Finite Group Databasebeta

← #13272496

Subgroups — #13272496

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×C7×#467C211yesmaximal, characteristic, complemented
2C22×#6402C211yesmaximal, characteristic, complemented
3C2×#89887C211yesmaximal, characteristic
4C2×#89359C211yesmaximal, characteristic, complemented
5C2×C22⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
6C2×#89286C211yesmaximal, characteristic, complemented
7#11725528C211yesmaximal, characteristic, complemented
8C22×C7×Q16C2211yescharacteristic
9C23×C7×C8C2211yescharacteristic
10C2×C7×C22⋊Q8—21no—
11C2×#6493—21no—
12C22×D56C2211yescharacteristic
13C7×#467—22no—
14C7×#467—22no—
15C2×C7×#107C2211yescharacteristic
16C2×C7×#103—21no—
17C2×#6402—22no—
18C2×#6402—22no—
19C2×#6402—22no—
20#89887—22no—
21#89887—22no—
22C2×#6395—21no—
23C2×#6392C2211yescharacteristic
24C22×C7⋊C16C2211yescharacteristic
25C22×C7⋊C16C2211yescharacteristic
26#89359—22no—
27C22⋊(C7⋊C16)—22no—
28#89286—22no—
29#13305—71nomaximal, Sylow
30C22×C7×Q8—21no—
31C23×C4×C7D411yescharacteristic
32C22×C22⋊D7—41no—
33C22×D28—21no—
34C2×C7×Q16—21no—
35C2×C7×Q16—21no—
36C2×C7×Q16—21no—
37C2×C7×Q16—21no—
38C2×C7×Q16—21no—
39C2×C7×Q16D412yescomplemented
40C22×C7×C8—21no—
41C22×C7×C8—21no—
42C22×C7×C8—21no—
43C22×C7×C8—21no—
44C22×C7×C8—21no—
45C22×C7×C8—21no—
46C22×C7×C8D411yescharacteristic
47C22×C7×C8C2311yescharacteristic
48C7×C22⋊Q8—24no—
49C7×C22⋊Q8—42no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—41no—
52C2×C7×C22⋊C4—41no—
53#6493—24no—
54#6493—42no—
55C2×(C2×D7)⋊C4—41no—
56C2×C4⋊Dic7—21no—
57C2×D56—21no—
58C2×D56—21no—
59C2×D56—21no—
60C2×D56—21no—
61C2×D56—21no—
62C2×D56D412yes—
63C7×#107—21no—
64C7×#107D412yescomplemented
65C7×#103—24no—
66#6402—28no—
67#6395—24no—
68#6392—21no—
69#6392D412yes—
70C2×C7⋊C16—21no—
71C2×C7⋊C16—21no—
72C2×C7⋊C16—21no—
73C2×C7⋊C16—21no—
74C2×C7⋊C16—21no—
75C2×C7⋊C16D412yescomplemented
76C22×SD32—71no—
77C2×#467D711yescharacteristic, complemented
78C2×#466—71no—
79C2×#358—71no—
80C2×#357—71no—
81C2×C22⋊C16—71no—
82#1380—71no—
83C24×C7D811yescharacteristic
84C23×D7—41no—
85C2×C7×Q8—22no—
86C2×C7×Q8—41no—
87C2×C7×Q8—41no—
88C2×C7×Q8—41no—
89C2×C7×Q8—41no—
90C2×C7×Q8—41no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—41no—
98C22×C4×C7—41no—
99C22×C4×C7D811yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C2×C22⋊D7—44no—
102C2×C22⋊D7—82no—
103C2×C22⋊D7—82no—
104C22×Dic7—41no—
105C2×D28—22no—
106C2×D28—41no—
107C2×D28—41no—
108C2×D28—41no—
109C2×D28—41no—
110C2×D28—41no—
111C7×Q16—22no—
112C7×Q16—22no—
113C7×Q16—41no—
114C7×Q16—41no—
115C2×C7×C8—21no—
116C2×C7×C8—21no—
117C2×C7×C8—21no—
118C2×C7×C8—41no—
119C2×C7×C8—41no—
120C2×C7×C8—41no—
121C2×C7×C8—41no—
122C2×C7×C8C2×D411yescharacteristic
123C2×C7×C8C2×D411yescharacteristic
124C2×C7×C8C2×D411yescharacteristic
125C2×C7×C8C2×D411yescharacteristic
126C2×C7×C8C2×D411yescharacteristic
127C2×C7×C8#4111yescharacteristic
128C7×C4⋊C4—22no—
129C7×C4⋊C4—44no—
130C7×C4⋊C4—41no—
131C7×C22⋊C4—44no—
132(C2×D7)⋊C4—44no—
133C4⋊Dic7—22no—
134C4⋊Dic7—41no—
135D56—22no—
136D56—22no—
137D56—41no—
138D56—41no—
139C7⋊C16—22no—
140C7⋊C16—41no—
141C22×Q16C2×D711yescharacteristic
142C22×D8—71no—
143C23×C8C2×D711yescharacteristic
144C2×C22⋊Q8—21no—
145C2×#121—141no—
146C2×SD32—142no—
147C2×SD32—142no—
148C2×SD32—142no—
149C22×C16—71no—
150C22×C16—71no—
151#467—22no—
152#467—22no—
153#466—142no—
154#466—142no—
155C2×#107—71no—
156C2×#107C2×D711yescharacteristic
157C2×#103—21no—
158C2×#102—141no—
159#358—142no—
160#357—142no—
161C22⋊C16—142no—
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—
177C7×Q8—42no—
178C7×Q8—42no—
179C7×Q8—81no—
180C7×Q8—81no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—81no—
191C2×C4×C7—81no—
192C2×C4×C7—81no—
193C2×C4×C7—81no—
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#13511yescharacteristic
198C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
199C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
200C22⋊D7—88no—
