Finite Group Databasebeta

← #13341820

Subgroups — #13341820

Order 1792.

742 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3169C211yesmaximal, characteristic
2#11740762C211yesmaximal, characteristic, complemented
3#11740728C211yesmaximal, characteristic, complemented
4#11740728C211yesmaximal, characteristic, complemented
5#11740716C211yesmaximal, characteristic, complemented
6C2×#89453C211yesmaximal, characteristic, complemented
7#11729118C212yesmaximal, complemented
8#11729032C212yesmaximal, complemented
9#11728937C212yesmaximal, complemented
10#11728935C212yesmaximal, complemented
11C8×#1136C211yesmaximal, characteristic, complemented
12C7×#532C2211yescharacteristic
13C7×#532C2211yescharacteristic
14C2×#6432C2211yescharacteristic
15C2×#6432C2211yescharacteristic
16C7×#500C2211yescharacteristic
17C7×#493C2211yescharacteristic
18C7×#486C2211yescharacteristic
19C7×#486C2211yescharacteristic
20C7×#466—22no—
21C4×C7×D8—22no—
22C2×C4×C7×C8C2211yescharacteristic
23#90004—22no—
24#90004—22no—
25C2×#6390—22no—
26#89865C2212yes—
27#89853C2212yes—
28#89853C2212yes—
29#89851C2212yes—
30#89848C2212yes—
31#89842C2212yes—
32#89842C2212yes—
33#89842C2212yes—
34#89827—22no—
35#89827—22no—
36#89812—22no—
37#89806—22no—
38#89791—22no—
39#89780—22no—
40#89697C2211yescharacteristic
41C4×#1136C2211yescharacteristic
42#89453—22no—
43#89453C2212yes—
44#89453C2212yes—
45#89249C2212yes—
46C8×C7⋊C8C2212yes—
47#19640—71nomaximal, Sylow
48C2×C7×#41—21no—
49C2×C7×#41—21no—
50C22×Dic14—21no—
51C22×Dic14—21no—
52C2×C7×D8—22no—
53C22×C7×C8D412yes—
54C7×C4⋊Q8C2311yescharacteristic
55C7×C4⋊Q8C2311yescharacteristic
56C7×#127C2311yescharacteristic
57C7×#127C2311yescharacteristic
58C7×#124—21no—
59C7×#124—21no—
60C7×#121—21no—
61C7×#121—21no—
62C7×#121—21no—
63C7×#121—21no—
64C4×C7×D4—21no—
65C4×C7×D4—21no—
66C4×C7×D4—21no—
67C4×C7×D4—21no—
68C2×C42×C7C2311yescharacteristic
69C2×#1142—24no—
70C2×C4⋊Dic7—21no—
71C2×C4⋊Dic7—21no—
72C2×C4⋊Dic7—21no—
73C2×C4⋊Dic7—21no—
74C2×#1136D412yes—
75#6432—21no—
76#6432—21no—
77#6432—21no—
78#6432—21no—
79#6432—21no—
80#6432—21no—
81#6432C2312yes—
82#6432C2312yes—
83C7×#107—22no—
84C7×#102—22no—
85C7×#102—22no—
86C4×C7×C8D412yes—
87C4×C7×C8C2311yescharacteristic
88C4×C7×C8C2311yescharacteristic
89#6409—24no—
90#6409—24no—
91#6406—24no—
92#6406—24no—
93#6406—24no—
94C22⋊(C7⋊C8)D412yes—
95#6390—24no—
96#6390—24no—
97#6384—24no—
98#6382—24no—
99C4⋊(C7⋊C8)D412yes—
100#6377D412yes—
101C4×C7⋊C8C2312yes—
102C4×C7⋊C8C2312yes—
103#3227—71no—
104#3206—71no—
105#3169D711yescharacteristic
106#3141—71no—
107#3141—71no—
108C2×#487—71no—
109#1718—72no—
110#1717—72no—
111#1711—72no—
112#1708—72no—
113C8×M16—71no—
114C7×#41—44no—
115C7×#41—44no—
116C2×C7×Q8—21no—
117C2×C7×Q8—21no—
118C2×C7×D4—21no—
119C2×C7×D4—21no—
120C2×C7×D4—21no—
121C2×C7×D4—21no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4—41no—
125C2×C7×D4—41no—
126C22×C4×C7—41no—
127C22×C4×C7—41no—
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7C2×D411yescharacteristic
130C22×C4×C7C2×D411yescharacteristic
131C22×Dic7—41no—
132C22×Dic7—41no—
133C2×Dic14—22no—
