Finite Group Databasebeta

← #13323002

Subgroups — #13323002

Order 1792.

572 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2989C211yesmaximal, characteristic
2#11739966C211yesmaximal, characteristic
3#11739958C211yesmaximal, characteristic
4#11728551C211yesmaximal, characteristic, complemented
5#11728550C211yesmaximal, characteristic, complemented
6#11728549C211yesmaximal, characteristic, complemented
7#11728548C211yesmaximal, characteristic, complemented
8#11728502C211yesmaximal, characteristic, complemented
9#11728501C211yesmaximal, characteristic, complemented
10#11728249C211yesmaximal, characteristic, complemented
11#11728247C211yesmaximal, characteristic, complemented
12#11728246C211yesmaximal, characteristic, complemented
13#11728244C211yesmaximal, characteristic, complemented
14#11728177C211yesmaximal, characteristic, complemented
15#11728175C211yesmaximal, characteristic, complemented
16C4×C7×#41C2211yescharacteristic
17C7×#446—21no—
18C7×#436—21no—
19C7×#435—21no—
20C7×#435—21no—
21C7×#432C2211yescharacteristic
22C7×#432C2211yescharacteristic
23C7×#431C2211yescharacteristic
24C7×#423C2211yescharacteristic
25C7×#423C2211yescharacteristic
26C2×C7×C4⋊C8C2211yescharacteristic
27#90006C412yes—
28#89796—21no—
29#89784—21no—
30#89783—21no—
31#89783—21no—
32#89772C412yes—
33#89772C412yes—
34#89771C412yes—
35#89698C2211yescharacteristic
36#89698C2211yescharacteristic
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89697C2211yescharacteristic
40#89697C2211yescharacteristic
41#89697C2211yescharacteristic
42#89696C2211yescharacteristic
43#89696C2211yescharacteristic
44C2×C4⋊(C7⋊C8)C2211yescharacteristic
45C4×#1136C2211yescharacteristic
46C2×C4×C7⋊C8C2211yescharacteristic
47#89275C2211yescharacteristic
48#89275C2211yescharacteristic
49#89275C2211yescharacteristic
50#89275C2211yescharacteristic
51#89275C2211yescharacteristic
52#89275C2211yescharacteristic
53#89275C2211yescharacteristic
54#89275C2211yescharacteristic
55#89275C2211yescharacteristic
56#89275C2211yescharacteristic
57#89274C2211yescharacteristic
58#89274C2211yescharacteristic
59#89274C2211yescharacteristic
60#89274C2211yescharacteristic
61#89274C2211yescharacteristic
62#89274C2211yescharacteristic
63#17824—71nomaximal, Sylow
64C2×C7×#41C2×C411yescharacteristic
65C2×C7×M16—21no—
66C2×C7×M16—21no—
67C22×C7×C8—21no—
68C4×C7×Q8D411yescharacteristic
69C4×C7×Q8Q811yescharacteristic
70C4×C7×D4D411yescharacteristic
71C4×C7×D4D411yescharacteristic
72C4×C7×D4D411yescharacteristic
73C4×C7×D4Q811yescharacteristic
74C4×C7×D4Q811yescharacteristic
75C4×C7×D4Q811yescharacteristic
76C7×#117C2×C411yescharacteristic