201C2×Dic7—42no—
202C2×Dic7—81no—
203C2×Dic7—81no—
204D28—42no—
205D28—42no—
206D28—81no—
207D28—81no—
208C7×C8—41no—
209C7×C8#12011yescharacteristic
210C7×C8#12111yescharacteristic
211C7×C8#12111yescharacteristic
212C7×C8#12711yescharacteristic
213C22×Q8—21no—
214C22×D4—141no—
215C22×D4—281no—
216C23×C4C22⋊D711yescharacteristic
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16C22⋊D712yescomplemented
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×C8C22⋊D711yescharacteristic
236C22×C8C22×D711yescharacteristic
237C22⋊Q8—24no—
238C22⋊Q8—42no—
239#121—144no—
240#121—282no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—141no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—281no—
246SD32—148no—
247C2×C16—72no—
248C2×C16—141no—
249C2×C16—141no—
250C2×C16—141no—
251C2×C16—141no—
252C2×C16—141no—
253#107—21no—
254#107—72no—
255#107—141no—
256#107C22⋊D712yescomplemented
257#103—24no—
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—82no—
282C4×C7—82no—
283C4×C7#39211yescharacteristic
284C4×C7#44911yescharacteristic
285C4×C7#44911yescharacteristic
286C4×C7#46911yescharacteristic
287Dic7—82no—
288C24—281no—
289C24#114111yescharacteristic
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
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—41no—
312C22×C4—41no—
313C22×C4—281no—
314C22×C4#114111yescharacteristic
315C22×C4C2×C22⋊D711yescharacteristic
316Q16—22no—
317Q16—22no—
318Q16—41no—
319Q16—41no—
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×C8D4×D711yescharacteristic
332C2×C8D4×D711yescharacteristic
333C2×C8D4×D711yescharacteristic
334C2×C8#115911yescharacteristic
335C2×C8C2×C22⋊D711yescharacteristic
336C2×C8C2×C22⋊D711yescharacteristic
337C4⋊C4—22no—
338C4⋊C4—44no—
339C4⋊C4—41no—
340C4⋊C4—142no—
341C4⋊C4—281no—
342C22⋊C4—44no—
343C22⋊C4—284no—
344C16—142no—
345C16—281no—
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#640212yescomplemented
369C23C2×#114111yescharacteristic
370Q8—42no—
371Q8—42no—
372Q8—81no—
373Q8—81no—
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—42no—
387C2×C4—42no—
388C2×C4—81no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—282no—
393C2×C4—561no—
394C2×C4—561no—
395C2×C4#649511yescharacteristic
396C2×C4#650011yescharacteristic
397C2×C4#650211yescharacteristic
398C2×C4#651211yescharacteristic
399C2×C4#651311yescharacteristic
400C2×C4#651611yescharacteristic
401C8—41no—
402C8#650011yescharacteristic
403C8#650111yescharacteristic
404C8#650211yescharacteristic
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#8980811yescharacteristic
422C22#8981011yescharacteristic
423C22C2×#640211yescharacteristic
424C22#8995111yescharacteristic
425C22#8996511yescharacteristic
426C22#8996611yescharacteristic
427C22#8998911yescharacteristic
428C4—41no—
429C4—82no—
430C4—82no—
431C4—562no—
432C4#8980911yescharacteristic
433C4#8981111yescharacteristic
434C4#8999511yescharacteristic
435C4#9001011yescharacteristic
436C2—42no—
437C2—1122no—
438C2#1172883611yescharacteristic
439C2#1172883811yescharacteristic
440C2#1172883911yescharacteristic
441C2#1172884011yescharacteristic
442C2#1172943811yescharacteristic
443C2#1172981711yescharacteristic
444C2#1172983611yescharacteristic
subgroupG/H
centerC23C2×#1141
commutator, cosocleC22×C7×C8C23
FrattiniC22×C8C22×D7
FittingC2×C7×#467C2
socleC23×C7C2×D8
prime radicalC2×#89887C2
hypercenterC2×#467D7
hypocenterC7#13305
LCSG ▷ C22×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×C7×#467C2semidirect
2. C22×#6402C2semidirect
4. C2×#89359C2semidirect
5. C2×C22⋊(C7⋊C16)C2semidirect
6. C2×#89286C2semidirect
7. #11725528C2semidirect
39. C2×C7×Q16D4semidirect
64. C7×#107D4semidirect
75. C2×C7⋊C16D4semidirect
77. C2×#467D7semidirect
168. C23×C7SD32semidirect
222. C2×Q16C22⋊D7semidirect
256. #107C22⋊D7semidirect
368. C23#6402semidirect
406. C7#13305semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4672
C22×#64022
C2×#898872
C2×#893592
C2×C22⋊(C7⋊C16)2
C2×#892862
#117255282
#133057