Finite Group Databasebeta

← #13364203

Subgroups — #13364203

Order 1792.

754 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×#2314C211yesmaximal, characteristic
2#11739676C211yesmaximal, characteristic
3C2×#89771C211yesmaximal, characteristic
4#11727322C211yesmaximal, characteristic, complemented
5#11727321C211yesmaximal, characteristic, complemented
6#11727294C211yesmaximal, characteristic, complemented
7#11727294C211yesmaximal, characteristic, complemented
8#11727267C211yesmaximal, characteristic
9#11727267C211yesmaximal, characteristic
10#11727266C211yesmaximal, characteristic
11#11727265C211yesmaximal, characteristic
12#11727222C211yesmaximal, characteristic, complemented
13#11727222C211yesmaximal, characteristic, complemented
14#11727221C211yesmaximal, characteristic, complemented
15#11727220C211yesmaximal, characteristic, complemented
16C2×C7×#117C2211yescharacteristic
17C22×#1136C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19C7×#388—21no—
20C7×#388—21no—
21C7×#387—21no—
22C7×#387—21no—
23C7×#384C2211yescharacteristic
24C7×#382C2211yescharacteristic
25C7×#380C2211yescharacteristic
26C7×#380C2211yescharacteristic
27C4×C7×C4⋊C4C2211yescharacteristic
28C4×C7×C4⋊C4C2211yescharacteristic
29#89984C2211yescharacteristic
30#89984C2211yescharacteristic
31#89771C418yescomplemented
32#89753—21no—
33#89753—21no—
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6382C2211yescharacteristic
37C2×#6382C2211yescharacteristic
38C2×#6381C2211yescharacteristic
39C2×#6381C2211yescharacteristic
40C2×#6381C2211yescharacteristic
41C2×#6381C2211yescharacteristic
42#89723—21no—
43#89723—21no—
44#89708C2211yescharacteristic
45#89707C2211yescharacteristic
46C4×C4⋊Dic7C2211yescharacteristic
47C4×C4⋊Dic7C2211yescharacteristic
48C2×C4⋊(C7⋊C8)C2211yescharacteristic
49C2×C4⋊(C7⋊C8)C2211yescharacteristic
50C2×C4⋊(C7⋊C8)C2211yescharacteristic
51C2×C4⋊(C7⋊C8)C2211yescharacteristic
52#89324C2211yescharacteristic
53#89324C2211yescharacteristic
54#89324C2211yescharacteristic
55#89324C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#21966—71nomaximal, Sylow
61C23×C4×C7C2311yescharacteristic
62C7×#117C2×C414yes—
63C7×#117Q814yes—
64C2×C7×C4⋊C4—21no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4D411yescharacteristic
67C2×C7×C4⋊C4D411yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C4⋊C4C2311yescharacteristic
72C2×C7×C22⋊C4—21no—
73C2×C7×C22⋊C4—21no—
74C2×C7×C22⋊C4D411yescharacteristic
75C2×C7×C22⋊C4D411yescharacteristic
76C2×C42×C7—21no—
77C2×C42×C7—21no—
78C2×C42×C7C2311yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C22⋊Dic7—21no—
81C2×C22⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7—21no—
84C2×C4⋊Dic7C2311yescharacteristic
85C2×C4⋊Dic7C2311yescharacteristic
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2311yescharacteristic
88C2×C4×Dic7—21no—
89C2×C4×Dic7—21no—
90C2×#1136—41no—
91C2×#1136—41no—
92C2×#1136C2×C414yes—
93C22×C7⋊C8—41no—
94C22×C7⋊C8—41no—
95C22×C7⋊C8C2×C414yes—
96C22×C7⋊C8C2311yescharacteristic
97C22×C7⋊C8C2311yescharacteristic
