C7C22×C411yescharacteristic
126C22×C4×C7C22×C411yescharacteristic
127C22×C4×C7C22×C411yescharacteristic
128C22×C4×C7C2411yescharacteristic
129C22×Dic7C22×C411yescharacteristic
130C22×Dic7C22×C411yescharacteristic
131C22×Dic7C22×C411yescharacteristic
132C22×Dic7C22×C411yescharacteristic
133C22×Dic7C22×C411yescharacteristic
134C22×Dic7C22×C411yescharacteristic
135C22×Dic7C22×C411yescharacteristic
136Q8×D7—232no—
137C2×C4×D7—22no—
138C2×C4×D7—22no—
139C2×C4×D7—22no—
140C2×C4×D7—22no—
141C2×C4×D7—22no—
142C2×C4×D7—22no—
143C2×C4×D7—22no—
144C2×C4×D7—22no—
145C2×C4×D7M1614yes—
146C2×Dic14—22no—
147C2×Dic14—22no—
148C2×Dic14—22no—
149C2×Dic14—22no—
150C2×Dic14—22no—
151C2×Dic14—22no—
152C2×Dic14—22no—
153C2×Dic14C22⋊C414yes—
154C2×Dic14M1614yes—
155C2×C7×C8—24no—
156C7×C4⋊C4—22no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—22no—
159C42×C7—21no—
160C42×C7—21no—
161C42×C7—21no—
162C42×C7—21no—
163C42×C7C2×D412yes—
164C42×C7C2×D412yes—
165(C2×D7)⋊C4—21no—
166(C2×D7)⋊C4—21no—
167(C2×D7)⋊C4—22no—
168(C2×D7)⋊C4—22no—
169(C2×D7)⋊C4—22no—
170C4⋊Dic7—22no—
171C4⋊Dic7—22no—
172C4⋊Dic7C22⋊C414yes—
173Dic7⋊C4—21no—
174Dic7⋊C4—21no—
175Dic7⋊C4—22no—
176Dic7⋊C4—22no—
177Dic7⋊C4—22no—
178Dic7⋊C4C22⋊C412yes—
179Dic7⋊C4C22⋊C412yes—
180Dic7⋊C4C22⋊C412yes—
181Dic7⋊C4C22⋊C412yes—
182C4×Dic7—22no—
183C4×Dic7—22no—
184C4×Dic7—22no—
185C4×Dic7—22no—
186C4×Dic7—22no—
187C4×Dic7—22no—
188C4×Dic7—22no—
189C4×Dic7C2×D414yes—
190C4×Dic7C2×D414yes—
191C2×C7⋊C8—24no—
192C23×Q8—71no—
193#519—144no—
194#519—144no—
195C2×C4×Q8—71no—
196C2×C4×Q8—71no—
197C2×C4×Q8—71no—
198C2×C4×Q8—71no—
199C2×C4×Q8—71no—
200C2×C4×Q8—71no—
201C2×C4×Q8—71no—
202C2×C4×Q8C2×D711yescharacteristic
203C2×#117—71no—
204C2×#117—71no—
205C2×#117—71no—
206C2×#117—71no—
207C2×#117—71no—
208C2×#117—71no—
209#446—144no—
210#446—144no—
211#432—72no—
212#432—74no—
213#432C2×D712yes—
214#431—74no—
215#324—72no—
216#324—72no—
217#324—74no—
218#324C2×D712yes—
219#324C2×D712yes—
220#323—74no—
221C23×C7C23×C411yescharacteristic
222C22×D7—21no—
223C22×D7—21no—
224C22×D7—41no—
225C22×D7—41no—
226C22×D7C2×M1612yes—
227C7×Q8—28no—
228C2×C4×C7—21no—
229C2×C4×C7—21no—
230C2×C4×C7—21no—
231C2×C4×C7—21no—
232C2×C4×C7—21no—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—41no—
237C2×C4×C7—41no—
238C2×C4×C7—41no—
239C2×C4×C7—41no—
240C2×C4×C7C2×C22⋊C412yes—
241C2×C4×C7C2×C22⋊C411yescharacteristic
242C2×C4×C7C2×C22⋊C411yescharacteristic
243C2×C4×C7C2×M1612yes—
244C2×C4×C7C2×M1612yes—
245C2×C4×C7C22×D412yes—
246C2×Dic7—21no—
247C2×Dic7—21no—
248C2×Dic7—21no—
249C2×Dic7—21no—
250C2×Dic7—21no—
251C2×Dic7—21no—
252C2×Dic7—41no—
253C2×Dic7—41no—
254C2×Dic7—41no—
255C2×Dic7—41no—
