Finite Group Databasebeta

← C2×#11740192 #13513317

Subgroups — C2×#11740192 #13513317

Order 1792.

3614 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×C22⋊Q8C211yesmaximal, characteristic
2C22×Q8×Dic7C211yesmaximal, characteristic, complemented
3C23×C4⋊Dic7C211yesmaximal, characteristic
4C2×#90002C211yesmaximal, characteristic, complemented
5C2×C4×C22⋊Dic7C211yesmaximal, characteristic
6#11740192C2116yesmaximal, direct product factor
7C2×#89757C211yesmaximal, characteristic, complemented
8C2×#89757C212yesmaximal, complemented
9C2×#89753C212yesmaximal, complemented
10C2×C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
11C2×#89722C212yesmaximal
12C2×C22⋊C4×Dic7C212yesmaximal
13C23×C7×Q8C2211yescharacteristic
14C24×C4×C7C2211yescharacteristic
15C24×Dic7C2212yes—
16C2×C7×C22⋊Q8C4116yescomplemented
17C2×C7×C22⋊Q8C2218yes—
18C22×C7×C4⋊C4C2211yescharacteristic
19C22×C7×C4⋊C4C2212yes—
20C22×C7×C22⋊C4C2212yes—
21C22×C22⋊Dic7C2211yescharacteristic
22C22×C22⋊Dic7C2211yescharacteristic
23C2×Q8×Dic7—28no—
24C2×Q8×Dic7C2218yescomplemented
25C22×C4⋊Dic7—28no—
26C22×C4⋊Dic7C2218yes—
27C22×C4⋊Dic7C2211yescharacteristic
28C22×C4⋊Dic7C2212yes—
29C22×C4⋊Dic7C2212yes—
30C22×C4⋊Dic7C2212yes—
31C22×C4⋊Dic7C2211yescharacteristic
32C22×C4×Dic7C2211yescharacteristic
33C22×C4×Dic7C2211yescharacteristic
34C22×C4×Dic7C2212yes—
35C22×C4×Dic7C2212yes—
36#90002C2218yescomplemented
37C4×C22⋊Dic7C2218yes—
38C2×C7⋊#95C2211yescharacteristic
39C2×C7⋊#95C2211yescharacteristic
40C2×C7⋊#95C2212yes—
41C2×C7⋊#95C2212yes—
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44#89757C22116yescomplemented
45#89757C2218yescomplemented
46#89753C22116yescomplemented
47C4⋊C4×Dic7C2218yescomplemented
48#89722C22116yes—
49C22⋊C4×Dic7C22116yes—
50C2×#2360—71nomaximal, Sylow
51C25×C7C2311yescharacteristic
52C22×C7×Q8—24no—
53C22×C7×Q8—24no—
54C22×C7×Q8C2×C418yescomplemented
55C22×C7×Q8C2314yes—
56C23×C4×C7—24no—
57C23×C4×C7—24no—
58C23×C4×C7C2×C418yes—
59C23×C4×C7C2312yes—
60C23×C4×C7C2312yes—
61C23×C4×C7C2311yescharacteristic
62C23×C4×C7C2314yes—
63C23×C4×C7C2311yescharacteristic
64C23×Dic7—28no—
65C23×Dic7—28no—
66C23×Dic7—44no—
67C23×Dic7C2311yescharacteristic
68C23×Dic7C2311yescharacteristic
69C23×Dic7C2312yes—
70C23×Dic7C2318yes—
71C23×Dic7C2312yes—
72C23×Dic7C2311yescharacteristic
73C23×Dic7C2311yescharacteristic
74C7×C22⋊Q8C2×C4164yescomplemented
75C2×C7×C4⋊C4C2×C418yescomplemented
76C2×C7×C4⋊C4C2×C4116yescomplemented
77C2×C7×C4⋊C4C2314yes—
78C2×C7×C4⋊C4C2318yes—
79C2×C7×C22⋊C4C2×C4116yes—
80C2×C7×C22⋊C4C2318yes—
81C2×C22⋊Dic7—24no—
82C2×C22⋊Dic7Q818yescomplemented
83C2×C22⋊Dic7C2314yes—
84C2×C22⋊Dic7C2314yes—
85Q8×Dic7—232no—
86C2×C4⋊Dic7—24no—
87C2×C4⋊Dic7—28no—
88C2×C4⋊Dic7—28no—
89C2×C4⋊Dic7—232no—
90C2×C4⋊Dic7—44no—
91C2×C4⋊Dic7D418yes—
92C2×C4⋊Dic7C2314yes—
