Finite Group Databasebeta

← #13371000

Subgroups — #13371000

Order 1792.

1006 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89885C211yesmaximal, characteristic
2#11731453C211yesmaximal, characteristic
3#11731327C211yesmaximal, characteristic, complemented
4C7×#1808C211yesmaximal, characteristic, complemented
5C8×C22⋊Dic7C211yesmaximal, characteristic, complemented
6#11728392C212yesmaximal, complemented
7#11728378C212yesmaximal, complemented
8#11727276C211yesmaximal, characteristic
9#11727257C211yesmaximal, characteristic, complemented
10#11726604C212yesmaximal, complemented
11#11726534C212yesmaximal, complemented
12C22×#1157C2211yescharacteristic
13C23×C7×C8C2211yescharacteristic
14C2×#6489C2211yescharacteristic
15C2×#6456C2212yes—
16C2×C7×#107C2212yes—
17C7×#380C2211yescharacteristic
18C7×#380C2211yescharacteristic
19C4×C22⋊Dic7C2211yescharacteristic
20C2×C22⋊(C7⋊C8)C2211yescharacteristic
21#89885C418yescomplemented
22C2×#6395C2211yescharacteristic
23C2×#6395C2211yescharacteristic
24C2×#6390C2211yescharacteristic
25C2×#6390C2211yescharacteristic
26C2×C8×Dic7C2212yes—
27C4⋊C4×Dic7C2212yes—
28#89746C2212yes—
29#89745C2212yes—
30C2×#6384C2212yes—
31C2×#6383C2212yes—
32#89728—22no—
33#89716—22no—
34#89436—21no—
35D7×#95—21no—
36#89423—21no—
37#89421—21no—
38C7×#340C2212yes—
39#89345C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#22576—71nomaximal, Sylow
43C23×C4×C7C2311yescharacteristic
44C22×C22⋊D7—21no—
45C2×#1157—22no—
46C2×#1157—24no—
47C2×#1157—42no—
48C2×#1157C2×C414yes—
49C22×D28C2311yescharacteristic
50C22×C4×D7—21no—
51C22×C4×D7—22no—
52C22×Dic14C2311yescharacteristic
53C22×C7×C8—42no—
54C22×C7×C8C2×C414yescomplemented
55C22×C7×C8C2311yescharacteristic
56C22×C7×C8C2311yescharacteristic
57C2×C7×C4⋊C4C2312yes—
58C2×C7×C4⋊C4C2312yes—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C22⋊Dic7—21no—
62C2×C22⋊Dic7D412yes—
63C2×(C2×D7)⋊C4—22no—
64#6489—22no—
65#6489C2×C414yescomplemented
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2311yescharacteristic
68C2×Dic7⋊C4—22no—
69C2×C4×Dic7—21no—
70C2×C4×Dic7—22no—
71C2×C4×Dic7C2312yes—
72C22×C7⋊C8C2312yes—
73#6456—24no—
74#6456—42no—
75C7×#107—42no—
76C7×#95—21no—
77C7×#95—21no—
78C7⋊#95—22no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95—22no—
82C7⋊#95D411yescharacteristic
83C7⋊#95D411yescharacteristic
84C22⋊(C7⋊C8)—22no—
85#6395C2×C414yes—
86#6395C2×C414yes—
87#6390C2×C414yescomplemented
88#6390C2×C414yescomplemented
89C8×Dic7—42no—
90#6384—42no—
91#6383—24no—
92C2×#416—71no—
93#2347—71no—
94#2339—71no—
95#1933—72no—
96#1928—72no—
97#1862—72no—
98#1825—71no—
99#1808D711yescharacteristic, complemented
100#1788—71no—
101C4×#102—72no—
102C8×C22⋊C4—71no—
103C24×C7C2×D411yescharacteristic
104C23×D7—21no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7C22×C412yes—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7#4112yes—
112C22×C4×C7C2411yescharacteristic
113C2×C22⋊D7—24no—
114C2×C22⋊D7—44no—
115C22×Dic7—21no—
116C22×Dic7—21no—
117C22×Dic7—21no—
118C22×Dic7C2×D412yes—
119C22×Dic7C2×D412yes—
120#1157—28no—
121#1157—44no—
122#1157—48no—
123C2×D28—21no—
124C2×D28—22no—
125C2×D28C22×C412yes—
126C2×D28#4114yes—
127C2×C4×D7—24no—
128C2×C4×D7—44no—
129C2×C4×D7—44no—
130C2×C4×D7—44no—
131C2×C4×D7—82no—
132C2×Dic14—22no—
133C2×Dic14—22no—
134C2×Dic14—21no—
135C2×Dic14C22×C412yes—
136C2×C7×C8—22no—
137C2×C7×C8—22no—
