Finite Group Databasebeta

← C2×#11731344 #13521629

Subgroups — C2×#11731344 #13521629

Order 1792.

3550 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C4⋊Dic7C211yesmaximal, characteristic
2C22×#6451C211yesmaximal, characteristic, complemented
3C22×#6441C212yesmaximal
4C2×C7×#380C211yesmaximal, characteristic
5C2×C4×C22⋊Dic7C211yesmaximal, characteristic
6C2×#89757C211yesmaximal, characteristic, complemented
7C2×#89745C211yesmaximal, characteristic, complemented
8C2×#89748C211yesmaximal, characteristic, complemented
9C2×#89718C212yesmaximal
10#11731344C2116yesmaximal, direct product factor
11C2×#89429C212yesmaximal, complemented
12C2×#89428C212yesmaximal, complemented
13C24×C4×C7C2211yescharacteristic
14C24×Dic7C2212yes—
15C23×Dic14C2212yes—
16C22×C7×C4⋊C4C2211yescharacteristic
17C22×C7×C4⋊C4C2211yescharacteristic
18C22×C7×C22⋊C4C2212yes—
19C22×C22⋊Dic7C2211yescharacteristic
20C22×C22⋊Dic7C2211yescharacteristic
21C22×C4⋊Dic7—28no—
22C22×C4⋊Dic7C2212yes—
23C22×C4⋊Dic7C2218yes—
24C22×C4⋊Dic7C2211yescharacteristic
25C22×C4⋊Dic7C2212yes—
26C22×C4⋊Dic7C2211yescharacteristic
27C22×Dic7⋊C4C2212yes—
28C22×Dic7⋊C4C2212yes—
29C22×Dic7⋊C4C2212yes—
30C22×Dic7⋊C4C2212yes—
31C22×C4×Dic7C2211yescharacteristic
32C22×C4×Dic7C2211yescharacteristic
33C2×#6451—28no—
34C2×#6451C2218yescomplemented
35C2×#6441—216no—
36C2×#6441C22116yes—
37C7×#380C2218yes—
38C2×C7×#95C2212yes—
39C4×C22⋊Dic7C2218yes—
40C2×C7⋊#95C2211yescharacteristic
41C2×C7⋊#95C2211yescharacteristic
42C2×C7⋊#95C2212yes—
43C2×C7⋊#95C2212yes—
44#89757C2218yescomplemented
45#89748C2218yescomplemented
46#89745C2218yescomplemented
47#89718C22116yes—
48#89429C22116yescomplemented
49#89428C22116yescomplemented
50C2×#2440—71nomaximal, Sylow
51C25×C7C2311yescharacteristic
52C23×C4×C7—24no—
53C23×C4×C7—24no—
54C23×C4×C7Q818yescomplemented
55C23×C4×C7C2312yes—
56C23×C4×C7C2312yes—
57C23×C4×C7C2311yescharacteristic
58C23×C4×C7C2314yes—
59C23×C4×C7C2311yescharacteristic
60C23×Dic7—28no—
61C23×Dic7—28no—
62C23×Dic7—44no—
63C23×Dic7C2312yes—
64C23×Dic7C2318yes—
65C23×Dic7C2311yescharacteristic
66C23×Dic7C2311yescharacteristic
67C23×Dic7C2312yes—
68C23×Dic7C2311yescharacteristic
69C23×Dic7C2311yescharacteristic
70C22×Dic14—28no—
71C22×Dic14—28no—
72C22×Dic14—44no—
73C22×Dic14C2318yes—
74C2×C7×C4⋊C4—24no—
75C2×C7×C4⋊C4—24no—
76C2×C7×C4⋊C4C2314yes—
77C2×C7×C4⋊C4C2314yes—
78C2×C7×C22⋊C4—28no—
79C2×C7×C22⋊C4C2318yes—
80C2×C22⋊Dic7D418yes—
81C2×C22⋊Dic7Q818yescomplemented
82C2×C22⋊Dic7C2314yes—
83C2×C22⋊Dic7C2314yes—
84C2×C4⋊Dic7—24no—
85C2×C4⋊Dic7—28no—
86C2×C4⋊Dic7—28no—
87C2×C4⋊Dic7—232no—
88C2×C4⋊Dic7D418yes—
89C2×C4⋊Dic7C2318yes—
90C2×C4⋊Dic7C2318yes—
91C2×C4⋊Dic7C2314yes—
92C2×C4⋊Dic7C2314yes—
