Finite Group Databasebeta

← #13371469

Subgroups — #13371469

Order 1792.

838 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D7×#106C211yesmaximal, characteristic
2#11733646C211yesmaximal, characteristic
3#11731212C211yesmaximal, characteristic
4C7×#1919C211yesmaximal, characteristic, complemented
5#11728311C211yesmaximal, characteristic, complemented
6#11727375C211yesmaximal, characteristic, complemented
7#11727235C211yesmaximal, characteristic, complemented
8#11726626C211yesmaximal, characteristic
9#11726625C211yesmaximal, characteristic
10#11726623C211yesmaximal, characteristic
11#11726615C211yesmaximal, characteristic, complemented
12#11726572C211yesmaximal, characteristic, complemented
13#11726540C211yesmaximal, characteristic, complemented
14#11726402C211yesmaximal, characteristic
15#11726377C211yesmaximal, characteristic, complemented
16C22×C8×D7C2211yescharacteristic
17C2×D7×C4⋊C4C2211yescharacteristic
18C2×D7×C4⋊C4C2211yescharacteristic
19C2×#6435—21no—
20C2×C7×#106C2211yescharacteristic
21C7×#389C2211yescharacteristic
22C4×C7×C4⋊C4C2211yescharacteristic
23C2×(C2×D7)⋊C8C2211yescharacteristic
24C2×#6391C2211yescharacteristic
25C2×#6389C2211yescharacteristic
26#89759—21no—
27#89759C2211yescharacteristic
28#89759C2211yescharacteristic
29#89757C2211yescharacteristic
30#89750—21no—
31C4⋊C4×Dic7C2211yescharacteristic
32C4⋊C4×Dic7C2211yescharacteristic
33#89745C2211yescharacteristic
34C2×#6382C2211yescharacteristic
35C2×#6382C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37C2×#6381C2211yescharacteristic
38C4×(C2×D7)⋊C4C2211yescharacteristic
39C4×C4⋊Dic7C2211yescharacteristic
40#89705—21no—
41#89700C2211yescharacteristic
42C2×#6377C2211yescharacteristic
43D7×#106C418yescomplemented
44#89438—21no—
45#89436—21no—
46#89427—21no—
47#89424—21no—
48C7×#340C2211yescharacteristic
49C7×#340C2211yescharacteristic
50C7×#340C2211yescharacteristic
51C7×#336C2211yescharacteristic
52#89346C2211yescharacteristic
53#89346C2211yescharacteristic
54#89346C2211yescharacteristic
55#89345C2211yescharacteristic
56#89324C2211yescharacteristic
57#89324C2211yescharacteristic
58#89324C2211yescharacteristic
59#89324C2211yescharacteristic
60#22615—71nomaximal, Sylow
61C22×C4×D7—21no—
62C22×C4×D7—21no—
63C22×C4×D7C2311yescharacteristic
64C22×C7×C8C2311yescharacteristic
65C22×C7×C8C2311yescharacteristic
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4C2311yescharacteristic
68C2×C7×C4⋊C4C2311yescharacteristic
69C2×C7×C4⋊C4C2311yescharacteristic
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C42×C7—21no—
72C2×C42×C7C2311yescharacteristic
73C2×(C2×D7)⋊C4—21no—
74C2×(C2×D7)⋊C4—21no—
75C2×(C2×D7)⋊C4D411yescharacteristic
76C2×(C2×D7)⋊C4D411yescharacteristic
77C2×C4⋊Dic7—21no—
78C2×C4⋊Dic7C2311yescharacteristic
79C2×C4⋊Dic7C2311yescharacteristic
80C2×C4⋊Dic7C2311yescharacteristic
81C2×C4⋊Dic7C2311yescharacteristic
82C2×Dic7⋊C4—21no—
83C2×Dic7⋊C4—21no—
84C2×Dic7⋊C4—21no—
85C2×Dic7⋊C4—21no—
86C2×C4×Dic7—21no—
87C2×C4×Dic7—21no—
88C2×C4×Dic7—21no—
89C2×C4×Dic7C2311yescharacteristic
90C22×C7⋊C8C2311yescharacteristic
91C22×C7⋊C8C2311yescharacteristic
92C2×C8×D7—22no—
93C2×C8×D7—22no—
94C2×C8×D7C2×C414yes—
95D7×C4⋊C4—22no—
96D7×C4⋊C4—22no—
97D7×C4⋊C4C2×C414yes—
98D7×C4⋊C4C2×C414yes—