93C2×C4⋊Dic7C2318yes—
94C2×C4⋊Dic7C2318yes—
95C2×C4⋊Dic7C2318yes—
96C2×C4⋊Dic7C2314yes—
97C2×C4×Dic7—24no—
98C2×C4×Dic7—24no—
99C2×C4×Dic7—28no—
100C2×C4×Dic7—44no—
101C2×C4×Dic7C2318yes—
102C2×C4×Dic7C2314yes—
103C2×C4×Dic7C2314yes—
104C2×C4×Dic7C2318yes—
105C7⋊#95C2318yes—
106C7⋊#95C2318yes—
107C7⋊#95C2318yes—
108C7⋊#95C2318yes—
109C7⋊#95C2314yes—
110C7⋊#95C2314yes—
111C22×C22⋊Q8D711yescharacteristic
112C22×C4×Q8—71no—
113C23×C4⋊C4—71no—
114#2360—716no—
115C2×#391—71no—
116C2×#389—72no—
117C2×#389—71no—
118C2×#387—72no—
119C2×#385—72no—
120C2×C4×C4⋊C4—71no—
121C2×C4×C22⋊C4—71no—
122C2×C4×C22⋊C4—72no—
123C24×C7—22no—
124C24×C7—22no—
125C24×C7—22no—
126C24×C7—22no—
127C24×C7C22×C414yes—
128C24×C7C2×Q814yes—
129C24×C7#4114yes—
130C24×C7C2412yes—
131C24×C7C2411yescharacteristic
132C2×C7×Q8—232no—
133C2×C7×Q8—28no—
134C2×C7×Q8—28no—
135C2×C7×Q8C22×C4116yescomplemented
136C22×C4×C7—24no—
137C22×C4×C7—22no—
138C22×C4×C7—22no—
139C22×C4×C7—24no—
140C22×C4×C7—24no—
141C22×C4×C7—24no—
142C22×C4×C7—232no—
143C22×C4×C7—28no—
144C22×C4×C7—28no—
145C22×C4×C7C22×C418yes—
146C22×C4×C7C22×C418yes—
147C22×C4×C7C22×C414yes—
148C22×C4×C7C22×C4116yes—
149C22×C4×C7C22×C414yes—
150C22×C4×C7C2×D414yes—
151C22×C4×C7#4114yes—
152C22×C4×C7C2414yes—
153C22×C4×C7C2414yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2412yes—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—24no—
159C22×Dic7—22no—
160C22×Dic7—24no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—24no—
164C22×Dic7—22no—
165C22×Dic7—22no—
166C22×Dic7—22no—
167C22×Dic7—216no—
168C22×Dic7—24no—
169C22×Dic7—22no—
170C22×Dic7—24no—
171C22×Dic7—22no—
172C22×Dic7—216no—
173C22×Dic7—48no—
174C22×Dic7—432no—
175C22×Dic7C2×D418yes—
176C22×Dic7C2×Q814yes—
177C22×Dic7C2×Q814yes—
178C22×Dic7C2412yes—
179C22×Dic7C2414yes—
180C22×Dic7C2412yes—
181C22×Dic7C2414yes—
182C22×Dic7C2412yes—
183C22×Dic7C2412yes—
184C7×C4⋊C4C22×C4116yescomplemented
185C7×C4⋊C4C22×C4132yescomplemented
186C7×C22⋊C4C22×C4132yes—
187C22⋊Dic7—28no—
188C22⋊Dic7C2×Q8116yescomplemented
189C4⋊Dic7—28no—
190C4⋊Dic7—216no—
191C4⋊Dic7—216no—
192C4⋊Dic7—48no—
193C4⋊Dic7C2×D4116yes—
194C4×Dic7—28no—
195C4×Dic7—28no—
196C4×Dic7—216no—
197C4×Dic7—48no—
198C23×Q8C2×D711yescharacteristic
199C24×C4—72no—
200C24×C4C2×D711yescharacteristic
201C2×C22⋊Q8Dic7116yescomplemented
202C2×C22⋊Q8C2×D718yes—
203C2×C4×Q8—78no—
204C2×C4×Q8—148no—
205C22×C4⋊C4—71no—
206C22×C4⋊C4—78no—
207C22×C4⋊C4—72no—
208C22×C4⋊C4—71no—
209C22×C4⋊C4—72no—
210C22×C4⋊C4—72no—
211C22×C4⋊C4—148no—
212C22×C4⋊C4C2×D711yescharacteristic
213C22×C4⋊C4C2×D712yes—
214C22×C22⋊C4—71no—
215C22×C22⋊C4—71no—
216C22×C22⋊C4C2×D712yes—
217C22×C42—72no—
218C22×C42—71no—
219C22×C42—71no—
220C22×C42—72no—
221#391—78no—
222#389—78no—
223#389—716no—
224#387—716no—
225#385—716no—
226C4×C4⋊C4—78no—
227C4×C22⋊C4—78no—