77C7×#117C2×C411yescharacteristic
78C7×#117C2×C411yescharacteristic
79C2×C42×C7C2311yescharacteristic
80C2×C42×C7C2311yescharacteristic
81C2×C42×C7C2311yescharacteristic
82C2×#1136—21no—
83C2×#1136—21no—
84C2×#1136C2×C412yes—
85C22×C7⋊C8—21no—
86C22×C7⋊C8C2×C412yes—
87C7×C4⋊C8—21no—
88C7×C4⋊C8—21no—
89C7×C4⋊C8—21no—
90C7×C4⋊C8C2311yescharacteristic
91C7×C4⋊C8C2311yescharacteristic
92C7×C4⋊C8C2311yescharacteristic
93C7×C4⋊C8C2311yescharacteristic
94C7×C22⋊C8—21no—
95C7×C22⋊C8—21no—
96C7×C22⋊C8—21no—
97C7×#97—21no—
98C7×#97—21no—
99C4×C7×C8—21no—
100C22⋊(C7⋊C8)—21no—
101C22⋊(C7⋊C8)—21no—
102C22⋊(C7⋊C8)—21no—
103C22⋊(C7⋊C8)C2×C412yes—
104C22⋊(C7⋊C8)C2×C412yes—
105C22⋊(C7⋊C8)C2×C412yes—
106C22⋊(C7⋊C8)C2×C412yes—
107C4⋊(C7⋊C8)—21no—
108C4⋊(C7⋊C8)—21no—
109C4⋊(C7⋊C8)—21no—
110C4⋊(C7⋊C8)C2×C412yes—
111C4⋊(C7⋊C8)C2×C412yes—
112C4⋊(C7⋊C8)C2×C412yes—
113C4⋊(C7⋊C8)C2×C412yes—
114C4⋊(C7⋊C8)C2311yescharacteristic
115C4⋊(C7⋊C8)C2311yescharacteristic
116C4⋊(C7⋊C8)C2311yescharacteristic
117C4⋊(C7⋊C8)C2311yescharacteristic
118#6377—21no—
119#6377—21no—
120#6377C2×C412yes—
121C4×C7⋊C8—21no—
122C4×C7⋊C8C2×C412yes—
123C4×C7⋊C8C2311yescharacteristic
124C4×C7⋊C8C2311yescharacteristic
125C4×C7⋊C8C2311yescharacteristic
126C4×C7⋊C8C2311yescharacteristic
127#2989—71no—
128#2989D711yescharacteristic
129#2922—71no—
130#1574—71no—
131#1574—71no—
132#1573—71no—
133#1573—71no—
134#1572—71no—
135#1572—71no—
136#1570—71no—
137#1570—71no—
138#1569—71no—
139#1569—71no—
140#1568—71no—
141#1566—71no—
142C7×#41—42no—
143C2×C7×Q8C4⋊C411yescharacteristic
144C2×C7×D4C4⋊C411yescharacteristic
145C2×C7×D4C4⋊C411yescharacteristic
146C2×C7×D4C4⋊C411yescharacteristic
147C22×C4×C7C22×C411yescharacteristic
148C22×C4×C7C22×C411yescharacteristic
149C22×C4×C7C22×C411yescharacteristic
150C22×C4×C7C2×D411yescharacteristic
151C22×C4×C7C2×D411yescharacteristic
152C22×C4×C7C2×D411yescharacteristic
153C22×C4×C7C2×Q811yescharacteristic
154C22×C4×C7C2×Q811yescharacteristic
155C22×C4×C7C2×Q811yescharacteristic
156C7×M16—81no—
157C7×M16—81no—
158C2×C7×C8—21no—
159C2×C7×C8—21no—
160C2×C7×C8—21no—
161C2×C7×C8—21no—
162C2×C7×C8—81no—
163C7×C4⋊C4C4⋊C411yescharacteristic
164C7×C4⋊C4C4⋊C411yescharacteristic
165C7×C4⋊C4C4⋊C411yescharacteristic
166C7×C4⋊C4C4⋊C411yescharacteristic
167C7×C4⋊C4C4⋊C411yescharacteristic
168C7×C4⋊C4C4⋊C411yescharacteristic
169C7×C22⋊C4C4⋊C411yescharacteristic
170C7×C22⋊C4C4⋊C411yescharacteristic
171C7×C22⋊C4C4⋊C411yescharacteristic
172C7×C22⋊C4C4⋊C411yescharacteristic
173C7×C22⋊C4C4⋊C411yescharacteristic
174C7×C22⋊C4C4⋊C411yescharacteristic
175C42×C7C22×C411yescharacteristic
176C42×C7C22×C411yescharacteristic
