Finite Group Databasebeta

← #13319643

Subgroups — #13319643

Order 1792.

1322 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
1C22×#6440C211yesmaximal, characteristic
2#11736064C212yesmaximal, complemented
3C7×#1462C211yesmaximal, characteristic
4#11729293C211yesmaximal, characteristic
5#11727725C212yesmaximal, complemented
6#11725856C214yesmaximal, complemented
7#11725854C214yesmaximal, complemented
8C24×Dic7C411yescharacteristic
9C22×C7×C22⋊C4C2211yescharacteristic
10C22×C22⋊Dic7C411yescharacteristic
11C22×Dic7⋊C4C411yescharacteristic
12C22×Dic7⋊C4C2211yescharacteristic
13C22×C4×Dic7C411yescharacteristic
14C22×C4×Dic7C2211yescharacteristic
15C2×#6440—22no—
16C2×#6440—22no—
17C2×#6440—22no—
18C2×#6440—22no—
19C2×#6440C412yes—
20C2×#6440C412yes—
21C2×#6440C2212yes—
22C2×#6440C2212yes—
23C7×#407C2212yes—
24#89979C2212yes—
25#89890—22no—
26M16×Dic7—22no—
27#89491—22no—
28#89491C2214yescomplemented
29#89490—22no—
30#89490C2214yescomplemented
31C7×#323C2214yes—
32#89320C2214yes—
33#89264C2218yescomplemented
34#17357—71nomaximal, Sylow
35C25×C7C2×C411yescharacteristic
36C23×C4×C7C2×C411yescharacteristic
37C23×C4×C7C2311yescharacteristic
38C23×Dic7—21no—
39C23×Dic7—21no—
40C23×Dic7—21no—
41C23×Dic7—21no—
42C23×Dic7—28no—
43C23×Dic7C2×C412yes—
44C23×Dic7C2×C411yescharacteristic
45C23×Dic7C2×C411yescharacteristic
46C23×Dic7C2×C412yes—
47C23×Dic7C2×C411yescharacteristic
48C23×Dic7C2×C411yescharacteristic
49C2×C7×M16—22no—
50C2×C7×C22⋊C4—22no—
51C2×C7×C22⋊C4—22no—
52C2×C7×C22⋊C4C2×C412yes—
53C2×C7×C22⋊C4C2312yes—
54C2×C22⋊Dic7—22no—
55C2×C22⋊Dic7—22no—
56C2×C22⋊Dic7C2×C412yes—
57C2×C22⋊Dic7C2×C412yes—
58C2×Dic7⋊C4—22no—
59C2×Dic7⋊C4—22no—
60C2×Dic7⋊C4—22no—
61C2×Dic7⋊C4—22no—
62C2×Dic7⋊C4C2×C412yes—
63C2×Dic7⋊C4C2×C412yes—
64C2×Dic7⋊C4C2×C412yes—
65C2×Dic7⋊C4C2312yes—
66C2×C4×Dic7—21no—
67C2×C4×Dic7—21no—
68C2×C4×Dic7—21no—
69C2×C4×Dic7—21no—
70C2×C4×Dic7C2×C412yes—
71C2×C4×Dic7C2×C412yes—
72C2×C4×Dic7C2×C411yescharacteristic
73C2×C4×Dic7C2×C411yescharacteristic
74C2×C4×Dic7D412yes—
75C2×C4×Dic7D412yes—
76C2×C4×Dic7D412yes—
77C2×C4×Dic7D412yes—
78C2×C4×Dic7C2311yescharacteristic
79C2×C4×Dic7C2311yescharacteristic
80C2×#1136—22no—
81#6440—24no—
82#6440—24no—
83#6440—24no—
84#6440—24no—
85#6440—24no—
86#6440—24no—
87#6440—24no—
88#6440—24no—
89C7×C22⋊C8—22no—
90C7×C22⋊C8C2314yes—
91C22⋊(C7⋊C8)—22no—
92C22⋊(C7⋊C8)C2314yes—
93#6389—24no—
94Dic7⋊C8—24no—
95Dic7⋊C8—24no—
96C8×Dic7—24no—
97C22×#117—71no—
98#2920—72no—
99#1464—74no—
100#1463—72no—
101#1462—71no—
102#1462D711yescharacteristic
103#1458—74no—
104C24×C7—21no—
105C24×C7—21no—
106C24×C7—24no—
107C24×C7—42no—
108C24×C7—42no—
109C24×C7C22×C411yescharacteristic
110C24×C7C22×C412yes—
