Finite Group Databasebeta

← #13321525

Subgroups — #13321525

Order 1792.

918 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
1#11742887C211yesmaximal, characteristic
2C7×#2356C211yesmaximal, characteristic, complemented
3#11739880C211yesmaximal, characteristic, complemented
4#11727319C214yesmaximal, complemented
5#11727300C211yesmaximal, characteristic, complemented
6#11727299C211yesmaximal, characteristic
7#11727225C211yesmaximal, characteristic, complemented
8#11727223C211yesmaximal, characteristic
9#11727185C214yesmaximal, complemented
10C2×C4×C7×Q8C2211yescharacteristic
11#90507—21no—
12#90276C412yescomplemented
13#90275C412yescomplemented
14#90272—22no—
15#90270C412yescomplemented
16#90259—22no—
17#90258C412yescomplemented
18#90206—22no—
19#90195—21no—
20#90170C2214yes—
21#90168C2211yescharacteristic
22#90161C2211yescharacteristic
23C7×#391—21no—
24C7×#387—22no—
25C7×#383C2212yes—
26C7×#382C2211yescharacteristic
27C4×C7×C4⋊C4—21no—
28C4×C7×C4⋊C4C2211yescharacteristic
29C7×#376C2212yes—
30Q8×C7⋊C8—22no—
31Q8×C7⋊C8—22no—
32#89697C2214yes—
33#89697C2211yescharacteristic
34#89697C2211yescharacteristic
35#89247C2212yes—
36#89247C2214yes—
37#89247C2212yes—
38#89246C2212yes—
39#89246C2214yes—
40#89246C2212yes—
41#17616—71nomaximal, Sylow
42C22×C7×Q8D411yescharacteristic
43C2×#1157—21no—
44C4×C7×Q8C2×C412yes—
45C4×C7×Q8C2×C412yes—
46C4×C7×Q8D412yes—
47C4×C7×Q8D412yes—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4D411yescharacteristic
51C2×C7×C4⋊C4D412yes—
52C2×C42×C7—22no—
53C2×C42×C7—21no—
54C2×C42×C7C2311yescharacteristic
55C2×C42×C7C2312yes—
56#6510—22no—
57#6509—22no—
58#6509—22no—
59#6507—22no—
60#6493—21no—
61#6493C2312yes—
62#6492—24no—
63C4×C22⋊D7—24no—
64#6489—21no—
65#6487—21no—
66#6487C2312yes—
67#6461—24no—
68#6459—24no—
69#6457—24no—
70#6456—22no—
71#6455—22no—
72D7×C4⋊C4—22no—
73#6452—24no—
74#6450—22no—
75#6439—24no—
76#6438—21no—
77#6438C2×C414yes—
78#6437C2×C411yescharacteristic
79C4×D28—21no—
80C4×D28C2×C412yes—
81C4×D28C2×C414yes—
82#6435—24no—
83#6433—21no—
84#6433C2×C414yes—
85#6432C2×C411yescharacteristic
86#6432C2×C411yescharacteristic
87#6432C2×C411yescharacteristic
88C4×Dic14—21no—
89C4×Dic14C2×C412yes—
90C4×Dic14C2×C414yes—
91C7×#95—22no—
92C7×#95—22no—
93C7×#95—21no—
94C7×#95—22no—
95C7×#95C2311yescharacteristic
96C7×#95C2312yes—
97C7×#95C2311yescharacteristic
98C22⋊(C7⋊C8)C2312yes—
99C22⋊(C7⋊C8)C2312yes—
100C4⋊(C7⋊C8)—21no—
101C4⋊(C7⋊C8)—21no—
102C4⋊(C7⋊C8)—24no—
103C4×C7⋊C8—21no—
104C4×C7⋊C8—21no—
105C4×C7⋊C8—24no—
106#3507—71no—
107#2982—71no—
108#2356D711yescharacteristic, complemented
109#1521—74no—
110#1518—71no—
111#1517—71no—
112#1503—71no—
113#1502—71no—
114#1500—74no—
115C2×C7×Q8—22no—
116C2×C7×Q8—22no—
117C2×C7×Q8C22⋊C412yes—
118C2×C7×Q8C22⋊C412yes—
119C22×C4×C7—21no—
120C22×C4×C7—22no—
121C22×C4×C7—21no—
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D412yes—
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2411yescharacteristic
127C2×C22⋊D7—22no—
128#1157—42no—
129#1157—42no—
130C2×D28—21no—
131C2×D28—21no—
132C2×D28C22×C412yes—
133C2×C4×D7—22no—
134C2×C4×D7—24no—
135C2×Dic14—21no—
136C2×Dic14—21no—
137C2×Dic14C22×C412yes—
138C7×C4⋊C4—42no—
139C7×C4⋊C4—42no—
140C7×C4⋊C4C22⋊C412yes—
141C7×C4⋊C4C22⋊C414yes—
142C7×C4⋊C4C22⋊C412yes—
143C7×C4⋊C4C22⋊C414yes—
144C42×C7—44no—
145C42×C7—42no—
