Finite Group Databasebeta

← #13272500

Subgroups — #13272500

Order 1792.

618 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89888C211yesmaximal, characteristic, complemented
2C2×#89887C211yesmaximal, characteristic
3C22×#6374C211yesmaximal, characteristic, complemented
4C2×C7×C22⋊C16C211yesmaximal, characteristic, complemented
5C2×#89298C211yesmaximal, characteristic, complemented
6C2×#89304C211yesmaximal, characteristic, complemented
7#11725612C211yesmaximal, characteristic, complemented
8C23×C7×C8C2211yescharacteristic
9C2×#6493—21no—
10C2×#6487—21no—
11C22×Dic28C2211yescharacteristic
12C22×D56C2211yescharacteristic
13C22×C7×C16C2211yescharacteristic
14C22×C7×C16C2211yescharacteristic
15#89888—22no—
16#89888—22no—
17#89887—22no—
18#89887—22no—
19C2×#6395—21no—
20C2×#6392C2211yescharacteristic
21C2×#6392C2211yescharacteristic
22C2×#6390—21no—
23C2×#6374—22no—
24C2×#6374—22no—
25C2×#6374—22no—
26C7×C22⋊C16—22no—
27#89304—22no—
28#89298—22no—
29#13305—71nomaximal, Sylow
30C23×C4×C7D411yescharacteristic
31C22×C22⋊D7—41no—
32C22×D28—21no—
33C22×Dic14—21no—
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C22×C7×C8—21no—
37C22×C7×C8—21no—
38C22×C7×C8—21no—
39C22×C7×C8—21no—
40C22×C7×C8D411yescharacteristic
41C22×C7×C8C2311yescharacteristic
42C2×C22⋊Dic7—41no—
43#6493—24no—
44#6493—42no—
45C2×(C2×D7)⋊C4—41no—
46C2×C4⋊Dic7—21no—
47C2×C4⋊Dic7—21no—
48#6487—24no—
49#6487—42no—
50C2×Dic7⋊C4—41no—
51C2×Dic28—21no—
52C2×Dic28—21no—
53C2×Dic28—21no—
54C2×Dic28—21no—
55C2×Dic28—21no—
56C2×Dic28D412yescomplemented
57C2×D56—21no—
58C2×D56—21no—
59C2×D56—21no—
60C2×D56—21no—
61C2×D56—21no—
62C2×D56D412yes—
63C2×C7×C16—21no—
64C2×C7×C16—21no—
65C2×C7×C16—21no—
66C2×C7×C16—21no—
67C2×C7×C16—21no—
68C2×C7×C16D412yescomplemented
69#6395—24no—
70#6392—21no—
71#6392—21no—
72#6392D412yes—
73#6392D412yescomplemented
74#6390—24no—
75#6374—28no—
76C22×SD32—71no—
77C2×#467—71no—
78C2×#466—71no—
79C2×#358—71no—
80C2×#357—71no—
81C2×C22⋊C16D711yescharacteristic, complemented
82#1380—71no—
83C24×C7D811yescharacteristic
84C23×D7—41no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7D811yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C2×C22⋊D7—44no—
94C2×C22⋊D7—82no—
95C2×C22⋊D7—82no—
96C22×Dic7—41no—
97C22×Dic7—41no—
98C22×Dic7—41no—
99C2×D28—22no—
100C2×D28—41no—
101C2×D28—41no—
102C2×D28—41no—
103C2×D28—41no—
104C2×D28—41no—
105C2×Dic14—22no—
106C2×Dic14—41no—
107C2×Dic14—41no—
108C2×Dic14—41no—
109C2×Dic14—41no—
110C2×Dic14—41no—
111C2×C7×C8—21no—
112C2×C7×C8—21no—
113C2×C7×C8—21no—
114C2×C7×C8—41no—
115C2×C7×C8—41no—
116C2×C7×C8—41no—
117C2×C7×C8—41no—
118C2×C7×C8C2×D411yescharacteristic
119C2×C7×C8C2×D411yescharacteristic
120C2×C7×C8C2×D411yescharacteristic
121C2×C7×C8C2×D411yescharacteristic
122C2×C7×C8C2×D411yescharacteristic
123C2×C7×C8#4111yescharacteristic
124C22⋊Dic7—44no—
125(C2×D7)⋊C4—44no—
126C4⋊Dic7—22no—
127C4⋊Dic7—22no—
128C4⋊Dic7—41no—
129C4⋊Dic7—41no—
130Dic7⋊C4—44no—
131Dic28—22no—
132Dic28—22no—
133Dic28—41no—
134Dic28—41no—
135D56—22no—
136D56—22no—
137D56—41no—
138D56—41no—
139C7×C16—22no—
140C7×C16—41no—
141C22×Q16—71no—
142C22×D8—71no—
143C23×C8C2×D711yescharacteristic
144C2×C22⋊Q8—141no—
