Finite Group Databasebeta

← #14224593

Subgroups — #14224593

Order 1792.

2130 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#536C211yesmaximal, characteristic, complemented
2#11744336C211yesmaximal, characteristic
3#11741519C211yesmaximal, characteristic, complemented
4#11741504C212yesmaximal, complemented
5#11740858C214yesmaximal, complemented
6#11740853C218yesmaximal, complemented
7#11740852C214yesmaximal, complemented
8#11740180C212yesmaximal, complemented
9#11740177C214yesmaximal, complemented
10#11740172C212yesmaximal, complemented
11#11739661C212yesmaximal, complemented
12C22×C7×#41C2211yescharacteristic
13C23×C7×Q8C2211yescharacteristic, complemented
14C22×#1157C2211yescharacteristic
15C7×#536—22no—
16C7×#536—22no—
17C7×#536C2214yes—
18C7×#536C2214yescomplemented
19C2×C7×C4⋊Q8—21no—
20C2×C7×#124—21no—
21C2×C7×#123—21no—
22C2×C7×#123—21no—
23C2×C7×C22⋊Q8C2212yes—
24C2×C7×C22⋊Q8C2212yescomplemented
25C2×C7×#117C2211yescharacteristic, complemented
26#90524C2214yes—
27#90523—22no—
28#90521C2212yes—
29#90519—22no—
30#90516C2212yes—
31#90507—21no—
32C2×#6505C2214yes—
33#90496—21no—
34#90495—21no—
35#90495—21no—
36#90482C2212yes—
37#90482C2212yes—
38C2×#6489C2211yescharacteristic
39C22×#1136C2212yescomplemented
40#90006C2214yes—
41#90006C2214yescomplemented
42#90000C2218yescomplemented
43#89999C2218yescomplemented
44#89997C2214yescomplemented
45#89979C2212yes—
46#89979C2212yescomplemented
47#89824C2218yescomplemented
48#89823C2218yescomplemented
49#89822C2214yescomplemented
50#89821C2218yescomplemented
51#89820C2218yescomplemented
52#89819C2218yescomplemented
53#89818C2214yescomplemented
54#89817C2218yescomplemented
55#89816C2214yescomplemented
56#89815C2214yescomplemented
57#89775C2218yescomplemented
58#89768C2218yescomplemented
59#89738C2214yes—
60#89736C2214yescomplemented
61#60754—71nomaximal, Sylow
62C2×C7×#41—44no—
63C2×C7×#41D412yes—
64C2×C7×#41D412yes—
65C2×C7×#41C2312yes—
66C2×C7×#41C2312yes—
67C22×C7×Q8—21no—
68C22×C7×Q8—22no—
69C22×C7×Q8—44no—
70C22×C7×Q8D411yescharacteristic
71C22×C7×Q8C2312yes—
72C22×C7×Q8C2314yes—
73C22×C7×D4D411yescharacteristic
74C22×C7×D4D411yescharacteristic
75C22×C7×D4D411yescharacteristic
76C23×C4×C7—21no—
77C23×C4×C7C2311yescharacteristic
78C23×C4×C7C2311yescharacteristic
79C23×C4×C7C2311yescharacteristic
80C22×C22⋊D7—21no—
81C2×#1157—21no—
82C2×#1157—21no—
83C2×#1157—44no—
84C2×#1157C2312yes—
85C2×#1157C2312yes—
86C22×D28C2311yescharacteristic
87C22×C4×D7—21no—
88C22×Dic14C2311yescharacteristic
89C7×C4⋊Q8—24no—