256C2×Dic7C2×C22⋊C412yes—
257C2×Dic7C2×C22⋊C412yes—
258C2×Dic7C2×C22⋊C412yes—
259C2×Dic7C2×C22⋊C412yes—
260C2×Dic7C2×M1612yes—
261C2×Dic7C2×M1612yes—
262C2×Dic7C2×M1612yes—
263C4×D7—28no—
264C4×D7—28no—
265C4×D7—28no—
266Dic14—28no—
267Dic14—28no—
268Dic14—28no—
269C7×C8—44no—
270C7⋊C8—44no—
271C22×Q8—71no—
272C22×Q8—71no—
273C22×Q8—71no—
274C22×Q8—144no—
275C22×Q8—144no—
276C22×Q8—144no—
277C22×Q8C4×D711yescharacteristic
278C23×C4—71no—
279C23×C4—71no—
280C23×C4—71no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—74no—
284C4×Q8—74no—
285C4×Q8—142no—
286C4×Q8—142no—
287C4×Q8—142no—
288C4×Q8—142no—
289C4×Q8—144no—
290C4×Q8—144no—
291C4×Q8—144no—
292C4×Q8—144no—
293#117—142no—
294#117—142no—
295#117—142no—
296#117—142no—
297#117—144no—
298#117—144no—
299#117—144no—
300#117—142no—
301#117—142no—
302C2×C4⋊C4—71no—
303C2×C4⋊C4—71no—
304C2×C4⋊C4—71no—
305C2×C4⋊C4—71no—
306C2×C4⋊C4—71no—
307C2×C4⋊C4—71no—
308C2×C4⋊C4—71no—
309C2×C4⋊C4—71no—
310C2×C4⋊C4—71no—
311C2×C4⋊C4C4×D711yescharacteristic
312C2×C4⋊C4C22×D711yescharacteristic
313C2×C4⋊C4C22×D711yescharacteristic
314C2×C22⋊C4—71no—
315C2×C22⋊C4—71no—
316C2×C22⋊C4—71no—
317C2×C22⋊C4—71no—
318C2×C42—71no—
319C2×C42—71no—
320C2×C42—71no—
321C2×C42—71no—
322C2×C42—71no—
323C2×C42—71no—
324C2×C42—71no—
325C2×C42—71no—
326C2×C42—71no—
327C2×C42C4×D711yescharacteristic
328C2×C42C4×D711yescharacteristic
329C2×C42C22×D711yescharacteristic
330C4⋊C8—22no—
331C4⋊C8—144no—
332C4⋊C8—144no—
333C4⋊C8—142no—
334C22⋊C8—74no—
335C22⋊C8C22×D714yes—
336#97—22no—
337#97—144no—
338#97—142no—
339C4×C8—144no—
340C22×C7—21no—
341C22×C7—21no—
342C22×C7C22×C22⋊C411yescharacteristic
343C22×C7C22×M1611yescharacteristic
344C22×C7C22×M1611yescharacteristic
345C2×D7—22no—
346C2×D7—42no—
347C2×D7—41no—
348C2×D7—41no—
349C2×D7—42no—
350C4×C7—22no—
351C4×C7—22no—
352C4×C7—22no—
353C4×C7—22no—
354C4×C7—21no—
355C4×C7—21no—
356C4×C7—41no—
357C4×C7—41no—
358Dic7—22no—
359Dic7—22no—
360Dic7—22no—
361Dic7—22no—
362Dic7—41no—
363Dic7—41no—
364Dic7#40714yes—
365C24—71no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—74no—
370C2×Q8—74no—
371C2×Q8—142no—
372C2×Q8—142no—
373C2×Q8—142no—
374C2×Q8—142no—
375C2×Q8—142no—
376C2×Q8—142no—
377C2×Q8—1432no—
378C2×Q8—142no—
379C22×C4—71no—
380C22×C4—71no—
381C22×C4—71no—
382C22×C4—71no—
383C22×C4—71no—
384C22×C4—74no—
385C22×C4—71no—
386C22×C4—71no—
387C22×C4—142no—
388C22×C4—142no—
389C22×C4—142no—
390C22×C4—142no—
391C22×C4—142no—
392C22×C4—142no—
393C22×C4—142no—
394C22×C4—142no—
395C22×C4C2×C4×D711yescharacteristic
396C22×C4C2×C4×D711yescharacteristic
397C22×C4C2×C4×D711yescharacteristic
398C22×C4C2×C4×D711yescharacteristic