93C2×Dic7⋊C4—28no—
94C2×Dic7⋊C4—28no—
95C2×Dic7⋊C4—28no—
96C2×Dic7⋊C4—44no—
97C2×Dic7⋊C4C2318yes—
98C2×Dic7⋊C4C2318yes—
99C2×Dic7⋊C4C2318yes—
100C2×Dic7⋊C4C2318yes—
101C2×C4×Dic7—24no—
102C2×C4×Dic7—24no—
103C2×C4×Dic7C2314yes—
104C2×C4×Dic7C2314yes—
105#6451—232no—
106#6441—264no—
107C7×#95C2318yes—
108C7⋊#95C2314yes—
109C7⋊#95C2318yes—
110C7⋊#95C2314yes—
111C7⋊#95C2318yes—
112C22×C4⋊Q8—71no—
113C22×C22⋊Q8—72no—
114C23×C4⋊C4—71no—
115#2440—716no—
116C2×#398—72no—
117C2×#395—72no—
118C2×#389—71no—
119C2×#389—71no—
120C2×#385—72no—
121C2×#384—71no—
122C2×#380D711yescharacteristic
123C2×C4×C22⋊C4—71no—
124C24×C7—22no—
125C24×C7—22no—
126C24×C7—22no—
127C24×C7—22no—
128C24×C7C2×D414yes—
129C24×C7C2×Q814yes—
130C24×C7C2×Q814yes—
131C24×C7C2412yes—
132C24×C7C2411yescharacteristic
133C22×C4×C7—22no—
134C22×C4×C7—22no—
135C22×C4×C7—232no—
136C22×C4×C7—28no—
137C22×C4×C7—28no—
138C22×C4×C7—24no—
139C22×C4×C7—24no—
140C22×C4×C7—24no—
141C22×C4×C7—24no—
142C22×C4×C7—24no—
143C22×C4×C7—24no—
144C22×C4×C7C2×D414yes—
145C22×C4×C7C2×Q814yes—
146C22×C4×C7C2×Q814yes—
147C22×C4×C7C2×Q8116yescomplemented
148C22×C4×C7#4114yes—
149C22×C4×C7C2414yes—
150C22×C4×C7C2414yes—
151C22×C4×C7C2412yes—
152C22×C4×C7C2412yes—
153C22×Dic7—22no—
154C22×Dic7—22no—
155C22×Dic7—24no—
156C22×Dic7—22no—
157C22×Dic7—22no—
158C22×Dic7—22no—
159C22×Dic7—22no—
160C22×Dic7—24no—
161C22×Dic7—22no—
162C22×Dic7—22no—
163C22×Dic7—216no—
164C22×Dic7—24no—
165C22×Dic7—24no—
166C22×Dic7—24no—
167C22×Dic7—216no—
168C22×Dic7—48no—
169C22×Dic7—432no—
170C22×Dic7C2×D414yes—
171C22×Dic7C2×D418yes—
172C22×Dic7C2×D414yes—
173C22×Dic7C2×Q814yes—
174C22×Dic7C2×Q814yes—
175C22×Dic7C2414yes—
176C22×Dic7C2412yes—
177C22×Dic7C2412yes—
178C22×Dic7C2414yes—
179C22×Dic7C2412yes—
180C22×Dic7C2412yes—
181C2×Dic14—216no—
182C2×Dic14—216no—
183C2×Dic14—48no—
184C2×Dic14—432no—
185C7×C4⋊C4—28no—
186C7×C4⋊C4—28no—
187C7×C22⋊C4—216no—
188C22⋊Dic7C2×D4116yes—
189C22⋊Dic7C2×Q8116yescomplemented
190C4⋊Dic7—28no—
191C4⋊Dic7—216no—
192C4⋊Dic7—216no—
193C4⋊Dic7C2×D4116yes—
194Dic7⋊C4—216no—
195Dic7⋊C4—216no—
196Dic7⋊C4—216no—
197Dic7⋊C4—48no—
198C4×Dic7—28no—
199C4×Dic7—28no—
200C23×Q8—72no—
201C24×C4—72no—
202C24×C4C2×D711yescharacteristic
203C2×C4⋊Q8—78no—
204C2×C4⋊Q8—148no—
205C2×C22⋊Q8—716no—
206C2×C22⋊Q8—1416no—
207C22×C4⋊C4—71no—
208C22×C4⋊C4—72no—
209C22×C4⋊C4—72no—
210C22×C4⋊C4—72no—
211C22×C4⋊C4—72no—
212C22×C4⋊C4—72no—
213C22×C4⋊C4—78no—
214C22×C4⋊C4—71no—
215C22×C4⋊C4—72no—
216C22×C4⋊C4—148no—
217C22×C4⋊C4C2×D711yescharacteristic
218C22×C4⋊C4C2×D711yescharacteristic
219C22×C22⋊C4—71no—
220C22×C22⋊C4—71no—
221C22×C22⋊C4C2×D712yes—
222C22×C42—71no—
223C22×C42—71no—
224#398—716no—
225#395—716no—
226#389—78no—