177C42×C7C22×C411yescharacteristic
178C42×C7C2×D411yescharacteristic
179C42×C7C2×D411yescharacteristic
180C42×C7C2×D411yescharacteristic
181C42×C7C2×Q811yescharacteristic
182C42×C7C2×Q811yescharacteristic
183C42×C7C2×Q811yescharacteristic
184C42×C7C2411yescharacteristic
185#1136—42no—
186#1136—81no—
187#1136—81no—
188C2×C7⋊C8—21no—
189C2×C7⋊C8—21no—
190C2×C7⋊C8—21no—
191C2×C7⋊C8—21no—
192C2×C7⋊C8—42no—
193C2×C7⋊C8—81no—
194C2×C7⋊C8C22×C412yes—
195C2×C7⋊C8C22×C412yes—
196C2×C7⋊C8C22×C412yes—
197C2×C7⋊C8C22×C412yes—
198C4×#41C2×D711yescharacteristic
199#446—21no—
200#446—141no—
201#436—21no—
202#436—141no—
203#435—21no—
204#435—21no—
205#435—141no—
206#435—141no—
207#433—72no—
208#433—72no—
209#432—71no—
210#432—71no—
211#432C2×D711yescharacteristic
212#432C2×D711yescharacteristic
213#431—71no—
214#431—71no—
215#431—71no—
216#431—71no—
217#431—71no—
218#431C2×D711yescharacteristic
219#424—72no—
220#423—71no—
221#423—71no—
222#423C2×D711yescharacteristic
223#423C2×D711yescharacteristic
224C2×C4⋊C8—71no—
225C2×C4⋊C8C2×D711yescharacteristic
226#408—72no—
227C4×M16—71no—
228C2×C4×C8—71no—
229#335—71no—
230#335—71no—
231#335—71no—
232#335—71no—
233#335—71no—
234#335—71no—
235#334—71no—
236#334—71no—
237#334—71no—
238#334—71no—
239#334—71no—
240#334—71no—
241#334—71no—
242#334—71no—
243#334—71no—
244#334—71no—
245C23×C7C2×C4⋊C411yescharacteristic
246C23×C7C2×C4⋊C411yescharacteristic
247C23×C7C2×C4⋊C411yescharacteristic
248C7×Q8—41no—
249C7×D4—41no—
250C7×D4—41no—
251C7×D4—41no—
252C2×C4×C7—22no—
253C2×C4×C7—22no—
254C2×C4×C7—22no—
255C2×C4×C7—21no—
256C2×C4×C7—21no—
257C2×C4×C7—21no—
258C2×C4×C7—21no—
259C2×C4×C7—21no—
260C2×C4×C7—21no—
261C2×C4×C7—41no—
262C2×C4×C7—41no—
263C2×C4×C7—41no—
264C2×C4×C7C2×C4⋊C411yescharacteristic
265C2×C4×C7C2×C4⋊C411yescharacteristic
266C2×C4×C7C2×C4⋊C411yescharacteristic
267C2×C4×C7C2×C4⋊C411yescharacteristic
268C2×C4×C7C2×C4⋊C411yescharacteristic
269C2×C4×C7C2×C4⋊C411yescharacteristic
270C2×C4×C7C2×C4⋊C411yescharacteristic
271C2×C4×C7C2×C4⋊C411yescharacteristic
272C2×C4×C7C2×C4⋊C411yescharacteristic
273C2×C4×C7C23×C411yescharacteristic
274C2×C4×C7C22×D411yescharacteristic
275C2×C4×C7C22×Q811yescharacteristic
276C7×C8—41no—
277C7×C8—41no—
278C7×C8—41no—
279C7×C8—41no—
280C7⋊C8—22no—
281C7⋊C8—22no—
282C7⋊C8—22no—
283C7⋊C8—22no—
284C7⋊C8—41no—
285C7⋊C8—41no—
286C7⋊C8—41no—
287C7⋊C8—41no—
288C2×#41C4×D711yescharacteristic
289C2×M16—21no—
290C2×M16—21no—
291C2×M16—72no—
292C2×M16—141no—
293C2×M16—141no—
294C22×C8—21no—
295C22×C8—72no—
296C22×C8—141no—
297C4×Q8Dic1411yescharacteristic
298C4×Q8C22⋊D711yescharacteristic
299C4×D4Dic1411yescharacteristic