90C7×C4⋊Q8—24no—
91C7×#124—24no—
92C7×#124—24no—
93C7×#123—24no—
94C7×#123—24no—
95C7×#123—24no—
96C7×#123—24no—
97C7×C22⋊Q8—22no—
98C7×C22⋊Q8—22no—
99C7×C22⋊Q8—24no—
100C7×C22⋊Q8C2314yes—
101C7×C22⋊Q8C2314yes—
102C7×C22⋊Q8C2318yes—
103C7×#117—22no—
104C7×#117C2314yes—
105C2×C7×C4⋊C4—22no—
106C2×C7×C4⋊C4—22no—
107C2×C7×C4⋊C4C2312yes—
108C2×C7×C22⋊C4—21no—
109C2×C7×C22⋊C4—22no—
110C2×C7×C22⋊C4—22no—
111C2×C42×C7—21no—
112#6516—22no—
113#6516—22no—
114C2×C22⋊Dic7—21no—
115#6510—22no—
116#6509—24no—
117#6507—22no—
118C2×#1144C2318yes—
119#6505—48no—
120C2×#1143C2318yes—
121#6503—22no—
122#6502—24no—
123#6502—24no—
124#6501—24no—
125#6500—22no—
126#6500—22no—
127#6499—22no—
128#6498—22no—
129#6498—22no—
130#6493C2314yes—
131#6493C2314yes—
132#6492—22no—
133#6492—22no—
134C4×C22⋊D7—24no—
135C4×C22⋊D7—24no—
136#6489—21no—
137#6489—21no—
138#6489C2312yes—
139#6489C2312yes—
140C2×C4⋊Dic7C2312yes—
141#6487C2314yes—
142#6487C2314yes—
143C2×C4×Dic7—21no—
144C2×#1136—44no—
145C2×#1136C2314yes—
146C2×#1136C2314yes—
147C22×C7⋊C8C2314yes—
148#6409C2318yes—
149#6409C2318yes—
150C22⋊(C7⋊C8)C2314yes—
151C22⋊(C7⋊C8)C2314yes—
152C22⋊(C7⋊C8)C2318yes—
153#6384C2318yes—
154#6383C2318yes—
155#6382C2318yes—
156#6381C2318yes—
157#3449—71no—
158C2×#536D711yescharacteristic, complemented
159#3202—74no—
160#3201—78no—
161#3200—74no—
162#3095—72no—
163#3093—72no—
164#3056—74no—
165#3007—72no—
166#3004—72no—
167#2879—71no—
168C24×C7C2×D411yescharacteristic
169C24×C7C2×D411yescharacteristic
170C24×C7C2×D411yescharacteristic
171C23×D7—21no—
172C7×#41—44no—
173C7×#41—44no—
174C7×#41—42no—
175C7×#41—42no—
176C7×#41—42no—
177C7×#41—42no—
178C2×C7×Q8—22no—
179C2×C7×Q8—24no—
180C2×C7×Q8—28no—
181C2×C7×Q8—42no—
182C2×C7×Q8—44no—
183C2×C7×Q8—44no—
184C2×C7×Q8—44no—
185C2×C7×Q8—48no—
186C2×C7×Q8C2×D412yes—
187C2×C7×Q8C2×D412yes—
188C2×C7×Q8C2414yes—
189C2×C7×D4—42no—
190C2×C7×D4—42no—
191C2×C7×D4—42no—
192C2×C7×D4C2×D412yes—
193C2×C7×D4C2×D412yes—
194C2×C7×D4C2×D412yes—
195C2×C7×D4C2×D412yes—
196C2×C7×D4C2×D412yes—
197C2×C7×D4C2×D412yes—
198C22×C4×C7—24no—
199C22×C4×C7—24no—
200C22×C4×C7—24no—
201C22×C4×C7—22no—
202C22×C4×C7—22no—
203C22×C4×C7—24no—
204C22×C4×C7—44no—
205C22×C4×C7C2×D411yescharacteristic
206C22×C4×C7C2×D411yescharacteristic
207C22×C4×C7C2×D411yescharacteristic
208C22×C4×C7C2×D412yes—
209C22×C4×C7C2×D412yes—
210C22×C4×C7C2×D411yescharacteristic
211C22×C4×C7C2×D411yescharacteristic
212C22×C4×C7C2412yes—
213C22×C4×C7C2411yescharacteristic
214C22×C4×C7C2411yescharacteristic
215C22×C4×C7C2412yes—
