Finite Group Databasebeta

← #13320049

Subgroups — #13320049

Order 1792.

938 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×#90233C211yesmaximal, characteristic
2#11736029C212yesmaximal, complemented
3C7×#1459C211yesmaximal, characteristic
4#11729356C211yesmaximal, characteristic
5#11727677C212yesmaximal, complemented
6#11725881C212yesmaximal, complemented
7#11725879C212yesmaximal, complemented
8#11725859C214yesmaximal, complemented
9C2×C4×C7×D4C2211yescharacteristic
10C2×#6499C412yes—
11C2×#6489C411yescharacteristic
12C2×#6489C2211yescharacteristic
13C2×#6487C411yescharacteristic
14C2×#6487C2211yescharacteristic
15#90233—22no—
16#90233—22no—
17#90233—24no—
18C2×#6450C2212yes—
19C2×#6442C412yes—
20C2×#6442C2212yes—
21C2×#6432C411yescharacteristic
22C2×C4×Dic14C411yescharacteristic
23C2×C7×C22⋊C8C2212yes—
24#89979C2212yes—
25#89874—22no—
26#89873—22no—
27#89492—24no—
28#89491C2214yes—
29#89490C2214yes—
30C7×#324C2211yescharacteristic
31C7×#324C2211yescharacteristic
32C7×#323C2212yes—
33#89323C2212yes—
34#89320C2212yes—
35#89264C2214yes—
36#89264C2214yes—
37#17420—71nomaximal, Sylow
38C22×C7×D4C2×C411yescharacteristic
39C23×C4×C7C2×C411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C22×Dic14C2×C412yes—
42C22×Dic14C2×C411yescharacteristic
43C22×C7×C8—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—
53C2×C22⋊Dic7C2×C412yes—
54#6499—24no—
55#6499—24no—
56#6489—22no—
57#6489—21no—
58#6489—21no—
59#6489D412yes—
60#6489D412yes—
61#6489D414yes—
62C2×C4⋊Dic7C2×C412yes—
63C2×C4⋊Dic7C2×C411yescharacteristic
64C2×C4⋊Dic7C2×C411yescharacteristic
65C2×C4⋊Dic7C2×C411yescharacteristic
66#6487—22no—
67#6487—22no—
68#6487—22no—
69#6487—22no—
70C2×Dic7⋊C4C2×C412yes—
71C2×Dic7⋊C4C2×C412yes—
72C2×Dic7⋊C4C2312yes—
73C2×C4×Dic7C2×C412yes—
74C2×C4×Dic7C2×C412yes—
75C2×C4×Dic7C2312yes—
76C2×#1136—22no—
77#6450—24no—
78#6450—24no—
79#6442—24no—
80#6442—24no—
81#6442—24no—
82#6442—24no—
83#6432—22no—
84#6432—22no—
85C4×Dic14—22no—
86C4×Dic14—21no—
87C4×Dic14—21no—
88C7×C22⋊C8—22no—
89C7×C22⋊C8C2312yes—
90C7×C22⋊C8C2312yes—
91C22⋊(C7⋊C8)—22no—
92C22⋊(C7⋊C8)C2314yes—
93#6389—24no—
94Dic7⋊C8—24no—
95Dic7⋊C8—24no—
96C8×Dic7—24no—
97C2×#556—71no—
98#2924—72no—
99#1473—74no—
100#1472—72no—
101#1467—72no—
102#1465—71no—
103#1461—72no—
104#1459D711yescharacteristic
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×Dic7C22×C412yes—
122C22×Dic7C22×C412yes—
123C22×Dic7C22×C412yes—
124C22×Dic7C22×C412yes—
125C2×Dic14—21no—
126C2×Dic14—21no—
127C2×Dic14—24no—
128C2×Dic14—24no—
129C2×Dic14—42no—
130C2×Dic14—42no—
131C2×C7×C8—22no—
132C2×C7×C8—22no—
133C2×C7×C8—44no—
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⋊Dic7—22no—
142C22⋊Dic7—22no—
143C22⋊Dic7C22⋊C414yes—
144C22⋊Dic7C22⋊C414yes—
145C4⋊Dic7—21no—
146C4⋊Dic7—21no—
147C4⋊Dic7—24no—
148C4⋊Dic7C22⋊C412yes—
149C4⋊Dic7C22⋊C411yescharacteristic
150C4⋊Dic7C22⋊C411yescharacteristic
151C4⋊Dic7C22⋊C412yes—
152C4⋊Dic7C22⋊C412yes—
153Dic7⋊C4—24no—
154Dic7⋊C4—22no—
