Finite Group Databasebeta

← C2×#11740974 #14228938

Subgroups — C2×#11740974 #14228938

Order 1792.

2742 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#547C211yesmaximal, characteristic
2C22×#6514C212yesmaximal, complemented
3C2×D4×Dic14C211yesmaximal, characteristic, complemented
4C2×#90011C212yesmaximal, complemented
5#11740974C2116yesmaximal, direct product factor
6C2×#89862C211yesmaximal, characteristic, complemented
7C2×#89864C211yesmaximal, characteristic, complemented
8C2×#89822C212yesmaximal, complemented
9C2×#89824C212yesmaximal, complemented
10C2×#89791C211yesmaximal, characteristic, complemented
11C2×C4×#1142C211yesmaximal, characteristic, complemented
12C2×#89784C211yesmaximal, characteristic, complemented
13C22×C7×#41C2212yes—
14C23×Dic14C2212yes—
15C7×#547—24no—
16C7×#547C2218yes—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C7×#123—22no—
19C2×C7×C22⋊Q8C2212yes—
20C2×C7×#121—21no—
21C2×C4×C7×D4C2211yescharacteristic
22C2×C4×C7×D4C2211yescharacteristic
23C22×C7×C4⋊C4—21no—
24C2×#6514—216no—
25C2×#6514C22116yescomplemented
26C22×#1144C2212yes—
27C22×#1144C2212yes—
28C22×#1144C2212yes—
29C2×D4×Dic7—21no—
30C22×#1142C2212yes—
31C22×#1142C2211yescharacteristic
32C2×#6487C2212yes—
33C22×#1136C2212yes—
34D4×Dic14—24no—
35D4×Dic14C2218yescomplemented
36C2×#6451—21no—
37C2×#6441—22no—
38C2×#6432C2211yescharacteristic
39C2×C4×Dic14C2211yescharacteristic
40#90011C22116yescomplemented
41C2×#6409C2212yes—
42C2×#6406C2211yescharacteristic
43C2×C22⋊(C7⋊C8)C2212yes—
44#89864C2218yescomplemented
45#89862C2218yescomplemented
46#89824C22116yescomplemented
47#89822C22116yescomplemented
48#89791C2218yescomplemented
49C4×#1142C2218yescomplemented
50#89784C2218yescomplemented
51C2×#6384C2212yes—
52C2×#6384C2211yescharacteristic
53C2×#6382C2211yescharacteristic
54C2×C4⋊(C7⋊C8)C2211yescharacteristic
55C2×C4×C7⋊C8C2211yescharacteristic
56C2×#3276—71nomaximal, Sylow
57C2×C7×#41—24no—
58C2×C7×#41—28no—
59C2×C7×#41—44no—
60C2×C7×#41C2318yes—
61C22×C7×Q8C2312yes—
62C22×C7×D4—22no—
63C22×C7×D4D411yescharacteristic
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—22no—
66C23×C4×C7—21no—
67C23×C4×C7C2312yes—
68C23×Dic7—22no—
69C22×Dic14—28no—
70C22×Dic14—22no—
71C22×Dic14—24no—
72C22×Dic14—44no—
73C22×Dic14C2312yes—
74C22×Dic14C2318yes—
75C22×Dic14C2311yescharacteristic
76C7×C4⋊Q8—22no—
77C7×C4⋊Q8C2314yes—
78C7×#123—216no—
79C7×C22⋊Q8—24no—
80C7×C22⋊Q8C2318yes—
81C7×#121—28no—
82C4×C7×D4—22no—
83C4×C7×D4D414yes—
84C4×C7×D4C2314yes—
85C4×C7×D4C2314yes—
86C2×C7×C4⋊C4—21no—
87C2×C7×C4⋊C4—22no—
88C2×C7×C4⋊C4—21no—
89C2×C7×C4⋊C4—28no—
90C2×C7×C4⋊C4D411yescharacteristic
