Finite Group Databasebeta

← C2×#11730910 #13513311

Subgroups — C2×#11730910 #13513311

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×#6487C211yesmaximal, characteristic
2C23×C4⋊Dic7C211yesmaximal, characteristic
3C22×C4×Dic14C211yesmaximal, characteristic, complemented
4C2×C4×C7×C22⋊C4C211yesmaximal, characteristic
5C2×#89718C212yesmaximal
6C2×C22⋊C4×Dic7C212yesmaximal
7C2×#89701C211yesmaximal, characteristic, complemented
8C2×#89700C211yesmaximal, characteristic, complemented
9C2×C4×C4⋊Dic7C211yesmaximal, characteristic, complemented
10#11730910C2116yesmaximal, direct product factor
11C2×#89419C212yesmaximal, complemented
12C2×#89427C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C24×Dic7C2212yes—
15C23×Dic14C2211yescharacteristic
16C22×C7×C22⋊C4C2211yescharacteristic
17C22×C7×C22⋊C4C2211yescharacteristic
18C22×C42×C7C2211yescharacteristic
19C22×C42×C7C2211yescharacteristic
20C22×C22⋊Dic7C2212yes—
21C22×C4⋊Dic7—28no—
22C22×C4⋊Dic7C2212yes—
23C22×C4⋊Dic7C2211yescharacteristic
24C22×C4⋊Dic7C2218yes—
25C22×C4⋊Dic7C2211yescharacteristic
26C22×C4⋊Dic7C2211yescharacteristic
27C2×#6487C4116yescomplemented
28C2×#6487C2218yes—
29C22×Dic7⋊C4C2212yes—
30C22×Dic7⋊C4C2212yes—
31C22×Dic7⋊C4C2212yes—
32C22×C4×Dic7C2212yes—
33C22×C4×Dic7C2212yes—
34C2×C4×Dic14—28no—
35C2×C4×Dic14C2218yescomplemented
36C4×C7×C22⋊C4C2218yes—
37C2×C7×#95C2211yescharacteristic
38C2×C7×#95C2211yescharacteristic
39C2×C7⋊#95C2212yes—
40C2×C7⋊#95C2212yes—
41C2×C7⋊#95C2212yes—
42C2×C7⋊#95C2212yes—
43#89718C22116yes—
44C22⋊C4×Dic7C22116yes—
45C4×C4⋊Dic7C2218yescomplemented
46#89701C2218yescomplemented
47#89700C2218yescomplemented
48#89427C22116yescomplemented
49#89419C22116yescomplemented
50C2×#2360—71nomaximal, Sylow
51C25×C7C2311yescharacteristic
52C23×C4×C7—24no—
53C23×C4×C7—24no—
54C23×C4×C7C2×C418yes—
55C23×C4×C7C2311yescharacteristic
56C23×C4×C7C2311yescharacteristic
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2314yes—
59C23×C4×C7C2311yescharacteristic
60C23×C4×C7C2311yescharacteristic
61C23×C4×C7C2311yescharacteristic
62C23×Dic7—28no—
63C23×Dic7—28no—
64C23×Dic7—44no—
65C23×Dic7C2312yes—
66C23×Dic7C2318yes—
67C23×Dic7C2312yes—
68C23×Dic7C2312yes—
69C23×Dic7C2312yes—
70C22×Dic14—24no—
71C22×Dic14—24no—
72C22×Dic14C2×C418yescomplemented
73C22×Dic14C2314yes—
74C2×C7×C22⋊C4—24no—
75C2×C7×C22⋊C4Q818yescomplemented
76C2×C7×C22⋊C4C2314yes—
77C2×C7×C22⋊C4C2314yes—
78C2×C42×C7—24no—
79C2×C42×C7—24no—
80C2×C42×C7C2314yes—
81C2×C42×C7C2314yes—