111C22×C4×C7—21no—
112C22×C4×C7—21no—
113C22×C4×C7—42no—
114C22×C4×C7—42no—
115C22×C4×C7C22×C412yes—
116C22×C4×C7C22×C411yescharacteristic
117C22×C4×C7C2×D412yes—
118C22×C4×C7C2×D412yes—
119C22×C4×C7C2411yescharacteristic
120C22×Dic7—28no—
121C22×Dic7—28no—
122C22×Dic7—28no—
123C22×Dic7—28no—
124C22×Dic7—21no—
125C22×Dic7—21no—
126C22×Dic7—24no—
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—24no—
132C22×Dic7—24no—
133C22×Dic7—24no—
134C22×Dic7—24no—
135C22×Dic7—21no—
136C22×Dic7—21no—
137C22×Dic7—42no—
138C22×Dic7—42no—
139C22×Dic7C22⋊C411yescharacteristic
140C22×Dic7C22⋊C411yescharacteristic
141C22×Dic7C22⋊C411yescharacteristic
142C22×Dic7C22⋊C411yescharacteristic
143C22×Dic7C22⋊C412yes—
144C22×Dic7C22⋊C412yes—
145C22×Dic7M1618yescomplemented
146C22×Dic7C22×C411yescharacteristic
147C22×Dic7C22×C412yes—
148C22×Dic7C22×C412yes—
149C22×Dic7C22×C411yescharacteristic
150C22×Dic7C22×C411yescharacteristic
151C22×Dic7C22×C411yescharacteristic
152C7×M16—44no—
153C2×C7×C8—24no—
154C7×C22⋊C4—24no—
155C7×C22⋊C4—24no—
156C22⋊Dic7—24no—
157C22⋊Dic7—24no—
158Dic7⋊C4—24no—
159Dic7⋊C4—24no—
160Dic7⋊C4—24no—
161Dic7⋊C4—24no—
162C4×Dic7—22no—
163C4×Dic7—22no—
164C4×Dic7—22no—
165C4×Dic7—22no—
166C4×Dic7C22⋊C412yes—
167C4×Dic7C22⋊C412yes—
168C4×Dic7C22⋊C412yes—
169C4×Dic7C22⋊C412yes—
170C4×Dic7C2×D412yes—
171C4×Dic7C2×D412yes—
172C4×Dic7C2×D412yes—
173C4×Dic7C2×D412yes—
174#1136—44no—
175C2×C7⋊C8—24no—
176C24×C4—71no—
177C2×#117—72no—
178C2×#117—72no—
179C2×#117—72no—
180C2×#117—72no—
181C2×#117—142no—
182C2×#117—142no—
183C2×#117—142no—
184C2×#117—142no—
185C22×C4⋊C4—71no—
186C22×C4⋊C4—71no—
187C22×C22⋊C4—71no—
188C22×C22⋊C4C2×D711yescharacteristic
189C22×C42—71no—
190C22×C42—71no—
191#432—74no—
192#432—142no—
193#431—74no—
194#431—142no—
195#423—142no—
196#407—72no—
197#407C2×D712yes—
198C4×M16—142no—
199#324—78no—
200#323—74no—
201#323C2×D714yes—
202C23×C7—22no—
203C23×C7—22no—
204C23×C7—22no—
205C23×C7—24no—
206C23×C7—24no—
207C23×C7—22no—
208C23×C7—22no—
209C23×C7—42no—
210C23×C7—42no—
211C23×C7—416no—
212C23×C7—42no—
213C23×C7—42no—
214C23×C7—42no—
215C23×C7—42no—
216C23×C7C2×C22⋊C411yescharacteristic
217C23×C7C2×C22⋊C411yescharacteristic
218C23×C7C2×M1614yes—
219C23×C7C23×C411yescharacteristic
220C2×C4×C7—24no—
221C2×C4×C7—22no—
222C2×C4×C7—41no—
223C2×C4×C7—42no—
224C2×C4×C7—41no—
225C2×C4×C7—42no—
226C2×C4×C7—42no—
227C2×C4×C7—42no—
228C2×C4×C7C2×C22⋊C412yes—
229C2×C4×C7C22×D412yes—
230C2×Dic7—28no—
231C2×Dic7—28no—
232C2×Dic7—24no—
233C2×Dic7—24no—
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—22no—
237C2×Dic7—24no—
238C2×Dic7—22no—
239C2×Dic7—22no—
240C2×Dic7—22no—
241C2×Dic7—22no—
242C2×Dic7—28no—
243C2×Dic7—28no—
244C2×Dic7—22no—
245C2×Dic7—42no—
