Finite Group Databasebeta

← #13411858

Subgroups — #13411858

Order 1792.

696 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
1C7×#2346C211yesmaximal, characteristic
2#11730940C211yesmaximal, characteristic
3#11729335C211yesmaximal, characteristic
4#11729329C211yesmaximal, characteristic
5#11727378C211yesmaximal, characteristic, complemented
6#11727378C211yesmaximal, characteristic, complemented
7#11727367C211yesmaximal, characteristic, complemented
8#11727360C211yesmaximal, characteristic, complemented
9#11727267C211yesmaximal, characteristic
10#11727259C211yesmaximal, characteristic
11#11727245C211yesmaximal, characteristic, complemented
12#11727222C211yesmaximal, characteristic, complemented
13#11727213C211yesmaximal, characteristic, complemented
14#11727213C211yesmaximal, characteristic, complemented
15#11727204C211yesmaximal, characteristic
16C2×C7×#117C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C22×#1136C2211yescharacteristic
19C2×#6433C2211yescharacteristic
20C7×#389C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C7×#388—21no—
23C7×#388—21no—
24C7×#382—21no—
25C7×#380C2211yescharacteristic
26C4×C7×C4⋊C4C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C22⋊C4C2211yescharacteristic
29C7×#376—21no—
30#89984C2211yescharacteristic
31C2×C22⋊(C7⋊C8)C2211yescharacteristic
32C2×#6382C2211yescharacteristic
33C2×#6382C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6381C2211yescharacteristic
36#89718—21no—
37#89714—21no—
38#89708C2211yescharacteristic
39#89700C2211yescharacteristic
40#89700C2211yescharacteristic
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89432—21no—
44#89429—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#26019—71nomaximal, Sylow
60C23×C4×C7C2311yescharacteristic
61C23×Dic7—21no—
62C7×#117—22no—
63C7×#117Q814yes—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4C2311yescharacteristic
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C22⋊C4—21no—
71C2×C7×C22⋊C4—21no—
72C2×C7×C22⋊C4D411yescharacteristic
73C2×C7×C22⋊C4D411yescharacteristic
74C2×C42×C7—21no—
75C2×C42×C7—21no—
76C2×C42×C7—21no—
77C2×C42×C7C2311yescharacteristic
78C2×C42×C7C2311yescharacteristic
79C2×C22⋊Dic7—21no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—41no—
83C2×C4⋊Dic7C2311yescharacteristic
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×Dic7⋊C4—21no—
88C2×Dic7⋊C4—21no—
89C2×Dic7⋊C4—21no—
90C2×Dic7⋊C4—21no—
91C2×#1136—22no—
92C2×#1136—22no—
93C2×#1136—41no—
94C22×C7⋊C8C2311yescharacteristic
95C22×C7⋊C8C2311yescharacteristic
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98#6433—22no—
99#6433—41no—
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95D411yescharacteristic
106C7×#95D411yescharacteristic
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112C22⋊(C7⋊C8)—22no—
113#6382—22no—
114#6382—22no—
115#6382—41no—
116#6381—41no—
117C4⋊(C7⋊C8)—22no—
118C4⋊(C7⋊C8)—41no—
119#2442—71no—
120#2346D711yescharacteristic
121#2101—71no—
122#2088—71no—
123#2088—71no—
124#2059—71no—
125#1989—71no—
126#1949—71no—
127#1919—71no—
128#1919—71no—
129#1853—71no—
130#1851—71no—
131#1825—71no—
132#1809—71no—
133#1736—71no—
134C24×C7C2×D411yescharacteristic
135C22×C4×C7—21no—
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×C7C2×D411yescharacteristic
144C22×C4×C7C2×D411yescharacteristic
145C22×C4×C7C2×D411yescharacteristic
146C22×C4×C7C2×D411yescharacteristic
147C22×C4×C7C2×D411yescharacteristic
148C22×C4×C7C2×Q812yes—
149C22×C4×C7#4112yes—
150C22×C4×C7C2411yescharacteristic
151C22×Dic7—21no—
152C22×Dic7—21no—
153C22×Dic7—21no—
154C22×Dic7—21no—
155C22×Dic7—42no—
156C22×Dic7—42no—
157C22×Dic7—81no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4—21no—
160C7×C4⋊C4—21no—
161C7×C4⋊C4—21no—
162C7×C4⋊C4—21no—
163C7×C4⋊C4—21no—
164C7×C4⋊C4—81no—
165C7×C4⋊C4—81no—
166C7×C4⋊C4C2×Q812yes—
167C7×C4⋊C4C2×Q812yes—