228C4×C22⋊C4—716no—
229C2×#95—71no—
230C2×#95—72no—
231C2×#95—71no—
232C2×#95—72no—
233C2×#95—72no—
234C2×#95—72no—
235C23×C7—28no—
236C23×C7—28no—
237C23×C7—28no—
238C23×C7—28no—
239C23×C7—28no—
240C23×C7—28no—
241C23×C7—22no—
242C23×C7—22no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7C4×Q8116yescomplemented
246C23×C7C23×C414yes—
247C23×C7C23×C412yes—
248C23×C7C22×D412yes—
249C23×C7C22×Q814yes—
250C23×C7C22×Q812yes—
251C23×C7C2×#4112yes—
252C23×C7C2×#4114yes—
253C23×C7C2×#4112yes—
254C23×C7H(5,𝔽2)12yes—
255C23×C72−1+412yes—
256C23×C7C2511yescharacteristic
257C7×Q8—232no—
258C2×C4×C7—28no—
259C2×C4×C7—216no—
260C2×C4×C7—216no—
261C2×C4×C7—24no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—232no—
268C2×C4×C7C4×D4116yes—
269C2×C4×C7C23×C418yes—
270C2×C4×C7C23×C418yes—
271C2×C4×C7C23×C414yes—
272C2×C4×C7C23×C414yes—
273C2×C4×C7C22×D414yes—
274C2×C4×C7C2×#4114yes—
275C2×Dic7—216no—
276C2×Dic7—28no—
277C2×Dic7—28no—
278C2×Dic7—22no—
279C2×Dic7—22no—
280C2×Dic7—24no—
281C2×Dic7—22no—
282C2×Dic7—24no—
283C2×Dic7—22no—
284C2×Dic7—22no—
285C2×Dic7—24no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—22no—
289C2×Dic7—24no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—22no—
293C2×Dic7—432no—
294C2×Dic7—44no—
295C2×Dic7—48no—
296C2×Dic7—44no—
297C2×Dic7C22×D418yes—
298C2×Dic7C22×Q814yes—
299C2×Dic7C22×Q814yes—
300C25C22×D711yescharacteristic
301C22×Q8—24no—
302C22×Q8—24no—
303C22×Q8C2×Dic718yescomplemented
304C22×Q8C22×D714yes—
305C23×C4—24no—
306C23×C4—24no—
307C23×C4—71no—
308C23×C4—71no—
309C23×C4—72no—
310C23×C4—71no—
311C23×C4—71no—
312C23×C4—72no—
313C23×C4—78no—
314C23×C4—148no—
315C23×C4—148no—
316C23×C4—284no—
317C23×C4C2×Dic718yes—
318C23×C4C22×D714yes—
319C23×C4C22×D712yes—
320C23×C4C22×D712yes—
321C23×C4C22×D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C22⋊Q8C2×Dic7164yescomplemented
324C4×Q8—1432no—
325C2×C4⋊C4—78no—
326C2×C4⋊C4—74no—
327C2×C4⋊C4—78no—
328C2×C4⋊C4—74no—
329C2×C4⋊C4—78no—
330C2×C4⋊C4—78no—
331C2×C4⋊C4—1432no—
332C2×C4⋊C4—148no—
333C2×C4⋊C4—148no—
334C2×C4⋊C4—144no—
335C2×C4⋊C4—284no—
336C2×C4⋊C4C2×Dic718yescomplemented
337C2×C4⋊C4C2×Dic7116yescomplemented
338C2×C4⋊C4C22×D714yes—
339C2×C4⋊C4C22×D718yes—
340C2×C22⋊C4—78no—
341C2×C22⋊C4—74no—
342C2×C22⋊C4—74no—
343C2×C22⋊C4—144no—
344C2×C22⋊C4C2×Dic7116yes—
345C2×C22⋊C4C22×D718yes—
346C2×C42—78no—
347C2×C42—74no—
348C2×C42—74no—
349C2×C42—78no—
350C2×C42—148no—
351C2×C42—144no—
352C2×C42—144no—
353C2×C42—284no—
354#95—74no—
355#95—78no—
356#95—74no—
357#95—78no—
358#95—78no—
359#95—78no—
360C22×C7—232no—
361C22×C7—24no—
362C22×C7—24no—
363C22×C7—24no—
364C22×C7—24no—
365C22×C7—24no—
366C22×C7—24no—
367C22×C7C2×C4×D414yes—
368C22×C7C2×C4×Q814yes—
369C22×C7C2×C4×Q818yescomplemented
370C22×C7#52014yes—
371C22×C7#54614yes—
372C22×C7D4×Q814yes—
