Finite Group Databasebeta

← #13264252

Subgroups — #13264252

Order 1792.

778 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2404C211yesmaximal, characteristic
2C2×#89321C211yesmaximal, characteristic, complemented
3#11729293C211yesmaximal, characteristic, complemented
4#11725452C212yesmaximal, complemented
5#11725451C212yesmaximal, complemented
6C22×C7×#41C411yescharacteristic, complemented
7C2×C7×#120—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C22×C22⋊Dic7—21no—
10C7×#393C411yescharacteristic, complemented
11C7×#393C2211yescharacteristic
12C7×#392C411yescharacteristic, complemented
13C7×#392C2211yescharacteristic
14C7×#386—21no—
15C7×#386—21no—
16C7×#386—21no—
17C7×#381C411yescharacteristic, complemented
18#89979—21no—
19#89321—21no—
20#89321C2212yes—
21#89320—21no—
22#89320C2212yes—
23#11753—71nomaximal, Sylow
24C25×C7D411yescharacteristic
25C2×C7×#41—41no—
26C2×C7×#41—41no—
27C2×C7×#41—44no—
28C22×C7×Q8C2×C411yescharacteristic, complemented
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C22×C7×D4—21no—
32C22×C7×D4C2×C411yescharacteristic, complemented
33C23×C4×C7—21no—
34C23×C4×C7C2×C411yescharacteristic
35C23×C4×C7D411yescharacteristic
36C23×Dic7—41no—
37C7×#120—44no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—41no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4C2311yescharacteristic
49C2×C22⋊Dic7—24no—
50C2×C22⋊Dic7—22no—
51C2×C22⋊Dic7—41no—
52C2×C22⋊Dic7—42no—
53C2×#1136—41no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95—21no—
57C7×#95C2×C411yescharacteristic, complemented
58C7×#95C2×C411yescharacteristic, complemented
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—41no—
61#2404D711yescharacteristic
62C2×#342—71no—
63#1462—71no—
64#1351—72no—
65#1343—72no—
66C24×C7—21no—
67C24×C7—21no—
68C24×C7—22no—
69C24×C7—21no—
70C24×C7—21no—
71C24×C7—21no—
72C24×C7—42no—
73C24×C7—81no—
74C24×C7C22⋊C411yescharacteristic
75C24×C7C22⋊C411yescharacteristic
76C24×C7C2×D411yescharacteristic
77C7×#41—44no—
78C7×#41—44no—
79C7×#41—44no—
80C7×#41—44no—
81C2×C7×Q8—42no—
82C2×C7×Q8—41no—
83C2×C7×D4—24no—
84C2×C7×D4—41no—
85C2×C7×D4—42no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—42no—
89C2×C7×D4—42no—
90C2×C7×D4—42no—
91C2×C7×D4—42no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C2×C7×D4—81no—
95C2×C7×D4—81no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7—21no—
103C22×C4×C7—42no—
104C22×C4×C7—42no—
105C22×C4×C7—42no—
106C22×C4×C7—42no—
107C22×C4×C7—81no—
108C22×C4×C7C22⋊C411yescharacteristic
109C22×C4×C7C22⋊C411yescharacteristic
110C22×C4×C7C22×C411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×Dic7—44no—
113C22×Dic7—42no—
114C22×Dic7—82no—
115C22×Dic7—82no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—42no—
121C7×C22⋊C4—42no—
122C7×C22⋊C4—42no—
123C7×C22⋊C4—81no—
124C7×C22⋊C4—81no—
125C22⋊Dic7—24no—
126C22⋊Dic7—44no—
127C22⋊Dic7—42no—
128#1136—82no—
129C2×C7⋊C8—42no—
130C22×#41Dic711yescharacteristic, complemented
131C2×#120—21no—
132C22×C22⋊C4—141no—
133C22×C22⋊C4C2×D711yescharacteristic
134#407—141no—
135#393Dic711yescharacteristic, complemented
136#393C2×D711yescharacteristic
137#392Dic711yescharacteristic, complemented
138#392C2×D711yescharacteristic
139#386—21no—
