Finite Group Databasebeta

← #13411785

Subgroups — #13411785

Order 1792.

732 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2441C211yesmaximal, characteristic
2#11730901C211yesmaximal, characteristic
3#11729493C211yesmaximal, characteristic, complemented
4#11729492C211yesmaximal, characteristic, complemented
5#11729335C211yesmaximal, characteristic
6#11729330C211yesmaximal, characteristic
7#11727378C211yesmaximal, characteristic, complemented
8#11727376C211yesmaximal, characteristic, complemented
9#11727357C211yesmaximal, characteristic, complemented
10#11727354C211yesmaximal, characteristic, complemented
11#11727326C211yesmaximal, characteristic, complemented
12#11727322C211yesmaximal, characteristic, complemented
13#11727267C211yesmaximal, characteristic
14#11727256C211yesmaximal, characteristic
15#11727204C211yesmaximal, characteristic
16C2×C7×#125C2211yescharacteristic
17C2×C7×#117C2211yescharacteristic
18C22×C4⋊Dic7C2211yescharacteristic
19C22×#1136C2211yescharacteristic
20C2×#6440—21no—
21C7×#399—21no—
22C7×#396—21no—
23C7×#389—21no—
24C7×#389C2211yescharacteristic
25C7×#389C2211yescharacteristic
26C7×#387—21no—
27C7×#384C2211yescharacteristic
28C7×#380C2211yescharacteristic
29C4×C7×C22⋊C4C2211yescharacteristic
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×#6382C2211yescharacteristic
33C2×#6382C2211yescharacteristic
34C2×#6381C2211yescharacteristic
35C2×#6381C2211yescharacteristic
36#89715—21no—
37C4×C4⋊Dic7C2211yescharacteristic
38C4×C4⋊Dic7C2211yescharacteristic
39#89700C2211yescharacteristic
40#89700C2211yescharacteristic
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89425—21no—
44#89425—21no—
45#89324C2211yescharacteristic
46#89324C2211yescharacteristic
47#89324C2211yescharacteristic
48#89324C2211yescharacteristic
49#89324C2211yescharacteristic
50#89324C2211yescharacteristic
51#89324C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89319C2211yescharacteristic
58#89319C2211yescharacteristic
59#26014—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C7×#125—22no—
63C7×#125—41no—
64C7×#117—22no—
65C7×#117—41no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—21no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C7×C4⋊C4C2311yescharacteristic
74C2×C7×C4⋊C4C2311yescharacteristic
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—21no—
77C2×C7×C22⋊C4—21no—
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C7×C22⋊C4D411yescharacteristic
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×C42×C7C2311yescharacteristic
83C2×C22⋊Dic7—21no—
84C2×C4⋊Dic7—21no—
85C2×C4⋊Dic7—22no—
86C2×C4⋊Dic7Q814yes—
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4⋊Dic7C2311yescharacteristic
89C2×C4⋊Dic7C2311yescharacteristic
90C2×C4⋊Dic7C2311yescharacteristic
91C2×Dic7⋊C4—21no—
92C2×Dic7⋊C4—21no—
93C2×C4×Dic7—21no—
94C2×C4×Dic7—21no—
95C2×#1136—22no—
96C2×#1136—22no—
97C2×#1136—41no—
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C22×C7⋊C8C2311yescharacteristic
101C22×C7⋊C8C2311yescharacteristic
102#6440—44no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7×#95D411yescharacteristic
107C7×#95D411yescharacteristic
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C7⋊#95—21no—
113C22⋊(C7⋊C8)—22no—
114#6382—41no—
115#6382—41no—
116#6381—22no—
117#6381—22no—
118C4⋊(C7⋊C8)—22no—
119C4⋊(C7⋊C8)—41no—
120#2441D711yescharacteristic
121#2344—71no—
122#2100—71no—
123#2088—71no—
124#2086—71no—
125#2059—71no—
126#1989—71no—
127#1948—71no—
128#1921—71no—
129#1918—71no—
130#1854—71no—
131#1851—71no—
132#1824—71no—
133#1809—71no—
134#1736—71no—
135C24×C7C2×D411yescharacteristic
136C22×C4×C7—21no—
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×D411yescharacteristic
149C22×C4×C7C2×Q812yes—
150C22×C4×C7#4112yes—
151C22×C4×C7C2411yescharacteristic
152C22×Dic7—24no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—21no—