91C2×C7×C4⋊C4C2311yescharacteristic
92C2×C7×C22⋊C4—22no—
93C2×C7×C22⋊C4—21no—
94C2×C7×C22⋊C4D412yes—
95C2×C42×C7C2311yescharacteristic
96C2×C22⋊Dic7—22no—
97#6514—264no—
98C2×#1144—28no—
99C2×#1144—28no—
100C2×#1144—44no—
101C2×#1144C2318yes—
102C2×#1144C2318yes—
103C2×#1144C2318yes—
104D4×Dic7—28no—
105C2×#1142—28no—
106C2×#1142D418yescomplemented
107C2×#1142C2318yes—
108C2×#1142C2314yes—
109C2×C4⋊Dic7—21no—
110C2×C4⋊Dic7—21no—
111C2×C4⋊Dic7C2311yescharacteristic
112#6487—24no—
113#6487C2318yes—
114C2×Dic7⋊C4—21no—
115C2×Dic7⋊C4—22no—
116C2×C4×Dic7—21no—
117C2×#1136—28no—
118C2×#1136C2318yes—
119C22×C7⋊C8C2311yescharacteristic
120C22×C7⋊C8C2312yes—
121C22×C7⋊C8C2311yescharacteristic
122#6451—28no—
123#6441—216no—
124#6432—22no—
125#6432C2314yes—
126C4×Dic14—22no—
127C4×Dic14C2314yes—
128#6409C2318yes—
129#6406C2314yes—
130C22⋊(C7⋊C8)C2318yes—
131#6384C2314yes—
132#6384C2318yes—
133#6382C2314yes—
134C4⋊(C7⋊C8)C2314yes—
135C4×C7⋊C8C2314yes—
136C22×#137—72no—
137C2×D4×Q8—71no—
138C2×#547D711yescharacteristic
139#3276—716no—
140C2×#494—71no—
141C2×#470—72no—
142C2×#462—71no—
143C2×#461—71no—
144C2×#452—72no—
145C2×#451—72no—
146C2×C4×SD16—71no—
147C2×#436—71no—
148C24×C7—21no—
149C24×C7C2×D412yes—
150C7×#41—216no—
151C7×#41—48no—
152C7×#41—416no—
153C2×C7×Q8—22no—
154C2×C7×Q8—24no—
155C2×C7×Q8—24no—
156C2×C7×Q8C2414yes—
157C2×C7×D4—28no—
158C2×C7×D4—24no—
159C2×C7×D4—21no—
160C2×C7×D4—22no—
161C2×C7×D4—48no—
162C2×C7×D4C2×D414yes—
163C2×C7×D4C2×D414yes—
164C2×C7×D4C2412yes—
165C22×C4×C7—24no—
166C22×C4×C7—24no—
167C22×C4×C7—21no—
168C22×C4×C7—22no—
169C22×C4×C7—24no—
170C22×C4×C7—28no—
171C22×C4×C7—48no—
172C22×C4×C7—44no—
173C22×C4×C7C2×D411yescharacteristic
174C22×C4×C7C2×D412yes—
175C22×C4×C7C2×D414yes—
176C22×C4×C7C2×D411yescharacteristic
177C22×C4×C7C2414yes—
178C22×C4×C7C2411yescharacteristic
179C22×Dic7—21no—
180C22×Dic7—28no—
181C22×Dic7—21no—
182C22×Dic7—22no—
183C22×Dic7—48no—
184C2×Dic14—21no—
185C2×Dic14—22no—
186C2×Dic14—24no—
187C2×Dic14—24no—
188C2×Dic14—216no—
189C2×Dic14—22no—
190C2×Dic14—28no—
191C2×Dic14—48no—
192C2×Dic14—416no—
193C2×Dic14—48no—
194C2×Dic14C2×D414yes—
195C2×Dic14C2414yes—
196C2×Dic14C2412yes—
197C7×C4⋊C4—24no—
198C7×C4⋊C4—28no—
199C7×C4⋊C4—24no—
200C7×C4⋊C4—21no—
201C7×C4⋊C4C2×D414yes—
202C7×C4⋊C4C2412yes—
203C7×C22⋊C4—28no—
204C7×C22⋊C4—24no—
205C7×C22⋊C4C2×D418yes—
206C42×C7C2×D412yes—
207C42×C7C2412yes—
208C22⋊Dic7—28no—
209#1144—216no—
210#1144—216no—
211#1144—48no—
212#1142—216no—
213#1142C2×D4116yescomplemented
214C4⋊Dic7—21no—
215C4⋊Dic7—24no—
216C4⋊Dic7—24no—
217C4⋊Dic7C2412yes—
218Dic7⋊C4—24no—
219Dic7⋊C4—28no—
220C4×Dic7—24no—
221#1136—216no—
