Finite Group Databasebeta

← #13320002

Subgroups — #13320002

Order 1792.

944 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
1C2×#90232C211yesmaximal, characteristic
2#11736066C212yesmaximal, complemented
3C7×#1465C211yesmaximal, characteristic
4#11729354C211yesmaximal, characteristic
5#11727726C212yesmaximal, complemented
6#11725882C214yesmaximal, complemented
7#11725856C212yesmaximal, complemented
8#11725854C212yesmaximal, complemented
9C2×C4×C7×D4C2211yescharacteristic
10C2×#6498C412yes—
11C2×#6489C2211yescharacteristic
12C22×C4⋊Dic7C411yescharacteristic
13C2×#6487C411yescharacteristic
14C2×#6487C2211yescharacteristic
15#90232—22no—
16#90232—22no—
17#90232—24no—
18C2×#6452C2212yes—
19C2×#6442C412yes—
20C2×#6440C2212yes—
21C2×#6433C411yescharacteristic
22C2×C4×Dic14C411yescharacteristic
23C7×#407C2212yes—
24C2×C22⋊(C7⋊C8)C2212yes—
25#89894—22no—
26#89891—22no—
27#89492—24no—
28#89491C2214yes—
29#89490C2214yes—
30C7×#324C2212yes—
31C7×#323C2212yes—
32#89323C2212yes—
33#89320C2211yescharacteristic
34#89320C2211yescharacteristic
35#89264C2214yes—
36#89264C2214yes—
37#17413—71nomaximal, Sylow
38C22×C7×D4C2×C411yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7C2×C412yes—
42C22×Dic14C2×C411yescharacteristic
43C2×C7×M16—22no—
44C4×C7×D4—22no—
45C4×C7×D4—22no—
46C2×C7×C4⋊C4C2311yescharacteristic
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4C2311yescharacteristic
49C2×C42×C7C2×C411yescharacteristic
50C2×C22⋊Dic7C2×C412yes—
51C2×C22⋊Dic7C2×C412yes—
52C2×C22⋊Dic7C2×C412yes—
53#6498—24no—
54#6498—24no—
55#6489D412yes—
56#6489D412yes—
57#6489D414yes—
58C2×C4⋊Dic7—22no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7C2×C411yescharacteristic
62C2×C4⋊Dic7C2×C412yes—
63C2×C4⋊Dic7C2×C411yescharacteristic
64C2×C4⋊Dic7C2×C411yescharacteristic
65#6487—22no—
66#6487—22no—
67#6487—22no—
68#6487—22no—
69C2×Dic7⋊C4C2×C412yes—
70C2×Dic7⋊C4C2×C412yes—
71C2×Dic7⋊C4C2×C412yes—
72C2×Dic7⋊C4C2×C412yes—
73C2×Dic7⋊C4C2312yes—
74C2×C4×Dic7C2×C412yes—
75C2×C4×Dic7C2312yes—
76C22×C7⋊C8—22no—
77#6452—24no—
78#6452—24no—
79#6442—24no—
80#6442—24no—
81#6440—24no—
82#6440—24no—
83#6433—22no—
84#6433—22no—
85C4×Dic14—22no—
86C4×Dic14—21no—
87C4×Dic14—21no—
88C7×C22⋊C8—22no—
89C7×C22⋊C8C2314yes—
90C22⋊(C7⋊C8)—22no—
91C22⋊(C7⋊C8)C2312yes—
92C22⋊(C7⋊C8)C2312yes—
93#6389—24no—
94Dic7⋊C8—24no—
95Dic7⋊C8—24no—
96C8×Dic7—24no—
97C2×#552—71no—
98#2924—72no—
99#1472—72no—
100#1470—74no—
101#1465D711yescharacteristic
102#1464—72no—
103#1459—71no—
104#1458—72no—
105C24×C7C22×C411yescharacteristic
106C24×C7C22×C411yescharacteristic
107C2×C7×D4—22no—
108C2×C7×D4—22no—
109C2×C7×D4—41no—
110C22×C4×C7—21no—
111C22×C4×C7—21no—
112C22×C4×C7—42no—
113C22×C4×C7—42no—
114C22×C4×C7C22×C411yescharacteristic
115C22×C4×C7C22×C411yescharacteristic
116C22×C4×C7C22×C411yescharacteristic
117C22×C4×C7C22×C411yescharacteristic
118C22×C4×C7C2×D412yes—
119C22×C4×C7C2×D412yes—
120C22×C4×C7C2411yescharacteristic
121C22×Dic7—24no—
122C22×Dic7—22no—
123C22×Dic7—22no—
124C22×Dic7—42no—
125C22×Dic7C22×C412yes—
126C22×Dic7C22×C412yes—
127C22×Dic7C22×C412yes—
128C22×Dic7C22×C412yes—
129C2×Dic14—21no—
130C2×Dic14—21no—
131C2×Dic14—42no—
132C7×M16—44no—
133C2×C7×C8—24no—
134C7×C4⋊C4—22no—
