Finite Group Databasebeta

← #13355166

Subgroups — #13355166

Order 1792.

1802 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6472C211yesmaximal, characteristic
2C22×#6450C211yesmaximal, characteristic, complemented
3#11739673C211yesmaximal, characteristic
4C2×#89776C211yesmaximal, characteristic, complemented
5C2×#89738C211yesmaximal, characteristic
6C7×#1736C211yesmaximal, characteristic, complemented
7#11729733C211yesmaximal, characteristic, complemented
8#11726663C214yesmaximal, complemented
9#11726662C214yesmaximal, complemented
10C22×#1157C2211yescharacteristic
11C23×Dic14C2211yescharacteristic
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×Dic7⋊C4—21no—
16C22×C4×Dic7—21no—
17C22×C4×Dic7C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6472—24no—
20C2×#6472—24no—
21C2×#6472C418yescomplemented
22C22×Dic28C2212yes—
23C22×#1132C2212yes—
24C2×#6450—24no—
25C2×#6450—24no—
26C2×#6450—22no—
27C2×#6450C2214yes—
28#89776—24no—
29#89760C2212yes—
30#89746C2212yes—
31C2×#6384C2214yes—
32C2×#6384C2212yes—
33#89738—24no—
34C2×#6383C2212yes—
35#89730—21no—
36#89730—21no—
37C4×(C2×D7)⋊C4—22no—
38C7×#340C2214yes—
39#89345C2214yes—
40#20926—71nomaximal, Sylow
41C23×C4×C7—21no—
42C23×C4×C7C2311yescharacteristic
43C22×C22⋊D7—21no—
44C23×Dic7—21no—
45C23×Dic7D411yescharacteristic
46C23×Dic7D411yescharacteristic
47C2×#1157—22no—
48C2×#1157—22no—
49C2×#1157—22no—
50C2×#1157—22no—
51C2×#1157—22no—
52C2×#1157C2×C414yes—
53C22×D28C2311yescharacteristic
54C22×C4×D7—21no—
55C22×Dic14—21no—
56C22×Dic14—22no—
57C22×Dic14—22no—
58C22×Dic14—22no—
59C22×Dic14C2×C414yes—
60C22×Dic14D414yes—
61C22×Dic14D414yes—
62C22×Dic14C2311yescharacteristic
63C22×Dic14C2312yes—
64C2×C7×M16—22no—
65C2×C7×M16—22no—
66C2×C7×M16C2×C414yes—
67C22×C7×C8C2312yes—
68C7×#117—22no—
69C7×#117—22no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4D414yes—
73C2×C7×C4⋊C4C2312yes—
74C2×C7×C4⋊C4C2312yes—
75C2×C7×C22⋊C4—21no—
76C2×C42×C7—21no—
77C2×(C2×D7)⋊C4—22no—
78C2×(C2×D7)⋊C4—22no—
79C2×Dic7⋊C4—28no—
80C2×Dic7⋊C4—24no—
81C2×C4×Dic7—28no—
82C2×C4×Dic7—22no—
83C2×C4×Dic7—22no—
84C2×C4×Dic7—24no—
85C2×C4×Dic7D412yes—
86C2×C4×Dic7C2312yes—
87C2×#1136—22no—
88C2×#1136—22no—
89C2×#1136—22no—
90C22×C7⋊C8C2312yes—
91#6472—232no—
92C2×Dic28—24no—
93C2×Dic28—24no—
94C2×Dic28C2×C418yescomplemented
95C2×#1132—24no—
96C2×#1132—24no—
97C2×#1132C2×C418yescomplemented
98#6450—216no—
99#6450—28no—
100#6450—28no—
101C7⋊#95—22no—
102C7⋊#95—22no—
103#6384—28no—
104#6384—24no—
105#6383—24no—
106C22×#137—71no—
107C22×C4×Q8—71no—
108C2×#417—71no—
109C2×#419—71no—
110#2297—71no—
111#1880—74no—
112#1878—74no—
113#1736D711yescharacteristic, complemented
114#1728—71no—
115C24×C7C2×D411yescharacteristic
116C23×D7—21no—
117C22×C4×C7—24no—
118C22×C4×C7—22no—
119C22×C4×C7—22no—
120C22×C4×C7—44no—
121C22×C4×C7C22×C412yes—
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D412yes—
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D412yes—
126C22×C4×C7#4112yes—
127C22×C4×C7#4112yes—
128C22×C4×C7C2411yescharacteristic
