Finite Group Databasebeta

← #13339288

Subgroups — #13339288

Order 1792.

694 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3039C211yesmaximal, characteristic, complemented
2#11740660C211yesmaximal, characteristic, complemented
3#11739970C211yesmaximal, characteristic
4#11739965C211yesmaximal, characteristic, complemented
5C2×#89342C211yesmaximal, characteristic, complemented
6#11728809C212yesmaximal, complemented
7#11728807C212yesmaximal, complemented
8#11728793C212yesmaximal, complemented
9#11728791C212yesmaximal, complemented
10#11728257C211yesmaximal, characteristic, complemented
11#11728248C211yesmaximal, characteristic, complemented
12C2×C7×#137—21no—
13C2×C7×C4⋊Q8C2211yescharacteristic
14C4×C7×#41C2211yescharacteristic
15C2×#6513—21no—
16#90168C2211yescharacteristic
17C7×#462—21no—
18C7×#462C2212yes—
19C7×#461—21no—
20C7×#461C2212yes—
21C7×#423C2211yescharacteristic
22C7×#419—21no—
23#89860C2212yes—
24#89859—21no—
25#89859—21no—
26#89850C2212yes—
27#89803C2211yescharacteristic
28#89800—21no—
29#89799C2211yescharacteristic
30Q8×C7⋊C8C2212yes—
31#89791C2211yescharacteristic
32#89788—21no—
33#89787C2211yescharacteristic
34D4×C7⋊C8—22no—
35D4×C7⋊C8C2212yes—
36#89774—21no—
37#89739—22no—
38#89697—21no—
39#89697C2211yescharacteristic
40C2×C4⋊(C7⋊C8)—21no—
41C2×C4⋊(C7⋊C8)C2211yescharacteristic
42C2×C4⋊(C7⋊C8)C2211yescharacteristic
43#89342C2212yes—
44#89342C2212yes—
45#89280C2212yes—
46#89277C2212yes—
47#89274C2212yes—
48#89274C2212yes—
49#19416—71nomaximal, Sylow
50C2×C7×#41D411yescharacteristic
51C22×C7×Q8—21no—
52C2×#1157—21no—
53C7×#137—44no—
54C2×C7×Q16—22no—
55C2×C7×SD16—22no—
56C2×C7×M16—21no—
57C7×C4⋊Q8—21no—
58C7×C4⋊Q8—21no—
59C7×C4⋊Q8—21no—
60C7×C4⋊Q8C2312yes—
61C4×C7×Q8D411yescharacteristic
62C4×C7×Q8C2311yescharacteristic
63C4×C7×D4—21no—
64C4×C7×D4—21no—
65C4×C7×D4D411yescharacteristic
66C4×C7×D4C2311yescharacteristic
67C7×#117—21no—
68C7×#117D411yescharacteristic
69C2×C7×C4⋊C4—21no—
70C2×C7×C4⋊C4—21no—
71C2×C42×C7—21no—
72C2×C42×C7C2311yescharacteristic
73#6513—44no—
74C2×#1144—21no—
75C2×#1143—21no—
76C2×#1142—21no—
77C2×#1141—21no—
78#6493—21no—
79#6493—21no—
80C22×C7⋊C8—21no—
81C22×C7⋊C8—21no—
82C22×C7⋊C8—22no—
83#6438—21no—
84#6437C2311yescharacteristic
85C4×D28—21no—
86C4×D28—21no—
87#6432C2311yescharacteristic
88C7×C4⋊C8—21no—
89C7×C4⋊C8C2312yes—
90C7×#103—22no—
91C7×#103—22no—
92C22⋊(C7⋊C8)—22no—
93C22⋊(C7⋊C8)D412yes—
94#6384—21no—
95#6384—21no—
96#6383—21no—
97#6383—21no—
98#6383—22no—
99#6383—22no—
100#6381—22no—
101#6381—22no—
102C4⋊(C7⋊C8)—21no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—22no—
105C4⋊(C7⋊C8)D411yescharacteristic
106C4⋊(C7⋊C8)D411yescharacteristic