82C2×C22⋊Dic7C2×C4116yes—
83C2×C22⋊Dic7C2318yes—
84C2×C4⋊Dic7—24no—
85C2×C4⋊Dic7—28no—
86C2×C4⋊Dic7—232no—
87C2×C4⋊Dic7C2×C418yescomplemented
88C2×C4⋊Dic7D418yes—
89C2×C4⋊Dic7C2318yes—
90C2×C4⋊Dic7C2314yes—
91C2×C4⋊Dic7C2314yes—
92C2×C4⋊Dic7C2314yes—
93#6487C2×C4164yescomplemented
94C2×Dic7⋊C4—28no—
95C2×Dic7⋊C4—44no—
96C2×Dic7⋊C4C2×C4116yescomplemented
97C2×Dic7⋊C4C2318yes—
98C2×Dic7⋊C4C2318yes—
99C2×Dic7⋊C4C2318yes—
100C2×C4×Dic7—28no—
101C2×C4×Dic7—44no—
102C2×C4×Dic7C2318yes—
103C2×C4×Dic7C2318yes—
104C4×Dic14—232no—
105C7×#95C2314yes—
106C7×#95C2314yes—
107C7⋊#95C2318yes—
108C7⋊#95C2318yes—
109C7⋊#95C2318yes—
110C7⋊#95C2318yes—
111C22×C22⋊Q8—71no—
112C22×C4×Q8—71no—
113C23×C4⋊C4—71no—
114#2360—716no—
115C2×#391—71no—
116C2×#389—71no—
117C2×#389—72no—
118C2×#387—72no—
119C2×#385—72no—
120C2×C4×C4⋊C4—71no—
121C2×C4×C22⋊C4—72no—
122C2×C4×C22⋊C4D711yescharacteristic
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
132C22×C4×C7—22no—
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—22no—
136C22×C4×C7—22no—
137C22×C4×C7—22no—
138C22×C4×C7—232no—
139C22×C4×C7—28no—
140C22×C4×C7—28no—
141C22×C4×C7—22no—
142C22×C4×C7—22no—
143C22×C4×C7—22no—
144C22×C4×C7—22no—
145C22×C4×C7—22no—
146C22×C4×C7—22no—
147C22×C4×C7C22×C414yes—
148C22×C4×C7C22×C4116yes—
149C22×C4×C7C22×C414yes—
150C22×C4×C7C2×D414yes—
151C22×C4×C7C2×Q814yes—
152C22×C4×C7C2×Q814yes—
153C22×C4×C7#4114yes—
154C22×C4×C7C2412yes—
155C22×C4×C7C2412yes—
156C22×C4×C7C2412yes—
157C22×C4×C7C2412yes—
158C22×C4×C7C2412yes—
159C22×C4×C7C2412yes—
160C22×Dic7—24no—
161C22×Dic7—24no—
162C22×Dic7—24no—
163C22×Dic7—24no—
164C22×Dic7—24no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—216no—
168C22×Dic7—24no—
169C22×Dic7—24no—
170C22×Dic7—216no—
171C22×Dic7—48no—
172C22×Dic7—432no—
173C22×Dic7C22×C418yes—
174C22×Dic7C22×C418yes—
175C22×Dic7C2×D418yes—
176C22×Dic7C2414yes—
177C22×Dic7C2414yes—
178C22×Dic7C2414yes—
179C22×Dic7C2414yes—
180C2×Dic14—232no—
181C2×Dic14—28no—
182C2×Dic14—28no—
183C2×Dic14C22×C4116yescomplemented
184C7×C22⋊C4—28no—
185C7×C22⋊C4C2×Q8116yescomplemented
186C42×C7—28no—
187C42×C7—28no—
188C22⋊Dic7C22×C4132yes—
189C4⋊Dic7—28no—
190C4⋊Dic7—216no—
191C4⋊Dic7C22×C4116yescomplemented
192C4⋊Dic7C2×D4116yes—
193Dic7⋊C4—216no—
194Dic7⋊C4—48no—
195Dic7⋊C4C22×C4132yescomplemented
196C4×Dic7—216no—
197C4×Dic7—48no—
198C23×Q8—71no—
199C24×C4—72no—
200C24×C4C2×D711yescharacteristic
201C2×C22⋊Q8—716no—
202C2×C22⋊Q8—78no—
203C2×C4×Q8—78no—
204C2×C4×Q8—148no—