373C22×C7#55214yes—
374C22×C7#55414yes—
375C22×C7C24×C411yescharacteristic
376C22×C7C23×D411yescharacteristic
377C22×C7C23×Q811yescharacteristic
378C22×C7C22×#4111yescharacteristic
379C22×C7C22×#4111yescharacteristic
380C22×C7C2×H(5,𝔽2)11yescharacteristic
381C22×C7C2×2−1+411yescharacteristic
382C4×C7—24no—
383C4×C7—28no—
384C4×C7—28no—
385C4×C7C2×C4×D418yes—
386Dic7—28no—
387Dic7—24no—
388Dic7—24no—
389Dic7—42no—
390Dic7—44no—
391Dic7—42no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24Q8×D714yes—
397C24#116114yes—
398C24C22×Dic714yes—
399C24C23×D712yes—
400C24C23×D711yescharacteristic
401C2×Q8—232no—
402C2×Q8—28no—
403C2×Q8—28no—
404C2×Q8C22×Dic7116yescomplemented
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—232no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—78no—
415C22×C4—74no—
416C22×C4—74no—
417C22×C4—72no—
418C22×C4—72no—
419C22×C4—74no—
420C22×C4—72no—
421C22×C4—74no—
422C22×C4—72no—
423C22×C4—1416no—
424C22×C4—142no—
425C22×C4—142no—
426C22×C4—144no—
427C22×C4—142no—
428C22×C4—144no—
429C22×C4—142no—
430C22×C4—142no—
431C22×C4—144no—
432C22×C4—142no—
433C22×C4—142no—
434C22×C4—142no—
435C22×C4—1416no—
436C22×C4—144no—
437C22×C4—142no—
438C22×C4—144no—
439C22×C4—142no—
440C22×C4—288no—
441C22×C4—2832no—
442C22×C4D4×D714yes—
443C22×C4#115914yes—
444C22×C4C22×Dic718yes—
445C22×C4C22×Dic718yes—
446C22×C4C22×Dic714yes—
447C22×C4C22×Dic7116yes—
448C22×C4C22×Dic714yes—
449C22×C4C23×D714yes—
450C22×C4C23×D714yes—
451C22×C4C23×D712yes—
452C22×C4C23×D712yes—
453C4⋊C4—716no—
454C4⋊C4—1416no—
455C4⋊C4—1416no—
456C4⋊C4—148no—
457C4⋊C4—288no—
458C4⋊C4C22×Dic7116yescomplemented
459C4⋊C4C22×Dic7132yescomplemented
460C22⋊C4—716no—
461C22⋊C4—148no—
462C22⋊C4C22×Dic7132yes—
463C42—1416no—
464C42—148no—
465C42—148no—
466C42—288no—
467C2×C7—28no—
468C2×C7#236018yescomplemented
469C2×C7C22×C4×D411yescharacteristic
470C2×C7C22×C4×Q811yescharacteristic
471C2×C7C2×#52011yescharacteristic
472C2×C7C2×#54611yescharacteristic
473C2×C7C2×D4×Q811yescharacteristic
474C2×C7C2×#55211yescharacteristic
475C2×C7C2×#55411yescharacteristic
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23Q8×Dic7116yescomplemented
487C23C2×D4×D712yes—
488C23C2×#115912yes—
489C23C2×Q8×D714yes—
490C23C2×Q8×D712yes—
491C23C2×#116114yes—
492C23C2×#116112yes—
493C23#655312yes—
494C23#655412yes—
495C23C23×Dic714yes—
496C23C23×Dic712yes—
497C23C24×D711yescharacteristic
498Q8—232no—
499C2×C4—28no—
500C2×C4—216no—
501C2×C4—216no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—232no—
509C2×C4—78no—
510C2×C4—74no—
511C2×C4—74no—
512C2×C4—1416no—
513C2×C4—148no—
514C2×C4—148no—
515C2×C4—142no—
516C2×C4—142no—
517C2×C4—144no—
518C2×C4—142no—
519C2×C4—144no—
520C2×C4—142no—
521C2×C4—142no—
522C2×C4—144no—
523C2×C4—142no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—144no—
