Finite Group Databasebeta

← C2×#11736324 #14223091

Subgroups — C2×#11736324 #14223091

Order 1792.

2182 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90523C211yesmaximal, characteristic
2C22×#6453C211yesmaximal, characteristic, complemented
3C2×C7×#417C211yesmaximal, characteristic, complemented
4C2×#90004C211yesmaximal, characteristic, complemented
5C2×#89905C211yesmaximal, characteristic, complemented
6C2×M16×Dic7C211yesmaximal, characteristic, complemented
7C2×#89738C211yesmaximal, characteristic, complemented
8#11736324C2116yesmaximal, direct product factor
9C2×#89569C212yesmaximal, complemented
10C2×#89572C212yesmaximal, complemented
11C2×#89536C212yesmaximal, complemented
12C2×#89540C212yesmaximal, complemented
13C22×C7×#41C2211yescharacteristic
14C22×#1157C2211yescharacteristic
15C22×C7×M16C2211yescharacteristic
16C22×C7×C4⋊C4C2211yescharacteristic
17#90523—24no—
18#90523C2218yes—
19C2×#6510C2211yescharacteristic
20C2×#6507C2211yescharacteristic
21C2×#6503C2211yescharacteristic
22C2×#6502—21no—
23C2×#6502—21no—
24C2×#6499C2211yescharacteristic
25C2×C4×C22⋊D7—21no—
26C2×C4×C22⋊D7—21no—
27C22×C4⋊Dic7—21no—
28C22×C4⋊Dic7C2211yescharacteristic
29C22×Dic7⋊C4—21no—
30C22×C4×Dic7C2211yescharacteristic
31C22×#1136C2211yescharacteristic
32C2×#6453—24no—
33C2×#6453—21no—
34C2×#6453—21no—
35C2×#6453C2212yes—
36C2×#6453C2212yes—
37C2×#6453C2218yescomplemented
38C7×#417C2218yescomplemented
39C2×C7×#103C2212yes—
40C2×C7×#102C2212yes—
41#90004C2218yescomplemented
42C2×#6409C2212yes—
43C2×#6406C2212yes—
44#89905C2218yescomplemented
45M16×Dic7C2218yescomplemented
46C2×#6395C2212yes—
47C2×#6390C2212yes—
48C2×#6389C2212yes—
49C2×C8×Dic7C2212yes—
50C2×#6384C2212yes—
51#89738C2218yescomplemented
52C2×#6383C2212yes—
53#89572C22116yescomplemented
54#89569C22116yescomplemented
55#89540C22116yescomplemented
56#89536C22116yescomplemented
57C2×#3142—71nomaximal, Sylow
58C2×C7×#41—22no—
59C2×C7×#41—24no—
60C2×C7×#41—24no—
61C2×C7×#41C2314yes—
62C22×C7×Q8C2311yescharacteristic
63C22×C7×D4—21no—
64C22×C7×D4C2311yescharacteristic
65C23×C4×C7—21no—
66C23×C4×C7—21no—
67C23×C4×C7C2311yescharacteristic
68C22×C22⋊D7—21no—
69C22×C22⋊D7—21no—
70C22×C22⋊D7—21no—
71C23×Dic7—21no—
72C23×Dic7D411yescharacteristic
73C23×Dic7D411yescharacteristic
74C2×#1157—22no—
75C2×#1157—24no—
76C2×#1157—24no—
77C2×#1157C2314yes—
78C22×D28C2311yescharacteristic
79C22×C4×D7—21no—
80C22×Dic14C2311yescharacteristic
81C2×C7×M16—24no—
82C2×C7×M16C2314yes—
83C22×C7×C8C2312yes—
84C2×C7×C4⋊C4—21no—
85C2×C7×C4⋊C4—22no—
86C2×C7×C4⋊C4C2312yes—
87C2×C7×C4⋊C4C2314yes—
88C2×C22⋊Dic7—21no—
89C2×C22⋊Dic7—21no—
90#6510—22no—
91#6510C2314yes—
92#6507—22no—
93#6507C2314yes—
94#6503—22no—
95#6503C2314yes—
96#6502—28no—
97#6502—28no—
98#6499—22no—
99#6499C2314yes—
100C4×C22⋊D7—28no—