138C2×C7×C8—44no—
139C2×C7×C8C22×C412yes—
140C2×C7×C8C22×C412yes—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C22⋊C4—81no—
146C7×C22⋊C4—81no—
147C22⋊Dic7—24no—
148C22⋊Dic7—24no—
149(C2×D7)⋊C4—44no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7C22×C412yes—
153C4⋊Dic7C22×C412yes—
154Dic7⋊C4—44no—
155C4×Dic7—24no—
156C4×Dic7—22no—
157C4×Dic7—22no—
158C4×Dic7—82no—
159C2×C7⋊C8—22no—
160C2×C7⋊C8—22no—
161C2×C7⋊C8—22no—
162C22×#41—71no—
163C23×C8C2×D711yescharacteristic
164C2×C4×D4—72no—
165C2×#117—71no—
166C2×#107C2×D712yes—
167#416—78no—
168C2×#103—71no—
169C2×#103—71no—
170C2×#103—72no—
171C2×#102—71no—
172C2×#102—71no—
173C2×#102—72no—
174C2×C22⋊C8—71no—
175C2×C4×C8—72no—
176#391—72no—
177#389—72no—
178#388—142no—
179#386—142no—
180#383—141no—
181#381—141no—
182#380C2×D711yescharacteristic
183#380C2×D711yescharacteristic
184C4×C4⋊C4—72no—
185C4×C4⋊C4—141no—
186C4×C22⋊C4—71no—
187C2×#95—141no—
188#340—72no—
189#340—72no—
190#340C2×D712yes—
191#336—72no—
192C23×C7—22no—
193C23×C7—22no—
194C23×C7—21no—
195C23×C7—21no—
196C23×C7C4×D412yes—
197C23×C7C22×D411yescharacteristic
198C22×D7—22no—
199C22×D7—24no—
200C22×D7—42no—
201C22×D7—41no—
202C22×D7—41no—
203C2×C4×C7—22no—
204C2×C4×C7—22no—
205C2×C4×C7—22no—
206C2×C4×C7—24no—
207C2×C4×C7—42no—
208C2×C4×C7—42no—
209C2×C4×C7—42no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7C4×D412yes—
214C2×C4×C7C23×C411yescharacteristic
215C2×C4×C7C2×#4111yescharacteristic
216C2×C4×C7C2×#4112yes—
217C2×C4×C7H(5,𝔽2)12yes—
218C22⋊D7—48no—
219C2×Dic7—22no—
220C2×Dic7—22no—
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—22no—
226C2×Dic7—22no—
227C2×Dic7—42no—
228C2×Dic7—42no—
229C2×Dic7—41no—
230C2×Dic7—41no—
231C2×Dic7#13514yes—
232D28—22no—
233D28—24no—
234D28#11714yes—
235C4×D7—48no—
236C4×D7—88no—
237Dic14—22no—
238Dic14—22no—
239Dic14—42no—
240C7×C8—22no—
241C7×C8—22no—
242C7⋊C8—42no—
243C2×#41—74no—
244C2×#41—144no—
245C2×#41—142no—
246C2×#41—282no—
247C22×Q8—71no—
248C22×D4—71no—
249C22×D4—141no—
250C23×C4—142no—
251C23×C4—141no—
252C23×C4C22×D711yescharacteristic
253C22×C8—42no—
254C22×C8—72no—
255C22×C8C4×D714yescomplemented
256C22×C8C22×D711yescharacteristic
257C22×C8C22×D711yescharacteristic
258C4×D4—144no—
259C4×D4—282no—
260#117—74no—
261#117—142no—
262C2×C4⋊C4—71no—
263C2×C4⋊C4—71no—
264C2×C4⋊C4—142no—
265C2×C4⋊C4C22×D712yes—
266C2×C4⋊C4C22×D712yes—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—72no—
270C2×C22⋊C4—141no—
271C2×C22⋊C4—142no—
272C2×C42—72no—
273C2×C42—141no—
274C2×C42—142no—
275#107—42no—
276#103—74no—
277#103—74no—
278#103—282no—
279#102—74no—
280#102—74no—
281#102—144no—
282C22⋊C8—142no—
283C4×C8—282no—
284#95—21no—
285#95—21no—
286#95—71no—
287#95—71no—
288#95—142no—
289#95—142no—
290#95—142no—
291#95—142no—
292C22×C7—22no—
293C22×C7—22no—
294C22×C7—22no—
295C22×C7—42no—
296C22×C7—42no—
297C22×C7C2×C4×D411yescharacteristic
298C22×C7#54612yes—
299C22×C7C2×#13512yes—
300C22×C7#57611yescharacteristic
301C22×C7#57811yescharacteristic
302C2×D7—24no—
303C2×D7—42no—
304C2×D7—42no—
305C2×D7—44no—
306C2×D7—44no—
307C4×C7—22no—
308C4×C7—82no—
309C4×C7—82no—
310C4×C7C2×#11711yescharacteristic
311C4×C7#51811yescharacteristic
312C4×C7#55112yes—
313Dic7—24no—
314Dic7—42no—
315Dic7—42no—
316Dic7—42no—
317Dic7—42no—
318C24—141no—
319C24D4×D711yescharacteristic
320#41—148no—
321#41—284no—
322#41—288no—
323C2×Q8—72no—
324C2×Q8—141no—