145C2×#121—141no—
146C2×SD32—142no—
147C2×SD32—142no—
148C2×SD32—142no—
149C22×C16C2×D711yescharacteristic
150C22×C16C2×D711yescharacteristic
151#467—142no—
152#467—142no—
153#466—142no—
154#466—142no—
155C2×#107—71no—
156C2×#107—71no—
157C2×#103—141no—
158C2×#102—141no—
159#358—142no—
160#357—142no—
161C22⋊C16—22no—
162C23×C7—21no—
163C23×C7—21no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—21no—
167C23×C7—21no—
168C23×C7SD3212yescomplemented
169C23×C7C2×D811yescharacteristic
170C22×D7—42no—
171C22×D7—81no—
172C22×D7—81no—
173C22×D7—81no—
174C22×D7—81no—
175C22×D7—81no—
176C22×D7—81no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7#12011yescharacteristic
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7#13511yescharacteristic
188C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
190C22⋊D7—88no—
191C2×Dic7—42no—
192C2×Dic7—42no—
193C2×Dic7—42no—
194C2×Dic7—81no—
195C2×Dic7—81no—
196C2×Dic7—81no—
197C2×Dic7—81no—
198C2×Dic7—81no—
199C2×Dic7—81no—
200D28—42no—
201D28—42no—
202D28—81no—
203D28—81no—
204Dic14—42no—
205Dic14—42no—
206Dic14—81no—
207Dic14—81no—
208C7×C8—41no—
209C7×C8#12011yescharacteristic
210C7×C8#12111yescharacteristic
211C7×C8#12111yescharacteristic
212C7×C8#12711yescharacteristic
213C22×Q8—141no—
214C22×D4—141no—
215C22×D4—281no—
216C23×C4D2811yescharacteristic
217C2×Q16—72no—
218C2×Q16—141no—
219C2×Q16—141no—
220C2×Q16—141no—
221C2×Q16—141no—
222C2×Q16—141no—
223C2×D8—72no—
224C2×D8—141no—
225C2×D8—141no—
226C2×D8—141no—
227C2×D8—141no—
228C2×D8—141no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8D2811yescharacteristic
236C22×C8C22×D711yescharacteristic
237C22⋊Q8—144no—
238C22⋊Q8—282no—
239#121—144no—
240#121—282no—
241C2×C4⋊C4—141no—
242C2×C4⋊C4—141no—
243C2×C4⋊C4—281no—
244C2×C22⋊C4—281no—
245C2×C22⋊C4—281no—
246SD32—148no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16C22⋊D712yescomplemented
253#107—72no—
254#107—72no—
255#107—141no—
256#107—141no—
257#103—144no—
258#102—144no—
259C22×C7—22no—
260C22×C7—22no—
261C22×C7—22no—
262C22×C7—41no—
263C22×C7—41no—
264C22×C7—41no—
265C22×C7—41no—
266C22×C7#44711yescharacteristic
267C22×C7#44711yescharacteristic
268C22×C7#46611yescharacteristic
269C22×C7C2×SD3211yescharacteristic
270C22×C7#50811yescharacteristic
271C22×C7#50911yescharacteristic
272C22×C7#51011yescharacteristic
273C2×D7—82no—
274C2×D7—82no—
275C2×D7—82no—
276C2×D7—161no—
277C2×D7—161no—
278C2×D7—161no—
279C2×D7—161no—
280C4×C7—41no—
281C4×C7#39211yescharacteristic
282C4×C7#44911yescharacteristic
283C4×C7#44911yescharacteristic
284C4×C7#46911yescharacteristic
285Dic7—82no—
286Dic7—82no—
287Dic7—82no—
288C24—281no—
289C24D5611yescharacteristic
290C2×Q8—142no—
291C2×Q8—281no—
292C2×Q8—281no—
293C2×Q8—281no—
294C2×Q8—281no—
295C2×Q8—281no—
296C2×D4—142no—
297C2×D4—284no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—281no—
301C2×D4—281no—
302C2×D4—281no—
303C2×D4—562no—
304C2×D4—562no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—281no—
312C22×C4—281no—
313C22×C4—281no—
314C22×C4D5611yescharacteristic
315C22×C4C2×D2811yescharacteristic
316Q16—142no—
317Q16—142no—
318Q16—281no—
319Q16—281no—
320D8—142no—
321D8—142no—
322D8—281no—