140#386—21no—
141#386—21no—
142#381Dic711yescharacteristic, complemented
143#342—72no—
144#342—141no—
145#323—72no—
146#323—141no—
147C23×C7—22no—
148C23×C7—22no—
149C23×C7—22no—
150C23×C7—22no—
151C23×C7—22no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7—21no—
155C23×C7—21no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7—22no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7—42no—
169C23×C7—41no—
170C23×C7—41no—
171C23×C7—41no—
172C23×C7—81no—
173C23×C7—81no—
174C23×C7—84no—
175C23×C7—82no—
176C23×C7—82no—
177C23×C7#9911yescharacteristic
178C23×C7#9911yescharacteristic
179C23×C7#10412yes—
180C23×C7C2×C22⋊C411yescharacteristic
181C23×C7AGL(1,ℤ/8)11yescharacteristic
182C23×C7#13711yescharacteristic
183C7×Q8—42no—
184C7×Q8—42no—
185C7×D4—44no—
186C7×D4—42no—
187C7×D4—42no—
188C7×D4—44no—
189C7×D4—84no—
190C7×D4—82no—
191C7×D4—82no—
192C2×C4×C7—22no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—21no—
197C2×C4×C7—42no—
198C2×C4×C7—41no—
199C2×C4×C7—42no—
200C2×C4×C7—48no—
201C2×C4×C7—42no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—41no—
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—41no—
208C2×C4×C7—41no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—81no—
214C2×Dic7—44no—
215C2×Dic7—88no—
216C2×Dic7—82no—
217C2×Dic7—82no—
218C7⋊C8—82no—
219C25C22⋊D711yescharacteristic
220C2×#41—41no—
221C2×#41—41no—
222C2×#41—44no—
223C22×Q8C2×Dic711yescharacteristic, complemented
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—21no—
227C22×D4C2×Dic711yescharacteristic, complemented
228C23×C4—21no—
229C23×C4—281no—
230C23×C4C2×Dic711yescharacteristic
231C23×C4C22⋊D711yescharacteristic
232C2×M16—281no—
233#120—44no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—144no—
242C2×C22⋊C4—142no—
243C2×C22⋊C4—281no—
244C2×C22⋊C4—282no—
245C2×C22⋊C4C2×Dic711yescharacteristic
246C2×C22⋊C4C22⋊D711yescharacteristic
247C2×C22⋊C4C22⋊D711yescharacteristic
248C2×C22⋊C4C22×D711yescharacteristic
249C22⋊C8—142no—
250C22⋊C8—281no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95C2×Dic711yescharacteristic, complemented
255#95C2×Dic711yescharacteristic, complemented
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—22no—
265C22×C7—21no—
266C22×C7—42no—
267C22×C7—42no—
268C22×C7—42no—
269C22×C7—41no—
270C22×C7—41no—
271C22×C7—44no—
272C22×C7—44no—
273C22×C7—42no—
274C22×C7—42no—
275C22×C7—41no—
276C22×C7—41no—
277C22×C7—41no—
278C22×C7—41no—
279C22×C7—42no—
280C22×C7—81no—
281C22×C7—81no—
282C22×C7—81no—
283C22×C7—81no—
284C22×C7—82no—
285C22×C7—82no—
286C22×C7—81no—
287C22×C7—81no—
288C22×C7AΣL(1,𝔽16)12yes—
289C22×C7C2×#9911yescharacteristic
290C22×C7#41711yescharacteristic
291C22×C7C2×#10411yescharacteristic
292C4×C7—41no—
293C4×C7—42no—
294C4×C7—41no—
295C4×C7—41no—
296C4×C7—41no—
297C4×C7—81no—
298C4×C7—81no—
299Dic7—84no—
300C24—21no—
301C24—21no—
302C24—22no—
303C24—21no—
304C24—21no—
305C24—21no—
306C24—42no—
307C24—81no—
308C24C22⋊Dic711yescharacteristic
309C24C22⋊Dic711yescharacteristic
310C24C2×C22⋊D711yescharacteristic
311#41—44no—
312#41—44no—
313#41—44no—
314#41—44no—
315C2×Q8—42no—
316C2×Q8—41no—
317C2×D4—24no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—81no—
