Finite Group Databasebeta

← #13338989

Subgroups — #13338989

Order 1792.

662 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3100C211yesmaximal, characteristic
2#11740763C211yesmaximal, characteristic, complemented
3#11739760C211yesmaximal, characteristic, complemented
4#11739746C211yesmaximal, characteristic, complemented
5#11729562C211yesmaximal, characteristic, complemented
6#11728856C212yesmaximal, complemented
7#11728855C212yesmaximal, complemented
8#11728761C212yesmaximal, complemented
9#11728758C212yesmaximal, complemented
10#11728531C211yesmaximal, characteristic, complemented
11#11728501C211yesmaximal, characteristic, complemented
12C7×#532C2211yescharacteristic
13C2×C4×C7×D4C2211yescharacteristic
14#90161C2211yescharacteristic
15C7×#480—21no—
16C7×#474C2212yes—
17C7×#459C2212yes—
18C7×#447—21no—
19C4×C7×D8—22no—
20C7×#431C2211yescharacteristic
21#90011—22no—
22#89990—21no—
23C2×C22⋊(C7⋊C8)—22no—
24#89864C2211yescharacteristic, complemented
25#89853C2211yescharacteristic, complemented
26#89848C2211yescharacteristic, complemented
27#89842C2211yescharacteristic, complemented
28#89826—22no—
29#89791C2212yes—
30#89790—22no—
31D4×C7⋊C8—22no—
32D4×C7⋊C8C2212yes—
33D4×C7⋊C8C2212yes—
34#89779C2212yes—
35#89742—21no—
36#89698C2211yescharacteristic
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89342C2212yescomplemented
40#89339C2212yescomplemented
41#89281C2212yescomplemented
42#89277C2212yescomplemented
43#89275C2212yescomplemented
44#89274C2212yescomplemented
45#19388—71nomaximal, Sylow
46C2×C7×#41—21no—
47C22×C7×D4D411yescharacteristic
48C23×C4×C7—21no—
49C2×C7×D8—22no—
50C7×C4⋊Q8C2311yescharacteristic
51C7×#127C2311yescharacteristic
52C7×#124—21no—
53C7×#121—21no—
54C7×#121—21no—
55C4×C7×D4—21no—
56C4×C7×D4—21no—
57C4×C7×D4—21no—
58C4×C7×D4—21no—
59C4×C7×D4D412yes—
60C4×C7×D4C2312yes—
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C22⋊C4—21no—
63C2×C42×C7C2311yescharacteristic
64C2×#1144—22no—
65C2×#1142—22no—
66C2×#1142—22no—
67#6489—21no—
68#6487—21no—
69#6487—21no—
70C22×C7⋊C8—22no—
71C22×C7⋊C8—22no—
72#6433—21no—
73#6432C2311yescharacteristic
74#6432C2311yescharacteristic
75C4×Dic14—21no—
76C4×Dic14—21no—
77C7×#107—22no—
78C7×C4⋊C8C2312yes—
79C7×#102—22no—
80C7×#102—22no—
81C7×C22⋊C8—21no—
82C4×C7×C8—21no—
83#6409—22no—
84#6406—22no—
85#6406—22no—
86C22⋊(C7⋊C8)—21no—
87C22⋊(C7⋊C8)—21no—
88C22⋊(C7⋊C8)—22no—
89C22⋊(C7⋊C8)—22no—
90C22⋊(C7⋊C8)D412yes—
91#6384—22no—
92#6381—22no—
93C4⋊(C7⋊C8)D412yes—
94C4⋊(C7⋊C8)C2312yes—
95C4⋊(C7⋊C8)C2312yes—
96#6377—21no—
97C4×C7⋊C8—21no—
98C4×C7⋊C8C2311yescharacteristic
99C4×C7⋊C8C2311yescharacteristic
100#3210—71no—
101#3110—71no—
102#3100D711yescharacteristic
103#2972—71no—
104#1681—72no—
105#1679—72no—
106#1669—72no—
107#1665—72no—
108#1572—71no—
109#1553—71no—
110#1544—71no—
111C24×C7—21no—
112C7×#41—82no—
113C2×C7×Q8—21no—
114C2×C7×D4—24no—
115C2×C7×D4—21no—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4—21no—
119C2×C7×D4—41no—
120C2×C7×D4C2×D412yes—
121C22×C4×C7—21no—
122C22×C4×C7—22no—
123C22×C4×C7—22no—
124C22×C4×C7—41no—
125C22×C4×C7—82no—
126C22×C4×C7C2×D411yescharacteristic
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7#4111yescharacteristic
