Finite Group Databasebeta

← #13245240

Subgroups — #13245240

Order 1792.

1276 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C7×C22⋊#95C211yesmaximal, characteristic
3#11729880C211yesmaximal, characteristic
4C2×#89247C214yesmaximal, complemented
5C24×C4×C7C2211yescharacteristic
6C23×Dic14C411yescharacteristic, complemented
7C22×C7×C22⋊C4—21no—
8C22×C22⋊Dic7—21no—
9C22×C4⋊Dic7C411yescharacteristic, complemented
10C2×#6487—21no—
11C2×#6487—21no—
12C2×#6487—28no—
13C2×#6487C412yescomplemented
14C2×#6487C2212yes—
15C22×Dic7⋊C4—21no—
16C2×C7×#95—21no—
17C2×C7×#95C2212yes—
18C2×C22⋊(C7⋊C8)—21no—
19C2×C22⋊(C7⋊C8)—21no—
20C2×C22⋊(C7⋊C8)C2212yes—
21#89247—28no—
22#8329—71nomaximal, Sylow
23C25×C7D411yescharacteristic
24C23×C4×C7—24no—
25C23×C4×C7—22no—
26C23×C4×C7—42no—
27C23×C4×C7—42no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7C2311yescharacteristic
34C23×Dic7—21no—
35C23×Dic7—21no—
36C22×Dic14—21no—
37C22×Dic14—24no—
38C22×Dic14—42no—
39C22×Dic14—42no—
40C22×Dic14C2×C412yes—
41C2×C7×C22⋊C4—24no—
42C2×C7×C22⋊C4—42no—
43C2×C7×C22⋊C4—42no—
44C2×C22⋊Dic7—28no—
45C2×C22⋊Dic7—22no—
46C2×C22⋊Dic7—22no—
47C2×C4⋊Dic7—21no—
48C2×C4⋊Dic7—24no—
49C2×C4⋊Dic7C2×C412yes—
50#6487—28no—
51#6487—28no—
52#6487—24no—
53#6487—24no—
54#6487—24no—
55#6487D418yes—
56C2×Dic7⋊C4—28no—
57C2×Dic7⋊C4—22no—
58C2×Dic7⋊C4—22no—
59C22×C7⋊C8—22no—
60C7×#95—24no—
61C7×#95—42no—
62C22⋊(C7⋊C8)—24no—
63C22⋊(C7⋊C8)—42no—
64C22⋊(C7⋊C8)—42no—
65C22×C22⋊Q8—71no—
66#1779—71no—
67C2×#328—74no—
68C22⋊#95D711yescharacteristic
69C24×C7—22no—
70C24×C7—22no—
71C24×C7—22no—
72C24×C7—22no—
73C24×C7—41no—
74C24×C7—41no—
75C24×C7C22⋊C411yescharacteristic
76C24×C7C22⋊C411yescharacteristic
77C24×C7SD1612yes—
78C24×C7Q1612yes—
79C24×C7C2×D411yescharacteristic
80C22×C4×C7—22no—
81C22×C4×C7—22no—
82C22×C4×C7—22no—
83C22×C4×C7—22no—
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—22no—
90C22×C4×C7—24no—
91C22×C4×C7—41no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7—41no—
97C22×C4×C7—41no—
98C22×C4×C7—416no—
99C22×C4×C7—42no—
100C22×C4×C7—42no—
101C22×C4×C7—42no—
102C22×C4×C7—42no—
103C22×C4×C7—42no—
104C22×C4×C7—42no—
105C22×C4×C7C22⋊C411yescharacteristic
106C22×C4×C7C22⋊C411yescharacteristic
107C22×C4×C7C22×C411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D414yes—
110C22×Dic7—24no—
111C22×Dic7—24no—
112C22×Dic7—22no—
113C22×Dic7—22no—
114C22×Dic7—42no—
115C22×Dic7—42no—
116C22×Dic7—42no—
117C22×Dic7—42no—
118C2×Dic14—24no—
119C2×Dic14—22no—
120C2×Dic14—42no—
121C2×Dic14—416no—
122C2×Dic14—42no—
123C2×Dic14—42no—
124C2×Dic14—42no—
125C2×Dic14C22⋊C414yescomplemented
126C7×C22⋊C4—48no—
127C22⋊Dic7—28no—
128C22⋊Dic7—28no—
129C4⋊Dic7—24no—
130C4⋊Dic7—22no—
131C4⋊Dic7C22⋊C414yescomplemented
132Dic7⋊C4—28no—
133Dic7⋊C4—28no—
134C2×C7⋊C8—42no—
135C2×C7⋊C8—42no—
136C2×C7⋊C8—42no—
137C23×Q8—71no—
138C24×C4C2×D711yescharacteristic
139C2×C22⋊Q8—72no—
140C2×C22⋊Q8—72no—
141C2×C22⋊Q8—148no—
142C2×C22⋊Q8—141no—
143C2×C22⋊Q8—141no—
144C22×C4⋊C4—71no—
145C22×C4⋊C4—141no—
146C22×C22⋊C4—21no—
147C22×C22⋊C4—141no—
148C2×C22⋊C8—72no—