129C2×C22⋊D7—24no—
130C2×C22⋊D7—42no—
131C2×C22⋊D7—42no—
132C22×Dic7—24no—
133C22×Dic7—22no—
134C22×Dic7—24no—
135C22×Dic7—24no—
136C22×Dic7—22no—
137C22×Dic7—24no—
138C22×Dic7—22no—
139C22×Dic7—21no—
140C22×Dic7—42no—
141C22×Dic7C2×D412yes—
142C22×Dic7C2×D412yes—
143#1157—28no—
144#1157—28no—
145#1157—44no—
146#1157—44no—
147C2×D28—21no—
148C2×D28—21no—
149C2×D28—21no—
150C2×D28—21no—
151C2×D28—21no—
152C2×D28C22×C412yes—
153C2×C4×D7—24no—
154C2×C4×D7—42no—
155C2×C4×D7—42no—
156C2×Dic14—28no—
157C2×Dic14—24no—
158C2×Dic14—24no—
159C2×Dic14—28no—
160C2×Dic14—24no—
161C2×Dic14—28no—
162C2×Dic14—22no—
163C2×Dic14—22no—
164C2×Dic14—22no—
165C2×Dic14—21no—
166C2×Dic14—21no—
167C2×Dic14—21no—
168C2×Dic14—21no—
169C2×Dic14—21no—
170C2×Dic14C22⋊C4116yescomplemented
171C2×Dic14C22×C412yes—
172C2×Dic14C22×C414yes—
173C2×Dic14C2×D414yes—
174C2×Dic14C2×D414yes—
175C7×M16—28no—
176C2×C7×C8—22no—
177C2×C7×C8—22no—
178C2×C7×C8C22×C414yes—
179C7×C4⋊C4—24no—
180C7×C4⋊C4—22no—
181C7×C4⋊C4—22no—
182C7×C4⋊C4—22no—
183C7×C4⋊C4C2×D414yes—
184C7×C22⋊C4—42no—
185C42×C7—42no—
186(C2×D7)⋊C4—44no—
187(C2×D7)⋊C4—44no—
188Dic7⋊C4—216no—
189C4×Dic7—216no—
190C4×Dic7—24no—
191C4×Dic7—22no—
192C4×Dic7—22no—
193#1136—44no—
194C2×C7⋊C8—22no—
195C2×C7⋊C8—22no—
196C2×C7⋊C8—22no—
197Dic28—216no—
198#1132—216no—
199C22×#41—71no—
200C23×Q8—71no—
201C2×#137—78no—
202C2×#137—144no—
203C2×#137—144no—
204C22×Q16—72no—
205C22×SD16—72no—
206C22×M16—71no—
207C22×M16C2×D711yescharacteristic
208C2×C4×Q8—74no—
209C2×C4×Q8—142no—
210C2×C4×Q8—144no—
211C2×C4×Q8—144no—
212C2×#117C2×D711yescharacteristic
213C22×C4⋊C4—141no—
214C22×C4⋊C4C2×D711yescharacteristic
215C22×C42—71no—
216C22×C42—141no—
217#419—144no—
218#417—144no—
219C2×#103—74no—
220C2×#103—72no—
221C2×#102—72no—
222#391—72no—
223#390—72no—
224#386—141no—
225#386—141no—
226C4×C22⋊C4—142no—
227#340C2×D714yes—
228#336—74no—
229C23×C7—22no—
230C23×C7—22no—
231C23×C7C4×D412yes—
232C23×C7#12012yes—
233C23×C7#12112yes—
234C23×C7C22×D411yescharacteristic
235C22×D7—22no—
236C22×D7—41no—
237C22×D7—41no—
238C22×D7—42no—
239C22×D7—42no—
240C2×C4×C7—24no—
241C2×C4×C7—48no—
242C2×C4×C7—44no—
243C2×C4×C7—42no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7C2×C22⋊C414yes—
247C2×C4×C7C4×D414yes—
248C2×C4×C7C4×D412yes—
249C2×C4×C7#12012yes—
250C2×C4×C7#12112yes—
251C2×C4×C7#12114yes—
252C2×C4×C7#12314yes—
253C2×C4×C7C23×C411yescharacteristic
254C2×C4×C7C22×D411yescharacteristic
255C2×C4×C7C22×D411yescharacteristic
256C2×C4×C7C22×D411yescharacteristic
257C2×C4×C7C2×#4111yescharacteristic
258C2×C4×C7C2×#4111yescharacteristic
259C22⋊D7—48no—
260C2×Dic7—28no—
261C2×Dic7—24no—
262C2×Dic7—22no—
263C2×Dic7—24no—
264C2×Dic7—24no—
265C2×Dic7—22no—
266C2×Dic7—216no—
267C2×Dic7—22no—
268C2×Dic7—22no—
269C2×Dic7—22no—
270C2×Dic7—22no—
271C2×Dic7—44no—
272C2×Dic7—41no—
273C2×Dic7—41no—
274C2×Dic7#13714yescomplemented
275D28—22no—
276D28—22no—
277D28—41no—
278D28—41no—
279C4×D7—48no—
280Dic14—24no—
281Dic14—24no—
282Dic14—24no—
283Dic14—216no—
284Dic14—22no—
285Dic14—22no—