129C2×Dic14—21no—
130C2×Dic14—21no—
131C7×D8—44no—
132C2×C7×C8—22no—
133C7×C4⋊C4—21no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4C2×D412yes—
137C7×C22⋊C4—22no—
138C7×C22⋊C4—22no—
139C7×C22⋊C4—41no—
140C7×C22⋊C4—41no—
141C42×C7C2×D411yescharacteristic
142C42×C7C2×D411yescharacteristic
143C42×C7#4111yescharacteristic
144C42×C7C2411yescharacteristic
145C22⋊Dic7—41no—
146#1144—82no—
147#1142—44no—
148#1142—82no—
149C4⋊Dic7—21no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7—21no—
153Dic7⋊C4—41no—
154Dic7⋊C4—41no—
155C4×Dic7—41no—
156C2×C7⋊C8—22no—
157C2×C7⋊C8—22no—
158C2×C7⋊C8—82no—
159C2×C7⋊C8—82no—
160C2×C7⋊C8C2×D412yes—
161C2×C7⋊C8C2×D412yes—
162#533—71no—
163#532C2×D711yescharacteristic
164C2×C4×D4C2×D711yescharacteristic
165#494—71no—
166#492—71no—
167#487—71no—
168#486—71no—
169#484—141no—
170#480—21no—
171#474—72no—
172#474C2×D712yes—
173#464—142no—
174#461—72no—
175#459C2×D712yes—
176#452—142no—
177#450—141no—
178#447—21no—
179#440—142no—
180C4×D8—22no—
181C8×D4—72no—
182C8×D4—72no—
183C8×D4—142no—
184#432—71no—
185#431—71no—
186#431—71no—
187#431C2×D711yescharacteristic
188C2×C22⋊C8—142no—
189#335—72no—
190#334—72no—
191#332—72no—
192#332—72no—
193#332—72no—
194#331—72no—
195C23×C7—24no—
196C23×C7—22no—
197C23×C7—41no—
198C23×C7—81no—
199C23×C7—81no—
200C23×C7#12111yescharacteristic
201C7×Q8—42no—
202C7×D4—24no—
203C7×D4—22no—
204C7×D4—42no—
205C7×D4—42no—
206C7×D4—81no—
207C7×D4—82no—
208C7×D4#13514yes—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—22no—
212C2×C4×C7—41no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—41no—
216C2×C4×C7—82no—
217C2×C4×C7—82no—
218C2×C4×C7—81no—
219C2×C4×C7#12111yescharacteristic
220C2×C4×C7#12111yescharacteristic
221C2×C4×C7#12111yescharacteristic
222C2×C4×C7C22×D411yescharacteristic
223C2×C4×C7C22×D411yescharacteristic
224C2×C4×C7C2×#4111yescharacteristic
225C2×Dic7—41no—
226C2×Dic7—41no—
227Dic14—42no—
228Dic14—42no—
229C7×C8—42no—
230C7⋊C8—22no—
231C7⋊C8—22no—
232C7⋊C8—42no—
233C7⋊C8—42no—
234C2×#41—21no—
235C22×D4C22⋊D711yescharacteristic
236C23×C4—21no—
237C2×Q16—142no—
238C2×SD16—142no—
239C2×SD16—142no—
240C2×D8—22no—
241C22×C8—142no—
242C22×C8—142no—
243C4⋊Q8—71no—
244C4⋊Q8—71no—
245C4⋊Q8C22×D711yescharacteristic
246#127C22×D711yescharacteristic
247#125—141no—
248#124—21no—
249C22⋊Q8—141no—
250C22⋊Q8—141no—
251#121—21no—
252#121—21no—
253C4×Q8—141no—
254C4×Q8—141no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4C22⋊D712yes—
260C4×D4C22×D712yes—
261#117—141no—
262C2×C4⋊C4C22⋊D711yescharacteristic
263C2×C22⋊C4—21no—
264C2×C42C22×D711yescharacteristic
265#107—22no—
266#107—142no—
267C4⋊C8—72no—
268C4⋊C8—72no—
269C4⋊C8—72no—
270C4⋊C8C22×D712yes—
271#103—142no—
272#103—142no—
273#102—22no—
274#102—22no—
275#102—142no—
276#102—142no—
277C22⋊C8—21no—
278C22⋊C8—72no—
279C22⋊C8—142no—
280C22⋊C8—142no—
281C22⋊C8—141no—
282C22⋊C8—141no—
283#97—141no—
284C4×C8—21no—
285C4×C8—71no—
286C4×C8—71no—
287C4×C8—141no—
288C22×C7—21no—
289C22×C7—21no—
290C22×C7—24no—
291C22×C7—21no—
292C22×C7—42no—
293C22×C7—42no—
294C22×C7—84no—
295C22×C7—81no—
296C22×C7—81no—
297C22×C7—81no—
298C22×C7C2×#12111yescharacteristic
299C4×C7—21no—