107C4⋊(C7⋊C8)C2312yes—
108C4⋊(C7⋊C8)C2311yescharacteristic
109C4⋊(C7⋊C8)C2311yescharacteristic
110C4×C7⋊C8—22no—
111C4×C7⋊C8C2312yes—
112#3226—71no—
113#3039D711yescharacteristic, complemented
114#3035—71no—
115#2988—71no—
116#1682—72no—
117#1681—72no—
118#1674—72no—
119#1673—72no—
120#1571—71no—
121#1547—71no—
122C2×#332—71no—
123C7×#41—22no—
124C7×#41D814yes—
125C2×C7×Q8—22no—
126C2×C7×Q8—41no—
127C2×C7×Q8—42no—
128C2×C7×Q8—42no—
129C2×C7×Q8C2×D411yescharacteristic
130C2×C7×D4—21no—
131C2×C7×D4C2×D411yescharacteristic
132C22×C4×C7—21no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7—41no—
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C22×C4×C7#4111yescharacteristic
139C2×C22⋊D7—41no—
140#1157—42no—
141#1157—42no—
142C2×D28—21no—
143C2×D28—21no—
144C2×D28—21no—
145C2×C4×D7—41no—
146C2×Dic14—21no—
147C7×Q16—44no—
148C7×SD16—44no—
149C7×M16—42no—
150C2×C7×C8—22no—
151C7×C4⋊C4—21no—
152C7×C4⋊C4—21no—
153C7×C4⋊C4—22no—
154C7×C4⋊C4—22no—
155C7×C4⋊C4—41no—
156C7×C4⋊C4—41no—
157C7×C4⋊C4C2×D411yescharacteristic
158C7×C4⋊C4C2×D411yescharacteristic
159C7×C22⋊C4—21no—
160C7×C22⋊C4—21no—
161C7×C22⋊C4—21no—
162C42×C7—21no—
163C42×C7—21no—
164C42×C7—21no—
165C42×C7C2×D411yescharacteristic
166C42×C7C2×D411yescharacteristic
167C42×C7#4111yescharacteristic
168C42×C7C2411yescharacteristic
169#1144—42no—
170#1143—42no—
171#1142—42no—
172#1141—42no—
173(C2×D7)⋊C4—41no—
174(C2×D7)⋊C4—41no—
175C4⋊Dic7—21no—
176C4⋊Dic7—21no—
177C2×C7⋊C8—22no—
178C2×C7⋊C8—21no—
179C2×C7⋊C8—21no—
180C2×C7⋊C8—42no—
181C2×C7⋊C8—82no—
182C2×C7⋊C8—81no—
183C2×C7⋊C8C2×D412yes—
184C2×C7⋊C8C2×D412yes—
185C2×#137—21no—
186C2×#135—141no—
187#532—71no—
188C2×C4⋊Q8C2×D711yescharacteristic
189C4×#41C2×D711yescharacteristic
190#500—72no—
191#486—72no—
192#480—142no—
193#474—141no—
194#474—141no—
195#464—141no—
196#463—141no—
197#462—21no—
198#462—71no—
199#462C2×D712yes—
200#461—21no—
201#461—71no—
202#461C2×D712yes—
203#460—71no—
204#459—71no—
205C8×Q8—72no—
206C8×D4—72no—
207C8×D4—142no—
208#431—71no—
209#431—141no—
210#423C2×D711yescharacteristic
211C2×C4⋊C8—71no—
212C2×C4⋊C8—71no—
213C2×C4⋊C8—141no—
214#419—21no—
215#416—141no—
216#335—72no—
217#335—72no—
218#332—72no—
219#332—72no—
220#332—72no—
221#331—72no—
222C23×C7—21no—
223C23×C7#12111yescharacteristic
224C22×D7—41no—
225C7×Q8—21no—
226C7×Q8—42no—
227C7×Q8—42no—
228C7×Q8—82no—
229C7×Q8C2×D812yes—
230C7×D4—24no—
231C7×D4—21no—
232C7×D4C2×D812yes—
233C2×C4×C7—21no—
234C2×C4×C7—21no—
235C2×C4×C7—21no—
236C2×C4×C7—24no—
237C2×C4×C7—21no—
238C2×C4×C7—21no—
239C2×C4×C7—41no—
240C2×C4×C7—41no—
241C2×C4×C7—41no—
242C2×C4×C7—42no—
243C2×C4×C7—81no—
244C2×C4×C7—82no—
245C2×C4×C7#12111yescharacteristic