300C4×D4Dic1411yescharacteristic
301C4×D4Dic1411yescharacteristic
302C4×D4C22⋊D711yescharacteristic
303C4×D4C22⋊D711yescharacteristic
304C4×D4C22⋊D711yescharacteristic
305#117C4×D711yescharacteristic
306#117C4×D711yescharacteristic
307#117C4×D711yescharacteristic
308C2×C42C22×D711yescharacteristic
309C2×C42C22×D711yescharacteristic
310C2×C42C22×D711yescharacteristic
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—72no—
315C4⋊C8—72no—
316C4⋊C8—72no—
317C4⋊C8—72no—
318C4⋊C8—71no—
319C4⋊C8—71no—
320C4⋊C8—71no—
321C4⋊C8—71no—
322C4⋊C8—141no—
323C4⋊C8—141no—
324C4⋊C8—141no—
325C4⋊C8C22×D711yescharacteristic
326C4⋊C8C22×D711yescharacteristic
327C4⋊C8C22×D711yescharacteristic
328C4⋊C8C22×D711yescharacteristic
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C22⋊C8—72no—
333C22⋊C8—72no—
334C22⋊C8—72no—
335C22⋊C8—72no—
336C22⋊C8—141no—
337C22⋊C8—141no—
338C22⋊C8—141no—
339#97—21no—
340#97—21no—
341#97—72no—
342#97—141no—
343#97—141no—
344C4×C8—21no—
345C4×C8—72no—
346C4×C8—71no—
347C4×C8—71no—
348C4×C8—71no—
349C4×C8—71no—
350C4×C8—141no—
351C22×C7—21no—
352C22×C7—21no—
353C22×C7—21no—
354C22×C7—21no—
355C22×C7—21no—
356C22×C7—21no—
357C22×C7—21no—
358C22×C7—21no—
359C22×C7—21no—
360C22×C7C22×C4⋊C411yescharacteristic
361C4×C7—21no—
362C4×C7—21no—
363C4×C7—21no—
364C4×C7—21no—
365C4×C7—21no—
366C4×C7—21no—
367C4×C7—21no—
368C4×C7—21no—
369C4×C7—21no—
370C4×C7—21no—
371C4×C7#56812yes—
372C4×C7#57711yescharacteristic
373C4×C7#57711yescharacteristic
374#41—42no—
375C2×Q8Dic7⋊C411yescharacteristic
376C2×D4Dic7⋊C411yescharacteristic
377C2×D4Dic7⋊C411yescharacteristic
378C2×D4Dic7⋊C411yescharacteristic
379C22×C4C2×Dic1411yescharacteristic
380C22×C4C2×Dic1411yescharacteristic
381C22×C4C2×Dic1411yescharacteristic
382C22×C4C2×C4×D711yescharacteristic
383C22×C4C2×C4×D711yescharacteristic
384C22×C4C2×C4×D711yescharacteristic
385C22×C4C2×C22⋊D711yescharacteristic
386C22×C4C2×C22⋊D711yescharacteristic
387C22×C4C2×C22⋊D711yescharacteristic
388M16—81no—
389M16—81no—
390M16—282no—
391M16—561no—
392M16—561no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—72no—
398C2×C8—72no—
399C2×C8—72no—
400C2×C8—72no—
401C2×C8—81no—
402C2×C8—141no—
403C2×C8—141no—
404C2×C8—141no—
405C2×C8—141no—
406C2×C8—282no—
407C2×C8—561no—
408C4⋊C4Dic7⋊C411yescharacteristic
409C4⋊C4Dic7⋊C411yescharacteristic
410C4⋊C4Dic7⋊C411yescharacteristic
411C4⋊C4Dic7⋊C411yescharacteristic
412C4⋊C4Dic7⋊C411yescharacteristic
413C4⋊C4Dic7⋊C411yescharacteristic
414C22⋊C4Dic7⋊C411yescharacteristic
415C22⋊C4Dic7⋊C411yescharacteristic
416C22⋊C4Dic7⋊C411yescharacteristic
417C22⋊C4Dic7⋊C411yescharacteristic
418C22⋊C4Dic7⋊C411yescharacteristic