101C4×C22⋊D7—28no—
102C2×(C2×D7)⋊C4—21no—
103C2×(C2×D7)⋊C4—21no—
104C2×C4⋊Dic7—21no—
105C2×C4⋊Dic7—22no—
106C2×C4⋊Dic7—22no—
107C2×C4⋊Dic7—22no—
108C2×C4⋊Dic7—28no—
109C2×C4⋊Dic7C2312yes—
110C2×C4⋊Dic7C2314yes—
111C2×Dic7⋊C4—22no—
112C2×Dic7⋊C4—22no—
113C2×Dic7⋊C4—28no—
114C2×Dic7⋊C4—21no—
115C2×Dic7⋊C4—21no—
116C2×C4×Dic7D411yescharacteristic
117C2×C4×Dic7D411yescharacteristic
118C2×C4×Dic7D414yes—
119C2×C4×Dic7C2311yescharacteristic
120C2×C4×Dic7C2311yescharacteristic
121C2×C4×Dic7C2314yes—
122C2×#1136—24no—
123C2×#1136C2314yes—
124C22×C7⋊C8C2312yes—
125#6453—24no—
126#6453—24no—
127#6453—28no—
128#6453—28no—
129#6453C2318yes—
130#6453C2318yes—
131C7×#103C2318yes—
132C7×#102C2318yes—
133#6409C2318yes—
134#6406C2318yes—
135#6395C2318yes—
136#6390C2318yes—
137#6389C2318yes—
138C8×Dic7C2318yes—
139#6384C2318yes—
140#6383C2318yes—
141C2×#532—71no—
142C22×#125—71no—
143#3142—716no—
144C2×#491—72no—
145C2×#490—72no—
146C2×#488—72no—
147C2×#488—72no—
148C2×#417—71no—
149C2×#417—71no—
150C2×#417—71no—
151C2×#417D711yescharacteristic, complemented
152C2×C4×M16—71no—
153C24×C7—21no—
154C24×C7C2×D411yescharacteristic
155C23×D7—21no—
156C7×#41—28no—
157C7×#41—28no—
158C7×#41—48no—
159C2×C7×Q8—21no—
160C2×C7×Q8—22no—
161C2×C7×Q8—22no—
162C2×C7×Q8C2412yes—
163C2×C7×D4—24no—
164C2×C7×D4—21no—
165C2×C7×D4—22no—
166C2×C7×D4—22no—
167C2×C7×D4—44no—
168C2×C7×D4C2412yes—
169C22×C4×C7—24no—
170C22×C4×C7—22no—
171C22×C4×C7—21no—
172C22×C4×C7—24no—
173C22×C4×C7—44no—
174C22×C4×C7—44no—
175C22×C4×C7C2×D411yescharacteristic
176C22×C4×C7C2×D412yes—
177C22×C4×C7#4114yes—
178C22×C4×C7#4114yes—
179C22×C4×C7C2411yescharacteristic
180C22×C4×C7C2412yes—
181C2×C22⋊D7—24no—
182C2×C22⋊D7—24no—
183C2×C22⋊D7—24no—
184C2×C22⋊D7—44no—
185C2×C22⋊D7—44no—
186C2×C22⋊D7—44no—
187C22×Dic7—24no—
188C22×Dic7—22no—
189C22×Dic7—21no—
190C22×Dic7—42no—
191C22×Dic7—42no—
192C22×Dic7—44no—
193C22×Dic7C2×D411yescharacteristic
194C22×Dic7C2×D411yescharacteristic
195C22×Dic7C2×D411yescharacteristic
196C22×Dic7C2×D411yescharacteristic
197C22×Dic7C2×D414yes—
198C22×Dic7C2×D414yes—
199#1157—28no—
200#1157—28no—
201#1157—48no—
202C2×D28—22no—
203C2×D28—22no—
204C2×D28—21no—
205C2×D28C2412yes—
206C2×C4×D7—24no—
207C2×C4×D7—44no—
208C2×Dic14—22no—
209C2×Dic14—22no—
210C2×Dic14—21no—
211C2×Dic14C2412yes—
212C7×M16—28no—
213C2×C7×C8—24no—
214C2×C7×C8C2414yes—
215C7×C4⋊C4—22no—
216C7×C4⋊C4—24no—
217C7×C4⋊C4C2414yes—
218C22⋊Dic7—24no—
219C22⋊Dic7—24no—
220(C2×D7)⋊C4—24no—
221(C2×D7)⋊C4—24no—
222C4⋊Dic7—22no—
223C4⋊Dic7—24no—
224C4⋊Dic7—28no—
225C4⋊Dic7—28no—
226C4⋊Dic7C2414yes—
227Dic7⋊C4—28no—
228Dic7⋊C4—28no—
229Dic7⋊C4—24no—
230Dic7⋊C4—24no—
231C4×Dic7C2×D414yes—
232C4×Dic7C2×D414yes—
233C4×Dic7C2×D412yes—
234C4×Dic7C2×D412yes—
235C4×Dic7C2412yes—
236C4×Dic7C2412yes—
237#1136—28no—
238C2×C7⋊C8—24no—