216C22×C4×C7C2411yescharacteristic
217C2×C22⋊D7—22no—
218C2×C22⋊D7—22no—
219C2×C22⋊D7—24no—
220C2×C22⋊D7—24no—
221C2×C22⋊D7—44no—
222C22×Dic7—21no—
223C22×Dic7—22no—
224#1157—28no—
225#1157—44no—
226#1157—44no—
227#1157—44no—
228C2×D28—21no—
229C2×D28—42no—
230C2×D28C2412yes—
231C2×C4×D7—22no—
232C2×C4×D7—22no—
233C2×C4×D7—44no—
234C2×Dic14—21no—
235C2×Dic14—42no—
236C2×Dic14C2412yes—
237C7×C4⋊C4—24no—
238C7×C4⋊C4—28no—
239C7×C4⋊C4—24no—
240C7×C4⋊C4—22no—
241C7×C4⋊C4C2414yes—
242C7×C22⋊C4—24no—
243C7×C22⋊C4—24no—
244C7×C22⋊C4—28no—
245C7×C22⋊C4—24no—
246C42×C7—24no—
247C22⋊Dic7—24no—
248C22⋊Dic7—24no—
249C22⋊Dic7—22no—
250C22⋊Dic7—22no—
251#1144—48no—
252#1143—48no—
253(C2×D7)⋊C4—24no—
254(C2×D7)⋊C4—24no—
255C4⋊Dic7—22no—
256C4⋊Dic7C2414yes—
257Dic7⋊C4—24no—
258Dic7⋊C4—24no—
259C4×Dic7—22no—
260C4×Dic7—22no—
261#1136—28no—
262#1136—44no—
263C2×C7⋊C8—44no—
264C2×C7⋊C8C2418yes—
265C22×#41—71no—
266C22×#41C2×D711yescharacteristic
267C23×Q8C2×D711yescharacteristic, complemented
268C2×#137—74no—
269C22×M16—72no—
270#537—74no—
271#536—22no—
272#536—22no—
273#536—72no—
274#536—141no—
275#536C2×D714yes—
276#536C2×D714yescomplemented
277#535—72no—
278#535—141no—
279#534—141no—
280#534—141no—
281#532—142no—
282C2×C4⋊Q8—21no—
283C2×#124—21no—
284#525—72no—
285#525—72no—
286#525—142no—
287C2×#123—21no—
288C2×#123—21no—
289C2×C22⋊Q8C2×D712yes—
290C2×C22⋊Q8C2×D712yescomplemented
291C2×#117—71no—
292C2×#117C2×D711yescharacteristic, complemented
293#485—78no—
294#485—78no—
295#484—74no—
296#483—74no—
297#482—78no—
298#470—78no—
299#468—78no—
300#467—78no—
301#465—78no—
302#452—78no—
303#451—78no—
304#451—74no—
305#450—74no—
306#448—78no—
307#428—74no—
308#419—74no—
309#417—74no—
310#408—74no—
311#408—74no—
312#407—72no—
313#407—72no—
314C23×C7—22no—
315C23×C7—22no—
316C23×C7—22no—
317C23×C7—42no—
318C23×C7—42no—
319C23×C7—42no—
320C23×C7C22×D411yescharacteristic
321C23×C7C22×D412yes—
322C23×C7C22×D412yes—
323C23×C7C22×D411yescharacteristic
324C23×C7C22×D411yescharacteristic
325C22×D7—22no—
326C22×D7—42no—
327C22×D7—82no—
328C7×Q8—28no—
329C7×Q8—42no—
330C7×Q8—42no—
331C7×Q8—48no—
332C7×Q8—44no—
333C7×D4—42no—
334C7×D4—42no—
335C7×D4—42no—
336C7×D4—42no—
337C7×D4—42no—
338C7×D4—42no—
339C2×C4×C7—22no—
340C2×C4×C7—22no—
341C2×C4×C7—22no—
342C2×C4×C7—22no—
343C2×C4×C7—22no—
344C2×C4×C7—24no—
345C2×C4×C7—24no—
346C2×C4×C7—22no—
347C2×C4×C7—22no—
348C2×C4×C7—22no—
349C2×C4×C7—22no—
350C2×C4×C7—22no—
351C2×C4×C7—22no—
352C2×C4×C7—24no—
353C2×C4×C7—24no—
354C2×C4×C7—44no—