222C2×C7⋊C8—24no—
223C2×C7⋊C8—22no—
224C2×C7⋊C8C2×D414yes—
225C2×C7⋊C8C2412yes—
226C2×C7⋊C8C2414yes—
227C2×C7⋊C8C2412yes—
228C22×#41C2×D712yes—
229C23×Q8—72no—
230C2×#137—716no—
231C2×#137—1416no—
232C22×Q16—72no—
233C22×Q16—72no—
234C22×Q16—72no—
235C22×SD16—71no—
236C22×SD16—72no—
237C22×M16—72no—
238D4×Q8—78no—
239D4×Q8—144no—
240#547—24no—
241#547C2×D718yes—
242C2×C4⋊Q8—71no—
243C2×C4⋊Q8—141no—
244C2×C4⋊Q8C2×D711yescharacteristic
245C2×#123—22no—
246C2×C22⋊Q8—72no—
247C2×C22⋊Q8—142no—
248C2×C22⋊Q8C2×D712yes—
249C2×#121—21no—
250C2×C4×Q8—71no—
251C2×C4×D4—141no—
252C2×C4×D4C2×D711yescharacteristic
253C2×C4×D4C2×D711yescharacteristic
254C22×C4⋊C4—21no—
255#494—78no—
256#470—716no—
257#462—78no—
258#461—78no—
259#452—716no—
260#451—716no—
261C4×SD16—78no—
262#436—78no—
263C2×#106—71no—
264C2×C4⋊C8—71no—
265C2×#103—72no—
266C2×#103—71no—
267C2×#103—72no—
268C2×#102—71no—
269C2×C22⋊C8—72no—
270C2×C4×C8—71no—
271C23×C7—24no—
272C23×C7—24no—
273C23×C7—24no—
274C23×C7—24no—
275C23×C7—22no—
276C23×C7—42no—
277C23×C7—42no—
278C23×C7C22×D414yes—
279C23×C7C22×D411yescharacteristic
280C7×Q8—24no—
281C7×Q8—24no—
282C7×Q8—44no—
283C7×D4—24no—
284C7×D4—28no—
285C7×D4—22no—
286C7×D4—416no—
287C7×D4C22×D414yes—
288C2×C4×C7—24no—
289C2×C4×C7—22no—
290C2×C4×C7—28no—
291C2×C4×C7—24no—
292C2×C4×C7—24no—
293C2×C4×C7—22no—
294C2×C4×C7—24no—
295C2×C4×C7—416no—
296C2×C4×C7—44no—
297C2×C4×C7—42no—
298C2×C4×C7—48no—
299C2×C4×C7#13714yescomplemented
300C2×C4×C7C22×D412yes—
301C2×C4×C7C22×D412yes—
302C2×C4×C7C22×D414yes—
303C2×C4×C7C22×D411yescharacteristic
304C2×C4×C7C22×D412yes—
305C2×C4×C7H(5,𝔽2)12yes—
306C2×C4×C7C2511yescharacteristic
307C2×Dic7—24no—
308C2×Dic7—22no—
309C2×Dic7—22no—
310C2×Dic7—24no—
311C2×Dic7—416no—
312C2×Dic7—44no—
313C2×Dic7—42no—
314Dic14—24no—
315Dic14—22no—
316Dic14—24no—
317Dic14—24no—
318Dic14—416no—
319Dic14—44no—
320Dic14C22×D414yes—
321C7⋊C8—24no—
322C7⋊C8—22no—
323C7⋊C8C22×D414yes—
324C2×#41—24no—
325C2×#41—28no—
326C2×#41—44no—
327C2×#41C22×D718yes—
328C22×Q8—71no—
329C22×Q8—72no—
330C22×Q8—78no—
331C22×Q8—148no—
332C22×Q8—142no—
333C22×Q8—144no—
334C22×Q8—284no—
335C22×Q8C22×D712yes—
336C22×D4—22no—
337C22×D4C22⋊D711yescharacteristic
338C22×D4C22×D711yescharacteristic
339C23×C4—21no—
340C23×C4—22no—
341C23×C4—142no—
342C23×C4C22×D712yes—
343#137—1464no—
344C2×Q16—78no—
345C2×Q16—78no—
346C2×Q16—78no—
347C2×Q16—148no—
348C2×Q16—148no—
349C2×Q16—284no—
350C2×SD16—78no—
351C2×SD16—74no—
352C2×SD16—78no—
353C2×SD16—148no—
354C2×M16—78no—
355C2×M16—148no—
356C22×C8—71no—
357C22×C8—72no—
358C22×C8—71no—
359C4⋊Q8—22no—
360C4⋊Q8—74no—
361C4⋊Q8—148no—
362C4⋊Q8—142no—
363C4⋊Q8C22×D714yes—