99#6435—44no—
100C7×#106C2×C414yes—
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⋊#95D411yescharacteristic
112C7⋊#95D411yescharacteristic
113(C2×D7)⋊C8—22no—
114#6391C2×C414yes—
115#6389—22no—
116#6382C2×C414yes—
117#6382C2×C414yes—
118#6381—22no—
119#6381—22no—
120#6377—22no—
121C22×#106—71no—
122#2345—71no—
123#2344—71no—
124#1944—71no—
125#1944—71no—
126#1920—71no—
127#1920—71no—
128#1919—71no—
129#1919D711yescharacteristic, complemented
130#1824—71no—
131#1823—71no—
132#1808—71no—
133#1776—71no—
134#1776—71no—
135#1752—71no—
136C23×D7C2×D411yescharacteristic
137C22×C4×C7—21no—
138C22×C4×C7—21no—
139C22×C4×C7—21no—
140C22×C4×C7—21no—
141C22×C4×C7C2×D411yescharacteristic
142C22×C4×C7C2×D411yescharacteristic
143C22×C4×C7C2411yescharacteristic
144C22×Dic7—21no—
145C22×Dic7—21no—
146C22×Dic7—21no—
147C22×Dic7—21no—
148C22×Dic7C2×D411yescharacteristic
149C22×Dic7C2×D411yescharacteristic
150C22×Dic7C2×D411yescharacteristic
151C2×C4×D7—21no—
152C2×C4×D7—21no—
153C2×C4×D7—21no—
154C2×C4×D7—21no—
155C2×C4×D7—24no—
156C2×C4×D7—24no—
157C2×C4×D7—42no—
158C2×C4×D7—42no—
159C2×C4×D7—81no—
160C2×C4×D7—81no—
161C2×C4×D7C22×C412yes—
162C2×C4×D7#4112yes—
163C2×C7×C8—21no—
164C2×C7×C8—21no—
165C2×C7×C8—21no—
166C2×C7×C8—21no—
167C2×C7×C8—21no—
168C2×C7×C8C22×C412yes—
169C7×C4⋊C4—21no—
170C7×C4⋊C4—21no—
171C7×C4⋊C4—21no—
172C7×C4⋊C4—21no—
173C7×C4⋊C4—21no—
174C7×C4⋊C4—21no—
175C7×C4⋊C4—81no—
176C7×C4⋊C4C22×C412yes—
177C7×C4⋊C4C22×C412yes—
178C42×C7—21no—
179C42×C7—21no—
180C42×C7—42no—
181(C2×D7)⋊C4—22no—
182(C2×D7)⋊C4—22no—
183(C2×D7)⋊C4—42no—
184(C2×D7)⋊C4—42no—
185C4⋊Dic7—21no—
186C4⋊Dic7—21no—
187C4⋊Dic7—21no—
188C4⋊Dic7—21no—
189C4⋊Dic7—21no—
190C4⋊Dic7—21no—
191C4⋊Dic7—81no—
192C4⋊Dic7C22×C412yes—
193C4⋊Dic7C22×C412yes—
194Dic7⋊C4—24no—
195Dic7⋊C4—24no—
196Dic7⋊C4—42no—
197Dic7⋊C4—42no—
198C4×Dic7—21no—
199C4×Dic7—21no—
200C4×Dic7—42no—
201C4×Dic7—81no—
202C4×Dic7—81no—
203C2×C7⋊C8—21no—
204C2×C7⋊C8—21no—
205C2×C7⋊C8—21no—
206C2×C7⋊C8C22×C412yes—
207C2×C7⋊C8#4112yes—
208C2×C7⋊C8#4112yes—
209C8×D7—28no—
210C23×C8—71no—
211C2×#117—141no—
212C22×C4⋊C4—71no—
213C22×C4⋊C4—71no—
214C2×#107—71no—
215C2×#107—71no—
216C2×#106—71no—
217C2×#106—71no—
218C2×#106—71no—
219C2×#106—78no—
220C2×#106C2×D711yescharacteristic
221C2×C22⋊C8—71no—
222C2×#97—71no—
223C2×#97—71no—
224#389—71no—
225#389—71no—
226#389—71no—
227#389C2×D711yescharacteristic
228#387—141no—
229#387—141no—
230#387—141no—
231#387—141no—
232#381—141no—
233#380—71no—
234#380—71no—
235#380—141no—
236C4×C4⋊C4—71no—
237C4×C4⋊C4—71no—
238C4×C4⋊C4—71no—
239C4×C4⋊C4C2×D711yescharacteristic
240C4×C22⋊C4—71no—
241C4×C22⋊C4—141no—
242#340—71no—
243#340—71no—
244#340—71no—
245#340—71no—
246#340—71no—
247#340—71no—
248#340—71no—
249#340C2×D711yescharacteristic
250#340C2×D711yescharacteristic
251#340C2×D711yescharacteristic
252#336—71no—
253#336C2×D711yescharacteristic
254C23×C7C22×D411yescharacteristic
255C22×D7—21no—
256C22×D7—21no—
257C22×D7—21no—
258C22×D7—21no—
259C22×D7C4×D412yes—
260C22×D7AGL(1,ℤ/8)12yes—
261C22×D7#13712yes—
262C2×C4×C7—22no—
263C2×C4×C7—22no—
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—41no—
271C2×C4×C7—41no—
272C2×C4×C7—41no—
273C2×C4×C7—41no—
274C2×C4×C7—41no—
275C2×C4×C7—41no—