146C42×C7C22×C411yescharacteristic
147C42×C7C22×C414yes—
148C42×C7C22×C411yescharacteristic
149C42×C7C2×D411yescharacteristic
150C42×C7C2×D411yescharacteristic
151C42×C7C2×D414yes—
152C22⋊Dic7—22no—
153(C2×D7)⋊C4—24no—
154(C2×D7)⋊C4—22no—
155(C2×D7)⋊C4—24no—
156(C2×D7)⋊C4—24no—
157(C2×D7)⋊C4—22no—
158C4⋊Dic7—21no—
159C4⋊Dic7—21no—
160C4⋊Dic7—21no—
161C4⋊Dic7—21no—
162C4⋊Dic7C22×C412yes—
163C4⋊Dic7C22×C412yes—
164Dic7⋊C4—22no—
165Dic7⋊C4—24no—
166Dic7⋊C4—24no—
167Dic7⋊C4—24no—
168Dic7⋊C4—22no—
169C4×Dic7—22no—
170C4×Dic7—24no—
171C2×C7⋊C8—22no—
172C2×C7⋊C8—22no—
173#559—72no—
174#557—72no—
175#556—72no—
176#556—142no—
177#554—72no—
178#552—142no—
179#539—142no—
180#536—141no—
181#533—71no—
182#532—71no—
183#529—74no—
184#520—141no—
185C2×C4×Q8C2×D711yescharacteristic
186C8×Q8—142no—
187C8×Q8—142no—
188#431—74no—
189#431—71no—
190#431—71no—
191#391—21no—
192#387—22no—
193#383C2×D712yes—
194#382C2×D711yescharacteristic
195C4×C4⋊C4—21no—
196C4×C4⋊C4C2×D711yescharacteristic
197#376C2×D712yes—
198#328—72no—
199#328—74no—
200#328—72no—
201#327—72no—
202#327—74no—
203#327—72no—
204C23×C7C22×D411yescharacteristic
205C22×D7—22no—
206C7×Q8—24no—
207C7×Q8—24no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—41no—
216C2×C4×C7—42no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—42no—
220C2×C4×C7—41no—
221C2×C4×C7—42no—
222C2×C4×C7C2×C22⋊C414yes—
223C2×C4×C7C2×C22⋊C411yescharacteristic
224C2×C4×C7C2×C22⋊C414yes—
225C2×C4×C7C2×C22⋊C411yescharacteristic
226C2×C4×C7C2×C22⋊C411yescharacteristic
227C2×C4×C7C2×C22⋊C411yescharacteristic
228C2×C4×C7#13512yes—
229C2×C4×C7#13512yes—
230C2×C4×C7C23×C411yescharacteristic
231C2×C4×C7C22×D411yescharacteristic
232C22⋊D7—42no—
233C22⋊D7—42no—
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—22no—
237D28—42no—
238D28—42no—
239D28—81no—
240C4×D7—44no—
241C4×D7—84no—
242Dic14—42no—
243Dic14—42no—
244Dic14—81no—
245C7⋊C8—42no—
246C7⋊C8—42no—
247C2×#41—141no—
248C22×Q8C22⋊D711yescharacteristic
249C4⋊Q8—71no—
250C4⋊Q8—71no—
251C4⋊Q8—71no—
252C4⋊Q8—142no—
253#127—71no—
254#126—144no—
255#126—144no—
256#125—74no—
257#125—144no—
258#125—141no—
259#124—74no—
260#124—142no—
261#124—141no—
262#123—144no—
263#123—144no—
264C22⋊Q8—72no—
265C22⋊Q8—142no—
266C22⋊Q8—142no—
267C22⋊Q8—141no—
268C22⋊Q8—144no—
269#121—72no—
270#121—141no—
271C4×Q8—74no—
272C4×Q8—72no—
273C4×Q8—142no—
274C4×Q8—141no—
275C4×Q8C4×D712yes—
276C4×Q8C4×D712yes—
277C4×Q8D2812yes—
278C4×Q8D2812yes—
279C4×D4—74no—
280C4×D4—72no—
281C4×D4—144no—
282C4×D4—142no—
283C4×D4—141no—
284#117—141no—
285#117—142no—
286#117—144no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—142no—
290C2×C4⋊C4C22⋊D711yescharacteristic
291C2×C4⋊C4C22⋊D712yes—
292C2×C42—22no—
293C2×C42—21no—
294C2×C42C22×D712yes—
295C2×C42C22×D711yescharacteristic
296C4⋊C8—144no—
297C4⋊C8—141no—
298C4⋊C8—141no—
299C22⋊C8—72no—
300C22⋊C8—72no—
301C4×C8—144no—
302C4×C8—141no—
303C4×C8—141no—
304#95—22no—
305#95—22no—
306#95—21no—
307#95—22no—
308#95C22×D711yescharacteristic
309#95C22×D712yes—
310#95C22×D711yescharacteristic
311C22×C7—21no—
312C22×C7—21no—
313C22×C7C22×C22⋊C411yescharacteristic
314C22×C7C2×#13511yescharacteristic
315C22×C7C2×#13511yescharacteristic
316C2×D7—42no—
317C2×D7—42no—
318C2×D7—42no—
319C4×C7—21no—
320C4×C7—21no—
321C4×C7—24no—
322C4×C7—21no—
323C4×C7—24no—
324C4×C7—21no—
325C4×C7—42no—
326C4×C7—81no—