419C22⋊C4Dic7⋊C411yescharacteristic
420C42C2×Dic1411yescharacteristic
421C42C2×Dic1411yescharacteristic
422C42C2×Dic1411yescharacteristic
423C42C2×C4×D711yescharacteristic
424C42C2×C4×D711yescharacteristic
425C42C2×C4×D711yescharacteristic
426C42C2×C22⋊D711yescharacteristic
427C42C2×C22⋊D711yescharacteristic
428C42C2×C22⋊D711yescharacteristic
429C42C23×D711yescharacteristic
430C2×C7—41no—
431C2×C7—41no—
432C2×C7—41no—
433C2×C7#290511yescharacteristic
434C2×C7#291211yescharacteristic
435C2×C7#291611yescharacteristic
436C23C2×Dic7⋊C411yescharacteristic
437C23C2×Dic7⋊C411yescharacteristic
438C23C2×Dic7⋊C411yescharacteristic
439Q8—41no—
440D4—41no—
441D4—41no—
442D4—41no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4C2×Dic7⋊C411yescharacteristic
456C2×C4C2×Dic7⋊C411yescharacteristic
457C2×C4C2×Dic7⋊C411yescharacteristic
458C2×C4C2×Dic7⋊C411yescharacteristic
459C2×C4C2×Dic7⋊C411yescharacteristic
460C2×C4C2×Dic7⋊C411yescharacteristic
461C2×C4C2×Dic7⋊C411yescharacteristic
462C2×C4C2×Dic7⋊C411yescharacteristic
463C2×C4C2×Dic7⋊C411yescharacteristic
464C2×C4C22×Dic1411yescharacteristic
465C2×C4C22×C4×D711yescharacteristic
466C2×C4C22×C22⋊D711yescharacteristic
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—142no—
472C8—142no—
473C8—142no—
474C8—142no—
475C8—281no—
476C8—281no—
477C8—281no—
478C8—281no—
479C7#1782411yescharacteristic, complemented, Sylow
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22C22×Dic7⋊C411yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4#9043612yes—
501C4#9052711yescharacteristic
502C4#9052811yescharacteristic
503C2—41no—
504C2—41no—
505C2—41no—
506C2#1174000611yescharacteristic
507C2#1174119511yescharacteristic
508C2#1174125511yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#2989C2
socleC22×C7C22×C4⋊C4
prime radicalC2×C7×#41C2×C4
cosocleC42×C7C24
hypercenter#2989D7
hypocenterC7#17824
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×#41 ◁ #2989

Complements

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

NH ≅ G/Ntype
4. #11728551C2semidirect
5. #11728550C2semidirect
6. #11728549C2semidirect
7. #11728548C2semidirect
8. #11728502C2semidirect
9. #11728501C2semidirect
10. #11728249C2semidirect
11. #11728247C2semidirect
12. #11728246C2semidirect
13. #11728244C2semidirect
14. #11728177C2semidirect
15. #11728175C2semidirect
479. C7#17824semidirect

Maximal subgroups

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

subgroupindex
C7×#29892
#117399662
#117399582
#117285512
#117285502
#117285492
#117285482
#117285022
#117285012
#117282492
#117282472
#117282462
#117282442
#117281772
#117281752
#178247