329C2×D4—81no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—81no—
342C22×C4—284no—
343C22×C4—282no—
344C22×C4—562no—
345C22×C4—562no—
346C22×C4C22⋊Dic711yescharacteristic
347C22×C4C22⋊Dic711yescharacteristic
348C22×C4C22×Dic711yescharacteristic
349C22×C4C2×C22⋊D711yescharacteristic
350M16—562no—
351C2×C8—282no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—41no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—81no—
360C22⋊C4—81no—
361C22⋊C4—144no—
362C22⋊C4—284no—
363C22⋊C4—282no—
364C2×C7—21no—
365C2×C7—21no—
366C2×C7—22no—
367C2×C7—41no—
368C2×C7—41no—
369C2×C7—42no—
370C2×C7—41no—
371C2×C7—82no—
372C2×C7#150411yescharacteristic
373C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
374C2×C7#212711yescharacteristic
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—22no—
384C23—21no—
385C23—21no—
386C23—22no—
387C23—41no—
388C23—42no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—81no—
401C23—81no—
402C23—84no—
403C23—82no—
404C23—82no—
405C23#640811yescharacteristic
406C23#640811yescharacteristic
407C23#641112yes—
408C23#651211yescharacteristic
409C23#651411yescharacteristic
410C23C2×C22⋊Dic711yescharacteristic
411Q8—42no—
412Q8—42no—
413D4—44no—
414D4—42no—
415D4—42no—
416D4—44no—
417D4—84no—
418D4—82no—
419D4—82no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—48no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—81no—
442C2×C4—284no—
443C2×C4—568no—
444C2×C4—562no—
445C2×C4—562no—
446C8—562no—
447C7#1175311yescharacteristic, complemented, Sylow
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—21no—
457C22—21no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—41no—
462C22—41no—
463C22—44no—
464C22—44no—
465C22—42no—
466C22—42no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—42no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—82no—
477C22—82no—
478C22—81no—
479C22—81no—
480C22#8933112yes—
481C22C2×#640811yescharacteristic
482C22#9000411yescharacteristic
483C22C2×#641111yescharacteristic
484C4—41no—
485C4—42no—
486C4—41no—
487C4—41no—
488C4—41no—
489C4—81no—
490C4—81no—
491C4—564no—
492C2—21no—
493C2—21no—
494C2—22no—
495C2—41no—
496C2—41no—
497C2—42no—
498C2—41no—
499C2—82no—
500C2C2×#8933111yescharacteristic
501C2#1172946211yescharacteristic
502C2#1172961411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2404C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2404D7
hypocenterC7#11753
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2404

Complements

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

NH ≅ G/Ntype
2. C2×#89321C2semidirect
3. #11729293C2semidirect
4. #11725452C2semidirect
5. #11725451C2semidirect
6. C22×C7×#41C4semidirect
10. C7×#393C4semidirect
12. C7×#392C4semidirect
17. C7×#381C4semidirect
28. C22×C7×Q8C2×C4semidirect
32. C22×C7×D4C2×C4semidirect
57. C7×#95C2×C4semidirect
58. C7×#95C2×C4semidirect
130. C22×#41Dic7semidirect
135. #393Dic7semidirect
137. #392Dic7semidirect
142. #381Dic7semidirect
223. C22×Q8C2×Dic7semidirect
227. C22×D4C2×Dic7semidirect
254. #95C2×Dic7semidirect
255. #95C2×Dic7semidirect
447. C7#11753semidirect

Maximal subgroups

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

subgroupindex
C7×#24042
C2×#893212
#117292932
#117254522
#117254512
#117537