135C7×C22⋊C4—22no—
136C7×C22⋊C4—22no—
137C42×C7—21no—
138C42×C7—21no—
139C22⋊Dic7—24no—
140C22⋊Dic7—24no—
141C22⋊Dic7C22⋊C414yes—
142C22⋊Dic7C22⋊C414yes—
143C4⋊Dic7—21no—
144C4⋊Dic7—21no—
145C4⋊Dic7—22no—
146C4⋊Dic7—22no—
147C4⋊Dic7C22⋊C412yes—
148C4⋊Dic7C22⋊C411yescharacteristic
149C4⋊Dic7C22⋊C411yescharacteristic
150C4⋊Dic7C22⋊C412yes—
151C4⋊Dic7C22⋊C412yes—
152Dic7⋊C4—24no—
153Dic7⋊C4—22no—
154Dic7⋊C4—22no—
155Dic7⋊C4—24no—
156Dic7⋊C4—24no—
157Dic7⋊C4—24no—
158C4×Dic7—22no—
159C4×Dic7—22no—
160C4×Dic7C2×D414yes—
161C4×Dic7C2×D414yes—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—22no—
164C2×C7⋊C8—44no—
165#552—142no—
166#552—142no—
167#552—144no—
168C2×#126—72no—
169C2×#125—72no—
170C2×#125—71no—
171C2×#123—72no—
172C2×C22⋊Q8—71no—
173C2×C22⋊Q8—71no—
174C2×C4×Q8—71no—
175C2×C4×D4C2×D711yescharacteristic
176C2×#117—71no—
177C2×#117—72no—
178C22×C4⋊C4—71no—
179#433—144no—
180#432—74no—
181#431—74no—
182#424—142no—
183#407C2×D712yes—
184C2×C22⋊C8—72no—
185#405—142no—
186#324—72no—
187#324—74no—
188#324—74no—
189#324C2×D712yes—
190#323—71no—
191#323—71no—
192#323C2×D712yes—
193C23×C7—21no—
194C23×C7—21no—
195C23×C7—21no—
196C23×C7—21no—
197C23×C7—41no—
198C23×C7—41no—
199C23×C7—41no—
200C23×C7—41no—
201C23×C7C2×C22⋊C412yes—
202C23×C7#13112yes—
203C23×C7C23×C411yescharacteristic
204C7×D4—44no—
205C2×C4×C7—24no—
206C2×C4×C7—22no—
207C2×C4×C7—21no—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7—41no—
217C2×C4×C7—41no—
218C2×C4×C7C2×C22⋊C411yescharacteristic
219C2×C4×C7C2×C22⋊C411yescharacteristic
220C2×C4×C7C22×D412yes—
221C2×Dic7—22no—
222C2×Dic7—22no—
223C2×Dic7—22no—
224C2×Dic7—22no—
225C2×Dic7—22no—
226C2×Dic7—22no—
227C2×Dic7—42no—
228C2×Dic7—44no—
229C2×Dic7—44no—
230C2×Dic7—42no—
231C2×Dic7C2×C22⋊C412yes—
232C2×Dic7C2×C22⋊C412yes—
233C2×Dic7C2×C22⋊C414yes—
234Dic14—42no—
235Dic14—42no—
236C7×C8—44no—
237C7⋊C8—42no—
238C7⋊C8—42no—
239C22×Q8—71no—
240C22×D4C4×D711yescharacteristic
241C23×C4—72no—
242C23×C4C4×D711yescharacteristic
243C23×C4C22×D711yescharacteristic
244C2×M16—22no—
245C22×C8—142no—
246#126—144no—
247#126—144no—
248#125—142no—
249#125—142no—
250#125—144no—
251#125—144no—
252#123—144no—
253#123—144no—
254C22⋊Q8—142no—
255C22⋊Q8—142no—
256C22⋊Q8—142no—
257C22⋊Q8—142no—
258C4×Q8—142no—
259C4×Q8—141no—
260C4×Q8—141no—
261C4×D4—22no—
262C4×D4—22no—
263#117—72no—
264#117—72no—
265#117—74no—
266#117—144no—
267#117—144no—
268C2×C4⋊C4—71no—
269C2×C4⋊C4—71no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—72no—
272C2×C4⋊C4—72no—
273C2×C4⋊C4—72no—
274C2×C4⋊C4—72no—
275C2×C4⋊C4—72no—
276C2×C4⋊C4—72no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4—141no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4C22×D711yescharacteristic
281C2×C22⋊C4—72no—
282C2×C22⋊C4—72no—
283C2×C22⋊C4—72no—
284C2×C22⋊C4C4×D711yescharacteristic
285C2×C22⋊C4C22×D711yescharacteristic
286C2×C42—72no—
287C2×C42—72no—
288C2×C42C4×D711yescharacteristic
289C4⋊C8—144no—
290C4⋊C8—144no—
291C22⋊C8—22no—
292C22⋊C8—72no—
293C22⋊C8—72no—
294C22⋊C8—142no—
295C22⋊C8C22×D714yes—
296#97—144no—
297C4×C8—144no—
298C22×C7—21no—
299C22×C7—21no—
300C22×C7—22no—
301C22×C7—22no—
302C22×C7—22no—
303C22×C7—22no—
304C22×C7—42no—
305C22×C7—42no—
306C22×C7—41no—
307C22×C7—41no—
