Finite Group Databasebeta

← #13337293

Subgroups — #13337293

Order 1792.

726 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3245C211yesmaximal, characteristic
2#11740733C211yesmaximal, characteristic, complemented
3#11740722C211yesmaximal, characteristic, complemented
4#11740715C211yesmaximal, characteristic, complemented
5#11729558C211yesmaximal, characteristic, complemented
6#11728859C212yesmaximal, complemented
7#11728855C212yesmaximal, complemented
8#11728762C212yesmaximal, complemented
9#11728758C212yesmaximal, complemented
10C2×#89281C211yesmaximal, characteristic, complemented
11#11728174C211yesmaximal, characteristic, complemented
12C7×#532C2211yescharacteristic
13C7×#532C2211yescharacteristic
14C2×#6514—21no—
15C2×#6514—21no—
16C2×#6511—21no—
17C2×#6432C2211yescharacteristic
18C7×#474—21no—
19C7×#474C2212yes—
20C7×#466—22no—
21C7×#459—21no—
22C7×#459C2212yes—
23C2×C7×C4⋊C8C2211yescharacteristic
24#90006—21no—
25#89864C2212yes—
26#89850—21no—
27#89848C2212yes—
28#89842—21no—
29#89832—22no—
30#89805—22no—
31#89784C2212yes—
32#89783—22no—
33D4×C7⋊C8C2212yes—
34#89779C2212yes—
35#89779C2212yes—
36#89697C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C4×#1136C2211yescharacteristic
39#89342C2211yescharacteristic
40#89342C2211yescharacteristic
41#89339C2211yescharacteristic
42#89339C2211yescharacteristic
43#89281C2212yes—
44#89281C2212yes—
45#89275C2212yes—
46#89274C2212yes—
47#19231—71nomaximal, Sylow
48C2×C7×#41—21no—
49C2×C7×#41—21no—
50C22×Dic14—21no—
51C2×C7×D8—22no—
52C22×C7×C8—21no—
53C7×C4⋊Q8D411yescharacteristic
54C7×C4⋊Q8C2311yescharacteristic
55C7×#127D411yescharacteristic
56C7×#127C2311yescharacteristic
57C7×#124—21no—
58C7×#124—21no—
59C7×#121—21no—
60C7×#121—21no—
61C7×#121—21no—
62C7×#121D412yes—
63C4×C7×D4—21no—
64C4×C7×D4—21no—
65C4×C7×D4—21no—
66C4×C7×D4C2312yes—
67C2×C42×C7C2311yescharacteristic
68#6514—44no—
69#6514—44no—
70#6511—44no—
71C2×#1144—22no—
72C2×#1144—22no—
73C2×#1142—22no—
74C2×#1142—22no—
75C2×C4⋊Dic7—21no—
76C2×C4⋊Dic7—21no—
77C2×#1136—22no—
78C2×#1136D411yescharacteristic
79C2×#1136D411yescharacteristic
80C22×C7⋊C8—21no—
81C22×C7⋊C8—22no—
82#6432—21no—
83#6432—21no—
84#6432—21no—
85#6432C2312yes—
86C7×#107—22no—
87C7×C4⋊C8—21no—
88C7×C4⋊C8C2312yes—
89C7×#102—22no—
90C7×#102—22no—
91#6406—22no—
92#6406—22no—
93C22⋊(C7⋊C8)—21no—
94C22⋊(C7⋊C8)—22no—
95C22⋊(C7⋊C8)—22no—
96#6381—22no—
97#6381—22no—
98C4⋊(C7⋊C8)—21no—
99C4⋊(C7⋊C8)C2311yescharacteristic
100C4⋊(C7⋊C8)C2311yescharacteristic
101C4⋊(C7⋊C8)C2311yescharacteristic
102C4⋊(C7⋊C8)C2311yescharacteristic
103#6377D412yes—
104C4×C7⋊C8—21no—
105C4×C7⋊C8C2312yes—
106#3245D711yescharacteristic
107#3226—71no—
108#3152—71no—
109#2935—71no—
110#1689—72no—
111#1685—72no—
112#1669—72no—
113#1665—72no—
114#1565—71no—
115#1540—71no—
116C2×#332—71no—
117C7×#41—42no—
118C7×#41—42no—
119C7×#41—42no—
120C7×#41—42no—
121C2×C7×Q8—21no—
122C2×C7×Q8—21no—
123C2×C7×D4—22no—
124C2×C7×D4—21no—
125C2×C7×D4—21no—
126C2×C7×D4—21no—
127C2×C7×D4—21no—
128C2×C7×D4—21no—
129C2×C7×D4—41no—
130C2×C7×D4C2×D412yes—
131C22×C4×C7—22no—
132C22×C4×C7—41no—
133C22×C4×C7C2×D411yescharacteristic
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7#4111yescharacteristic
136C22×Dic7—41no—
137C2×Dic14—22no—
138C2×Dic14—42no—
139C2×Dic14—42no—
140C2×Dic14—41no—
141C7×D8—82no—