239C2×C7⋊C8C2414yes—
240C22×#41—71no—
241C22×#41C2×D711yescharacteristic
242C22×M16—71no—
243C22×M16C2×D711yescharacteristic
244#532—78no—
245#532—144no—
246C2×C4⋊Q8—71no—
247C2×#127—71no—
248C2×#125—72no—
249C2×#125—72no—
250C2×#125—78no—
251C2×#125—141no—
252C2×#125—141no—
253C2×#125—144no—
254C2×#124—71no—
255C2×#124—71no—
256C2×#121—141no—
257C2×#121—141no—
258C2×C4×D4—141no—
259C2×C4×D4—141no—
260C22×C4⋊C4—71no—
261C22×C4⋊C4—141no—
262C22×C4⋊C4—141no—
263C22×C4⋊C4C2×D711yescharacteristic
264C22×C42—71no—
265#491—716no—
266#490—716no—
267#488—716no—
268#488—716no—
269#417—78no—
270#417—78no—
271#417—78no—
272#417C2×D718yescomplemented
273C2×#103—72no—
274C2×#103—72no—
275C2×#103—72no—
276C2×#103C2×D712yes—
277C2×#102—72no—
278C2×#102—72no—
279C2×#102—72no—
280C2×#102C2×D712yes—
281C4×M16—78no—
282C2×#97—72no—
283C2×C4×C8—72no—
284C23×C7—22no—
285C23×C7—22no—
286C23×C7—22no—
287C23×C7—42no—
288C23×C7—42no—
289C23×C7—42no—
290C23×C7#12414yes—
291C23×C7C22×D412yes—
292C23×C7C22×D411yescharacteristic
293C22×D7—22no—
294C22×D7—42no—
295C22×D7—42no—
296C22×D7—42no—
297C7×Q8—22no—
298C7×Q8—22no—
299C7×Q8—42no—
300C7×D4—22no—
301C7×D4—22no—
302C7×D4—48no—
303C7×D4—42no—
304C2×C4×C7—24no—
305C2×C4×C7—22no—
306C2×C4×C7—48no—
307C2×C4×C7—42no—
308C2×C4×C7—44no—
309C2×C4×C7—48no—
310C2×C4×C7#12414yes—
311C2×C4×C7#12418yes—
312C2×C4×C7C22×D411yescharacteristic
313C2×C4×C7C22×D412yes—
314C2×C4×C7C2×#4112yes—
315C2×C4×C7C2×#4114yes—
316C2×C4×C7C2×#4112yes—
317C2×C4×C7C2×#4114yes—
318C2×C4×C7C2511yescharacteristic
319C22⋊D7—48no—
320C22⋊D7—48no—
321C22⋊D7—48no—
322C2×Dic7—22no—
323C2×Dic7—22no—
324C2×Dic7—22no—
325C2×Dic7—22no—
326C2×Dic7—24no—
327C2×Dic7—22no—
328C2×Dic7—44no—
329C2×Dic7—44no—
330C2×Dic7—48no—
331C2×Dic7—42no—
332C2×Dic7—44no—
333C2×Dic7C22×D412yes—
334C2×Dic7C22×D412yes—
335C2×Dic7C22×D412yes—
336C2×Dic7C22×D412yes—
337D28—22no—
338D28—22no—
339D28—42no—
340C4×D7—48no—
341Dic14—22no—
342Dic14—22no—
343Dic14—42no—
344C7×C8—24no—
345C7⋊C8—24no—
346C2×#41—22no—
347C2×#41—24no—
348C2×#41—24no—
349C2×#41—74no—
350C2×#41—142no—
351C2×#41—144no—
352C2×#41—144no—
353C2×#41C22×D714yes—
354C22×Q8—71no—
355C22×Q8C22×D711yescharacteristic
356C22×D4—21no—
357C22×D4—71no—
358C22×D4—141no—
359C22×D4—141no—
360C22×D4—141no—
361C22×D4C22×D711yescharacteristic
362C23×C4—21no—
363C23×C4—21no—
364C23×C4—71no—
365C23×C4—71no—
366C23×C4—141no—
367C23×C4—141no—
368C23×C4C22×D711yescharacteristic
369C2×M16—24no—
370C2×M16—74no—
371C2×M16—144no—
372C2×M16C22×D714yes—
373C22×C8—72no—
374C22×C8C22×D712yes—
375C4⋊Q8—74no—
376C4⋊Q8—142no—
377#127—74no—
378#127—142no—
379#125—78no—
380#125—78no—
381#125—148no—
382#125—148no—
383#125—144no—
384#125—144no—
385#124—74no—
386#124—74no—
387#124—142no—
388#124—142no—
389#121—148no—
390#121—148no—
391C4×D4—148no—
