Finite Group Databasebeta

← #13388708

Subgroups — #13388708

Order 1792.

1084 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
1C2×C7×#557C211yesmaximal, characteristic, complemented
2C2×#89855C211yesmaximal, characteristic
3#11739946C211yesmaximal, characteristic
4C2×#89798C211yesmaximal, characteristic
5#11729545C212yesmaximal
6#11729540C211yesmaximal, characteristic, complemented
7#11729535C212yesmaximal, complemented
8#11729389C211yesmaximal, characteristic, complemented
9#11729387C212yesmaximal
10#11729382C212yesmaximal, complemented
11#11729381C211yesmaximal, characteristic, complemented
12C7×#557—24no—
13C7×#557—24no—
14C2×C7×C4⋊Q8C2212yes—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—22no—
17C2×C7×#125—21no—
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8C2212yes—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22C22×#1143C2211yescharacteristic
23C22×#1143C2212yes—
24C2×C4×D28C2211yescharacteristic
25#90003—21no—
26C2×#6409C2211yescharacteristic
27C2×#6409C2212yes—
28C2×#6409C2212yes—
29C2×#6409C2212yes—
30C2×#6409C2212yes—
31#89855—24no—
32#89798—24no—
33#89764—21no—
34#89761—22no—
35C2×#6383C2211yescharacteristic
36C2×#6381C2211yescharacteristic
37#89711C2212yes—
38#89708C2211yescharacteristic
39#89700C2212yes—
40C2×C4⋊(C7⋊C8)C2211yescharacteristic
41C2×#6377C2211yescharacteristic
42#89324C2212yes—
43#89319C2212yes—
44#89319C2212yes—
45#24199—71nomaximal, Sylow
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8C2312yes—
48C22×C7×Q8C2311yescharacteristic
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C7×C4⋊Q8—24no—
52C7×C4⋊Q8—24no—
53C7×C4⋊Q8—22no—
54C7×C4⋊Q8D414yes—
55C7×#125—216no—
56C7×#125—28no—
57C4×C7×Q8—28no—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—22no—
61C4×C7×Q8—22no—
62C4×C7×Q8—22no—
63C4×C7×Q8D414yes—
64C2×C7×C4⋊C4—22no—
65C2×C7×C4⋊C4—21no—
66C2×C7×C4⋊C4—22no—
67C2×C7×C4⋊C4—21no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4D412yes—
71C2×C7×C4⋊C4D411yescharacteristic
72C2×C7×C4⋊C4C2311yescharacteristic
73C2×C42×C7—22no—
74C2×C42×C7—21no—
75C2×C42×C7C2312yes—
76C2×C42×C7C2311yescharacteristic
77C2×#1143—22no—
78C2×#1143—22no—
79C2×#1143—24no—
80C2×#1143—24no—
81C2×#1143—22no—
82C2×#1143—42no—
83C2×(C2×D7)⋊C4—21no—
84C2×(C2×D7)⋊C4—22no—
85C2×(C2×D7)⋊C4—22no—
86C2×C4⋊Dic7C2311yescharacteristic
87C2×C4⋊Dic7C2312yes—
88C2×Dic7⋊C4—22no—
89C22×C7⋊C8C2311yescharacteristic
90C22×C7⋊C8C2312yes—
91C22×C7⋊C8C2311yescharacteristic
92C4×D28—22no—
93C4×D28—22no—
94#6409—22no—
95#6409—24no—
96#6409—24no—
97#6409—42no—
98#6409—42no—
99C7⋊#95—22no—
100C7⋊#95—21no—
101C7⋊#95—21no—
102#6383—22no—
103#6381—22no—
104C4⋊(C7⋊C8)—22no—
105#6377—22no—
106C2×#557D711yescharacteristic, complemented
107C2×#461—71no—
108C2×#440—71no—
109#2580—71no—
110#2043—72no—
111#2013—72no—
112#1985—71no—
113#1960—71no—
114#1880—72no—
115#1863—72no—
116#1805—71no—
117C23×D7—21no—
118C2×C7×Q8—22no—
119C2×C7×Q8—24no—
120C2×C7×Q8—22no—
121C2×C7×Q8—21no—
122C2×C7×Q8—22no—
123C2×C7×Q8—21no—
124C2×C7×Q8—22no—
125C2×C7×Q8—21no—
126C2×C7×Q8—21no—
127C2×C7×Q8—22no—
128C2×C7×Q8—21no—
129C2×C7×Q8—22no—
130C2×C7×Q8C2×D414yes—
131C2×C7×Q8C2×D412yes—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7—21no—
135C22×C4×C7C2×D412yes—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7C2×D412yes—
139C22×C4×C7C2411yescharacteristic
140C22×Dic7—22no—
141C22×Dic7—21no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×D28—21no—
146C2×D28—21no—
147C2×D28#4112yes—
148C2×C4×D7—42no—