142C2×C7×C8—22no—
143C2×C7×C8—81no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4C2×D412yes—
148C7×C22⋊C4—22no—
149C7×C22⋊C4—22no—
150C7×C22⋊C4—41no—
151C7×C22⋊C4—41no—
152C42×C7C2×D411yescharacteristic
153C42×C7C2×D411yescharacteristic
154C42×C7#4111yescharacteristic
155C42×C7C2411yescharacteristic
156#1144—44no—
157#1144—44no—
158#1142—44no—
159#1142—44no—
160C4⋊Dic7—22no—
161C4⋊Dic7—22no—
162C4⋊Dic7—41no—
163C4⋊Dic7—41no—
164#1136—22no—
165#1136—44no—
166#1136D814yescomplemented
167C2×C7⋊C8—21no—
168C2×C7⋊C8—21no—
169C2×C7⋊C8—21no—
170C2×C7⋊C8—21no—
171C2×C7⋊C8—42no—
172C2×C7⋊C8—44no—
173C2×C7⋊C8C2×D412yes—
174C2×C7⋊C8C2×D412yes—
175C2×#137—141no—
176C2×#137—141no—
177C2×#131—141no—
178#532C2×D711yescharacteristic
179#532C2×D711yescharacteristic
180C2×C4⋊Q8—71no—
181#500—141no—
182#497—142no—
183#494—72no—
184#492—72no—
185#486—141no—
186#480—142no—
187#474—21no—
188#474—72no—
189#474—72no—
190#474C2×D712yes—
191#466—22no—
192#459—21no—
193#459C2×D712yes—
194#436—72no—
195#435—142no—
196C8×D4—72no—
197#431—71no—
198C2×C4⋊C8—71no—
199C2×C4⋊C8C2×D711yescharacteristic
200#408—141no—
201C4×M16—71no—
202#335—72no—
203#334—72no—
204#332—71no—
205#332—71no—
206#332—72no—
207#332—72no—
208#331—71no—
209#331—71no—
210C23×C7—22no—
211C23×C7—41no—
212C23×C7#12111yescharacteristic
213C7×Q8—42no—
214C7×Q8—41no—
215C7×Q8—41no—
216C7×D4—22no—
217C7×D4—22no—
218C7×D4—44no—
219C7×D4—41no—
220C7×D4—41no—
221C7×D4—41no—
222C7×D4—41no—
223C7×D4—42no—
224C7×D4—81no—
225C7×D4—81no—
226C7×D4—82no—
227C2×C4×C7—21no—
228C2×C4×C7—21no—
229C2×C4×C7—22no—
230C2×C4×C7—41no—
231C2×C4×C7—42no—
232C2×C4×C7—42no—
233C2×C4×C7—41no—
234C2×C4×C7—82no—
235C2×C4×C7#12111yescharacteristic
236C2×C4×C7#12111yescharacteristic
237C2×C4×C7#12111yescharacteristic
238C2×C4×C7C2×D812yes—
239C2×C4×C7AGL(1,ℤ/8)12yes—
240C2×C4×C7C22×D411yescharacteristic
241C2×C4×C7C22×D411yescharacteristic
242C2×C4×C7C2×#4111yescharacteristic
243C2×Dic7—42no—
244C2×Dic7—82no—
245Dic14—42no—
246Dic14—42no—
247Dic14—82no—
248C7×C8—42no—
249C7⋊C8—22no—
250C7⋊C8—41no—
251C7⋊C8—41no—
252C7⋊C8—41no—
253C7⋊C8—41no—
254C7⋊C8C2×D814yes—
255C2×#41—21no—
256C2×#41—21no—
257C22×Q8—141no—
258#137—284no—
259#137—284no—
260C2×Q16—142no—
261C2×Q16—142no—
262C2×SD16—142no—
263C2×SD16—142no—
264C2×D8—22no—
265#131—284no—
266C2×M16—71no—
267C2×M16—71no—
268C2×M16—142no—
269C22×C8—21no—
270C22×C8—142no—
271C22×C8—141no—
272C4⋊Q8—72no—
273C4⋊Q8—141no—
274C4⋊Q8—141no—
275C4⋊Q8—141no—
276C4⋊Q8C22⋊D711yescharacteristic
277C4⋊Q8C22×D711yescharacteristic
278#127C22⋊D711yescharacteristic
279#127C22×D711yescharacteristic
280#124—21no—
281#124—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121C22⋊D712yes—
286C4×D4—21no—
287C4×D4—21no—
288C4×D4—21no—
289C4×D4C22×D712yes—
290C2×C4⋊C4—141no—
291C2×C4⋊C4—141no—
292C2×C42C22×D711yescharacteristic
293#107—22no—
294#107—142no—
295#107—142no—
296C4⋊C8—21no—
297C4⋊C8—71no—
298C4⋊C8—71no—
299C4⋊C8—71no—
300C4⋊C8—71no—
301C4⋊C8—141no—
302C4⋊C8C22×D712yes—
303#102—22no—
304#102—22no—
305#102—142no—
306#102—142no—
307C22⋊C8—142no—
308C22⋊C8—142no—
309C22⋊C8—141no—
310#97—72no—
311C4×C8—72no—
312C4×C8—141no—
313C22×C7—21no—
314C22×C7—21no—
315C22×C7—42no—
316C22×C7—42no—
317C22×C7—42no—
318C22×C7—81no—
319C22×C7—81no—
320C22×C7—81no—
321C22×C7#45912yes—
322C22×C7C2×#12111yescharacteristic