276C2×C4×C7—41no—
277C2×C4×C7—41no—
278C2×C4×C7—41no—
279C2×C4×C7—41no—
280C2×C4×C7C23×C411yescharacteristic
281C2×C4×C7C2×#4111yescharacteristic
282C2×C4×C7C2×#4111yescharacteristic
283C2×C4×C7C2×#4111yescharacteristic
284C2×C4×C72−1+411yescharacteristic
285C2×C4×C72−1+411yescharacteristic
286C2×Dic7—21no—
287C2×Dic7—21no—
288C2×Dic7—22no—
289C2×Dic7—22no—
290C2×Dic7—21no—
291C2×Dic7—21no—
292C2×Dic7—21no—
293C2×Dic7—21no—
294C2×Dic7—21no—
295C2×Dic7—21no—
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×Dic7C4×D412yes—
307C4×D7—22no—
308C4×D7—22no—
309C4×D7—22no—
310C4×D7—22no—
311C4×D7—48no—
312C4×D7—84no—
313C7×C8—22no—
314C7×C8—41no—
315C7⋊C8—41no—
316C7⋊C8#11714yescomplemented
317C23×C4—71no—
318C23×C4—141no—
319C23×C4—141no—
320C22×C8—71no—
321C22×C8—74no—
322C22×C8—71no—
323C22×C8—142no—
324C22×C8—142no—
325C22×C8C22×D711yescharacteristic
326C22×C8C22×D711yescharacteristic
327#117—284no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—71no—
330C2×C4⋊C4—71no—
331C2×C4⋊C4—74no—
332C2×C4⋊C4—74no—
333C2×C4⋊C4—71no—
334C2×C4⋊C4—71no—
335C2×C4⋊C4—142no—
336C2×C4⋊C4—142no—
337C2×C4⋊C4—141no—
338C2×C4⋊C4—141no—
339C2×C4⋊C4—141no—
340C2×C4⋊C4—141no—
341C2×C4⋊C4—141no—
342C2×C4⋊C4C22×D711yescharacteristic
343C2×C4⋊C4C22×D711yescharacteristic
344C2×C4⋊C4C22×D711yescharacteristic
345C2×C4⋊C4C22×D711yescharacteristic
346C2×C22⋊C4—71no—
347C2×C22⋊C4—71no—
348C2×C22⋊C4—141no—
349C2×C22⋊C4—141no—
350C2×C42—21no—
351C2×C42—71no—
352C2×C42—141no—
353C2×C42—141no—
354C2×C42—141no—
355C2×C42C22×D711yescharacteristic
356#107—142no—
357#107—142no—
358#106—74no—
359#106—74no—
360#106—74no—
361#106C4×D714yes—
362C22⋊C8—142no—
363#97—142no—
364#97—142no—
365#95—21no—
366#95—21no—
367#95—71no—
368#95—71no—
369#95—141no—
370#95—141no—
371#95—141no—
372#95—141no—
373#95—141no—
374#95—141no—
375#95—141no—
376#95—141no—
377C22×C7C2×C4×D411yescharacteristic
378C22×C7#54711yescharacteristic
379C22×C7#54711yescharacteristic
380C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
381C22×C7C2×#13711yescharacteristic
382C22×C7#57711yescharacteristic
383C22×C7#57711yescharacteristic
384C2×D7—22no—
385C2×D7—22no—
386C2×D7—22no—
387C2×D7—22no—
388C2×D7—22no—
389C2×D7—22no—
390C2×D7#44014yes—
391C4×C7—41no—
392C4×C7—41no—
393C4×C7—42no—
394C4×C7—42no—
395C4×C7—81no—
396C4×C7—81no—
397C4×C7C2×#11711yescharacteristic
398C4×C7#51911yescharacteristic
399C4×C7#55211yescharacteristic
400C4×C7#55211yescharacteristic
401Dic7—22no—
402Dic7—41no—
403Dic7—41no—
404Dic7—42no—
405Dic7—42no—
406Dic7—81no—
407Dic7—81no—
408C24—71no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—71no—
414C22×C4—72no—
415C22×C4—72no—
416C22×C4—71no—
417C22×C4—71no—
418C22×C4—144no—
419C22×C4—144no—
420C22×C4—141no—
421C22×C4—141no—
422C22×C4—141no—
423C22×C4—141no—
424C22×C4—141no—
425C22×C4—141no—
426C22×C4—141no—
427C22×C4—141no—
428C22×C4—282no—
429C22×C4—282no—
430C22×C4—561no—
431C22×C4—561no—
432C22×C4D4×D711yescharacteristic
433C22×C4D4×D711yescharacteristic
434C22×C4C23×D711yescharacteristic
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—72no—
441C2×C8—72no—
442C2×C8—72no—
443C2×C8—148no—
444C2×C8—141no—
445C2×C8—141no—
446C2×C8—141no—
447C2×C8C2×C4×D712yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—72no—