149C2×C4×D7—82no—
150C7×C4⋊C4—28no—
151C7×C4⋊C4—28no—
152C7×C4⋊C4—28no—
153C7×C4⋊C4—24no—
154C7×C4⋊C4—24no—
155C7×C4⋊C4—24no—
156C7×C4⋊C4—24no—
157C7×C4⋊C4—22no—
158C7×C4⋊C4—21no—
159C7×C4⋊C4C2×D412yes—
160C42×C7—28no—
161C42×C7—24no—
162C42×C7—22no—
163C42×C7—22no—
164C42×C7C2×D412yes—
165C42×C7C2×D412yes—
166#1143—28no—
167#1143—48no—
168(C2×D7)⋊C4—42no—
169(C2×D7)⋊C4—44no—
170(C2×D7)⋊C4—82no—
171C4⋊Dic7—21no—
172C4⋊Dic7—21no—
173C4⋊Dic7—22no—
174C4⋊Dic7—22no—
175Dic7⋊C4—82no—
176C2×C7⋊C8—21no—
177C2×C7⋊C8—21no—
178C2×C7⋊C8—22no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—22no—
181C2×C7⋊C8—21no—
182C2×C7⋊C8—22no—
183C2×C7⋊C8—21no—
184C2×C7⋊C8#4112yes—
185C22×SD16—71no—
186C22×SD16—72no—
187#557—24no—
188#557—24no—
189C2×C4⋊Q8C2×D712yes—
190C2×C4⋊Q8C2×D711yescharacteristic
191C2×#125—22no—
192C2×#125—21no—
193C2×C4×Q8—21no—
194C2×C4×Q8C2×D712yes—
195C2×C4×Q8C2×D711yescharacteristic
196C2×C4×Q8C2×D711yescharacteristic
197C2×C4×D4—71no—
198#461—144no—
199#440—144no—
200C2×#107—71no—
201C2×C4⋊C8—71no—
202C2×#103—72no—
203C2×#103—72no—
204C2×#103—71no—
205C2×#103—72no—
206C2×#103—72no—
207C2×#102—71no—
208C2×#97—71no—
209#399—141no—
210#393—141no—
211#390—72no—
212#389—72no—
213#385—142no—
214#382—71no—
215#340—72no—
216#336—72no—
217#336—72no—
218C23×C7C22×D411yescharacteristic
219C22×D7—22no—
220C22×D7—41no—
221C22×D7—41no—
222C22×D7—42no—
223C22×D7—42no—
224C7×Q8—28no—
225C7×Q8—22no—
226C7×Q8—22no—
227C7×Q8—24no—
228C7×Q8—24no—
229C7×Q8—42no—
230C7×Q8—42no—
231C7×Q8—41no—
232C7×Q8—41no—
233C2×C4×C7—22no—
234C2×C4×C7—22no—
235C2×C4×C7—22no—
236C2×C4×C7—22no—
237C2×C4×C7—22no—
238C2×C4×C7—22no—
239C2×C4×C7—24no—
240C2×C4×C7—22no—
241C2×C4×C7—22no—
242C2×C4×C7—22no—
243C2×C4×C7—41no—
244C2×C4×C7—42no—
245C2×C4×C7—42no—
246C2×C4×C7—41no—
247C2×C4×C7—42no—
248C2×C4×C7#12012yes—
249C2×C4×C7#12012yes—
250C2×C4×C7#13714yes—
251C2×C4×C7C22×D411yescharacteristic
252C2×C4×C7C22×D411yescharacteristic
253C2×C4×C7C2×#4111yescharacteristic
254C2×C4×C7C2×#4111yescharacteristic
255C2×C4×C7H(5,𝔽2)11yescharacteristic
256C2×C4×C72−1+411yescharacteristic
257C2×Dic7—41no—
258C2×Dic7—42no—
259C2×Dic7—42no—
260C2×Dic7—42no—
261C2×Dic7—42no—
262D28—22no—
263D28—22no—
264D28—41no—
265D28—41no—
266C4×D7—84no—
267C7⋊C8—22no—
268C7⋊C8—41no—
269C7⋊C8—42no—
270C22×Q8C22⋊D711yescharacteristic
271C22×Q8C22×D712yes—
272C22×Q8C22×D711yescharacteristic
273C22×D4—71no—
274C23×C4—141no—
275C2×SD16—142no—
276C2×SD16—142no—
277C2×SD16—142no—
278C2×SD16—144no—
279C2×SD16—144no—
280C2×SD16—282no—
281C22×C8—71no—
282C22×C8—72no—
283C22×C8—71no—
284C4⋊Q8—24no—
285C4⋊Q8—24no—
286C4⋊Q8—22no—
287C4⋊Q8C22⋊D714yes—
288#125—216no—
289#125—28no—
290C4×Q8—28no—
291C4×Q8—24no—
292C4×Q8—24no—
293C4×Q8—22no—
294C4×Q8—22no—
295C4×Q8—22no—
296C4×Q8C22⋊D714yes—
297C4×D4—142no—
298C4×D4—142no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—71no—
306C2×C4⋊C4—72no—
307C2×C4⋊C4—142no—
308C2×C4⋊C4C22⋊D712yes—
309C2×C4⋊C4C22⋊D711yescharacteristic
310C2×C4⋊C4C22×D711yescharacteristic
311C2×C22⋊C4—141no—
312C2×C22⋊C4—142no—
313C2×C22⋊C4—142no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42C22×D712yes—
317C2×C42C22×D711yescharacteristic
318#107—142no—
319C4⋊C8—142no—
320#103—144no—
321#103—142no—
322#103—144no—
323#103—282no—
324#103—282no—
325#102—142no—
326#97—142no—
327#95—141no—
328#95—141no—
329#95—142no—
330C22×C7C2×#12011yescharacteristic
331C22×C7#54811yescharacteristic
332C22×C7#55011yescharacteristic
333C22×C7C2×#13712yes—
334C22×C7#57712yes—
335C2×D7—44no—