286Dic14—41no—
287Dic14—41no—
288Dic14C2×C22⋊C418yes—
289C7×C8—24no—
290C7⋊C8—42no—
291C2×#41—74no—
292C2×#41—142no—
293C2×#41—142no—
294C2×#41—142no—
295C2×#41—142no—
296C2×#41—142no—
297C22×Q8—74no—
298C22×Q8—74no—
299C22×Q8—74no—
300C22×Q8—72no—
301C22×Q8—71no—
302C22×Q8—142no—
303C22×Q8—142no—
304C22×Q8—141no—
305C22×Q8—142no—
306C22×D4—71no—
307C22×D4—141no—
308C23×C4—21no—
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—141no—
312C23×C4—141no—
313C23×C4C22×D711yescharacteristic
314#137—1432no—
315C2×Q16—78no—
316C2×Q16—144no—
317C2×Q16—144no—
318C2×SD16—78no—
319C2×SD16—144no—
320C2×SD16—144no—
321C2×M16—22no—
322C2×M16—22no—
323C2×M16—142no—
324C2×M16—142no—
325C2×M16—142no—
326C2×M16C4×D714yes—
327C22×C8—72no—
328C22×C8C22×D712yes—
329C4×Q8—148no—
330C4×Q8—1416no—
331C4×Q8—148no—
332#117—22no—
333#117—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—144no—
337C2×C4⋊C4—148no—
338C2×C4⋊C4D2814yes—
339C2×C4⋊C4C22×D712yes—
340C2×C4⋊C4C22×D712yes—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—142no—
343C2×C22⋊C4—142no—
344C2×C42—21no—
345C2×C42—72no—
346C2×C42—72no—
347C2×C42—144no—
348C2×C42—148no—
349C2×C42—142no—
350C2×C42—142no—
351#103—148no—
352#103—144no—
353#102—144no—
354#95—142no—
355#95—142no—
356C22×C7—24no—
357C22×C7—22no—
358C22×C7—22no—
359C22×C7—24no—
360C22×C7#38614yes—
361C22×C7C2×C4×D411yescharacteristic
362C22×C7C2×#12011yescharacteristic
363C22×C7C2×#12111yescharacteristic
364C22×C7C2×#13712yes—
365C22×C7#57512yes—
366C2×D7—44no—
367C2×D7—42no—
368C2×D7—81no—
369C2×D7—81no—
370C2×D7—82no—
371C4×C7—44no—
372C4×C7—82no—
373C4×C7#38614yes—
374C4×C7C22×C22⋊C411yescharacteristic
375C4×C7C2×C4×D411yescharacteristic
376C4×C7C2×#12111yescharacteristic
377C4×C7C2×#12311yescharacteristic
378Dic7—24no—
379Dic7—28no—
380Dic7—42no—
381Dic7—42no—
382C24—141no—
383C24D4×D711yescharacteristic
384#41—148no—
385#41—148no—
386#41—284no—
387#41—284no—
388C2×Q8—716no—
389C2×Q8—74no—
390C2×Q8—72no—
391C2×Q8—74no—
392C2×Q8—74no—
393C2×Q8—148no—
394C2×Q8—144no—
395C2×Q8—144no—
396C2×Q8—144no—
397C2×Q8—148no—
398C2×Q8—148no—
399C2×Q8—142no—
400C2×Q8—142no—
401C2×Q8—142no—
402C2×Q8—141no—
403C2×Q8—141no—
404C2×Q8—141no—
405C2×Q8—141no—
406C2×Q8—141no—
407C2×D4—72no—
408C2×D4—144no—
409C2×D4—141no—
410C2×D4—141no—
411C2×D4—141no—
412C2×D4—141no—
413C2×D4—141no—
414C2×D4—282no—
415C2×D4—282no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—44no—
420C22×C4—72no—
421C22×C4—72no—
422C22×C4—144no—
423C22×C4—142no—
424C22×C4—144no—
425C22×C4—144no—
426C22×C4—144no—
427C22×C4—142no—
428C22×C4—142no—
429C22×C4—141no—
430C22×C4—144no—
431C22×C4—282no—
432C22×C4—282no—
433C22×C4—282no—
434C22×C4C2×C4×D712yes—
435C22×C4C2×D2812yes—
436C22×C4#115712yes—
437C22×C4D4×D712yes—
438C22×C4D4×D712yes—
439C22×C4D4×D711yescharacteristic
440C22×C4#116112yes—
441C22×C4C23×D711yescharacteristic
442Q16—1416no—
443SD16—1416no—
444M16—28no—
445M16—284no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—142no—
449C2×C8—142no—
450C2×C8—142no—