205C22×C4⋊C4—71no—
206C22×C4⋊C4—72no—
207C22×C4⋊C4—71no—
208C22×C4⋊C4—78no—
209C22×C4⋊C4—72no—
210C22×C4⋊C4—71no—
211C22×C4⋊C4—72no—
212C22×C4⋊C4—72no—
213C22×C4⋊C4—148no—
214C22×C22⋊C4—72no—
215C22×C22⋊C4C2×D711yescharacteristic
216C22×C22⋊C4C2×D711yescharacteristic
217C22×C42—72no—
218C22×C42—72no—
219C22×C42C2×D711yescharacteristic
220C22×C42C2×D711yescharacteristic
221#391—78no—
222#389—78no—
223#389—716no—
224#387—716no—
225#385—716no—
226C4×C4⋊C4—78no—
227C4×C22⋊C4—716no—
228C4×C22⋊C4C2×D718yes—
229C2×#95—72no—
230C2×#95—72no—
231C2×#95—72no—
232C2×#95—72no—
233C2×#95C2×D711yescharacteristic
234C2×#95C2×D711yescharacteristic
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×#4114yes—
252C23×C7C2×#4112yes—
253C23×C7C2×#4112yes—
254C23×C7H(5,𝔽2)12yes—
255C23×C72−1+412yes—
256C23×C7C2511yescharacteristic
257C2×C4×C7—28no—
258C2×C4×C7—28no—
259C2×C4×C7—28no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—232no—
267C2×C4×C7—22no—
268C2×C4×C7—22no—
269C2×C4×C7—22no—
270C2×C4×C7—22no—
271C2×C4×C7—22no—
272C2×C4×C7—22no—
273C2×C4×C7—44no—
274C2×C4×C7—44no—
275C2×C4×C7C4×D4116yes—
276C2×C4×C7C23×C414yes—
277C2×C4×C7C23×C414yes—
278C2×C4×C7C22×D414yes—
279C2×C4×C7C22×Q814yes—
280C2×C4×C7C22×Q814yes—
281C2×C4×C7C2×#4114yes—
282C2×Dic7—216no—
283C2×Dic7—216no—
284C2×Dic7—216no—
285C2×Dic7—24no—
286C2×Dic7—24no—
287C2×Dic7—24no—
288C2×Dic7—24no—
289C2×Dic7—24no—
290C2×Dic7—24no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—432no—
295C2×Dic7—48no—
296C2×Dic7C23×C418yes—
297C2×Dic7C23×C418yes—
298C2×Dic7C22×D418yes—
299Dic14—232no—
300C25C22×D711yescharacteristic
301C22×Q8—78no—
302C22×Q8—74no—
303C22×Q8—144no—
304C22×Q8—144no—
305C23×C4—24no—
306C23×C4—24no—
307C23×C4—72no—
308C23×C4—72no—
309C23×C4—72no—
310C23×C4—72no—
311C23×C4—78no—
312C23×C4—148no—
313C23×C4—148no—
314C23×C4—284no—
315C23×C4C4×D718yes—
316C23×C4C22×D711yescharacteristic
317C23×C4C22×D711yescharacteristic
318C23×C4C22×D711yescharacteristic
319C23×C4C22×D711yescharacteristic
320C23×C4C22×D714yes—
321C23×C4C22×D711yescharacteristic
322C23×C4C22×D711yescharacteristic
323C22⋊Q8—764no—
324C4×Q8—1432no—
325C2×C4⋊C4—716no—
326C2×C4⋊C4—78no—
327C2×C4⋊C4—78no—
328C2×C4⋊C4—74no—
329C2×C4⋊C4—78no—
330C2×C4⋊C4—74no—
331C2×C4⋊C4—78no—
332C2×C4⋊C4—74no—
333C2×C4⋊C4—78no—
334C2×C4⋊C4—78no—
335C2×C4⋊C4—1432no—
336C2×C4⋊C4—148no—
337C2×C4⋊C4—148no—
338C2×C4⋊C4—144no—
339C2×C4⋊C4—284no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—716no—
342C2×C22⋊C4—78no—