364#123—216no—
365C22⋊Q8—24no—
366C22⋊Q8—78no—
367C22⋊Q8—144no—
368C22⋊Q8—1416no—
369C22⋊Q8C22×D718yes—
370#121—28no—
371C4×Q8—74no—
372C4×Q8—142no—
373C4×D4—22no—
374C4×D4—148no—
375C4×D4C22⋊D714yes—
376C4×D4C22×D714yes—
377C4×D4C22×D714yes—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—28no—
382C2×C4⋊C4—71no—
383C2×C4⋊C4—141no—
384C2×C4⋊C4—141no—
385C2×C4⋊C4—142no—
386C2×C4⋊C4—141no—
387C2×C4⋊C4C22⋊D711yescharacteristic
388C2×C4⋊C4C22×D711yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—21no—
391C2×C22⋊C4—142no—
392C2×C22⋊C4C22⋊D712yes—
393C2×C42—141no—
394C2×C42C22×D711yescharacteristic
395#106—74no—
396C4⋊C8—74no—
397#103—78no—
398#103—74no—
399#103—78no—
400#102—74no—
401C22⋊C8—78no—
402C4×C8—74no—
403C22×C7—24no—
404C22×C7—24no—
405C22×C7—24no—
406C22×C7—24no—
407C22×C7—48no—
408C22×C7—48no—
409C22×C7—42no—
410C22×C7—42no—
411C22×C7D4212yes—
412C22×C7C2×#13712yes—
413C22×C7#57812yes—
414C22×C7C23×D411yescharacteristic
415C4×C7—24no—
416C4×C7—22no—
417C4×C7—44no—
418C4×C7—42no—
419C4×C7D4212yes—
420C4×C7C2×#13714yescomplemented
421C4×C7C23×D411yescharacteristic
422C4×C7C2×H(5,𝔽2)11yescharacteristic
423Dic7—24no—
424Dic7—44no—
425Dic7—42no—
426C24—21no—
427C24C2×C22⋊D712yes—
428#41—216no—
429#41—48no—
430#41—416no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—74no—
435C2×Q8—72no—
436C2×Q8—74no—
437C2×Q8—1416no—
438C2×Q8—142no—
439C2×Q8—141no—
440C2×Q8—148no—
441C2×Q8—142no—
442C2×Q8—144no—
443C2×Q8—144no—
444C2×Q8—288no—
445C2×Q8—288no—
446C2×Q8—2816no—
447C2×Q8C23×D714yes—
448C2×D4—28no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—48no—
453C2×D4D4×D714yes—
454C2×D4C2×C22⋊D714yes—
455C2×D4C23×D712yes—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—141no—
465C22×C4—148no—
466C22×C4—141no—
467C22×C4—142no—
468C22×C4—288no—
469C22×C4C2×C22⋊D714yes—
470C22×C4C2×C22⋊D711yescharacteristic
471C22×C4C2×C22⋊D712yes—
472C22×C4C2×C22⋊D711yescharacteristic
473C22×C4C23×D714yes—
474C22×C4C23×D711yescharacteristic
475Q16—1416no—
476Q16—1416no—
477Q16—288no—
478SD16—716no—
479SD16—1416no—
480M16—1416no—
481C2×C8—74no—
482C2×C8—72no—
483C2×C8—72no—
484C2×C8—74no—
485C2×C8—144no—
486C2×C8—142no—
487C4⋊C4—24no—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—72no—
492C4⋊C4—144no—
493C4⋊C4—148no—
494C4⋊C4—141no—
495C4⋊C4—144no—
496C4⋊C4—144no—
497C4⋊C4C2×C22⋊D714yes—
498C4⋊C4C23×D712yes—
499C22⋊C4—28no—
500C22⋊C4—24no—
501C22⋊C4—148no—
502C22⋊C4C2×C22⋊D718yes—
503C42—144no—
504C42C2×C22⋊D712yes—
505C42C23×D712yes—
506C2×C7—44no—
507C2×C7—44no—
508C2×C7#327614yescomplemented
509C2×C7C2×D4211yescharacteristic
510C2×C7C22×#13711yescharacteristic