246C2×Dic7—41no—
247C2×Dic7—41no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7C2×C22⋊C411yescharacteristic
252C2×Dic7C2×C22⋊C411yescharacteristic
253C2×Dic7C2×C22⋊C412yes—
254C2×Dic7C2×C22⋊C412yes—
255C2×Dic7C2×C22⋊C411yescharacteristic
256C2×Dic7C2×C22⋊C411yescharacteristic
257C2×Dic7C2×M1614yes—
258C2×Dic7C2×M1614yes—
259C7×C8—44no—
260C7⋊C8—44no—
261C25C4×D711yescharacteristic
262C23×C4—72no—
263C23×C4—71no—
264C23×C4—71no—
265C23×C4—71no—
266C23×C4—72no—
267C23×C4—71no—
268C23×C4—141no—
269C23×C4—141no—
270C23×C4—141no—
271C23×C4—141no—
272C23×C4—148no—
273C23×C4C4×D711yescharacteristic
274C23×C4C22×D711yescharacteristic
275C2×M16—22no—
276C2×M16—142no—
277#117—144no—
278#117—144no—
279#117—144no—
280#117—144no—
281#117—144no—
282#117—144no—
283#117—144no—
284#117—144no—
285C2×C4⋊C4—72no—
286C2×C4⋊C4—72no—
287C2×C4⋊C4—72no—
288C2×C4⋊C4—72no—
289C2×C4⋊C4—142no—
290C2×C4⋊C4—142no—
291C2×C4⋊C4—142no—
292C2×C4⋊C4—142no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—72no—
296C2×C22⋊C4—72no—
297C2×C22⋊C4—142no—
298C2×C22⋊C4—142no—
299C2×C22⋊C4C4×D712yes—
300C2×C22⋊C4C22×D712yes—
301C2×C42—71no—
302C2×C42—71no—
303C2×C42—72no—
304C2×C42—72no—
305C2×C42—72no—
306C2×C42—72no—
307C2×C42—71no—
308C2×C42—71no—
309C2×C42—72no—
310C2×C42—72no—
311C2×C42—141no—
312C2×C42—141no—
313C2×C42—141no—
314C2×C42—141no—
315C4⋊C8—144no—
316C4⋊C8—144no—
317C22⋊C8—22no—
318C22⋊C8—74no—
319C22⋊C8—142no—
320C22⋊C8C22×D714yes—
321#97—144no—
322C4×C8—144no—
323C22×C7—28no—
324C22×C7—21no—
325C22×C7—21no—
326C22×C7—21no—
327C22×C7—21no—
328C22×C7—21no—
329C22×C7—21no—
330C22×C7—24no—
331C22×C7—44no—
332C22×C7—44no—
333C22×C7—48no—
334C22×C7—48no—
335C22×C7—42no—
336C22×C7—42no—
337C22×C7#41012yes—
338C22×C7#41312yes—
339C22×C7C22×C22⋊C411yescharacteristic
340C22×C7C22×M1612yes—
341C4×C7—22no—
342C4×C7—22no—
343C4×C7—42no—
344Dic7—21no—
345Dic7—21no—
346Dic7—24no—
347Dic7—22no—
348Dic7—22no—
349Dic7—42no—
350Dic7#40712yes—
351Dic7#40712yes—
352C24—21no—
353C24—21no—
354C24—24no—
355C24—42no—
356C24—42no—
357C24C2×C4×D711yescharacteristic
358C24C2×C4×D712yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—71no—
364C22×C4—71no—
365C22×C4—71no—
366C22×C4—71no—
367C22×C4—71no—
368C22×C4—72no—
369C22×C4—78no—
370C22×C4—71no—
371C22×C4—72no—
372C22×C4—72no—
373C22×C4—71no—
374C22×C4—71no—
375C22×C4—72no—
376C22×C4—142no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—142no—
380C22×C4—144no—
381C22×C4—144no—
382C22×C4—144no—
383C22×C4—144no—
384C22×C4—148no—
385C22×C4—148no—
386C22×C4—148no—
387C22×C4—148no—
388C22×C4—141no—
389C22×C4—141no—
390C22×C4—144no—
391C22×C4—141no—
392C22×C4—141no—
393C22×C4—282no—
394C22×C4—282no—
395C22×C4C2×C4×D712yes—