355C2×C4×C7—48no—
356C2×C4×C7—44no—
357C2×C4×C7C22×D411yescharacteristic
358C2×C4×C7C22×D412yes—
359C2×C4×C7C22×D412yes—
360C2×C4×C7C22×D411yescharacteristic
361C2×C4×C7C22×D411yescharacteristic
362C2×C4×C7H(5,𝔽2)14yes—
363C2×C4×C7C2511yescharacteristic
364C22⋊D7—44no—
365C22⋊D7—44no—
366C22⋊D7—84no—
367C22⋊D7—84no—
368C2×Dic7—24no—
369C2×Dic7—22no—
370C2×Dic7—42no—
371C2×Dic7—82no—
372D28—24no—
373D28—42no—
374C4×D7—44no—
375C4×D7—44no—
376Dic14—24no—
377Dic14—42no—
378C7⋊C8—28no—
379C2×#41—44no—
380C2×#41—72no—
381C2×#41—72no—
382C2×#41—141no—
383C2×#41—141no—
384C2×#41—284no—
385C2×#41C22⋊D712yes—
386C2×#41C22⋊D712yes—
387C2×#41C22×D712yes—
388C2×#41C22×D712yes—
389C22×Q8—21no—
390C22×Q8—22no—
391C22×Q8—44no—
392C22×Q8—71no—
393C22×Q8C22⋊D711yescharacteristic
394C22×Q8C22×D712yes—
395C22×Q8C22×D714yes—
396C22×D4—71no—
397C22×D4—141no—
398C22×D4C22⋊D711yescharacteristic
399C22×D4C22⋊D711yescharacteristic
400C22×D4C22⋊D711yescharacteristic
401C23×C4—21no—
402C23×C4—141no—
403C23×C4C22×D711yescharacteristic
404C23×C4C22×D711yescharacteristic
405C23×C4C22×D711yescharacteristic
406#137—288no—
407C2×Q16—78no—
408C2×SD16—78no—
409C2×M16—74no—
410C2×M16—74no—
411C2×M16—284no—
412C22×C8—74no—
413C4⋊Q8—24no—
414C4⋊Q8—24no—
415C4⋊Q8—142no—
416#127—142no—
417#124—24no—
418#124—24no—
419#124—142no—
420#124—142no—
421#123—24no—
422#123—24no—
423#123—24no—
424#123—24no—
425#123—142no—
426#123—142no—
427#123—142no—
428#123—142no—
429C22⋊Q8—22no—
430C22⋊Q8—22no—
431C22⋊Q8—24no—
432C22⋊Q8—74no—
433C22⋊Q8—74no—
434C22⋊Q8—144no—
435C22⋊Q8C22×D714yes—
436C22⋊Q8C22×D714yes—
437C22⋊Q8C22×D718yes—
438#121—74no—
439#121—74no—
440#121—144no—
441#121—144no—
442#121—144no—
443#120—142no—
444#120—142no—
445#120—142no—
446#120—142no—
447C4×D4—144no—
448C4×D4—144no—
449#117—22no—
450#117—72no—
451#117—72no—
452#117—141no—
453#117—141no—
454#117C22×D714yes—
455C2×C4⋊C4—22no—
456C2×C4⋊C4—22no—
457C2×C4⋊C4—72no—
458C2×C4⋊C4C22×D712yes—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—22no—
461C2×C22⋊C4—22no—
462C2×C22⋊C4—141no—
463C2×C42—21no—
464C2×C42—141no—
465#107—78no—
466#106—78no—
467#103—78no—
468#103—78no—
469#103—78no—
470#102—78no—
471C22⋊C8—78no—
472C22⋊C8—74no—
473C22⋊C8—74no—
474C22×C7—22no—
475C22×C7—22no—
476C22×C7—24no—
477C22×C7—24no—
478C22×C7—22no—
479C22×C7—21no—
480C22×C7—21no—
481C22×C7—22no—
482C22×C7—22no—
483C22×C7—44no—
484C22×C7—82no—
485C22×C7—82no—
486C22×C7#53412yes—
487C22×C7C23×D411yescharacteristic
488C2×D7—44no—
489C2×D7—42no—
490C2×D7—84no—
491C4×C7—21no—