134C2×Dic14—22no—
135C2×Dic14—44no—
136C2×Dic14—41no—
137C2×Dic14—41no—
138C2×Dic14—82no—
139C7×D8—44no—
140C2×C7×C8—42no—
141C2×C7×C8C2×D412yes—
142C2×C7×C8C2×D412yes—
143C7×C4⋊C4—21no—
144C7×C4⋊C4—21no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C22⋊C4—41no—
148C7×C22⋊C4—41no—
149C7×C22⋊C4—41no—
150C7×C22⋊C4—41no—
151C42×C7C2×D411yescharacteristic
152C42×C7C2×D411yescharacteristic
153C42×C7C2×D411yescharacteristic
154C42×C7C2411yescharacteristic
155#1142—84no—
156C4⋊Dic7—22no—
157C4⋊Dic7—22no—
158C4⋊Dic7—22no—
159C4⋊Dic7—22no—
160C4⋊Dic7—41no—
161C4⋊Dic7—41no—
162C4⋊Dic7—41no—
163C4⋊Dic7—41no—
164#1136—24no—
165C2×C7⋊C8C2×D414yes—
166C2×C7⋊C8C2×D414yes—
167#532C2×D711yescharacteristic
168#532C2×D711yescharacteristic
169C2×C4⋊Q8—71no—
170C2×C4⋊Q8—71no—
171#500C2×D711yescharacteristic
172#498—72no—
173#493C2×D711yescharacteristic
174#492—72no—
175#489—142no—
176#489—142no—
177#487—72no—
178#487—72no—
179#487—72no—
180#487—72no—
181#487—72no—
182#487—142no—
183#486—72no—
184#486—72no—
185#486—72no—
186#486C2×D711yescharacteristic
187#486C2×D711yescharacteristic
188#481—142no—
189#476—142no—
190#466—22no—
191#461—142no—
192#448—142no—
193C4×D8—22no—
194#431—71no—
195#417—142no—
196#417—142no—
197C2×#103—142no—
198C4×M16—71no—
199C2×C4×C8C2×D711yescharacteristic
200#322—72no—
201C82—72no—
202C23×C7—41no—
203C23×C7—41no—
204C23×C7#12711yescharacteristic
205C7×Q8—42no—
206C7×Q8—42no—
207C7×D4—42no—
208C7×D4—42no—
209C7×D4—42no—
210C7×D4—42no—
211C7×D4—42no—
212C7×D4—42no—
213C7×D4—82no—
214C7×D4—82no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—41no—
218C2×C4×C7—41no—
219C2×C4×C7—82no—
220C2×C4×C7—82no—
221C2×C4×C7#12711yescharacteristic
222C2×C4×C7#12711yescharacteristic
223C2×C4×C7#12711yescharacteristic
224C2×C4×C7#13514yes—
225C2×C4×C7C22×D411yescharacteristic
226C2×C4×C7C22×D411yescharacteristic
227C2×C4×C7C22×D411yescharacteristic
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7—82no—
231C2×Dic7—82no—
232Dic14—44no—
233Dic14—44no—
234Dic14—82no—
235Dic14—82no—
236C7×C8—22no—
237C7×C8—22no—
238C7⋊C8—24no—
239C7⋊C8—24no—
240C2×#41—21no—
241C2×#41—21no—
242C22×Q8—141no—
243C22×Q8—141no—
244C2×SD16—144no—
245C2×D8—22no—
246C2×M16—72no—
247C22×C8C22⋊D712yes—
248C4⋊Q8—72no—
249C4⋊Q8—72no—
250C4⋊Q8—141no—
251C4⋊Q8—141no—
252C4⋊Q8—141no—
253C4⋊Q8—141no—
254C4⋊Q8—141no—
255C4⋊Q8—141no—
256C4⋊Q8C22×D711yescharacteristic
257C4⋊Q8C22×D711yescharacteristic
258#127C22×D711yescharacteristic
259#127C22×D711yescharacteristic
260#124—21no—
261#124—21no—
262#121—21no—
263#121—21no—
264#121—21no—
265#121—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C2×C4⋊C4—141no—
271C2×C4⋊C4—141no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4—141no—
274C2×C42C22×D711yescharacteristic
275#107—22no—
276#106—144no—
277C4⋊C8—72no—
278#103—144no—
279#103—144no—
280#103—144no—
281#103—144no—
282#103—144no—
283#102—22no—
284#102—22no—
285#102—144no—
286#102—144no—
287#102—144no—
288C22⋊C8—72no—
289#97—72no—
290C4×C8—72no—
291C4×C8—72no—
292C4×C8C22⋊D712yes—
293C4×C8C22×D711yescharacteristic
294C4×C8C22×D711yescharacteristic