396C22×C4C2×C4×D711yescharacteristic
397C22×C4D4×D712yes—
398C22×C4D4×D712yes—
399C22×C4C23×D711yescharacteristic
400M16—44no—
401M16—284no—
402C2×C8—24no—
403C2×C8—144no—
404C4⋊C4—144no—
405C4⋊C4—144no—
406C4⋊C4—144no—
407C4⋊C4—144no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—144no—
411C22⋊C4—144no—
412C42—72no—
413C42—72no—
414C42—72no—
415C42—72no—
416C42—72no—
417C42—72no—
418C42—72no—
419C42—72no—
420C42—142no—
421C42—142no—
422C42—142no—
423C42—142no—
424C2×C7—22no—
425C2×C7—22no—
426C2×C7—22no—
427C2×C7—44no—
428C2×C7C2×#40711yescharacteristic
429C2×C7C2×#41011yescharacteristic
430C2×C7C2×#41311yescharacteristic
431C23—22no—
432C23—22no—
433C23—22no—
434C23—24no—
435C23—24no—
436C23—22no—
437C23—22no—
438C23—42no—
439C23—42no—
440C23—416no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23D7×C22⋊C411yescharacteristic
446C23D7×C22⋊C411yescharacteristic
447C23D7×M1614yes—
448C23C22×C4×D711yescharacteristic
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—71no—
458C2×C4—71no—
459C2×C4—74no—
460C2×C4—71no—
461C2×C4—71no—
462C2×C4—72no—
463C2×C4—72no—
464C2×C4—74no—
465C2×C4—142no—
466C2×C4—142no—
467C2×C4—142no—
468C2×C4—142no—
469C2×C4—148no—
470C2×C4—148no—
471C2×C4—144no—
472C2×C4—144no—
473C2×C4—148no—
474C2×C4—148no—
475C2×C4—142no—
476C2×C4—142no—
477C2×C4—142no—
478C2×C4—144no—
479C2×C4—142no—
480C2×C4—282no—
481C2×C4—281no—
482C2×C4—281no—
483C2×C4—282no—
484C2×C4—282no—
485C2×C4—282no—
486C2×C4D7×C22⋊C412yes—
487C2×C4C2×D4×D712yes—
488C8—44no—
489C8—284no—
490C7#1735711yescharacteristic, complemented, Sylow
491C22—28no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—24no—
499C22—44no—
500C22—44no—
501C22—48no—
502C22—48no—
503C22—42no—
504C22—42no—
505C22#8951212yes—
506C22#8951712yes—
507C22C2×D7×C22⋊C411yescharacteristic
508C22C2×D7×M1612yes—
509C4—22no—
510C4—22no—
511C4—42no—
512C4—72no—
513C4—72no—
514C4—141no—
515C4—141no—
516C4—144no—
517C4—142no—
518C4—142no—
519C4—282no—
520C2—22no—
521C2—22no—
522C2—22no—
523C2—44no—
524C2D7×#40711yescharacteristic
525C2C2×#8951211yescharacteristic
526C2C2×#8951711yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1462C2
socleC22×C7C22×C22⋊C4
prime radicalC25×C7C2×C4
cosocleC22×C4×C7C24
hypercenter#1462D7
hypocenterC7#17357
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #11736064C2semidirect
5. #11727725C2semidirect
6. #11725856C2semidirect
7. #11725854C2semidirect
28. #89491C22semidirect
30. #89490C22semidirect
33. #89264C22semidirect
145. C22×Dic7M16semidirect
490. C7#17357semidirect

Maximal subgroups

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

subgroupindex
C22×#64402
#117360642
C7×#14622
#117292932
#117277252
#117258562
#117258542
#173577