156C22×Dic7—21no—
157C22×Dic7—42no—
158C22×Dic7—42no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—21no—
165C7×C4⋊C4—21no—
166C7×C4⋊C4—42no—
167C7×C4⋊C4—42no—
168C7×C4⋊C4—42no—
169C7×C4⋊C4—42no—
170C7×C4⋊C4—81no—
171C7×C4⋊C4—81no—
172C7×C4⋊C4#4112yes—
173C7×C22⋊C4—22no—
174C7×C22⋊C4—41no—
175C7×C22⋊C4—42no—
176C7×C22⋊C4—81no—
177C42×C7—21no—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—21no—
181C42×C7—42no—
182C22⋊Dic7—42no—
183C4⋊Dic7—24no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7C2×Q812yes—
191C4⋊Dic7C2×Q812yes—
192Dic7⋊C4—42no—
193Dic7⋊C4—42no—
194C4×Dic7—42no—
195C4×Dic7—42no—
196#1136—44no—
197C2×C7⋊C8—21no—
198C2×C7⋊C8—21no—
199C2×C7⋊C8—21no—
200C2×C7⋊C8—21no—
201C2×C7⋊C8—21no—
202C2×C7⋊C8—21no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8—21no—
207C2×C7⋊C8—21no—
208C2×C7⋊C8#4112yes—
209C22×M16—71no—
210C2×#125C2×D711yescharacteristic
211C2×#117—141no—
212C2×#117C2×D711yescharacteristic
213C22×C4⋊C4—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×#106—71no—
218C2×C4⋊C8—71no—
219C2×C4⋊C8—71no—
220C2×C22⋊C8—71no—
221#399—21no—
222#396—21no—
223#389—21no—
224#389—71no—
225#389—71no—
226#389C2×D711yescharacteristic
227#389C2×D711yescharacteristic
228#387—21no—
229#387—141no—
230#387—141no—
231#384C2×D711yescharacteristic
232#381—141no—
233#380—71no—
234#380C2×D711yescharacteristic
235C4×C4⋊C4—71no—
236C4×C4⋊C4—71no—
237C4×C22⋊C4C2×D711yescharacteristic
238#340—71no—
239#340—71no—
240#340—71no—
241#340—71no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#336—71no—
251#336—71no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7C22⋊Q812yes—
259C23×C7C22×D411yescharacteristic
260C2×C4×C7—21no—
261C2×C4×C7—21no—
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—22no—
265C2×C4×C7—22no—
266C2×C4×C7—22no—
267C2×C4×C7—22no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—22no—
271C2×C4×C7—21no—
272C2×C4×C7—21no—
273C2×C4×C7—21no—
274C2×C4×C7—21no—
275C2×C4×C7—21no—
276C2×C4×C7—21no—
277C2×C4×C7—21no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7—41no—
281C2×C4×C7—41no—
282C2×C4×C7—41no—
283C2×C4×C7—41no—
284C2×C4×C7—41no—
285C2×C4×C7—41no—
286C2×C4×C7—41no—
287C2×C4×C7—41no—
288C2×C4×C7—41no—
289C2×C4×C7C22⋊Q812yes—
290C2×C4×C7AGL(1,ℤ/8)12yes—
291C2×C4×C7C22×Q811yescharacteristic
292C2×C4×C7C2×#4111yescharacteristic
293C2×C4×C7C2×#4111yescharacteristic
294C2×C4×C7C2×#4111yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C72−1+411yescharacteristic
297C2×Dic7—22no—
298C2×Dic7—22no—
299C2×Dic7—48no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—41no—
307C2×Dic7—41no—
308C2×Dic7—41no—
309C2×Dic7—41no—
310C7⋊C8—22no—
311C7⋊C8—41no—
312C7⋊C8—41no—
313C7⋊C8—41no—
314C23×C4—141no—
315C23×C4C22×D711yescharacteristic
316C2×M16—142no—
317C2×M16—142no—
318C2×M16—281no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8—71no—
322C22×C8—71no—
323#125—22no—
324#125—41no—
325#117—22no—
326#117—41no—
327#117—284no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—74no—
335C2×C4⋊C4—71no—
336C2×C4⋊C4—71no—
337C2×C4⋊C4—71no—
338C2×C4⋊C4—71no—
339C2×C4⋊C4—142no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4—141no—
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C4⋊C4C22×D711yescharacteristic
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—141no—
350C2×C22⋊C4C22⋊D711yescharacteristic
351C2×C22⋊C4C22⋊D711yescharacteristic
352C2×C42—21no—
353C2×C42—141no—
354C2×C42—141no—
355C2×C42C22×D711yescharacteristic
356C2×C42C22×D711yescharacteristic
357#107—142no—
358#107—142no—
359#106—281no—
360#106—281no—
361C4⋊C8—142no—
362C4⋊C8—281no—
363C22⋊C8—142no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—141no—
368#95—141no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95C22⋊D711yescharacteristic
373#95C22⋊D711yescharacteristic
374C22×C7—22no—
375C22×C7—22no—
376C22×C7—22no—