149C2×C22⋊C8—141no—
150C2×C22⋊C8—141no—
151C2×#95—21no—
152C2×#95C2×D712yes—
153#328—148no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—21no—
162C23×C7—24no—
163C23×C7—24no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—21no—
168C23×C7—21no—
169C23×C7—21no—
170C23×C7—24no—
171C23×C7—24no—
172C23×C7—42no—
173C23×C7—42no—
174C23×C7—44no—
175C23×C7—44no—
176C23×C7—44no—
177C23×C7—44no—
178C23×C7#9912yes—
179C23×C7#10312yes—
180C23×C7#10312yes—
181C23×C7#10412yes—
182C23×C7C2×C22⋊C411yescharacteristic
183C23×C7#12012yes—
184C23×C7C2×SD1611yescharacteristic
185C23×C7C2×Q1611yescharacteristic
186C23×C7#13712yes—
187C2×C4×C7—28no—
188C2×C4×C7—21no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—44no—
198C2×C4×C7—44no—
199C2×C4×C7—44no—
200C2×C4×C7—44no—
201C2×C4×C7—44no—
202C2×C4×C7—44no—
203C2×C4×C7—44no—
204C2×C4×C7—44no—
205C2×C4×C7—82no—
206C2×C4×C7—82no—
207C2×C4×C7—82no—
208C2×C4×C7—82no—
209C2×C4×C7#9912yes—
210C2×C4×C7C2×C22⋊C412yes—
211C2×C4×C7#12012yes—
212C2×Dic7—24no—
213C2×Dic7—24no—
214C2×Dic7—48no—
215C2×Dic7—48no—
216C2×Dic7—42no—
217C2×Dic7—42no—
218C2×Dic7—42no—
219C2×Dic7—42no—
220Dic14—48no—
221Dic14—44no—
222Dic14—44no—
223C7⋊C8—82no—
224C25C22⋊D711yescharacteristic
225C22×Q8—72no—
226C22×Q8—144no—
227C22×Q8—141no—
228C22×Q8—282no—
229C22×Q8—282no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—141no—
235C23×C4—141no—
236C23×C4C4×D711yescharacteristic
237C23×C4C4×D711yescharacteristic
238C23×C4D2811yescharacteristic
239C23×C4D2811yescharacteristic
240C23×C4C22⋊D711yescharacteristic
241C23×C4C22×D711yescharacteristic
242C22×C8—142no—
243C22⋊Q8—78no—
244C22⋊Q8—144no—
245C22⋊Q8—144no—
246C22⋊Q8—148no—
247C22⋊Q8—148no—
248C22⋊Q8—144no—
249C2×C4⋊C4—72no—
250C2×C4⋊C4—144no—
251C2×C4⋊C4—148no—
252C2×C4⋊C4—141no—
253C2×C4⋊C4—142no—
254C2×C4⋊C4—142no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—148no—
259C2×C22⋊C4—142no—
260C2×C22⋊C4—142no—
261C22⋊C8—144no—
262C22⋊C8—282no—
263C22⋊C8—282no—
264#95—24no—
265#95—42no—
266C22×C7—21no—
267C22×C7—21no—
268C22×C7—22no—
269C22×C7—22no—
270C22×C7—22no—
271C22×C7—22no—
272C22×C7—21no—
273C22×C7—22no—
274C22×C7—21no—
275C22×C7—22no—
276C22×C7—22no—
277C22×C7—22no—
278C22×C7—22no—
279C22×C7—22no—
280C22×C7—22no—
281C22×C7—22no—
282C22×C7—22no—
283C22×C7—48no—
284C22×C7—48no—
285C22×C7—41no—
286C22×C7—41no—
287C22×C7—41no—
288C22×C7—41no—
289C22×C7#32814yescomplemented
290C22×C7#37911yescharacteristic
291C22×C7C2×#9911yescharacteristic
292C22×C7C2×#9911yescharacteristic
293C22×C7C2×#10311yescharacteristic
294C22×C7#41911yescharacteristic
295C22×C7C2×#10411yescharacteristic
296C22×C7#42111yescharacteristic
297C22×C7#44812yes—
298C22×C7#45112yes—
299C4×C7—22no—
300C4×C7—22no—
301C4×C7—22no—
302C4×C7—41no—
303C4×C7—84no—
304Dic7—44no—
305Dic7—44no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—41no—
311C24—41no—
312C24(C2×D7)⋊C411yescharacteristic
313C24(C2×D7)⋊C411yescharacteristic
314C24#114212yes—
315C24#114412yes—
316C24C2×C22⋊D711yescharacteristic
317C2×Q8—74no—
318C2×Q8—144no—
319C2×Q8—142no—
320C2×Q8—2816no—
321C2×Q8—282no—