246C2×C4×C7#12111yescharacteristic
247C2×C4×C7#12111yescharacteristic
248C2×C4×C7C2×D812yes—
249C2×C4×C7#13512yes—
250C2×C4×C7C22×D411yescharacteristic
251C2×C4×C7C22×D411yescharacteristic
252C2×C4×C7C2×#4111yescharacteristic
253C22⋊D7—82no—
254C2×Dic7—41no—
255D28—41no—
256D28—41no—
257D28—41no—
258D28—41no—
259D28—81no—
260C4×D7—82no—
261Dic14—41no—
262Dic14—41no—
263C7×C8—42no—
264C7⋊C8—22no—
265C7⋊C8—22no—
266C7⋊C8—41no—
267C7⋊C8—41no—
268C7⋊C8—42no—
269C2×#41—141no—
270C2×#41C22⋊D711yescharacteristic
271C22×Q8—21no—
272#137—44no—
273#135—284no—
274C2×Q16—22no—
275C2×Q16—141no—
276C2×SD16—22no—
277C2×SD16—141no—
278C2×SD16—141no—
279C2×D8—141no—
280C2×M16—21no—
281C22×C8—142no—
282C22×C8—141no—
283C22×C8—141no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8—71no—
288C4⋊Q8C22×D712yes—
289#127—71no—
290#124—141no—
291#121—141no—
292#121—141no—
293C4×Q8C22⋊D711yescharacteristic
294C4×Q8C22×D711yescharacteristic
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—141no—
298C4×D4—141no—
299C4×D4C22⋊D711yescharacteristic
300C4×D4C22×D711yescharacteristic
301#117—21no—
302#117C22⋊D711yescharacteristic
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C42—21no—
306C2×C42C22×D711yescharacteristic
307#107—142no—
308#107—142no—
309C4⋊C8—21no—
310C4⋊C8—71no—
311C4⋊C8—71no—
312C4⋊C8—72no—
313C4⋊C8—71no—
314C4⋊C8—71no—
315C4⋊C8—142no—
316C4⋊C8—141no—
317C4⋊C8—141no—
318C4⋊C8C22×D712yes—
319#103—22no—
320#103—22no—
321#103—141no—
322#103—141no—
323#102—142no—
324#102—142no—
325#102—141no—
326#102—141no—
327C22⋊C8—72no—
328C22⋊C8—142no—
329C4×C8—72no—
330C4×C8—142no—
331C22×C7—21no—
332C22×C7—21no—
333C22×C7—22no—
334C22×C7—41no—
335C22×C7—41no—
336C22×C7#46612yes—
337C22×C7C2×#12111yescharacteristic
338C2×D7—81no—
339C2×D7—81no—
340C2×D7—81no—
341C4×C7—21no—
342C4×C7—22no—
343C4×C7—21no—
344C4×C7—21no—
345C4×C7—41no—
346C4×C7—41no—
347C4×C7—82no—
348C4×C7#46612yes—
349C4×C7C22×D811yescharacteristic
350C4×C7C2×#13511yescharacteristic
351C4×C7#57711yescharacteristic
352C4×C7#57811yescharacteristic
353Dic7—81no—
354#41—22no—
355#41—282no—
356#41—282no—
357#41#114114yes—
358C2×Q8—22no—
359C2×Q8—41no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—141no—
363C2×Q8C2×C22⋊D711yescharacteristic
364C2×D4—21no—
365C2×D4—141no—
366C2×D4—141no—
367C2×D4—141no—
368C2×D4—281no—
369C2×D4C2×C22⋊D711yescharacteristic
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—41no—
374C22×C4—281no—
375C22×C4D4×D711yescharacteristic
376C22×C4#115911yescharacteristic
377C22×C4C2×C22⋊D711yescharacteristic
378Q16—44no—
379Q16—282no—
380SD16—44no—
381SD16—282no—
382SD16—282no—
383D8—282no—
384M16—42no—
385C2×C8—22no—
386C2×C8—72no—
387C2×C8—72no—