511C2×C7C2×#57811yescharacteristic
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—42no—
518C23—42no—
519C23C22×C22⋊D714yes—
520C23C22×C22⋊D711yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—44no—
524Q8—74no—
525Q8—144no—
526Q8—142no—
527Q8—144no—
528Q8—144no—
529Q8—2816no—
530Q8—284no—
531D4—24no—
532D4—28no—
533D4—22no—
534D4—416no—
535D4C2×D4×D714yes—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—416no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—48no—
547C2×C4—144no—
548C2×C4—142no—
549C2×C4—142no—
550C2×C4—144no—
551C2×C4—2816no—
552C2×C4—284no—
553C2×C4—282no—
554C2×C4#651414yescomplemented
555C2×C4C2×D4×D712yes—
556C2×C4#654812yes—
557C2×C4C22×C22⋊D712yes—
558C2×C4C22×C22⋊D714yes—
559C2×C4C22×C22⋊D711yescharacteristic
560C2×C4C22×C22⋊D712yes—
561C2×C4C24×D711yescharacteristic
562C8—74no—
563C8—144no—
564C8—142no—
565C7C2×#327611yescharacteristic, complemented, Sylow
566C22—24no—
567C22—24no—
568C22—24no—
569C22—24no—
570C22—48no—
571C22—48no—
572C22—42no—
573C22—42no—
574C22D4×C22⋊D712yes—
575C22C2×#651412yes—
576C22#9052912yes—
577C22C23×C22⋊D711yescharacteristic
578C4—24no—
579C4—22no—
580C4—44no—
581C4—42no—
582C4—144no—
583C4—284no—
584C4—282no—
585C4D4×C22⋊D712yes—
586C4C2×#651414yescomplemented
587C4C22×D4×D711yescharacteristic
588C4C2×#654811yescharacteristic
589C2—44no—
590C2—44no—
591C2#1174097414yesdirect product factor
592C2C2×D4×C22⋊D711yescharacteristic
593C2C22×#651411yescharacteristic
594C2C2×#9052911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
Fitting, prime radicalC2×C7×#547C2
socleC23×C7C22×D4
hypercenterC2×#547D7
hypocenterC7C2×#3276
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
2. C22×#6514C2semidirect
3. C2×D4×Dic14C2semidirect
4. C2×#90011C2semidirect
5. #11740974C2direct (N × H)
6. C2×#89862C2semidirect
7. C2×#89864C2semidirect
8. C2×#89822C2semidirect
9. C2×#89824C2semidirect
10. C2×#89791C2semidirect
11. C2×C4×#1142C2semidirect
12. C2×#89784C2semidirect
25. C2×#6514C22semidirect
35. D4×Dic14C22semidirect
40. #90011C22semidirect
44. #89864C22semidirect
45. #89862C22semidirect
46. #89824C22semidirect
47. #89822C22semidirect
48. #89791C22semidirect
49. C4×#1142C22semidirect
50. #89784C22semidirect
106. C2×#1142D4semidirect
213. #1142C2×D4semidirect
299. C2×C4×C7#137semidirect
420. C4×C7C2×#137semidirect
508. C2×C7#3276semidirect
554. C2×C4#6514semidirect
565. C7C2×#3276semidirect
586. C4C2×#6514semidirect
591. C2#11740974direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5472
C22×#65142
C2×D4×Dic142
C2×#900112
#117409742
C2×#898622
C2×#898642
C2×#898222
C2×#898242
C2×#897912
C2×C4×#11422
C2×#897842
C2×#32767