492C4×C7—21no—
493C4×C7—22no—
494C4×C7—22no—
495C4×C7—44no—
496C4×C7—44no—
497C4×C7#53412yes—
498C4×C7C2×H(5,𝔽2)12yes—
499Dic7—44no—
500Dic7—42no—
501C24—141no—
502C24C2×C22⋊D711yescharacteristic
503C24C2×C22⋊D711yescharacteristic
504C24C2×C22⋊D711yescharacteristic
505#41—44no—
506#41—44no—
507#41—42no—
508#41—42no—
509#41—42no—
510#41—42no—
511#41—148no—
512#41—284no—
513#41—284no—
514#41—284no—
515C2×Q8—22no—
516C2×Q8—24no—
517C2×Q8—28no—
518C2×Q8—42no—
519C2×Q8—44no—
520C2×Q8—44no—
521C2×Q8—44no—
522C2×Q8—48no—
523C2×Q8—72no—
524C2×Q8—141no—
525C2×Q8—282no—
526C2×Q8C2×C22⋊D712yes—
527C2×Q8C2×C22⋊D712yes—
528C2×Q8C23×D714yes—
529C2×D4—42no—
530C2×D4—42no—
531C2×D4—42no—
532C2×D4—72no—
533C2×D4—142no—
534C2×D4—142no—
535C2×D4—144no—
536C2×D4—144no—
537C2×D4—141no—
538C2×D4—282no—
539C2×D4—284no—
540C2×D4C2×C22⋊D712yes—
541C2×D4C2×C22⋊D712yes—
542C2×D4C2×C22⋊D712yes—
543C2×D4C2×C22⋊D712yes—
544C2×D4C2×C22⋊D712yes—
545C2×D4C2×C22⋊D712yes—
546C22×C4—24no—
547C22×C4—24no—
548C22×C4—24no—
549C22×C4—22no—
550C22×C4—22no—
551C22×C4—24no—
552C22×C4—44no—
553C22×C4—141no—
554C22×C4—142no—
555C22×C4—142no—
556C22×C4—142no—
557C22×C4—284no—
558C22×C4C2×C22⋊D711yescharacteristic
559C22×C4C2×C22⋊D711yescharacteristic
560C22×C4C2×C22⋊D711yescharacteristic
561C22×C4C2×C22⋊D711yescharacteristic
562C22×C4C2×C22⋊D712yes—
563C22×C4C2×C22⋊D711yescharacteristic
564C22×C4C2×C22⋊D712yes—
565C22×C4C23×D711yescharacteristic
566C22×C4C23×D711yescharacteristic
567C22×C4C23×D712yes—
568C22×C4C23×D712yes—
569C22×C4C23×D711yescharacteristic
570Q16—288no—
571SD16—288no—
572M16—148no—
573M16—284no—
574C2×C8—78no—
575C2×C8—284no—
576C4⋊C4—24no—
577C4⋊C4—28no—
578C4⋊C4—24no—
579C4⋊C4—22no—
580C4⋊C4—74no—
581C4⋊C4—144no—
582C4⋊C4—144no—
583C4⋊C4—142no—
584C4⋊C4C23×D714yes—
585C22⋊C4—24no—
586C22⋊C4—24no—
587C22⋊C4—28no—
588C22⋊C4—24no—
589C22⋊C4—142no—
590C22⋊C4—142no—
591C22⋊C4—144no—
592C22⋊C4—144no—
593C22⋊C4—144no—
594C22⋊C4—144no—
595C42—24no—
596C42—142no—
597C42—142no—
598C2×C7—22no—
599C2×C7—41no—
600C2×C7—42no—
601C2×C7—41no—
602C2×C7—42no—
603C2×C7C2×#53411yescharacteristic
604C2×C7#358612yes—
605D7—82no—
606C23—22no—
607C23—22no—
608C23—22no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—142no—
613C23—282no—
614C23—562no—
615C23C22×C22⋊D711yescharacteristic
616C23C22×C22⋊D712yes—
617C23C22×C22⋊D711yescharacteristic
618C23C22×C22⋊D712yes—
619C23C22×C22⋊D711yescharacteristic
620Q8—28no—
621Q8—42no—
622Q8—42no—
623Q8—48no—
624Q8—44no—
625Q8—144no—
626Q8—282no—
627D4—42no—