388C2×C8—142no—
389C2×C8—141no—
390C2×C8—141no—
391C2×C8—282no—
392C2×C8—562no—
393C2×C8—561no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—141no—
401C4⋊C4—141no—
402C4⋊C4C2×C22⋊D711yescharacteristic
403C4⋊C4C2×C22⋊D711yescharacteristic
404C22⋊C4—21no—
405C22⋊C4—21no—
406C22⋊C4—21no—
407C22⋊C4—281no—
408C22⋊C4—281no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42D4×D711yescharacteristic
413C42#115911yescharacteristic
414C42C2×C22⋊D711yescharacteristic
415C42C23×D711yescharacteristic
416C2×C7—22no—
417C2×C7—42no—
418C2×C7#304611yescharacteristic
419C2×C7C2×#46611yescharacteristic
420C2×C7#306911yescharacteristic
421D7—161no—
422C23—21no—
423C23—281no—
424C23#650211yescharacteristic
425Q8—21no—
426Q8—42no—
427Q8—42no—
428Q8—82no—
429Q8—281no—
430Q8—281no—
431Q8C2×#114112yes—
432D4—24no—
433D4—21no—
434D4—281no—
435D4—281no—
436D4—281no—
437D4—281no—
438D4—561no—
439D4—562no—
440D4C2×#114112yes—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—281no—
454C2×C4—562no—
455C2×C4C2×#114112yes—
456C2×C4#650211yescharacteristic
457C2×C4#650211yescharacteristic
458C2×C4#650211yescharacteristic
459C2×C4#651312yes—
460C2×C4C2×D4×D711yescharacteristic
461C2×C4C2×#115911yescharacteristic
462C2×C4C22×C22⋊D711yescharacteristic
463C8—42no—
464C8—142no—
465C8—142no—
466C8—282no—
467C8—281no—
468C8—281no—
469C7#1941611yescharacteristic, complemented, Sylow
470C22—21no—
471C22—21no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—561no—
476C22—561no—
477C22—561no—
478C22#8981012yes—
479C22C2×#650211yescharacteristic
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—82no—
487C4—561no—
488C4#8981012yes—
489C4#9046911yescharacteristic
490C4#9047011yescharacteristic
491C4C22×#114111yescharacteristic
492C4C2×#651311yescharacteristic
493C2—22no—
494C2—42no—
495C2—1121no—
496C2C2×#8981011yescharacteristic
497C2#1174141611yescharacteristic
498C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3039C2
socleC22×C7C2×#121
prime radical#11739970C2
hypercenter#3039D7
hypocenterC7#19416
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3039

Complements

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

NH ≅ G/Ntype
1. C7×#3039C2semidirect
2. #11740660C2semidirect
4. #11739965C2semidirect
5. C2×#89342C2semidirect
6. #11728809C2semidirect
7. #11728807C2semidirect
8. #11728793C2semidirect
9. #11728791C2semidirect
10. #11728257C2semidirect
11. #11728248C2semidirect
113. #3039D7semidirect
469. C7#19416semidirect

Maximal subgroups

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

subgroupindex
C7×#30392
#117406602
#117399702
#117399652
C2×#893422
#117288092
#117288072
#117287932
#117287912
#117282572
#117282482
#194167