336C2×D7—42no—
337C2×D7—82no—
338C2×D7—81no—
339C2×D7—81no—
340C4×C7—22no—
341C4×C7—24no—
342C4×C7—22no—
343C4×C7—42no—
344C4×C7—44no—
345C4×C7—42no—
346C4×C7—42no—
347C4×C7#52511yescharacteristic
348C4×C7#53711yescharacteristic
349C4×C7#54611yescharacteristic
350C4×C7#54711yescharacteristic
351Dic7—82no—
352Dic7—81no—
353C24—141no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—21no—
366C2×Q8C2×C22⋊D714yes—
367C2×Q8C2×C22⋊D712yes—
368C2×D4—72no—
369C2×D4—141no—
370C2×D4—141no—
371C2×D4—141no—
372C2×D4—141no—
373C2×D4—141no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—142no—
378C22×C4—141no—
379C22×C4—282no—
380C22×C4—562no—
381C22×C4C2×C22⋊D712yes—
382C22×C4C2×C22⋊D712yes—
383C22×C4C2×C22⋊D711yescharacteristic
384C22×C4C2×C22⋊D711yescharacteristic
385C22×C4C23×D711yescharacteristic
386SD16—148no—
387SD16—288no—
388C2×C8—72no—
389C2×C8—141no—
390C2×C8—141no—
391C2×C8—142no—
392C2×C8—141no—
393C2×C8—142no—
394C2×C8—141no—
395C2×C8—141no—
396C2×C8—142no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—28no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—142no—
407C4⋊C4—142no—
408C4⋊C4—141no—
409C4⋊C4—141no—
410C4⋊C4—562no—
411C4⋊C4C2×C22⋊D712yes—
412C22⋊C4—284no—
413C22⋊C4—282no—
414C22⋊C4—562no—
415C42—28no—
416C42—24no—
417C42—22no—
418C42—22no—
419C42C2×C22⋊D712yes—
420C42C2×C22⋊D712yes—
421C2×C7#240711yescharacteristic
422C2×C7#300411yescharacteristic
423C2×C7#301011yescharacteristic
424C2×C7#337212yes—
425C2×C7#340012yes—
426D7—82no—
427C23—142no—
428C23—282no—
429C23—281no—
430C23—282no—
431C23—281no—
432C23C22×C22⋊D711yescharacteristic
433Q8—28no—
434Q8—24no—
435Q8—22no—
436Q8—24no—
437Q8—22no—
438Q8—41no—
439Q8—42no—
440Q8—41no—
441Q8—42no—
442D4—142no—
443D4—142no—
444D4—281no—
445D4—281no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—22no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—282no—
462C2×C4—282no—
463C2×C4—282no—
464C2×C4—282no—
465C2×C4—281no—
466C2×C4—564no—
467C2×C4#650514yes—
468C2×C4#651612yes—
469C2×C4#651612yes—
470C2×C4D7×#4111yescharacteristic
471C2×C4D7×#4111yescharacteristic
472C2×C4#655311yescharacteristic
473C2×C4#655411yescharacteristic
474C2×C4C22×C22⋊D711yescharacteristic
475C2×C4C22×C22⋊D711yescharacteristic
476C8—142no—
477C8—282no—
478C8—281no—
479C7#2419911yescharacteristic, complemented, Sylow
480C22—282no—
481C22—284no—
482C22—561no—
483C22—561no—
484C22—562no—
485C22C2×#650512yes—
486C22#9052211yescharacteristic
487C22#9052511yescharacteristic
488C22#9052812yes—
489C22C2×#651611yescharacteristic
490C4—24no—
491C4—22no—
492C4—22no—
493C4—44no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—561no—
498C4—562no—
499C4#9051511yescharacteristic
500C4#9051911yescharacteristic
501C4#9052011yescharacteristic
502C4#9052411yescharacteristic
503C2—562no—
504C2#1174113212yes—
505C2#1174113712yes—
506C2#1174150411yescharacteristic
507C2#1174154211yescharacteristic
508C2#1174155811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#557C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenterC2×#557D7
hypocenterC7#24199
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
1. C2×C7×#557C2semidirect
6. #11729540C2semidirect
7. #11729535C2semidirect
8. #11729389C2semidirect
10. #11729382C2semidirect
11. #11729381C2semidirect
106. C2×#557D7semidirect
479. C7#24199semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5572
C2×#898552
#117399462
C2×#897982
#117295452
#117295402
#117295352
#117293892
#117293872
#117293822
#117293812
#241997