168C7×C22⋊C4—24no—
169C7×C22⋊C4—41no—
170C7×C22⋊C4—41no—
171C7×C22⋊C4—81no—
172C42×C7—24no—
173C42×C7—21no—
174C42×C7—21no—
175C42×C7—21no—
176C42×C7—21no—
177C42×C7—42no—
178C42×C7—42no—
179C22⋊Dic7—42no—
180C4⋊Dic7—21no—
181C4⋊Dic7—21no—
182C4⋊Dic7—21no—
183C4⋊Dic7—21no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—42no—
188C4⋊Dic7#4112yes—
189Dic7⋊C4—42no—
190Dic7⋊C4—42no—
191Dic7⋊C4—81no—
192Dic7⋊C4—81no—
193#1136—44no—
194C2×C7⋊C8—21no—
195C2×C7⋊C8—21no—
196C2×C7⋊C8—21no—
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#4112yes—
206C22×M16—71no—
207C2×#125—71no—
208C2×#117C2×D711yescharacteristic
209C22×C4⋊C4—71no—
210C2×#107—71no—
211C2×#106—71no—
212C2×#106—71no—
213C2×#106—71no—
214C2×C4⋊C8—71no—
215C2×C4⋊C8—71no—
216C2×C22⋊C8—71no—
217#400—141no—
218#398—141no—
219#389—71no—
220#389—71no—
221#389C2×D711yescharacteristic
222#389C2×D711yescharacteristic
223#388—21no—
224#388—21no—
225#385—141no—
226#382—21no—
227#382—71no—
228#380—71no—
229#380—141no—
230#380C2×D711yescharacteristic
231C4×C4⋊C4C2×D711yescharacteristic
232C4×C4⋊C4C2×D711yescharacteristic
233C4×C22⋊C4C2×D711yescharacteristic
234#376—21no—
235#340—71no—
236#340—71no—
237#340—71no—
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#336—71no—
248#336—71no—
249C23×C7—21no—
250C23×C7—21no—
251C23×C7—21no—
252C23×C7—21no—
253C23×C7—21no—
254C23×C7—21no—
255C23×C7C22⋊Q812yes—
256C23×C7C22×D411yescharacteristic
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—22no—
260C2×C4×C7—22no—
261C2×C4×C7—22no—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
264C2×C4×C7—22no—
265C2×C4×C7—21no—
266C2×C4×C7—21no—
267C2×C4×C7—21no—
268C2×C4×C7—21no—
269C2×C4×C7—21no—
270C2×C4×C7—21no—
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—41no—
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×C7C22⋊Q812yes—
288C2×C4×C7C22×Q811yescharacteristic
289C2×C4×C7C2×#4111yescharacteristic
290C2×C4×C7C2×#4111yescharacteristic
291C2×C4×C7C2×#4111yescharacteristic
292C2×C4×C7H(5,𝔽2)11yescharacteristic
293C2×C4×C72−1+411yescharacteristic
294C2×Dic7—22no—
295C2×Dic7—41no—
296C2×Dic7—41no—
297C2×Dic7—41no—
298C2×Dic7—41no—
299C2×Dic7—41no—
300C2×Dic7—41no—
301C2×Dic7—41no—
302C2×Dic7—41no—
303C2×Dic7—41no—
304C2×Dic7—41no—
305C2×Dic7—41no—
306C2×Dic7—84no—
307C7⋊C8—22no—
308C7⋊C8—41no—
309C7⋊C8—41no—
310C7⋊C8—41no—
311C23×C4—141no—
312C23×C4C22×D711yescharacteristic
313C2×M16—142no—
314C2×M16—142no—
315C2×M16—281no—
316C22×C8—71no—
317C22×C8—71no—
318C22×C8—71no—
319C22×C8—71no—
320#125—142no—
321#125—281no—
322#117—22no—
323#117Dic1414yes—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—71no—
327C2×C4⋊C4—71no—
328C2×C4⋊C4—71no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—142no—
331C2×C4⋊C4—141no—
332C2×C4⋊C4—141no—
333C2×C4⋊C4—141no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4—281no—
337C2×C4⋊C4C22×D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—141no—
344C2×C22⋊C4C22⋊D711yescharacteristic
345C2×C22⋊C4C22⋊D711yescharacteristic
346C2×C42—21no—
347C2×C42—21no—
348C2×C42—21no—
349C2×C42C22×D711yescharacteristic
350C2×C42C22×D711yescharacteristic
351#107—281no—
352#106—142no—
353#106—142no—
354#106—281no—
355C4⋊C8—142no—
356C4⋊C8—281no—
357C22⋊C8—142no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—141no—
364#95—141no—
365#95—141no—
366#95—141no—
367#95—141no—
368#95C22⋊D711yescharacteristic
369#95C22⋊D711yescharacteristic
370C22×C7—22no—
371C22×C7—22no—
372C22×C7—22no—
373C22×C7—41no—
374C22×C7—41no—
375C22×C7—41no—
376C22×C7—41no—
377C22×C7C2×C22⋊Q811yescharacteristic
378C22×C7#54611yescharacteristic
379C22×C7#54711yescharacteristic
380C22×C7#57511yescharacteristic
381C22×C7#57511yescharacteristic
382C22×C7#57711yescharacteristic
383C22×C7#57711yescharacteristic
384C4×C7—22no—
385C4×C7—42no—
386C4×C7—42no—
387C4×C7—41no—
388C4×C7—41no—
389C4×C7—41no—