451C2×C8C2×C4×D714yes—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—22no—
456C4⋊C4—1416no—
457C4⋊C4C2×D2814yes—
458C22⋊C4—42no—
459C22⋊C4—284no—
460C22⋊C4—284no—
461C42—42no—
462C42—1416no—
463C42—144no—
464C42—142no—
465C42—142no—
466C2×C7—24no—
467C2×C7C2×#38611yescharacteristic
468C2×C7C4×#13712yes—
469C2×C7#300411yescharacteristic
470C2×C7#300511yescharacteristic
471C2×C7#303712yes—
472D7—82no—
473C23—22no—
474C23—22no—
475C23—142no—
476C23—281no—
477C23—281no—
478C23—282no—
479C23—282no—
480C23#644512yes—
481C23#644712yes—
482C23#645612yes—
483C23C2×D4×D711yescharacteristic
484Q8—78no—
485Q8—144no—
486Q8—144no—
487Q8—144no—
488Q8—1416no—
489Q8—142no—
490Q8—142no—
491Q8—281no—
492Q8—281no—
493D4—142no—
494D4—142no—
495D4—288no—
496D4—281no—
497D4—281no—
498C2×C4—24no—
499C2×C4—48no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—74no—
505C2×C4—148no—
506C2×C4—142no—
507C2×C4—144no—
508C2×C4—144no—
509C2×C4—144no—
510C2×C4—1416no—
511C2×C4—142no—
512C2×C4—142no—
513C2×C4—142no—
514C2×C4—142no—
515C2×C4—142no—
516C2×C4—284no—
517C2×C4—281no—
518C2×C4—281no—
519C2×C4—288no—
520C2×C4C4×D2814yes—
521C2×C4D7×C22⋊C414yes—
522C2×C4#644512yes—
523C2×C4#644712yes—
524C2×C4#645612yes—
525C2×C4#645714yes—
526C2×C4#645814yes—
527C2×C4C22×C4×D711yescharacteristic
528C2×C4C22×D2811yescharacteristic
529C2×C4C2×#115711yescharacteristic
530C2×C4C2×D4×D711yescharacteristic
531C2×C4C2×D4×D711yescharacteristic
532C2×C4C2×#116111yescharacteristic
533C8—24no—
534C8—282no—
535C7#2092611yescharacteristic, complemented, Sylow
536C22—24no—
537C22—22no—
538C22—24no—
539C22—22no—
540C22—284no—
541C22—282no—
542C22—562no—
543C22—561no—
544C22—561no—
545C22#8943714yes—
546C22C2×#644511yescharacteristic
547C22C2×#644711yescharacteristic
548C22C2×#645611yescharacteristic
549C22#9046412yes—
550C22D7×#13712yes—
551C4—44no—
552C4—82no—
553C4—144no—
554C4—148no—
555C4—282no—
556C4—282no—
557C4#8943714yes—
558C4C2×C4×D2811yescharacteristic
559C4C2×D7×C22⋊C411yescharacteristic
560C4C2×#645711yescharacteristic
561C4C2×#645811yescharacteristic
562C2—24no—
563C2—562no—
564C2C2×#8943711yescharacteristic
565C2#1173635812yes—
566C2#1173637211yescharacteristic
567C2#1173640311yescharacteristic
568C2#1173664512yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1736C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#1736D7
hypocenterC7#20926
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1736

Complements

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

NH ≅ G/Ntype
2. C22×#6450C2semidirect
4. C2×#89776C2semidirect
6. C7×#1736C2semidirect
7. #11729733C2semidirect
8. #11726663C2semidirect
9. #11726662C2semidirect
21. C2×#6472C4semidirect
94. C2×Dic28C2×C4semidirect
97. C2×#1132C2×C4semidirect
113. #1736D7semidirect
170. C2×Dic14C22⋊C4semidirect
274. C2×Dic7#137semidirect
535. C7#20926semidirect

Maximal subgroups

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

subgroupindex
C22×#64722
C22×#64502
#117396732
C2×#897762
C2×#897382
C7×#17362
#117297332
#117266632
#117266622
#209267