155Dic7⋊C4—22no—
156Dic7⋊C4—24no—
157C4×Dic7—22no—
158C4×Dic7—22no—
159C4×Dic7—22no—
160C4×Dic7—22no—
161C4×Dic7C2×D414yes—
162C4×Dic7C2×D414yes—
163#1136—44no—
164C2×C7⋊C8—24no—
165#556—142no—
166#556—142no—
167#556—144no—
168C2×C4⋊Q8—71no—
169C2×#126—72no—
170C2×#126—72no—
171C2×#124—72no—
172C2×C22⋊Q8—71no—
173C2×C22⋊Q8—71no—
174C2×C4×Q8—72no—
175C2×C4×Q8—71no—
176C2×C4×D4C2×D711yescharacteristic
177C2×#117—71no—
178C2×#117—71no—
179#433—144no—
180#432—74no—
181#431—74no—
182#424—142no—
183#407—72no—
184C2×C22⋊C8C2×D712yes—
185#405—142no—
186#324—72no—
187#324—74no—
188#324—74no—
189#324C2×D711yescharacteristic
190#324C2×D711yescharacteristic
191#323—72no—
192#323C2×D712yes—
193C23×C7—21no—
194C23×C7—21no—
195C23×C7—21no—
196C23×C7—21no—
197C23×C7—21no—
198C23×C7—41no—
199C23×C7—41no—
200C23×C7—41no—
201C23×C7—41no—
202C23×C7C2×C22⋊C412yes—
203C23×C7C23×C411yescharacteristic
204C7×D4—44no—
205C2×C4×C7—21no—
206C2×C4×C7—21no—
207C2×C4×C7—24no—
208C2×C4×C7—22no—
209C2×C4×C7—21no—
210C2×C4×C7—21no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—41no—
216C2×C4×C7—41no—
217C2×C4×C7C2×C22⋊C411yescharacteristic
218C2×C4×C7C2×C22⋊C411yescharacteristic
219C2×C4×C7#13112yes—
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—42no—
229C2×Dic7C2×C22⋊C412yes—
230C2×Dic7C2×C22⋊C412yes—
231C2×Dic7C2×C22⋊C414yes—
232Dic14—48no—
233Dic14—42no—
234Dic14—42no—
235C7×C8—42no—
236C7×C8—42no—
237C7⋊C8—44no—
238C22×Q8—72no—
239C22×Q8—71no—
240C22×D4C4×D711yescharacteristic
241C23×C4C4×D711yescharacteristic
242C23×C4C22×D711yescharacteristic
243C2×M16—142no—
244C22×C8—22no—
245C4⋊Q8—142no—
246C4⋊Q8—142no—
247#126—144no—
248#126—144no—
249#126—144no—
250#126—144no—
251#124—144no—
252#124—144no—
253C22⋊Q8—142no—
254C22⋊Q8—142no—
255C22⋊Q8—142no—
256C22⋊Q8—142no—
257C4×Q8—142no—
258C4×Q8—141no—
259C4×Q8—141no—
260C4×Q8—144no—
261C4×Q8—144no—
262C4×D4—22no—
263C4×D4—22no—
264#117—72no—
265#117—72no—
266#117—74no—
267#117—142no—
268#117—141no—
269#117—141no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—71no—
273C2×C4⋊C4—72no—
274C2×C4⋊C4—72no—
275C2×C4⋊C4—72no—
276C2×C4⋊C4—72no—
277C2×C4⋊C4C22×D711yescharacteristic
278C2×C22⋊C4—72no—
279C2×C22⋊C4—72no—
280C2×C22⋊C4—72no—
281C2×C22⋊C4—72no—
282C2×C22⋊C4C4×D711yescharacteristic
283C2×C22⋊C4C22×D711yescharacteristic
284C2×C42—72no—
285C2×C42—72no—
286C2×C42—72no—
287C2×C42C4×D711yescharacteristic
288C4⋊C8—144no—
289C4⋊C8—144no—
290C22⋊C8—22no—
291C22⋊C8—74no—
292C22⋊C8—142no—
293C22⋊C8C22×D712yes—
294C22⋊C8C22×D712yes—
295#97—144no—
296C4×C8—144no—
297C22×C7—21no—
298C22×C7—21no—
299C22×C7—22no—
300C22×C7—22no—
301C22×C7—22no—
302C22×C7—41no—
303C22×C7—41no—
304C22×C7—41no—
305C22×C7—81no—
306C22×C7—81no—
307C22×C7—81no—
308C22×C7—81no—
309C22×C7C22×C22⋊C411yescharacteristic
310C22×C7C2×#13111yescharacteristic
311C22×C7#56811yescharacteristic
312C4×C7—22no—
313C4×C7—22no—
314C4×C7—21no—
315C4×C7—41no—
316C4×C7—41no—
317C4×C7#41012yes—
318Dic7—24no—
319Dic7—22no—