377C22×C7—41no—
378C22×C7—41no—
379C22×C7—41no—
380C22×C7—41no—
381C22×C7C2×C22⋊Q811yescharacteristic
382C22×C7#54611yescharacteristic
383C22×C7#54711yescharacteristic
384C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
385C22×C7#57511yescharacteristic
386C22×C7#57711yescharacteristic
387C22×C7#57811yescharacteristic
388C4×C7—22no—
389C4×C7—22no—
390C4×C7—41no—
391C4×C7—42no—
392C4×C7—41no—
393C4×C7—41no—
394C4×C7—81no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7#53311yescharacteristic
398C4×C7#54411yescharacteristic
399C4×C7#55111yescharacteristic
400C4×C7#55211yescharacteristic
401Dic7—42no—
402Dic7—42no—
403Dic7—81no—
404Dic7—81no—
405C24C2×C22⋊D711yescharacteristic
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—144no—
415C22×C4—141no—
416C22×C4—141no—
417C22×C4—141no—
418C22×C4—141no—
419C22×C4—282no—
420C22×C4—282no—
421C22×C4Q8×D712yes—
422C22×C4#116112yes—
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C23×D711yescharacteristic
429M16—284no—
430C2×C8—72no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—72no—
454C4⋊C4—72no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—144no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—282no—
465C4⋊C4—282no—
466C4⋊C4#115912yes—
467C22⋊C4—22no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—81no—
471C22⋊C4—282no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—42no—
477C42—282no—
478C42—282no—
479C2×C7—42no—
480C2×C7#243811yescharacteristic
481C2×C7#307711yescharacteristic
482C2×C7#308211yescharacteristic
483C2×C7#331711yescharacteristic
484C2×C7#333011yescharacteristic
485C2×C7#334511yescharacteristic
486C2×C7#335211yescharacteristic
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23#650912yes—
494C23C22×C22⋊D711yescharacteristic
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—142no—
525C2×C4—142no—
526C2×C4—288no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4#650912yes—
538C2×C4#651212yes—
539C2×C4C2×#115911yescharacteristic
540C2×C4#654811yescharacteristic
541C2×C4C2×Q8×D711yescharacteristic
542C2×C4C2×#116111yescharacteristic
543C2×C4D7×#4111yescharacteristic
544C2×C4#655411yescharacteristic
545C8—142no—
546C8—281no—
547C8—281no—
548C8—281no—
549C7#2601411yescharacteristic, complemented, Sylow
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22#9049711yescharacteristic
558C22C2×#650911yescharacteristic
559C22C2×#651211yescharacteristic
560C22#9051311yescharacteristic
561C22#9051511yescharacteristic
562C22#9052711yescharacteristic
563C22#9052911yescharacteristic
564C4—22no—
565C4—22no—
566C4—41no—
567C4—42no—
568C4—41no—
569C4—41no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—282no—
574C4—282no—
575C4—561no—
576C4—561no—
577C4#9031311yescharacteristic
578C4#9031411yescharacteristic
579C4#9032811yescharacteristic
580C4#9033311yescharacteristic
581C2—42no—
582C2#1174020711yescharacteristic
583C2#1174079511yescharacteristic
584C2#1174092511yescharacteristic
585C2#1174095911yescharacteristic
586C2#1174101811yescharacteristic
587C2#1174104211yescharacteristic
588C2#1174105311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2441C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2441D7
hypocenterC7#26014
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2441

Complements

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

NH ≅ G/Ntype
3. #11729493C2semidirect
4. #11729492C2semidirect
7. #11727378C2semidirect
8. #11727376C2semidirect
9. #11727357C2semidirect
10. #11727354C2semidirect
11. #11727326C2semidirect
12. #11727322C2semidirect
549. C7#26014semidirect

Maximal subgroups

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

subgroupindex
C7×#24412
#117309012
#117294932
#117294922
#117293352
#117293302
#117273782
#117273762
#117273572
#117273542
#117273262
#117273222
#117272672
#117272562
#117272042
#260147