323C4×C7—21no—
324C4×C7—21no—
325C4×C7—21no—
326C4×C7—42no—
327C4×C7—81no—
328C4×C7#45912yes—
329C4×C7C22×D811yescharacteristic
330C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
331C4×C7#57711yescharacteristic
332C4×C7#57811yescharacteristic
333Dic7—82no—
334#41—42no—
335#41—42no—
336#41—42no—
337#41—42no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—142no—
341C2×Q8—281no—
342C2×Q8—282no—
343C2×Q8—282no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—41no—
351C2×D4C2×C22⋊D712yes—
352C22×C4—22no—
353C22×C4—41no—
354C22×C4—281no—
355C22×C4D4×D711yescharacteristic
356C22×C4#115911yescharacteristic
357C22×C4C2×C22⋊D711yescharacteristic
358Q16—284no—
359Q16—284no—
360SD16—284no—
361SD16—284no—
362D8—82no—
363M16—74no—
364M16—142no—
365M16—284no—
366C2×C8—22no—
367C2×C8—72no—
368C2×C8—72no—
369C2×C8—81no—
370C2×C8—141no—
371C2×C8—141no—
372C2×C8—141no—
373C2×C8—141no—
374C2×C8—284no—
375C2×C8—282no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—142no—
380C4⋊C4—142no—
381C4⋊C4—281no—
382C4⋊C4—281no—
383C4⋊C4C2×C22⋊D712yes—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C42D4×D711yescharacteristic
389C42#115911yescharacteristic
390C42C2×C22⋊D711yescharacteristic
391C42C23×D711yescharacteristic
392C2×C7—22no—
393C2×C7—82no—
394C2×C7—161no—
395C2×C7C2×#45911yescharacteristic
396C2×C7#306311yescharacteristic
397C2×C7#306911yescharacteristic
398C23—22no—
399C23—41no—
400C23#650211yescharacteristic
401Q8—42no—
402Q8—41no—
403Q8—41no—
404Q8—282no—
405Q8—282no—
406Q8—562no—
407D4—22no—
408D4—22no—
409D4—44no—
410D4—42no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—41no—
415D4—81no—
416D4—81no—
417D4—82no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—82no—
426C2×C4—282no—
427C2×C4—562no—
428C2×C4D7×D812yes—
429C2×C4#647412yes—
430C2×C4#650211yescharacteristic
431C2×C4#650211yescharacteristic
432C2×C4#650211yescharacteristic
433C2×C4C2×D4×D711yescharacteristic
434C2×C4C2×#115911yescharacteristic
435C2×C4C22×C22⋊D711yescharacteristic
436C8—42no—
437C8—74no—
438C8—142no—
439C8—281no—
440C8—281no—
441C8—281no—
442C8—281no—
443C7#1923111yescharacteristic, complemented, Sylow
444C22—21no—
445C22—21no—
446C22—42no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#8992212yes—
453C22C2×#650211yescharacteristic
454C4—21no—
455C4—21no—
456C4—21no—
457C4—42no—
458C4—81no—
459C4—562no—
460C4#8992212yes—
461C4C2×D7×D811yescharacteristic
462C4C2×#647411yescharacteristic
463C4#9052711yescharacteristic
464C4#9052911yescharacteristic
465C2—22no—
466C2—82no—
467C2—161no—
468C2#1174086811yescharacteristic
469C2C2×#8992211yescharacteristic
470C2#1174144811yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3245C2
socleC22×C7C2×#121
hypercenter#3245D7
hypocenterC7#19231
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
2. #11740733C2semidirect
3. #11740722C2semidirect
4. #11740715C2semidirect
5. #11729558C2semidirect
6. #11728859C2semidirect
7. #11728855C2semidirect
8. #11728762C2semidirect
9. #11728758C2semidirect
10. C2×#89281C2semidirect
11. #11728174C2semidirect
166. #1136D8semidirect
443. C7#19231semidirect

Maximal subgroups

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

subgroupindex
C7×#32452
#117407332
#117407222
#117407152
#117295582
#117288592
#117288552
#117287622
#117287582
C2×#892812
#117281742
#192317