399C22×C4C2×C4×D711yescharacteristic
400C22×C4C2×C4×D711yescharacteristic
401C22×C4C23×D711yescharacteristic
402C2×C8—24no—
403C2×C8—144no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—72no—
408C4⋊C4—72no—
409C4⋊C4—72no—
410C4⋊C4—72no—
411C4⋊C4—74no—
412C4⋊C4—142no—
413C4⋊C4—142no—
414C4⋊C4—142no—
415C4⋊C4—142no—
416C4⋊C4—141no—
417C4⋊C4—141no—
418C4⋊C4—142no—
419C22⋊C4—142no—
420C22⋊C4—142no—
421C22⋊C4—141no—
422C22⋊C4—141no—
423C22⋊C4—142no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—74no—
429C42—74no—
430C42—142no—
431C42—142no—
432C42—142no—
433C42—142no—
434C42—142no—
435C42—142no—
436C42—142no—
437C42D4×D712yes—
438C42D4×D712yes—
439C2×C7—21no—
440C2×C7—21no—
441C2×C7C2×#40711yescharacteristic
442C2×C7#288411yescharacteristic
443C2×C7#288911yescharacteristic
444D7—42no—
445C23—72no—
446C23—141no—
447C23—141no—
448C23—281no—
449C23—281no—
450C23C22×C4×D711yescharacteristic
451Q8—28no—
452Q8—148no—
453Q8—148no—
454Q8—148no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—72no—
468C2×C4—72no—
469C2×C4—72no—
470C2×C4—72no—
471C2×C4—72no—
472C2×C4—72no—
473C2×C4—72no—
474C2×C4—148no—
475C2×C4—148no—
476C2×C4—148no—
477C2×C4—141no—
478C2×C4—141no—
479C2×C4—141no—
480C2×C4—141no—
481C2×C4—141no—
482C2×C4—141no—
483C2×C4—281no—
484C2×C4—281no—
485C2×C4—281no—
486C2×C4—281no—
487C2×C4D7×C22⋊C411yescharacteristic
488C2×C4D7×C22⋊C411yescharacteristic
489C2×C4D7×C22⋊C412yes—
490C2×C4D7×M1612yes—
491C2×C4D7×M1612yes—
492C2×C4C2×D4×D712yes—
493C8—44no—
494C8—284no—
495C7#1737111yescharacteristic, complemented, Sylow
496C22—21no—
497C22—21no—
498C22—142no—
499C22—282no—
500C22—282no—
501C22—281no—
502C22—281no—
503C22C2×D7×C22⋊C411yescharacteristic
504C22C2×D7×M1611yescharacteristic
505C22C2×D7×M1611yescharacteristic
506C4—22no—
507C4—22no—
508C4—22no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—41no—
514C4—74no—
515C4—142no—
516C4—142no—
517C4—142no—
518C4—142no—
519C4—281no—
520C4—281no—
521C2—21no—
522C2—21no—
523C2—282no—
524C2D7×#40711yescharacteristic
525C2#1173613411yescharacteristic
526C2#1173615111yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1467C2
socleC22×C7C22×C22⋊C4
prime radicalC23×D7C22×C4
cosocleC22×C4×C7C24
hypercenter#1467D7
hypocenterC7#17371
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #11736759C2semidirect
3. C7×#1467C2semidirect
4. #11729379C2semidirect
6. #11725881C2semidirect
8. #11725879C2semidirect
113. #1467D7semidirect
495. C7#17371semidirect

Maximal subgroups

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

subgroupindex
C2×#902632
#117367592
C7×#14672
#117293792
#117285802
#117258812
#117258802
#117258792
#117258782
#173717