98C22×C7⋊C8C2311yescharacteristic
99C22×C7⋊C8C2311yescharacteristic
100C7×#95—21no—
101C7×#95—21no—
102C7×#95—21no—
103C7×#95—21no—
104C7×#95—21no—
105C7×#95—21no—
106C7⋊#95—21no—
107C7⋊#95—21no—
108C7⋊#95—21no—
109C7⋊#95—21no—
110C7⋊#95—21no—
111C7⋊#95—21no—
112#6382—41no—
113#6382—41no—
114#6382—41no—
115#6382—41no—
116#6381—41no—
117#6381—41no—
118#6381—41no—
119#6381—41no—
120C4⋊(C7⋊C8)C2×C414yes—
121C4⋊(C7⋊C8)C2×C414yes—
122C4⋊(C7⋊C8)C2×C414yes—
123C4⋊(C7⋊C8)C2×C414yes—
124C2×#424—71no—
125#2314—71no—
126#2314D711yescharacteristic
127#1949—71no—
128#1949—71no—
129#1949—71no—
130#1949—71no—
131#1948—71no—
132#1948—71no—
133#1947—71no—
134#1947—71no—
135#1809—71no—
136#1809—71no—
137#1808—71no—
138#1807—71no—
139C24×C7C2×D411yescharacteristic
140C22×C4×C7—21no—
141C22×C4×C7—21no—
142C22×C4×C7—21no—
143C22×C4×C7—21no—
144C22×C4×C7—21no—
145C22×C4×C7—21no—
146C22×C4×C7—21no—
147C22×C4×C7—21no—
148C22×C4×C7C22×C412yes—
149C22×C4×C7C2×D411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×D411yescharacteristic
154C22×C4×C7C2×Q812yes—
155C22×C4×C7C2411yescharacteristic
156C22×Dic7—21no—
157C22×Dic7—21no—
158C22×Dic7—21no—
159C22×Dic7—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—21no—
167C7×C4⋊C4—21no—
168C7×C4⋊C4—81no—
169C7×C4⋊C4—81no—
170C7×C4⋊C4C4⋊C414yes—
171C7×C4⋊C4C4⋊C414yes—
172C7×C22⋊C4—81no—
173C7×C22⋊C4—81no—
174C7×C22⋊C4C4⋊C414yes—
175C7×C22⋊C4C4⋊C414yes—
176C42×C7—81no—
177C42×C7—81no—
178C42×C7C22×C412yes—
179C42×C7C22×C412yes—
180C42×C7C2×Q812yes—
181C42×C7C2×Q812yes—
182C22⋊Dic7—81no—
183C22⋊Dic7—81no—
184C4⋊Dic7—21no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—21no—
192C4⋊Dic7—81no—
193C4⋊Dic7—81no—
194C4×Dic7—81no—
195C4×Dic7—81no—
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—44no—
206C2×C7⋊C8C22×C412yes—
207C2×C7⋊C8C22×C412yes—
208C2×C7⋊C8C22×C412yes—
209C2×C7⋊C8C22×C412yes—
210C22×M16—71no—
211C23×C8—71no—
212C2×#117C2×D711yescharacteristic
213C2×#107—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#107—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×#106—71no—
220C2×#106—71no—
221#424—78no—
222C2×C4⋊C8—71no—
223C2×C4⋊C8—71no—
224C2×C4⋊C8—71no—
225C2×C4⋊C8—71no—
226#388—21no—
227#388—21no—
228#388—141no—
229#388—141no—
230#387—21no—
231#387—21no—
232#387—141no—
233#387—141no—
234#384—71no—
235#384C2×D711yescharacteristic
236#382—71no—
237#382C2×D711yescharacteristic
238#380—71no—
239#380—71no—
240#380C2×D711yescharacteristic
241#380C2×D711yescharacteristic
242C4×C4⋊C4—71no—
243C4×C4⋊C4—71no—
244C4×C4⋊C4C2×D711yescharacteristic
245C4×C4⋊C4C2×D711yescharacteristic
246#340—71no—
247#340—71no—
248#340—71no—
249#340—71no—
250#340—71no—
251#340—71no—
252#340—71no—
253#340—71no—
254C23×C7—21no—
255C23×C7—21no—
256C23×C7—21no—
257C23×C7—21no—
258C23×C7—21no—
259C23×C7—21no—
260C23×C7C2×C4⋊C412yes—
261C23×C7C22×D411yescharacteristic
262C2×C4×C7—21no—
263C2×C4×C7—21no—
264C2×C4×C7—21no—
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—22no—
273C2×C4×C7—22no—
274C2×C4×C7—22no—
275C2×C4×C7—22no—
276C2×C4×C7—41no—