322C2×Q8—282no—
323C2×Q8—282no—
324C2×Q8—282no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—416no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—144no—
351C22×C4—144no—
352C22×C4—142no—
353C22×C4—142no—
354C22×C4—282no—
355C22×C4—282no—
356C22×C4—282no—
357C22×C4—282no—
358C22×C4(C2×D7)⋊C411yescharacteristic
359C22×C4(C2×D7)⋊C411yescharacteristic
360C22×C4C2×C4×D711yescharacteristic
361C22×C4C2×D2811yescharacteristic
362C22×C4D4×D714yes—
363C2×C8—282no—
364C2×C8—282no—
365C2×C8—282no—
366C4⋊C4—74no—
367C4⋊C4—148no—
368C4⋊C4—144no—
369C4⋊C4—142no—
370C4⋊C4—148no—
371C22⋊C4—48no—
372C22⋊C4—148no—
373C22⋊C4—148no—
374C2×C7—22no—
375C2×C7—24no—
376C2×C7—22no—
377C2×C7—42no—
378C2×C7C2×#32811yescharacteristic
379C2×C7#150811yescharacteristic
380C2×C7#151111yescharacteristic
381C2×C7#152011yescharacteristic
382C2×C7#178111yescharacteristic
383C2×C7#178711yescharacteristic
384C2×C7#178911yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—24no—
398C23—24no—
399C23—21no—
400C23—21no—
401C23—24no—
402C23—24no—
403C23—42no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23#637912yes—
410C23#638012yes—
411C23#638412yes—
412C23#638412yes—
413C23C2×(C2×D7)⋊C411yescharacteristic
414C23C2×#114211yescharacteristic
415C23#650012yes—
416C23C2×#114411yescharacteristic
417C23#651412yes—
418Q8—284no—
419Q8—284no—
420Q8—288no—
421C2×C4—28no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—144no—
444C2×C4—144no—
445C2×C4—288no—
446C2×C4—288no—
447C2×C4—282no—
448C2×C4—282no—
449C2×C4—282no—
450C2×C4—282no—
451C2×C4#638012yes—
452C2×C4D7×C22⋊C412yes—
453C2×C4#644512yes—
454C8—562no—
455C7#832911yescharacteristic, complemented, Sylow
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—48no—
474C22—48no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22#8924714yescomplemented
480C22C2×#637911yescharacteristic
481C22C2×#638011yescharacteristic
482C22C2×#638011yescharacteristic
483C22#8972711yescharacteristic
484C22C2×#638411yescharacteristic
485C22#8977611yescharacteristic
486C22#8977711yescharacteristic
487C22#8981212yes—
488C22#8982212yes—
489C4—22no—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—84no—
494C4—284no—
495C4—284no—
496C2—22no—
497C2—24no—
498C2—22no—
499C2—42no—
500C2C2×#8924711yescharacteristic
501C2#1172719911yescharacteristic
502C2#1172720811yescharacteristic
503C2#1172725211yescharacteristic
504C2#1172726411yescharacteristic
505C2#1172727911yescharacteristic
506C2#1172728711yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×C22⋊#95C2
socleC23×C7C2×C22⋊C4
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenterC22⋊#95D7
hypocenterC7#8329
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. C2×#89247C2semidirect
6. C23×Dic14C4semidirect
9. C22×C4⋊Dic7C4semidirect
13. C2×#6487C4semidirect
125. C2×Dic14C22⋊C4semidirect
131. C4⋊Dic7C22⋊C4semidirect
289. C22×C7#328semidirect
455. C7#8329semidirect
479. C22#89247semidirect

Maximal subgroups

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

subgroupindex
C22×#64872
C7×C22⋊#952
#117298802
C2×#892472
#83297