327C4×C7—82no—
328C4×C7#41612yes—
329C4×C7#41612yes—
330Dic7—42no—
331Dic7—42no—
332Dic7—42no—
333#41—282no—
334#41—282no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—72no—
338C2×Q8—141no—
339C2×Q8—141no—
340C2×Q8(C2×D7)⋊C412yes—
341C2×Q8(C2×D7)⋊C412yes—
342C2×D4—72no—
343C2×D4—142no—
344C2×D4—141no—
345C2×D4—141no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—142no—
350C22×C4—144no—
351C22×C4C2×C22⋊D712yes—
352C22×C4C2×C22⋊D712yes—
353C22×C4C2×C22⋊D711yescharacteristic
354C22×C4C2×C22⋊D711yescharacteristic
355C22×C4C23×D711yescharacteristic
356C2×C8—142no—
357C2×C8—142no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—72no—
361C4⋊C4—72no—
362C4⋊C4—144no—
363C4⋊C4—144no—
364C4⋊C4—142no—
365C4⋊C4—144no—
366C4⋊C4—141no—
367C4⋊C4—141no—
368C4⋊C4—141no—
369C4⋊C4—141no—
370C4⋊C4—142no—
371C4⋊C4(C2×D7)⋊C412yes—
372C4⋊C4(C2×D7)⋊C412yes—
373C4⋊C4(C2×D7)⋊C414yes—
374C4⋊C4(C2×D7)⋊C414yes—
375C22⋊C4—142no—
376C22⋊C4—144no—
377C22⋊C4—144no—
378C22⋊C4—142no—
379C22⋊C4—144no—
380C22⋊C4—142no—
381C42—44no—
382C42—42no—
383C42—144no—
384C42—142no—
385C42C2×C4×D711yescharacteristic
386C42C2×C4×D714yes—
387C42C2×C4×D711yescharacteristic
388C42C2×D2814yes—
389C42C2×D2811yescharacteristic
390C42C2×D2811yescharacteristic
391C2×C7—21no—
392C2×C7—21no—
393C2×C7#288411yescharacteristic
394C2×C7C2×#41611yescharacteristic
395C2×C7#290111yescharacteristic
396D7—82no—
397C23—142no—
398C23C22×C22⋊D711yescharacteristic
399Q8—24no—
400Q8—24no—
401Q8—282no—
402Q8—282no—
403Q8—561no—
404D4—282no—
405D4—282no—
406D4—282no—
407D4—282no—
408D4—561no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—142no—
424C2×C4—142no—
425C2×C4—142no—
426C2×C4—284no—
427C2×C4—564no—
428C2×C4C2×(C2×D7)⋊C414yes—
429C2×C4C2×(C2×D7)⋊C414yes—
430C2×C4C2×(C2×D7)⋊C411yescharacteristic
431C2×C4C2×(C2×D7)⋊C411yescharacteristic
432C2×C4C2×(C2×D7)⋊C411yescharacteristic
433C2×C4C2×(C2×D7)⋊C411yescharacteristic
434C2×C4#651312yes—
435C2×C4#651312yes—
436C2×C4C22×C4×D711yescharacteristic
437C2×C4C22×D2811yescharacteristic
438C8—282no—
439C8—282no—
440C7#1761611yescharacteristic, complemented, Sylow
441C22—21no—
442C22—21no—
443C22—282no—
444C22—282no—
445C22—282no—
446C22C22×(C2×D7)⋊C411yescharacteristic
447C22C2×#651311yescharacteristic
448C22C2×#651311yescharacteristic
449C4—24no—
450C4—24no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—21no—
455C4—42no—
456C4—81no—
457C4—82no—
458C4—282no—
459C4—282no—
460C4—282no—
461C4#8977412yes—
462C4#8977412yes—
463C2—21no—
464C2—21no—
465C2—562no—
466C2C2×#8977411yescharacteristic
467C2#1173998111yescharacteristic
468C2#1174000111yescharacteristic
subgroupG/H
centerC22C22×(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#2356C2
socleC22×C7C22×C22⋊C4
prime radical#90168C22
cosocleC22×C4×C7C24
hypercenter#2356D7
hypocenterC7#17616
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
2. C7×#2356C2semidirect
3. #11739880C2semidirect
4. #11727319C2semidirect
5. #11727300C2semidirect
7. #11727225C2semidirect
9. #11727185C2semidirect
12. #90276C4semidirect
13. #90275C4semidirect
15. #90270C4semidirect
17. #90258C4semidirect
108. #2356D7semidirect
440. C7#17616semidirect

Maximal subgroups

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

subgroupindex
#117428872
C7×#23562
#117398802
#117273192
#117273002
#117272992
#117272252
#117272232
#117271852
#176167