295C22×C7—21no—
296C22×C7—21no—
297C22×C7—81no—
298C22×C7—81no—
299C22×C7—81no—
300C22×C7—81no—
301C22×C7—81no—
302C22×C7—81no—
303C22×C7#49512yes—
304C22×C7C2×#12711yescharacteristic
305C4×C7—21no—
306C4×C7—21no—
307C4×C7—81no—
308C4×C7—81no—
309C4×C7#49512yes—
310C4×C7C2×#13512yes—
311C4×C7#57512yes—
312C4×C7#57712yes—
313Dic7—82no—
314Dic7—82no—
315#41—44no—
316#41—44no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—142no—
320C2×Q8—142no—
321C2×Q8—284no—
322C2×Q8—281no—
323C2×Q8—281no—
324C2×Q8—562no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—281no—
336C22×C4—281no—
337C22×C4D4×D711yescharacteristic
338C22×C4D4×D711yescharacteristic
339C22×C4C2×C22⋊D711yescharacteristic
340SD16—564no—
341D8—44no—
342M16—144no—
343C2×C8—42no—
344C2×C8—74no—
345C2×C8—74no—
346C2×C8C2×C22⋊D712yes—
347C2×C8C2×C22⋊D712yes—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—142no—
353C4⋊C4—142no—
354C4⋊C4—142no—
355C4⋊C4—142no—
356C4⋊C4—281no—
357C4⋊C4—281no—
358C4⋊C4—281no—
359C4⋊C4—281no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C42D4×D711yescharacteristic
365C42D4×D711yescharacteristic
366C42C2×C22⋊D711yescharacteristic
367C42C23×D711yescharacteristic
368C2×C7—22no—
369C2×C7—161no—
370C2×C7—161no—
371C2×C7C2×#49511yescharacteristic
372C2×C7#315311yescharacteristic
373C2×C7#315411yescharacteristic
374C23—41no—
375C23—41no—
376C23#650311yescharacteristic
377Q8—42no—
378Q8—42no—
379Q8—284no—
380Q8—284no—
381Q8—562no—
382Q8—562no—
383D4—42no—
384D4—42no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—82no—
390D4—82no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—282no—
398C2×C4—282no—
399C2×C4—562no—
400C2×C4—562no—
401C2×C4#647514yes—
402C2×C4#650311yescharacteristic
403C2×C4#650311yescharacteristic
404C2×C4#650311yescharacteristic
405C2×C4C2×D4×D711yescharacteristic
406C2×C4C2×D4×D711yescharacteristic
407C2×C4C22×C22⋊D711yescharacteristic
408C8—22no—
409C8—22no—
410C8—144no—
411C8—144no—
412C7#1964011yescharacteristic, complemented, Sylow
413C22—21no—
414C22—21no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22#8992712yes—
422C22C2×#650311yescharacteristic
423C4—21no—
424C4—21no—
425C4—81no—
426C4—81no—
427C4—562no—
428C4—562no—
429C4#8992712yes—
430C4C2×#647512yes—
431C4#9046112yes—
432C4#9051312yes—
433C2—22no—
434C2—161no—
435C2—161no—
436C2#1174073411yescharacteristic
437C2C2×#8992711yescharacteristic
438C2#1174135011yescharacteristic
subgroupG/H
centerC22C2×#6503
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3169C2
socleC22×C7C2×#127
hypercenter#3169D7
hypocenterC7#19640
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
2. #11740762C2semidirect
3. #11740728C2semidirect
4. #11740728C2semidirect
5. #11740716C2semidirect
6. C2×#89453C2semidirect
7. #11729118C2semidirect
8. #11729032C2semidirect
9. #11728937C2semidirect
10. #11728935C2semidirect
11. C8×#1136C2semidirect
412. C7#19640semidirect

Maximal subgroups

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

subgroupindex
C7×#31692
#117407622
#117407282
#117407162
C2×#894532
#117291182
#117290322
#117289372
#117289352
C8×#11362
#196407