390C4×C7—41no—
391C4×C7—81no—
392C4×C7—81no—
393C4×C7#53311yescharacteristic
394C4×C7#54411yescharacteristic
395C4×C7#55111yescharacteristic
396C4×C7#55211yescharacteristic
397Dic7—42no—
398Dic7—81no—
399Dic7—81no—
400Dic7—81no—
401C24C2×C22⋊D711yescharacteristic
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—141no—
411C22×C4—141no—
412C22×C4—141no—
413C22×C4—141no—
414C22×C4—282no—
415C22×C4—282no—
416C22×C4—561no—
417C22×C4C2×Dic1412yes—
418C22×C4#115712yes—
419C22×C4C2×C22⋊D711yescharacteristic
420C22×C4C2×C22⋊D711yescharacteristic
421C22×C4C2×C22⋊D711yescharacteristic
422C22×C4C2×C22⋊D711yescharacteristic
423C22×C4C2×C22⋊D711yescharacteristic
424C22×C4C23×D711yescharacteristic
425M16—284no—
426C2×C8—72no—
427C2×C8—141no—
428C2×C8—141no—
429C2×C8—141no—
430C2×C8—141no—
431C2×C8—141no—
432C2×C8—141no—
433C2×C8—141no—
434C2×C8—141no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—72no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—141no—
448C4⋊C4—141no—
449C4⋊C4—141no—
450C4⋊C4—141no—
451C4⋊C4—141no—
452C4⋊C4—141no—
453C4⋊C4—141no—
454C4⋊C4—282no—
455C4⋊C4—282no—
456C4⋊C4—282no—
457C4⋊C4—561no—
458C4⋊C4—561no—
459C4⋊C4C2×Dic1412yes—
460C4⋊C4C2×Dic1412yes—
461C22⋊C4—24no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—81no—
465C22⋊C4—282no—
466C42—24no—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—42no—
472C42—42no—
473C2×C7—42no—
474C2×C7#243811yescharacteristic
475C2×C7#308011yescharacteristic
476C2×C7#308411yescharacteristic
477C2×C7#332111yescharacteristic
478C2×C7#332311yescharacteristic
479C2×C7#334811yescharacteristic
480C2×C7#335111yescharacteristic
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23#648712yes—
488C23C22×C22⋊D711yescharacteristic
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—142no—
520C2×C4—281no—
521C2×C4—281no—
522C2×C4—281no—
523C2×C4—281no—
524C2×C4—281no—
525C2×C4—281no—
526C2×C4—281no—
527C2×C4—281no—
528C2×C4—281no—
529C2×C4—281no—
530C2×C4—281no—
531C2×C4—564no—
532C2×C4#648712yes—
533C2×C4C22×Dic1411yescharacteristic
534C2×C4C2×#115711yescharacteristic
535C2×C4D7×#4111yescharacteristic
536C2×C4D7×#4111yescharacteristic
537C2×C4#655311yescharacteristic
538C2×C4#655411yescharacteristic
539C8—142no—
540C8—281no—
541C8—281no—
542C8—281no—
543C7#2601911yescharacteristic, complemented, Sylow
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22C2×#648711yescharacteristic
552C22#9051311yescharacteristic
553C22#9051311yescharacteristic
554C22#9051511yescharacteristic
555C22#9052011yescharacteristic
556C22#9052711yescharacteristic
557C22#9052811yescharacteristic
558C4—22no—
559C4—42no—
560C4—42no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—282no—
568C4—561no—
569C4—561no—
570C4—561no—
571C4#9020811yescharacteristic
572C4#9021011yescharacteristic
573C4#9021811yescharacteristic
574C4#9022211yescharacteristic
575C2—42no—
576C2#1173969211yescharacteristic
577C2#1173995411yescharacteristic
578C2#1174000511yescharacteristic
579C2#1174104111yescharacteristic
580C2#1174104611yescharacteristic
581C2#1174109411yescharacteristic
582C2#1174110111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2346C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenter#2346D7
hypocenterC7#26019
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2346

Complements

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

NH ≅ G/Ntype
5. #11727378C2semidirect
6. #11727378C2semidirect
7. #11727367C2semidirect
8. #11727360C2semidirect
11. #11727245C2semidirect
12. #11727222C2semidirect
13. #11727213C2semidirect
14. #11727213C2semidirect
543. C7#26019semidirect

Maximal subgroups

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

subgroupindex
C7×#23462
#117309402
#117293352
#117293292
#117273782
#117273672
#117273602
#117272672
#117272592
#117272452
#117272222
#117272132
#117272042
#260197