392C4×D4—148no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—72no—
396C2×C4⋊C4—74no—
397C2×C4⋊C4—148no—
398C2×C4⋊C4—142no—
399C2×C4⋊C4—142no—
400C2×C4⋊C4—148no—
401C2×C4⋊C4—142no—
402C2×C4⋊C4—142no—
403C2×C4⋊C4—141no—
404C2×C4⋊C4—141no—
405C2×C4⋊C4—142no—
406C2×C4⋊C4—141no—
407C2×C4⋊C4C22×D712yes—
408C2×C4⋊C4C22×D714yes—
409C2×C22⋊C4—141no—
410C2×C22⋊C4—141no—
411C2×C22⋊C4—141no—
412C2×C22⋊C4—141no—
413C2×C42—71no—
414C2×C42—71no—
415C2×C42—74no—
416C2×C42—71no—
417C2×C42—71no—
418C2×C42—74no—
419#103—78no—
420#103—78no—
421#103—78no—
422#103C22×D718yes—
423#102—78no—
424#102—78no—
425#102—78no—
426#102C22×D718yes—
427#97—78no—
428C4×C8—78no—
429C22×C7—28no—
430C22×C7—22no—
431C22×C7—22no—
432C22×C7—42no—
433C22×C7—42no—
434C22×C7—42no—
435C22×C7—84no—
436C22×C7C2×#12414yes—
437C22×C7C2×#12412yes—
438C22×C7#57512yes—
439C22×C7#57512yes—
440C22×C7C23×D411yescharacteristic
441C2×D7—42no—
442C2×D7—42no—
443C2×D7—42no—
444C2×D7—84no—
445C4×C7—42no—
446C4×C7—44no—
447C4×C7C2×#12412yes—
448C4×C7C2×#12414yes—
449C4×C7C22×#4111yescharacteristic
450C4×C7C22×#4111yescharacteristic
451Dic7—22no—
452Dic7—22no—
453Dic7—22no—
454Dic7—22no—
455Dic7—42no—
456Dic7—44no—
457C24—21no—
458C24—141no—
459C24D4×D711yescharacteristic
460#41—28no—
461#41—28no—
462#41—48no—
463#41—148no—
464#41—148no—
465#41—288no—
466C2×Q8—21no—
467C2×Q8—22no—
468C2×Q8—22no—
469C2×Q8—72no—
470C2×Q8—141no—
471C2×Q8—142no—
472C2×Q8—142no—
473C2×Q8C23×D712yes—
474C2×D4—24no—
475C2×D4—22no—
476C2×D4—21no—
477C2×D4—22no—
478C2×D4—44no—
479C2×D4—72no—
480C2×D4—144no—
481C2×D4—144no—
482C2×D4—144no—
483C2×D4—142no—
484C2×D4—141no—
485C2×D4—142no—
486C2×D4—284no—
487C2×D4—284no—
488C2×D4—284no—
489C2×D4C23×D712yes—
490C22×C4—24no—
491C22×C4—21no—
492C22×C4—22no—
493C22×C4—24no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—71no—
497C22×C4—71no—
498C22×C4—71no—
499C22×C4—71no—
500C22×C4—74no—
501C22×C4—74no—
502C22×C4—144no—
503C22×C4—141no—
504C22×C4—142no—
505C22×C4—144no—
506C22×C4—282no—
507C22×C4—282no—
508C22×C4—284no—
509C22×C4—284no—
510C22×C4#115714yes—
511C22×C4D4×D712yes—
512C22×C4D4×D711yescharacteristic
513C22×C4#115914yes—
514C22×C4C23×D712yes—
515C22×C4C23×D711yescharacteristic
516M16—28no—
517M16—148no—
518C2×C8—24no—
519C2×C8—74no—
520C2×C8—144no—
521C2×C8C23×D714yes—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—74no—
525C4⋊C4—148no—
526C4⋊C4—148no—
527C4⋊C4—144no—
528C4⋊C4—144no—
529C4⋊C4—148no—
530C4⋊C4—148no—
531C4⋊C4—144no—
532C4⋊C4—142no—
533C4⋊C4C23×D714yes—
534C22⋊C4—144no—
535C22⋊C4—144no—
536C22⋊C4—144no—
537C22⋊C4—144no—
538C42—74no—
539C42—74no—
540C42—72no—
541C42—72no—
542C42—72no—
543C42—72no—
544C2×C7—24no—
545C2×C7—82no—
546C2×C7#314214yescomplemented
547C2×C7C22×#12411yescharacteristic
548C2×C7C2×#57511yescharacteristic