628D4—42no—
629D4—42no—
630D4—42no—
631D4—42no—
632D4—42no—
633D4—144no—
634D4—282no—
635D4—284no—
636D4—284no—
637D4—564no—
638D4—564no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—44no—
655C2×C4—48no—
656C2×C4—44no—
657C2×C4—144no—
658C2×C4—142no—
659C2×C4—284no—
660C2×C4—282no—
661C2×C4—284no—
662C2×C4—562no—
663C2×C4#654814yes—
664C2×C4C22×C22⋊D712yes—
665C2×C4C22×C22⋊D711yescharacteristic
666C2×C4C22×C22⋊D712yes—
667C2×C4C22×C22⋊D711yescharacteristic
668C2×C4C22×C22⋊D711yescharacteristic
669C2×C4C24×D711yescharacteristic
670C8—148no—
671C7#6075411yescharacteristic, complemented, Sylow
672C22—22no—
673C22—22no—
674C22—21no—
675C22—21no—
676C22—22no—
677C22—22no—
678C22—24no—
679C22—24no—
680C22—22no—
681C22—44no—
682C22—82no—
683C22—82no—
684C22—282no—
685C22—284no—
686C22—564no—
687C22#9049512yes—
688C22C23×C22⋊D711yescharacteristic
689C4—21no—
690C4—21no—
691C4—22no—
692C4—22no—
693C4—44no—
694C4—44no—
695C4—282no—
696C4—284no—
697C4#9049512yes—
698C4C2×#654812yes—
699C2—22no—
700C2—42no—
701C2—41no—
702C2—41no—
703C2—42no—
704C2—562no—
705C2C2×#9049511yescharacteristic
706C2#1174435512yes—
subgroupG/H
centerC22C23×C22⋊D7
commutator, cosocleC2×C4×C7C25
FrattiniC2×C4C24×D7
FittingC2×C7×#536C2
socleC22×C7C23×D4
prime radical#11744336C2
hypercenterC2×#536D7
hypocenterC7#60754
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
1. C2×C7×#536C2semidirect
3. #11741519C2semidirect
4. #11741504C2semidirect
5. #11740858C2semidirect
6. #11740853C2semidirect
7. #11740852C2semidirect
8. #11740180C2semidirect
9. #11740177C2semidirect
10. #11740172C2semidirect
11. #11739661C2semidirect
13. C23×C7×Q8C22semidirect
18. C7×#536C22semidirect
24. C2×C7×C22⋊Q8C22semidirect
25. C2×C7×#117C22semidirect
39. C22×#1136C22semidirect
41. #90006C22semidirect
42. #90000C22semidirect
43. #89999C22semidirect
44. #89997C22semidirect
46. #89979C22semidirect
47. #89824C22semidirect
48. #89823C22semidirect
49. #89822C22semidirect
50. #89821C22semidirect
51. #89820C22semidirect
52. #89819C22semidirect
53. #89818C22semidirect
54. #89817C22semidirect
55. #89816C22semidirect
56. #89815C22semidirect
57. #89775C22semidirect
58. #89768C22semidirect
60. #89736C22semidirect
158. C2×#536D7semidirect
267. C23×Q8C2×D7semidirect
276. #536C2×D7semidirect
290. C2×C22⋊Q8C2×D7semidirect
292. C2×#117C2×D7semidirect
671. C7#60754semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5362
#117443362
#117415192
#117415042
#117408582
#117408532
#117408522
#117401802
#117401772
#117401722
#117396612
#607547