308C22×C7—81no—
309C22×C7—81no—
310C22×C7C22×C22⋊C411yescharacteristic
311C22×C7C2×#13111yescharacteristic
312C22×C7#56811yescharacteristic
313C4×C7—22no—
314C4×C7—21no—
315C4×C7—41no—
316C4×C7—41no—
317C4×C7—41no—
318C4×C7#41012yes—
319Dic7—22no—
320Dic7—22no—
321Dic7—24no—
322Dic7—42no—
323Dic7—42no—
324C24C2×C4×D711yescharacteristic
325C24C2×C4×D711yescharacteristic
326C2×Q8—141no—
327C2×Q8—141no—
328C2×Q8—282no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—41no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—72no—
337C22×C4—72no—
338C22×C4—72no—
339C22×C4—72no—
340C22×C4—142no—
341C22×C4—142no—
342C22×C4—144no—
343C22×C4—282no—
344C22×C4C2×C4×D711yescharacteristic
345C22×C4C2×C4×D711yescharacteristic
346C22×C4C2×C4×D711yescharacteristic
347C22×C4C2×C4×D711yescharacteristic
348C22×C4D4×D712yes—
349C22×C4D4×D712yes—
350C22×C4C23×D711yescharacteristic
351M16—44no—
352C2×C8—24no—
353C2×C8—142no—
354C2×C8—142no—
355C2×C8—284no—
356C4⋊C4—22no—
357C4⋊C4—72no—
358C4⋊C4—71no—
359C4⋊C4—71no—
360C4⋊C4—72no—
361C4⋊C4—72no—
362C4⋊C4—144no—
363C4⋊C4—142no—
364C4⋊C4—142no—
365C4⋊C4—144no—
366C4⋊C4—144no—
367C4⋊C4—142no—
368C4⋊C4—142no—
369C4⋊C4—141no—
370C4⋊C4—141no—
371C4⋊C4—144no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—74no—
375C22⋊C4—74no—
376C22⋊C4—144no—
377C22⋊C4—144no—
378C42—21no—
379C42—21no—
380C42—74no—
381C42—74no—
382C42—142no—
383C42—142no—
384C2×C7—21no—
385C2×C7—21no—
386C2×C7—42no—
387C2×C7—42no—
388C2×C7#288011yescharacteristic
389C2×C7C2×#41011yescharacteristic
390C2×C7#288911yescharacteristic
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23D7×C22⋊C412yes—
400C23#647012yes—
401C23C22×C4×D711yescharacteristic
402Q8—282no—
403Q8—282no—
404D4—44no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—72no—
419C2×C4—72no—
420C2×C4—74no—
421C2×C4—142no—
422C2×C4—142no—
423C2×C4—142no—
424C2×C4—142no—
425C2×C4—142no—
426C2×C4—142no—
427C2×C4—282no—
428C2×C4—284no—
429C2×C4—284no—
430C2×C4—282no—
431C2×C4D7×C22⋊C411yescharacteristic
432C2×C4D7×C22⋊C411yescharacteristic
433C2×C4C2×D4×D712yes—
434C8—44no—
435C8—282no—
436C8—282no—
437C7#1741311yescharacteristic, complemented, Sylow
438C22—21no—
439C22—21no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—41no—
448C22—81no—
449C22—81no—
450C22C2×D7×C22⋊C411yescharacteristic
451C22C2×#647011yescharacteristic
452C22#9043611yescharacteristic
453C4—22no—
454C4—21no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—142no—
459C4—142no—
460C4—144no—
461C4—282no—
462C4—282no—
463C4#8951212yes—
464C2—21no—
465C2—21no—
466C2—42no—
467C2—42no—
468C2#1173607611yescharacteristic
469C2C2×#8951211yescharacteristic
470C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1465C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4C2×C4
cosocleC22×C4×C7C24
hypercenter#1465D7
hypocenterC7#17413
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
2. #11736066C2semidirect
5. #11727726C2semidirect
6. #11725882C2semidirect
7. #11725856C2semidirect
8. #11725854C2semidirect
437. C7#17413semidirect

Maximal subgroups

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

subgroupindex
C2×#902322
#117360662
C7×#14652
#117293542
#117277262
#117258822
#117258562
#117258542
#174137