227#389—78no—
228#385—716no—
229#384—78no—
230#380C2×D718yes—
231C4×C22⋊C4—78no—
232C2×#95—71no—
233C2×#95—72no—
234C2×#95—71no—
235C2×#95—72no—
236C2×#95C2×D712yes—
237C23×C7—28no—
238C23×C7—28no—
239C23×C7—28no—
240C23×C7—28no—
241C23×C7—28no—
242C23×C7—28no—
243C23×C7—22no—
244C23×C7—22no—
245C23×C7—22no—
246C23×C7—22no—
247C23×C7C4⋊Q8116yescomplemented
248C23×C7C22×D414yes—
249C23×C7C22×D412yes—
250C23×C7C22×D412yes—
251C23×C7C22×Q814yes—
252C23×C7C22×Q812yes—
253C23×C7C22×Q814yes—
254C23×C7C22×Q812yes—
255C23×C7C2×#4112yes—
256C23×C7H(5,𝔽2)12yes—
257C23×C72−1+412yes—
258C23×C7C2511yescharacteristic
259C2×C4×C7—28no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—232no—
263C2×C4×C7—24no—
264C2×C4×C7—24no—
265C2×C4×C7—24no—
266C2×C4×C7—24no—
267C2×C4×C7—24no—
268C2×C4×C7—24no—
269C2×C4×C7—48no—
270C2×C4×C7—48no—
271C2×C4×C7C22⋊Q8116yescomplemented
272C2×C4×C7C22×D414yes—
273C2×C4×C7C22×Q814yes—
274C2×C4×C7C22×Q814yes—
275C2×C4×C7C2×#4114yes—
276C2×Dic7—28no—
277C2×Dic7—216no—
278C2×Dic7—28no—
279C2×Dic7—28no—
280C2×Dic7—28no—
281C2×Dic7—22no—
282C2×Dic7—22no—
283C2×Dic7—24no—
284C2×Dic7—22no—
285C2×Dic7—22no—
286C2×Dic7—22no—
287C2×Dic7—22no—
288C2×Dic7—24no—
289C2×Dic7—22no—
290C2×Dic7—22no—
291C2×Dic7—24no—
292C2×Dic7—24no—
293C2×Dic7—24no—
294C2×Dic7—432no—
295C2×Dic7—48no—
296C2×Dic7C22×D414yes—
297C2×Dic7C22×D418yes—
298C2×Dic7C22×D414yes—
299C2×Dic7C22×Q814yes—
300C2×Dic7C22×Q814yes—
301Dic14—432no—
302C25C22×D711yescharacteristic
303C22×Q8—78no—
304C22×Q8—148no—
305C22×Q8—148no—
306C22×Q8—284no—
307C23×C4—24no—
308C23×C4—24no—
309C23×C4—71no—
310C23×C4—71no—
311C23×C4—71no—
312C23×C4—71no—
313C23×C4—72no—
314C23×C4—72no—
315C23×C4—78no—
316C23×C4—148no—
317C23×C4—148no—
318C23×C4—284no—
319C23×C4Dic1418yescomplemented
320C23×C4C22×D712yes—
321C23×C4C22×D712yes—
322C23×C4C22×D711yescharacteristic
323C23×C4C22×D714yes—
324C23×C4C22×D711yescharacteristic
325C4⋊Q8—1432no—
326C22⋊Q8—1464no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—24no—
329C2×C4⋊C4—78no—
330C2×C4⋊C4—74no—
331C2×C4⋊C4—74no—
332C2×C4⋊C4—78no—
333C2×C4⋊C4—78no—
334C2×C4⋊C4—78no—
335C2×C4⋊C4—78no—
336C2×C4⋊C4—78no—
337C2×C4⋊C4—78no—
338C2×C4⋊C4—1432no—
339C2×C4⋊C4—148no—
340C2×C4⋊C4—148no—
341C2×C4⋊C4—148no—
342C2×C4⋊C4—148no—
343C2×C4⋊C4—148no—
344C2×C4⋊C4—144no—
345C2×C4⋊C4—284no—
346C2×C4⋊C4C22×D714yes—
347C2×C4⋊C4C22×D714yes—
348C2×C22⋊C4—28no—
349C2×C22⋊C4—78no—
350C2×C22⋊C4—78no—
351C2×C22⋊C4—74no—
352C2×C22⋊C4—74no—
353C2×C22⋊C4C22×D718yes—
354C2×C42—74no—
355C2×C42—74no—
356C2×C42—144no—
357C2×C42—144no—
358#95—74no—
359#95—74no—
360#95—78no—
361#95—78no—
362#95C22×D718yes—
363C22×C7—24no—
364C22×C7—24no—
365C22×C7—24no—
366C22×C7—232no—
367C22×C7—24no—