549C2×C7C2×#57511yescharacteristic
550D7—82no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—142no—
558C23—282no—
559C23—282no—
560C23—282no—
561C23#644814yes—
562C23C2×D4×D712yes—
563C23C2×D4×D711yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—42no—
567Q8—142no—
568Q8—142no—
569Q8—282no—
570D4—22no—
571D4—22no—
572D4—48no—
573D4—42no—
574D4—142no—
575D4—142no—
576D4—288no—
577D4—288no—
578D4—288no—
579D4—282no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—48no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—48no—
586C2×C4—72no—
587C2×C4—72no—
588C2×C4—72no—
589C2×C4—72no—
590C2×C4—142no—
591C2×C4—142no—
592C2×C4—142no—
593C2×C4—142no—
594C2×C4—142no—
595C2×C4—144no—
596C2×C4—284no—
597C2×C4—284no—
598C2×C4—288no—
599C2×C4—282no—
600C2×C4—284no—
601C2×C4—288no—
602C2×C4#644814yes—
603C2×C4#644818yes—
604C2×C4C2×#115712yes—
605C2×C4C2×#115714yes—
606C2×C4C2×D4×D712yes—
607C2×C4C2×D4×D711yescharacteristic
608C2×C4C2×#115912yes—
609C2×C4C2×#115914yes—
610C2×C4C24×D711yescharacteristic
611C8—24no—
612C8—144no—
613C7C2×#314211yescharacteristic, complemented, Sylow
614C22—22no—
615C22—28no—
616C22—22no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—84no—
621C22—282no—
622C22—282no—
623C22—282no—
624C22—564no—
625C22C2×#644814yes—
626C22C2×#644812yes—
627C22#9046412yes—
628C22#9046812yes—
629C22C22×D4×D711yescharacteristic
630C4—44no—
631C4—42no—
632C4—142no—
633C4—142no—
634C4—142no—
635C4—142no—
636C4—284no—
637C4—282no—
638C4C2×#644814yes—
639C4C2×#644812yes—
640C4C22×#115711yescharacteristic
641C4C22×#115911yescharacteristic
642C2—24no—
643C2—82no—
644C2—562no—
645C2#1173632414yesdirect product factor
646C2C22×#644811yescharacteristic
647C2C2×#9046411yescharacteristic
648C2C2×#9046811yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#417C2
socleC23×C7C22×D4
prime radicalC2×#90523C2
hypercenterC2×#417D7
hypocenterC7C2×#3142
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C22×#6453C2semidirect
3. C2×C7×#417C2semidirect
4. C2×#90004C2semidirect
5. C2×#89905C2semidirect
6. C2×M16×Dic7C2semidirect
7. C2×#89738C2semidirect
8. #11736324C2direct (N × H)
9. C2×#89569C2semidirect
10. C2×#89572C2semidirect
11. C2×#89536C2semidirect
12. C2×#89540C2semidirect
37. C2×#6453C22semidirect
38. C7×#417C22semidirect
41. #90004C22semidirect
44. #89905C22semidirect
45. M16×Dic7C22semidirect
51. #89738C22semidirect
53. #89572C22semidirect
54. #89569C22semidirect
55. #89540C22semidirect
56. #89536C22semidirect
151. C2×#417D7semidirect
272. #417C2×D7semidirect
546. C2×C7#3142semidirect
613. C7C2×#3142semidirect
645. C2#11736324direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#905232
C22×#64532
C2×C7×#4172
C2×#900042
C2×#899052
C2×M16×Dic72
C2×#897382
#117363242
C2×#895692
C2×#895722
C2×#895362
C2×#895402
C2×#31427