323D8—281no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8#115711yescharacteristic
332C2×C8D4×D711yescharacteristic
333C2×C8D4×D711yescharacteristic
334C2×C8D4×D711yescharacteristic
335C2×C8D4×D711yescharacteristic
336C2×C8C2×C22⋊D711yescharacteristic
337C4⋊C4—142no—
338C4⋊C4—142no—
339C4⋊C4—284no—
340C4⋊C4—281no—
341C4⋊C4—281no—
342C22⋊C4—284no—
343C22⋊C4—284no—
344C16—22no—
345C16—41no—
346C2×C7—42no—
347C2×C7#199911yescharacteristic
348C2×C7#218511yescharacteristic
349C2×C7#218611yescharacteristic
350C2×C7#218711yescharacteristic
351C2×C7#218911yescharacteristic
352C2×C7#221711yescharacteristic
353C2×C7#222011yescharacteristic
354D7—162no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—282no—
362C23—561no—
363C23—561no—
364C23—561no—
365C23—561no—
366C23—561no—
367C23—561no—
368C23#637412yescomplemented
369C23C2×D5611yescharacteristic
370Q8—282no—
371Q8—282no—
372Q8—561no—
373Q8—561no—
374D4—282no—
375D4—282no—
376D4—568no—
377D4—561no—
378D4—561no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—282no—
387C2×C4—282no—
388C2×C4—282no—
389C2×C4—561no—
390C2×C4—561no—
391C2×C4—561no—
392C2×C4—561no—
393C2×C4—561no—
394C2×C4—561no—
395C2×C4#644511yescharacteristic
396C2×C4#644511yescharacteristic
397C2×C4#646711yescharacteristic
398C2×C4#647111yescharacteristic
399C2×C4#647111yescharacteristic
400C2×C4#649311yescharacteristic
401C8—41no—
402C8#644711yescharacteristic
403C8#644711yescharacteristic
404C8#650011yescharacteristic
405C8#650311yescharacteristic
406C7#1330511yescharacteristic, complemented, Sylow
407C22—22no—
408C22—22no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—562no—
415C22—562no—
416C22—562no—
417C22—1121no—
418C22—1121no—
419C22—1121no—
420C22—1121no—
421C22#8950411yescharacteristic
422C22#8950411yescharacteristic
423C22C2×#637411yescharacteristic
424C22#8967611yescharacteristic
425C22#8968011yescharacteristic
426C22#8968111yescharacteristic
427C22#8988711yescharacteristic
428C4—41no—
429C4—562no—
430C4—562no—
431C4—562no—
432C4#8950611yescharacteristic
433C4#8950611yescharacteristic
434C4#8973211yescharacteristic
435C4#8990711yescharacteristic
436C2—42no—
437C2—1122no—
438C2#1172686311yescharacteristic
439C2#1172686411yescharacteristic
440C2#1172686511yescharacteristic
441C2#1172686611yescharacteristic
442C2#1172771511yescharacteristic
443C2#1172916811yescharacteristic
444C2#1172919811yescharacteristic
subgroupG/H
centerC23C2×D56
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC2×C7×C22⋊C16C2
socleC23×C7C2×D8
prime radicalC2×#89887C2
hypercenterC2×C22⋊C16D7
hypocenterC7#13305
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
1. C2×#89888C2semidirect
3. C22×#6374C2semidirect
4. C2×C7×C22⋊C16C2semidirect
5. C2×#89298C2semidirect
6. C2×#89304C2semidirect
7. #11725612C2semidirect
56. C2×Dic28D4semidirect
68. C2×C7×C16D4semidirect
73. #6392D4semidirect
81. C2×C22⋊C16D7semidirect
168. C23×C7SD32semidirect
252. C2×C16C22⋊D7semidirect
368. C23#6374semidirect
406. C7#13305semidirect

Maximal subgroups

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

subgroupindex
C2×#898882
C2×#898872
C22×#63742
C2×C7×C22⋊C162
C2×#892982
C2×#893042
#117256122
#133057