368C22×C7—24no—
369C22×C7—24no—
370C22×C7C2×C22⋊Q814yes—
371C22×C7C2×C4⋊Q814yes—
372C22×C7C2×C4⋊Q818yescomplemented
373C22×C7#54414yes—
374C22×C7D4214yes—
375C22×C7#54714yes—
376C22×C7D4×Q814yes—
377C22×C7#55414yes—
378C22×C7C23×D411yescharacteristic
379C22×C7C23×D411yescharacteristic
380C22×C7C23×Q811yescharacteristic
381C22×C7C23×Q811yescharacteristic
382C22×C7C22×#4111yescharacteristic
383C22×C7C2×H(5,𝔽2)11yescharacteristic
384C22×C7C2×2−1+411yescharacteristic
385C4×C7—24no—
386C4×C7—44no—
387C4×C7—44no—
388C4×C7C2×C22⋊Q818yescomplemented
389Dic7—24no—
390Dic7—28no—
391Dic7—24no—
392Dic7—24no—
393Dic7—24no—
394Dic7—44no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—22no—
399C24C2×Dic1414yes—
400C24D4×D714yes—
401C24Q8×D714yes—
402C24C23×D712yes—
403C24C23×D711yescharacteristic
404C2×Q8—1416no—
405C2×Q8—1416no—
406C2×Q8—288no—
407C2×Q8—2832no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—232no—
411C22×C4—28no—
412C22×C4—28no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—74no—
420C22×C4—74no—
421C22×C4—78no—
422C22×C4—74no—
423C22×C4—74no—
424C22×C4—72no—
425C22×C4—72no—
426C22×C4—72no—
427C22×C4—72no—
428C22×C4—74no—
429C22×C4—74no—
430C22×C4—1416no—
431C22×C4—142no—
432C22×C4—142no—
433C22×C4—144no—
434C22×C4—142no—
435C22×C4—142no—
436C22×C4—142no—
437C22×C4—144no—
438C22×C4—142no—
439C22×C4—142no—
440C22×C4—142no—
441C22×C4—144no—
442C22×C4—144no—
443C22×C4—1416no—
444C22×C4—144no—
445C22×C4—288no—
446C22×C4—2832no—
447C22×C4C2×Dic1414yes—
448C22×C4C2×Dic1414yes—
449C22×C4C2×Dic14116yescomplemented
450C22×C4D4×D714yes—
451C22×C4#115914yes—
452C22×C4C23×D714yes—
453C22×C4C23×D714yes—
454C22×C4C23×D712yes—
455C22×C4C23×D712yes—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—716no—
459C4⋊C4—1416no—
460C4⋊C4—1416no—
461C4⋊C4—1416no—
462C4⋊C4—1416no—
463C4⋊C4—1416no—
464C4⋊C4—148no—
465C4⋊C4—288no—
466C22⋊C4—216no—
467C22⋊C4—716no—
468C22⋊C4—716no—
469C42—148no—
470C42—148no—
471C2×C7—28no—
472C2×C7#244018yescomplemented
473C2×C7C22×C22⋊Q811yescharacteristic
474C2×C7C22×C4⋊Q811yescharacteristic
475C2×C7C2×#54411yescharacteristic
476C2×C7C2×D4211yescharacteristic
477C2×C7C2×#54711yescharacteristic
478C2×C7C2×D4×Q811yescharacteristic
479C2×C7C2×#55411yescharacteristic
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23#6451116yescomplemented
491C23C22×Dic1414yes—
492C23C22×Dic1412yes—
493C23C2×D4×D714yes—
494C23C2×D4×D712yes—
495C23C2×D4×D712yes—
496C23C2×#115912yes—
497C23C2×Q8×D714yes—
498C23C2×Q8×D712yes—
499C23#655312yes—
500C23#655412yes—
501C23C24×D711yescharacteristic
502Q8—2832no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—232no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—74no—