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×C7—41no—
288C2×C4×C7C2×C4⋊C412yes—
289C2×C4×C7C2×C4⋊C412yes—
290C2×C4×C7C2×C4⋊C412yes—
291C2×C4×C7C2×C4⋊C412yes—
292C2×C4×C7C2×C4⋊C412yes—
293C2×C4×C7C23×C411yescharacteristic
294C2×C4×C7C22×Q811yescharacteristic
295C2×C4×C7H(5,𝔽2)11yescharacteristic
296C2×C4×C7H(5,𝔽2)11yescharacteristic
297C2×C4×C72−1+411yescharacteristic
298C2×C4×C72−1+411yescharacteristic
299C2×Dic7—41no—
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—
310C2×Dic7—41no—
311C7⋊C8—22no—
312C7⋊C8—22no—
313C7⋊C8—22no—
314C7⋊C8—22no—
315C23×C4C22×D711yescharacteristic
316C2×M16—74no—
317C2×M16—281no—
318C2×M16—281no—
319C22×C8—71no—
320C22×C8—71no—
321C22×C8—71no—
322C22×C8—71no—
323C22×C8—74no—
324C22×C8—281no—
325C22×C8—281no—
326#117Dic1414yes—
327#117C4×D714yes—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—71no—
332C2×C4⋊C4—71no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—141no—
335C2×C4⋊C4—141no—
336C2×C4⋊C4C22⋊D711yescharacteristic
337C2×C4⋊C4C22⋊D711yescharacteristic
338C2×C4⋊C4C22×D711yescharacteristic
339C2×C4⋊C4C22×D711yescharacteristic
340C2×C4⋊C4C22×D711yescharacteristic
341C2×C4⋊C4C22×D711yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—141no—
345C2×C22⋊C4—141no—
346C2×C22⋊C4C22⋊D711yescharacteristic
347C2×C22⋊C4C22⋊D711yescharacteristic
348C2×C42—21no—
349C2×C42—21no—
350C2×C42—141no—
351C2×C42—141no—
352C2×C42C22×D711yescharacteristic
353C2×C42C22×D711yescharacteristic
354#107—281no—
355#107—281no—
356#107—281no—
357#107—281no—
358#106—281no—
359#106—281no—
360#106—281no—
361#106—281no—
362C4⋊C8—74no—
363C4⋊C8—74no—
364C4⋊C8—74no—
365C4⋊C8—74no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377#95—141no—
378C22×C7—22no—
379C22×C7—22no—
380C22×C7—22no—
381C22×C7—41no—
382C22×C7—41no—
383C22×C7—41no—
384C22×C7—41no—
385C22×C7C22×C4⋊C411yescharacteristic
386C22×C7#53711yescharacteristic
387C22×C7#53711yescharacteristic
388C22×C7#57611yescharacteristic
389C22×C7#57711yescharacteristic
390C22×C7#57711yescharacteristic
391C22×C7#57811yescharacteristic
392C4×C7—22no—
393C4×C7—22no—
394C4×C7—22no—
395C4×C7—22no—
396C4×C7—22no—
397C4×C7—81no—
398C4×C7—81no—
399C4×C7—81no—
400C4×C7—81no—
401C4×C7#51811yescharacteristic
402C4×C7#51911yescharacteristic
403C4×C7#54411yescharacteristic
404C4×C7#54411yescharacteristic
405Dic7—81no—
406Dic7—81no—
407Dic7—81no—
408Dic7—81no—
409C24C2×C22⋊D711yescharacteristic
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—141no—
419C22×C4—141no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4C2×Dic1412yes—
423C22×C4C2×C4×D712yes—
424C22×C4C2×C22⋊D711yescharacteristic
425C22×C4C2×C22⋊D711yescharacteristic
426C22×C4C2×C22⋊D711yescharacteristic
427C22×C4C2×C22⋊D711yescharacteristic
428C22×C4C2×C22⋊D711yescharacteristic
429C22×C4C23×D711yescharacteristic
430M16—284no—
431C2×C8—72no—
432C2×C8—72no—
433C2×C8—72no—
434C2×C8—72no—
435C2×C8—141no—
436C2×C8—141no—
437C2×C8—141no—
438C2×C8—141no—
439C2×C8—141no—
440C2×C8—141no—
441C2×C8—141no—
442C2×C8—141no—
443C2×C8—284no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—141no—