300C4×C7—21no—
301C4×C7—21no—
302C4×C7—21no—
303C4×C7—42no—
304C4×C7—81no—
305C4×C7C2×#13512yes—
306C4×C7#57612yes—
307Dic7—81no—
308Dic7—81no—
309C24—21no—
310#41—82no—
311C2×Q8—21no—
312C2×Q8—141no—
313C2×Q8—141no—
314C2×D4—24no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4C2×C22⋊D712yes—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—41no—
325C22×C4—82no—
326C22×C4D4×D711yescharacteristic
327C22×C4#115911yescharacteristic
328C22×C4C2×C22⋊D711yescharacteristic
329Q16—562no—
330SD16—284no—
331SD16—562no—
332D8—44no—
333C2×C8—22no—
334C2×C8—72no—
335C2×C8—72no—
336C2×C8—142no—
337C2×C8—142no—
338C2×C8—562no—
339C2×C8—562no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—141no—
344C4⋊C4—141no—
345C4⋊C4—141no—
346C4⋊C4—141no—
347C4⋊C4—281no—
348C4⋊C4—281no—
349C4⋊C4C2×C22⋊D712yes—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—41no—
353C22⋊C4—41no—
354C22⋊C4—281no—
355C42—281no—
356C42D4×D711yescharacteristic
357C42#115911yescharacteristic
358C42C2×C22⋊D711yescharacteristic
359C42C23×D711yescharacteristic
360C2×C7—41no—
361C2×C7—44no—
362C2×C7—161no—
363C2×C7#304111yescharacteristic
364C2×C7#305011yescharacteristic
365C2×C7#306911yescharacteristic
366C23—24no—
367C23—22no—
368C23—41no—
369C23—81no—
370C23—81no—
371C23#650211yescharacteristic
372Q8—42no—
373Q8—282no—
374Q8—282no—
375D4—24no—
376D4—22no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—82no—
381D4#651314yes—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—81no—
392C2×C4—281no—
393C2×C4—281no—
394C2×C4#650211yescharacteristic
395C2×C4#650211yescharacteristic
396C2×C4#650211yescharacteristic
397C2×C4C2×D4×D711yescharacteristic
398C2×C4C2×#115911yescharacteristic
399C2×C4C22×C22⋊D711yescharacteristic
400C8—42no—
401C8—142no—
402C8—142no—
403C8—282no—
404C8—282no—
405C7#1938811yescharacteristic, complemented, Sylow
406C22—21no—
407C22—21no—
408C22—21no—
409C22—24no—
410C22—42no—
411C22—42no—
412C22—84no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22C2×#650211yescharacteristic
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—42no—
422C4—81no—
423C4—561no—
424C4—561no—
425C4#9044812yes—
426C4C2×#651312yes—
427C2—41no—
428C2—44no—
429C2—161no—
430C2#1174036711yescharacteristic
431C2#1174127111yescharacteristic
432C2#1174144811yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3100C2
socleC22×C7C2×#121
hypercenter#3100D7
hypocenterC7#19388
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. #11740763C2semidirect
3. #11739760C2semidirect
4. #11739746C2semidirect
5. #11729562C2semidirect
6. #11728856C2semidirect
7. #11728855C2semidirect
8. #11728761C2semidirect
9. #11728758C2semidirect
10. #11728531C2semidirect
11. #11728501C2semidirect
24. #89864C22semidirect
25. #89853C22semidirect
26. #89848C22semidirect
27. #89842C22semidirect
39. #89342C22semidirect
40. #89339C22semidirect
41. #89281C22semidirect
42. #89277C22semidirect
43. #89275C22semidirect
44. #89274C22semidirect
405. C7#19388semidirect

Maximal subgroups

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

subgroupindex
C7×#31002
#117407632
#117397602
#117397462
#117295622
#117288562
#117288552
#117287612
#117287582
#117285312
#117285012
#193887