516C2×C4—74no—
517C2×C4—78no—
518C2×C4—74no—
519C2×C4—74no—
520C2×C4—1416no—
521C2×C4—148no—
522C2×C4—148no—
523C2×C4—148no—
524C2×C4—148no—
525C2×C4—142no—
526C2×C4—142no—
527C2×C4—144no—
528C2×C4—142no—
529C2×C4—142no—
530C2×C4—142no—
531C2×C4—144no—
532C2×C4—142no—
533C2×C4—142no—
534C2×C4—142no—
535C2×C4—144no—
536C2×C4—144no—
537C2×C4—144no—
538C2×C4—2832no—
539C2×C4—288no—
540C2×C4#6441116yescomplemented
541C2×C4C22×Dic1414yes—
542C2×C4C22×Dic1414yes—
543C2×C4C2×D4×D714yes—
544C2×C4C2×#115914yes—
545C7C2×#244011yescharacteristic, complemented, Sylow
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—232no—
553C22C2×#644114yes—
554C22C2×#645114yes—
555C22C2×#645118yescomplemented
556C22#9021014yes—
557C22#9030014yes—
558C22#9030214yes—
559C22#9032314yes—
560C22#9032614yes—
561C22C23×Dic1411yescharacteristic
562C22C22×D4×D711yescharacteristic
563C22C22×D4×D711yescharacteristic
564C22C22×#115911yescharacteristic
565C22C22×Q8×D711yescharacteristic
566C22C2×#655311yescharacteristic
567C22C2×#655411yescharacteristic
568C4—24no—
569C4—44no—
570C4—44no—
571C4—144no—
572C4—148no—
573C4—144no—
574C4—144no—
575C4—144no—
576C4—284no—
577C4C2×#644118yescomplemented
578C2—28no—
579C2#1173134418yesdirect product factor
580C2C22×#644111yescharacteristic
581C2C22×#645111yescharacteristic
582C2C2×#9021011yescharacteristic
583C2C2×#9030211yescharacteristic
584C2C2×#9030011yescharacteristic
585C2C2×#9032311yescharacteristic
586C2C2×#9032611yescharacteristic
subgroupG/H
centerC24C23×D7
commutator, cosocleC23×C7C25
FrattiniC23C24×D7
FittingC2×C7×#380C2
socleC24×C7C24
prime radicalC25×C7C23
hypercenterC2×#380D7
hypocenterC7C2×#2440
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×#6451C2semidirect
6. C2×#89757C2semidirect
7. C2×#89745C2semidirect
8. C2×#89748C2semidirect
10. #11731344C2direct (N × H)
11. C2×#89429C2semidirect
12. C2×#89428C2semidirect
34. C2×#6451C22semidirect
44. #89757C22semidirect
45. #89748C22semidirect
46. #89745C22semidirect
48. #89429C22semidirect
49. #89428C22semidirect
54. C23×C4×C7Q8semidirect
81. C2×C22⋊Dic7Q8semidirect
147. C22×C4×C7C2×Q8semidirect
189. C22⋊Dic7C2×Q8semidirect
247. C23×C7C4⋊Q8semidirect
271. C2×C4×C7C22⋊Q8semidirect
319. C23×C4Dic14semidirect
372. C22×C7C2×C4⋊Q8semidirect
388. C4×C7C2×C22⋊Q8semidirect
449. C22×C4C2×Dic14semidirect
472. C2×C7#2440semidirect
490. C23#6451semidirect
540. C2×C4#6441semidirect
545. C7C2×#2440semidirect
555. C22C2×#6451semidirect
577. C4C2×#6441semidirect
579. C2#11731344direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C4⋊Dic72
C22×#64512
C22×#64412
C2×C7×#3802
C2×C4×C22⋊Dic72
C2×#897572
C2×#897452
C2×#897482
C2×#897182
#117313442
C2×#894292
C2×#894282
C2×#24407