455C4⋊C4—141no—
456C4⋊C4—141no—
457C4⋊C4—141no—
458C4⋊C4—141no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—561no—
463C4⋊C4—561no—
464C4⋊C4Dic7⋊C414yes—
465C4⋊C4Dic7⋊C414yes—
466C22⋊C4—81no—
467C22⋊C4—81no—
468C22⋊C4—561no—
469C22⋊C4—561no—
470C22⋊C4Dic7⋊C414yes—
471C22⋊C4Dic7⋊C414yes—
472C42—81no—
473C42—81no—
474C42—561no—
475C42—561no—
476C42C2×Dic1412yes—
477C42C2×Dic1412yes—
478C42C2×C4×D712yes—
479C42C2×C4×D712yes—
480C2×C7—42no—
481C2×C7#231611yescharacteristic
482C2×C7#291211yescharacteristic
483C2×C7#291311yescharacteristic
484C2×C7#325711yescharacteristic
485C2×C7#325711yescharacteristic
486C2×C7#325811yescharacteristic
487C2×C7#325811yescharacteristic
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23C2×Dic7⋊C412yes—
495C23C22×C22⋊D711yescharacteristic
496C2×C4—21no—
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—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
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—41no—
520C2×C4—41no—
521C2×C4—41no—
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—281no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4C2×Dic7⋊C412yes—
535C2×C4C2×Dic7⋊C412yes—
536C2×C4C2×Dic7⋊C412yes—
537C2×C4C2×Dic7⋊C412yes—
538C2×C4C2×Dic7⋊C412yes—
539C2×C4C22×Dic1411yescharacteristic
540C2×C4C22×C4×D711yescharacteristic
541C2×C4#655311yescharacteristic
542C2×C4#655311yescharacteristic
543C2×C4#655411yescharacteristic
544C2×C4#655411yescharacteristic
545C8—142no—
546C8—142no—
547C8—142no—
548C8—142no—
549C7#2196611yescharacteristic, complemented, Sylow
550C22—22no—
551C22—22no—
552C22—22no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22C22×Dic7⋊C411yescharacteristic
558C22#9052411yescharacteristic
559C22#9052411yescharacteristic
560C22#9052611yescharacteristic
561C22#9052711yescharacteristic
562C22#9052811yescharacteristic
563C22#9052911yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—81no—
570C4—81no—
571C4—81no—
572C4—81no—
573C4—561no—
574C4—561no—
575C4—561no—
576C4—561no—
577C4#9020711yescharacteristic
578C4#9021011yescharacteristic
579C4#9021011yescharacteristic
580C4#9021211yescharacteristic
581C2—42no—
582C2#1173967711yescharacteristic
583C2#1174000611yescharacteristic
584C2#1174000711yescharacteristic
585C2#1174088211yescharacteristic
586C2#1174089411yescharacteristic
587C2#1174091011yescharacteristic
588C2#1174091311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2314C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenter#2314D7
hypocenterC7#21966
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
4. #11727322C2semidirect
5. #11727321C2semidirect
6. #11727294C2semidirect
7. #11727294C2semidirect
12. #11727222C2semidirect
13. #11727222C2semidirect
14. #11727221C2semidirect
15. #11727220C2semidirect
31. #89771C4semidirect
549. C7#21966semidirect

Maximal subgroups

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

subgroupindex
C7×#23142
#117396762
C2×#897712
#117273222
#117273212
#117272942
#117272672
#117272662
#117272652
#117272222
#117272212
#117272202
#219667