343C2×C22⋊C4Dic1418yescomplemented
344C2×C22⋊C4C22×D714yes—
345C2×C22⋊C4C22×D714yes—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—78no—
349C2×C42—78no—
350C2×C42—148no—
351C2×C42—284no—
352C2×C42C22×D714yes—
353C2×C42C22×D714yes—
354#95—78no—
355#95—78no—
356#95—78no—
357#95—78no—
358#95C22×D714yes—
359#95C22×D714yes—
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—24no—
384C4×C7—24no—
385C4×C7—42no—
386C4×C7—42no—
387C4×C7C2×C4×D418yes—
388Dic7—28no—
389Dic7—28no—
390Dic7—28no—
391Dic7—44no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24C2×Dic1414yes—
397C24C2×C4×D714yes—
398C24#115714yes—
399C24C23×D712yes—
400C24C23×D711yescharacteristic
401C2×Q8—716no—
402C2×Q8—1432no—
403C2×Q8—148no—
404C2×Q8—148no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—232no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—78no—
421C22×C4—78no—
422C22×C4—78no—
423C22×C4—74no—
424C22×C4—74no—
425C22×C4—74no—
426C22×C4—74no—
427C22×C4—1416no—
428C22×C4—144no—
429C22×C4—144no—
430C22×C4—144no—
431C22×C4—144no—
432C22×C4—144no—
433C22×C4—144no—
434C22×C4—1416no—
435C22×C4—144no—
436C22×C4—144no—
437C22×C4—144no—
438C22×C4—288no—
439C22×C4—2832no—
440C22×C4C2×Dic1414yes—
441C22×C4C2×Dic1414yes—
442C22×C4C2×C4×D714yes—
443C22×C4C2×C4×D7116yes—
444C22×C4C2×C4×D714yes—
445C22×C4D4×D714yes—
446C22×C4#115914yes—
447C22×C4C23×D712yes—
448C22×C4C23×D712yes—
449C22×C4C23×D712yes—
450C22×C4C23×D712yes—
451C22×C4C23×D712yes—
452C22×C4C23×D712yes—
453C4⋊C4—732no—
454C4⋊C4—716no—
455C4⋊C4—716no—
456C4⋊C4—1416no—
457C4⋊C4—1416no—
458C4⋊C4—148no—
459C4⋊C4—288no—
460C22⋊C4—28no—
461C22⋊C4—732no—
462C22⋊C4C2×Dic14116yescomplemented
463C42—28no—
464C42—28no—
465C42—1416no—
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—
486C23C4×Dic14116yescomplemented
487C23C22×Dic1414yes—
488C23C22×Dic1412yes—
489C23C22×C4×D714yes—
490C23C22×C4×D712yes—
491C23C2×#115714yes—
492C23C2×#115712yes—
493C23C2×D4×D712yes—
494C23C2×#115912yes—
495C23#655312yes—
496C23#655412yes—
497C23C24×D711yescharacteristic
498Q8—1432no—
499C2×C4—28no—
500C2×C4—28no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—232no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—78no—
518C2×C4—78no—
519C2×C4—78no—
520C2×C4—1416no—
521C2×C4—1416no—
522C2×C4—1416no—
523C2×C4—144no—
524C2×C4—144no—
525C2×C4—144no—
526C2×C4—144no—
527C2×C4—144no—
528C2×C4—144no—
529C2×C4—144no—
530C2×C4—144no—
531C2×C4—144no—
532C2×C4—2832no—
533C2×C4—288no—
534C2×C4(C22⋊D7)⋊C4116yes—