455C4⋊C4—72no—
456C4⋊C4—81no—
457C4⋊C4—144no—
458C4⋊C4—144no—
459C4⋊C4—141no—
460C4⋊C4—141no—
461C4⋊C4—141no—
462C4⋊C4—141no—
463C4⋊C4—141no—
464C4⋊C4—141no—
465C4⋊C4—282no—
466C4⋊C4—282no—
467C4⋊C4—561no—
468C4⋊C4C2×C4×D712yes—
469C4⋊C4C2×C4×D712yes—
470C22⋊C4—142no—
471C22⋊C4—142no—
472C22⋊C4—282no—
473C22⋊C4—282no—
474C42—21no—
475C42—21no—
476C42—42no—
477C42—141no—
478C42—141no—
479C42—282no—
480C42—561no—
481C42—561no—
482C2×C7#234411yescharacteristic
483C2×C7C2×#44011yescharacteristic
484C2×C7#294911yescharacteristic
485C2×C7#334411yescharacteristic
486C2×C7#334411yescharacteristic
487C2×C7#334511yescharacteristic
488C2×C7#334511yescharacteristic
489D7—24no—
490C23—72no—
491C23—72no—
492C23—72no—
493C23—141no—
494C23—141no—
495C23—141no—
496C23—141no—
497C23C2×D4×D711yescharacteristic
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—72no—
517C2×C4—141no—
518C2×C4—141no—
519C2×C4—142no—
520C2×C4—142no—
521C2×C4—142no—
522C2×C4—142no—
523C2×C4—141no—
524C2×C4—141no—
525C2×C4—141no—
526C2×C4—141no—
527C2×C4—142no—
528C2×C4—142no—
529C2×C4—141no—
530C2×C4—141no—
531C2×C4—288no—
532C2×C4—281no—
533C2×C4—281no—
534C2×C4—281no—
535C2×C4—281no—
536C2×C4—281no—
537C2×C4—281no—
538C2×C4—281no—
539C2×C4—281no—
540C2×C4—281no—
541C2×C4—281no—
542C2×C4—564no—
543C2×C4C22×C4×D711yescharacteristic
544C2×C4#655111yescharacteristic
545C2×C4D7×#4111yescharacteristic
546C2×C4D7×#4111yescharacteristic
547C2×C4D7×#4111yescharacteristic
548C2×C4#655411yescharacteristic
549C8—22no—
550C8—41no—
551C8—74no—
552C8—281no—
553C7#2261511yescharacteristic, complemented, Sylow
554C22—74no—
555C22—142no—
556C22—142no—
557C22—142no—
558C22—142no—
559C22—142no—
560C22—142no—
561C22C4×D4×D711yescharacteristic
562C22#9032211yescharacteristic
563C22#9034611yescharacteristic
564C22#9045311yescharacteristic
565C22#9046111yescharacteristic
566C22D7×AGL(1,ℤ/8)11yescharacteristic
567C22D7×#13711yescharacteristic
568C4—42no—
569C4—41no—
570C4—42no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—142no—
575C4—282no—
576C4—281no—
577C4—281no—
578C4—282no—
579C4—561no—
580C4—561no—
581C4D7×#11711yescharacteristic
582C4#9026311yescharacteristic
583C4#9038411yescharacteristic
584C4#9039911yescharacteristic
585C2—144no—
586C2#1173240611yescharacteristic
587C2#1173709411yescharacteristic
588C2D7×#44011yescharacteristic
589C2#1173847511yescharacteristic
590C2#1173864211yescharacteristic
591C2#1173878011yescharacteristic
592C2#1173897911yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1919C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1919D7
hypocenterC7#22615
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. C7×#1919C2semidirect
5. #11728311C2semidirect
6. #11727375C2semidirect
7. #11727235C2semidirect
11. #11726615C2semidirect
12. #11726572C2semidirect
13. #11726540C2semidirect
15. #11726377C2semidirect
43. D7×#106C4semidirect
129. #1919D7semidirect
316. C7⋊C8#117semidirect
553. C7#22615semidirect

Maximal subgroups

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

subgroupindex
C2×D7×#1062
#117336462
#117312122
C7×#19192
#117283112
#117273752
#117272352
#117266262
#117266252
#117266232
#117266152
#117265722
#117265402
#117264022
#117263772
#226157