325C2×Q8—142no—
326C2×Q8—142no—
327C2×D4—72no—
328C2×D4—74no—
329C2×D4—144no—
330C2×D4—142no—
331C2×D4—141no—
332C2×D4—284no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—72no—
338C22×C4—72no—
339C22×C4—141no—
340C22×C4—141no—
341C22×C4—141no—
342C22×C4—144no—
343C22×C4—284no—
344C22×C4—284no—
345C22×C4—284no—
346C22×C4—562no—
347C22×C4C2×C4×D712yes—
348C22×C4D4×D711yescharacteristic
349C22×C4#116112yes—
350C22×C4C23×D711yescharacteristic
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—44no—
354C2×C8—142no—
355C2×C8—142no—
356C2×C8—142no—
357C2×C8C2×C4×D712yes—
358C2×C8C2×C4×D712yes—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—72no—
364C4⋊C4—72no—
365C4⋊C4—141no—
366C4⋊C4—141no—
367C4⋊C4—284no—
368C22⋊C4—81no—
369C22⋊C4—81no—
370C22⋊C4—144no—
371C22⋊C4—144no—
372C22⋊C4—284no—
373C42—144no—
374C42—142no—
375C42—142no—
376C42—562no—
377C2×C7—42no—
378C2×C7#234211yescharacteristic
379C2×C7C4×#13511yescharacteristic
380C2×C7#295111yescharacteristic
381C2×C7#329712yes—
382C2×C7#330212yes—
383D7—44no—
384C23—21no—
385C23—21no—
386C23—22no—
387C23—22no—
388C23—142no—
389C23—144no—
390C23—281no—
391C23—281no—
392C23—282no—
393C23#645612yes—
394C23C2×D4×D711yescharacteristic
395Q8—142no—
396Q8—142no—
397Q8—282no—
398D4—74no—
399D4—142no—
400D4—144no—
401D4—288no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—74no—
413C2×C4—142no—
414C2×C4—142no—
415C2×C4—142no—
416C2×C4—142no—
417C2×C4—142no—
418C2×C4—142no—
419C2×C4—142no—
420C2×C4—142no—
421C2×C4—288no—
422C2×C4—281no—
423C2×C4—281no—
424C2×C4—282no—
425C2×C4—282no—
426C2×C4—568no—
427C2×C4#645612yes—
428C2×C4C22×C4×D711yescharacteristic
429C2×C4#654812yes—
430C2×C4C2×#116111yescharacteristic
431C2×C4D7×#4112yes—
432C8—22no—
433C8—22no—
434C8—282no—
435C7#2257611yescharacteristic, complemented, Sylow
436C22—22no—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—42no—
441C22—144no—
442C22—282no—
443C22—282no—
444C22—284no—
445C22—284no—
446C22C2×#645611yescharacteristic
447C22#9030112yes—
448C22#9044811yescharacteristic
449C22#9045711yescharacteristic
450C22D7×#13512yes—
451C4—22no—
452C4—82no—
453C4—82no—
454C4—144no—
455C4—282no—
456C4—282no—
457C4—282no—
458C4—282no—
459C4#9017711yescharacteristic
460C4D7×#11711yescharacteristic
461C4#9032812yes—
462C2—42no—
463C2—284no—
464C2#1173145411yescharacteristic
465C2#1173658111yescharacteristic
466C2#1173661011yescharacteristic
467C2#1173817012yes—
468C2#1173822112yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1808C2
socleC23×C7C22×D4
prime radicalC22×#1157C22
cosocleC22×C4×C7C24
hypercenter#1808D7
hypocenterC7#22576
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1808

Complements

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

NH ≅ G/Ntype
3. #11731327C2semidirect
4. C7×#1808C2semidirect
5. C8×C22⋊Dic7C2semidirect
6. #11728392C2semidirect
7. #11728378C2semidirect
9. #11727257C2semidirect
10. #11726604C2semidirect
11. #11726534C2semidirect
21. #89885C4semidirect
54. C22×C7×C8C2×C4semidirect
65. #6489C2×C4semidirect
87. #6390C2×C4semidirect
88. #6390C2×C4semidirect
99. #1808D7semidirect
255. C22×C8C4×D7semidirect
435. C7#22576semidirect

Maximal subgroups

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

subgroupindex
C2×#898852
#117314532
#117313272
C7×#18082
C8×C22⋊Dic72
#117283922
#117283782
#117272762
#117272572
#117266042
#117265342
#225767