527C2×C4—142no—
528C2×C4—144no—
529C2×C4—142no—
530C2×C4—2832no—
531C2×C4—284no—
532C2×C4—288no—
533C2×C4—284no—
534C2×C4D4×Dic7116yes—
535C2×C4C2×D4×D714yes—
536C2×C4C2×#115914yes—
537C2×C4C23×Dic718yes—
538C2×C4C23×Dic718yes—
539C2×C4C23×Dic714yes—
540C2×C4C23×Dic714yes—
541C7C2×#236011yescharacteristic, complemented, Sylow
542C22—232no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22#9032114yes—
550C22#9032314yes—
551C22#9032614yes—
552C22#9033314yes—
553C22C2×D4×Dic714yes—
554C22C2×Q8×Dic714yes—
555C22C2×Q8×Dic718yescomplemented
556C22#9051814yes—
557C22C22×D4×D711yescharacteristic
558C22C22×#115911yescharacteristic
559C22C22×Q8×D711yescharacteristic
560C22C22×#116111yescharacteristic
561C22C2×#655311yescharacteristic
562C22C2×#655411yescharacteristic
563C22C24×Dic711yescharacteristic
564C4—24no—
565C4—28no—
566C4—28no—
567C4—144no—
568C4—144no—
569C4—148no—
570C4—282no—
571C4—284no—
572C4—282no—
573C4C2×D4×Dic718yes—
574C2—28no—
575C2#1174019218yesdirect product factor
576C2C2×#9032311yescharacteristic
577C2C2×#9033311yescharacteristic
578C2C2×#9032611yescharacteristic
579C2C2×#9032111yescharacteristic
580C2C22×D4×Dic711yescharacteristic
581C2C22×Q8×Dic711yescharacteristic
582C2C2×#9051811yescharacteristic
subgroupG/H
centerC24C23×D7
commutatorC22×C7C24×C4
FrattiniC23C24×D7
FittingC22×C7×C22⋊Q8C2
socleC24×C7C24
prime radicalC25×C7C23
cosocleC23×C7C25
hypercenterC22×C22⋊Q8D7
hypocenterC7C2×#2360
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C22×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C22×Q8×Dic7C2semidirect
4. C2×#90002C2semidirect
6. #11740192C2direct (N × H)
7. C2×#89757C2semidirect
8. C2×#89757C2semidirect
9. C2×#89753C2semidirect
10. C2×C4⋊C4×Dic7C2semidirect
16. C2×C7×C22⋊Q8C4semidirect
24. C2×Q8×Dic7C22semidirect
36. #90002C22semidirect
44. #89757C22semidirect
45. #89757C22semidirect
46. #89753C22semidirect
47. C4⋊C4×Dic7C22semidirect
54. C22×C7×Q8C2×C4semidirect
74. C7×C22⋊Q8C2×C4semidirect
75. C2×C7×C4⋊C4C2×C4semidirect
76. C2×C7×C4⋊C4C2×C4semidirect
82. C2×C22⋊Dic7Q8semidirect
135. C2×C7×Q8C22×C4semidirect
184. C7×C4⋊C4C22×C4semidirect
185. C7×C4⋊C4C22×C4semidirect
188. C22⋊Dic7C2×Q8semidirect
201. C2×C22⋊Q8Dic7semidirect
245. C23×C7C4×Q8semidirect
303. C22×Q8C2×Dic7semidirect
323. C22⋊Q8C2×Dic7semidirect
336. C2×C4⋊C4C2×Dic7semidirect
337. C2×C4⋊C4C2×Dic7semidirect
369. C22×C7C2×C4×Q8semidirect
404. C2×Q8C22×Dic7semidirect
458. C4⋊C4C22×Dic7semidirect
459. C4⋊C4C22×Dic7semidirect
468. C2×C7#2360semidirect
486. C23Q8×Dic7semidirect
541. C7C2×#2360semidirect
555. C22C2×Q8×Dic7semidirect
575. C2#11740192direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C7×C22⋊Q82
C22×Q8×Dic72
C23×C4⋊Dic72
C2×#900022
C2×C4×C22⋊Dic72
#117401922
C2×#897572
C2×#897532
C2×C4⋊C4×Dic72
C2×#897222
C2×C22⋊C4×Dic72
C2×#23607