535C2×C4C22×Dic1414yes—
536C2×C4C22×Dic1414yes—
537C2×C4C22×C4×D714yes—
538C2×C4C22×C4×D714yes—
539C2×C4C2×D4×D714yes—
540C2×C4C2×#115914yes—
541C7C2×#236011yescharacteristic, complemented, Sylow
542C22—24no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—232no—
549C22C2×C4×Dic1414yes—
550C22C2×C4×Dic1418yescomplemented
551C22C2×(C22⋊D7)⋊C414yes—
552C22#9021414yes—
553C22D4×Dic1414yes—
554C22#9023214yes—
555C22#9023414yes—
556C22#9024114yes—
557C22C23×Dic1411yescharacteristic
558C22C23×C4×D711yescharacteristic
559C22C22×#115711yescharacteristic
560C22C22×D4×D711yescharacteristic
561C22C22×#115911yescharacteristic
562C22C2×#655311yescharacteristic
563C22C2×#655411yescharacteristic
564C4—24no—
565C4—24no—
566C4—24no—
567C4—42no—
568C4—42no—
569C4—148no—
570C4—148no—
571C4—148no—
572C4—284no—
573C4C2×(C22⋊D7)⋊C418yes—
574C2—28no—
575C2#1173091018yesdirect product factor
576C2C22×C4×Dic1411yescharacteristic
577C2C22×(C22⋊D7)⋊C411yescharacteristic
578C2C2×#9021411yescharacteristic
579C2C2×D4×Dic1411yescharacteristic
580C2C2×#9023211yescharacteristic
581C2C2×#9023411yescharacteristic
582C2C2×#9024111yescharacteristic
subgroupG/H
centerC24C23×D7
commutatorC22×C7C24×C4
FrattiniC23C24×D7
FittingC2×C4×C7×C22⋊C4C2
socleC24×C7C24
prime radicalC25×C7C23
cosocleC23×C7C25
hypercenterC2×C4×C22⋊C4D7
hypocenterC7C2×#2360
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C22×C4×Dic14C2semidirect
7. C2×#89701C2semidirect
8. C2×#89700C2semidirect
9. C2×C4×C4⋊Dic7C2semidirect
10. #11730910C2direct (N × H)
11. C2×#89419C2semidirect
12. C2×#89427C2semidirect
27. C2×#6487C4semidirect
35. C2×C4×Dic14C22semidirect
45. C4×C4⋊Dic7C22semidirect
46. #89701C22semidirect
47. #89700C22semidirect
48. #89427C22semidirect
49. #89419C22semidirect
72. C22×Dic14C2×C4semidirect
75. C2×C7×C22⋊C4Q8semidirect
87. C2×C4⋊Dic7C2×C4semidirect
93. #6487C2×C4semidirect
96. C2×Dic7⋊C4C2×C4semidirect
183. C2×Dic14C22×C4semidirect
185. C7×C22⋊C4C2×Q8semidirect
191. C4⋊Dic7C22×C4semidirect
195. Dic7⋊C4C22×C4semidirect
245. C23×C7C4×Q8semidirect
343. C2×C22⋊C4Dic14semidirect
369. C22×C7C2×C4×Q8semidirect
462. C22⋊C4C2×Dic14semidirect
468. C2×C7#2360semidirect
486. C23C4×Dic14semidirect
541. C7C2×#2360semidirect
550. C22C2×C4×Dic14semidirect
575. C2#11730910direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C23×C4⋊Dic72
C22×C4×Dic142
C2×C4×C7×C22⋊C42
C2×#897182
C2×C22⋊C4×Dic72
C2×#897012
C2×#897002
C2×C4×C4⋊Dic72
#117309102
C2×#894192
C2×#894272
C2×#23607