320Dic7—22no—
321Dic7—42no—
322Dic7—42no—
323C24C2×C4×D711yescharacteristic
324C24C2×C4×D711yescharacteristic
325C2×Q8—144no—
326C2×Q8—141no—
327C2×Q8—141no—
328C2×Q8—144no—
329C2×Q8—282no—
330C2×Q8—282no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—41no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—72no—
339C22×C4—72no—
340C22×C4—72no—
341C22×C4—72no—
342C22×C4C2×C4×D711yescharacteristic
343C22×C4C2×C4×D711yescharacteristic
344C22×C4C2×C4×D711yescharacteristic
345C22×C4C2×C4×D711yescharacteristic
346C22×C4D4×D712yes—
347C22×C4D4×D712yes—
348C22×C4C23×D711yescharacteristic
349M16—284no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—44no—
353C2×C8—144no—
354C4⋊C4—22no—
355C4⋊C4—72no—
356C4⋊C4—71no—
357C4⋊C4—71no—
358C4⋊C4—72no—
359C4⋊C4—72no—
360C4⋊C4—144no—
361C4⋊C4—142no—
362C4⋊C4—142no—
363C4⋊C4—144no—
364C4⋊C4—141no—
365C4⋊C4—141no—
366C4⋊C4—144no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—74no—
370C22⋊C4—74no—
371C22⋊C4—142no—
372C22⋊C4—142no—
373C22⋊C4—144no—
374C22⋊C4—144no—
375C42—21no—
376C42—21no—
377C42—74no—
378C42—74no—
379C42—142no—
380C42—142no—
381C42—142no—
382C42—142no—
383C2×C7—21no—
384C2×C7—21no—
385C2×C7—42no—
386C2×C7—81no—
387C2×C7#288011yescharacteristic
388C2×C7C2×#41011yescharacteristic
389C2×C7#288911yescharacteristic
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23D7×C22⋊C412yes—
400C23C22×C4×D711yescharacteristic
401Q8—288no—
402Q8—282no—
403Q8—282no—
404D4—44no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—72no—
418C2×C4—72no—
419C2×C4—74no—
420C2×C4—142no—
421C2×C4—142no—
422C2×C4—142no—
423C2×C4—142no—
424C2×C4—142no—
425C2×C4—142no—
426C2×C4—282no—
427C2×C4—282no—
428C2×C4D7×C22⋊C411yescharacteristic
429C2×C4D7×C22⋊C411yescharacteristic
430C2×C4#646412yes—
431C2×C4C2×D4×D712yes—
432C8—42no—
433C8—42no—
434C8—284no—
435C7#1742011yescharacteristic, complemented, Sylow
436C22—21no—
437C22—21no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22C2×D7×C22⋊C411yescharacteristic
449C22C2×#646411yescharacteristic
450C22#9043611yescharacteristic
451C4—22no—
452C4—21no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—142no—
457C4—142no—
458C4—144no—
459C4—282no—
460C4—282no—
461C4#8951212yes—
462C2—21no—
463C2—21no—
464C2—42no—
465C2—81no—
466C2#1173604511yescharacteristic
467C2C2×#8951211yescharacteristic
468C2#1173616011yescharacteristic
subgroupG/H
centerC22C2×D7×C22⋊C4
commutatorC23×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1459C2
socleC22×C7C22×C22⋊C4
prime radicalC22×C7×D4C2×C4
cosocleC22×C4×C7C24
hypercenter#1459D7
hypocenterC7#17420
LCSG ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #11736029C2semidirect
5. #11727677C2semidirect
6. #11725881C2semidirect
7. #11725879C2semidirect
8. #11725859C2semidirect
435. C7#17420semidirect

Maximal subgroups

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

subgroupindex
C2×#902332
#117360292
C7×#14592
#117293562
#117276772
#117258812
#117258792
#117258592
#174207