Finite Group Databasebeta

← #13339116

Subgroups — #13339116

Order 1792.

632 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3103C211yesmaximal, characteristic, complemented
2#11740748C211yesmaximal, characteristic
3#11739889C211yesmaximal, characteristic, complemented
4#11739880C211yesmaximal, characteristic, complemented
5#11729562C211yesmaximal, characteristic, complemented
6#11728873C212yesmaximal, complemented
7#11728870C212yesmaximal, complemented
8#11728776C212yesmaximal, complemented
9#11728775C212yesmaximal, complemented
10#11728527C211yesmaximal, characteristic, complemented
11#11728501C211yesmaximal, characteristic, complemented
12C7×#532C2211yescharacteristic
13C2×C4×C7×Q8C2211yescharacteristic
14#90168C2211yescharacteristic
15C7×#481—21no—
16C7×#475C2212yes—
17C7×#460C2212yes—
18C7×#448—21no—
19C4×C7×SD16—22no—
20C7×#431C2211yescharacteristic
21#90010—22no—
22#89999—21no—
23#89857C2211yescharacteristic
24#89853C2211yescharacteristic
25#89845C2211yescharacteristic
26#89842C2211yescharacteristic
27#89829—22no—
28#89799C2212yes—
29#89798—22no—
30Q8×C7⋊C8—22no—
31Q8×C7⋊C8C2212yes—
32Q8×C7⋊C8C2212yes—
33#89793C2212yes—
34#89739—21no—
35#89698C2211yescharacteristic
36#89697—22no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89342C2212yes—
40#89339C2212yes—
41#89280C2212yes—
42#89276C2212yes—
43#89275C2212yes—
44#89274C2212yes—
45#19401—71nomaximal, Sylow
46C2×C7×#41—21no—
47C22×C7×Q8D411yescharacteristic
48C2×#1157—21no—
49C2×C7×SD16—22no—
50C7×C4⋊Q8C2311yescharacteristic
51C7×#127C2311yescharacteristic
52C7×#124—21no—
53C7×#121—21no—
54C7×#121—21no—
55C4×C7×Q8—21no—
56C4×C7×Q8—21no—
57C4×C7×Q8D412yes—
58C4×C7×Q8C2312yes—
59C4×C7×D4—21no—
60C4×C7×D4—21no—
61C2×C7×C4⋊C4—21no—
62C2×C7×C4⋊C4D411yescharacteristic
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65C2×#1143—22no—
66C2×#1143—22no—
67C2×#1141—22no—
68#6493—21no—
69#6493—21no—
70#6438—21no—
71#6437C2311yescharacteristic
72C4×D28—21no—
73C4×D28—21no—
74#6432C2311yescharacteristic
75C7×#106—22no—
76C7×C4⋊C8C2312yes—
77C7×#103—22no—
78C7×#102—22no—
79C7×C22⋊C8—21no—
80C4×C7×C8—21no—
81#6409—22no—
82#6409—22no—
83#6406—22no—
84C22⋊(C7⋊C8)—21no—
85C22⋊(C7⋊C8)—21no—
86C22⋊(C7⋊C8)D412yes—
87#6383—22no—
88#6381—22no—
89C4⋊(C7⋊C8)—22no—
90C4⋊(C7⋊C8)—22no—
91C4⋊(C7⋊C8)D412yes—
92C4⋊(C7⋊C8)C2312yes—
93C4⋊(C7⋊C8)C2312yes—
94#6377—21no—
95C4×C7⋊C8—21no—
96C4×C7⋊C8—22no—
97C4×C7⋊C8—22no—
98C4×C7⋊C8C2311yescharacteristic
99C4×C7⋊C8C2311yescharacteristic
100#3207—71no—
101#3112—71no—
102#3103D711yescharacteristic, complemented
103#2982—71no—
104#1684—72no—
105#1678—72no—
106#1668—72no—
107#1666—72no—
108#1572—71no—
109#1553—71no—
110#1543—71no—
111C7×#41—82no—
112C2×C7×Q8—24no—
113C2×C7×Q8—21no—
114C2×C7×Q8—21no—
115C2×C7×Q8C2×D412yes—
116C2×C7×D4—21no—
117C2×C7×D4—21no—
118C2×C7×D4—21no—
119C2×C7×D4—41no—
120C22×C4×C7—21no—
121C22×C4×C7—21no—
122C22×C4×C7—41no—
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7#4111yescharacteristic
126C2×C22⋊D7—41no—
127#1157—82no—
128C2×D28—21no—
129C2×D28—21no—
130C2×D28—21no—
131C2×C4×D7—41no—
132C2×Dic14—21no—
133C7×SD16—44no—
134C2×C7×C8—22no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—22no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—21no—
140C7×C4⋊C4C2×D412yes—
141C7×C22⋊C4—41no—
142C7×C22⋊C4—41no—
143C42×C7—22no—
144C42×C7—22no—
145C42×C7C2×D411yescharacteristic
146C42×C7C2×D411yescharacteristic
147C42×C7#4111yescharacteristic
148C42×C7C2411yescharacteristic
149#1143—44no—
150#1143—82no—
151#1141—82no—
152(C2×D7)⋊C4—41no—
153(C2×D7)⋊C4—41no—
154C4⋊Dic7—21no—
155C4⋊Dic7—21no—
156C2×C7⋊C8—22no—
157C2×C7⋊C8—22no—
158C2×C7⋊C8C2×D412yes—
159C2×C7⋊C8C2×D412yes—
160#532—71no—
161#532C2×D711yescharacteristic
162C2×C4×Q8C2×D711yescharacteristic
163#493—71no—
164#492—71no—
165#487—71no—
166#486—71no—
167#481—21no—
168#480—141no—
169#477—72no—
170#475C2×D712yes—
171#463—142no—
172#460—72no—
173#460C2×D712yes—
174#449—142no—
175#448—21no—
176#448—141no—
177C8×Q8—72no—
178C8×Q8—72no—
179C8×Q8—142no—
180#440—142no—
181C4×SD16—22no—
182#432—71no—
183#431—71no—
184#431—71no—
185#431—142no—
186#431C2×D711yescharacteristic
187#335—72no—
188#334—72no—
189#333—72no—
190#332—72no—
191#331—72no—
192#331—72no—
193C23×C7—41no—
194C23×C7#12111yescharacteristic
195C22×D7—41no—
196C7×Q8—24no—
197C7×Q8—22no—
198C7×Q8—42no—
199C7×Q8#13514yes—
200C7×D4—42no—
201C7×D4—42no—
202C7×D4—81no—
203C7×D4—82no—
204C2×C4×C7—24no—
205C2×C4×C7—22no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—41no—
210C2×C4×C7—41no—
211C2×C4×C7—82no—
212C2×C4×C7—81no—
213C2×C4×C7#12111yescharacteristic
214C2×C4×C7#12111yescharacteristic
215C2×C4×C7#12111yescharacteristic
216C2×C4×C7C22×D411yescharacteristic
217C2×C4×C7C22×D411yescharacteristic
218C2×C4×C7C2×#4111yescharacteristic
219C22⋊D7—82no—
220C2×Dic7—41no—
221D28—42no—
222D28—42no—
223D28—81no—
224C4×D7—82no—
225Dic14—81no—
226C7×C8—42no—
227C7⋊C8—22no—
228C7⋊C8—22no—
229C7⋊C8—42no—
230C7⋊C8—42no—
231C2×#41—21no—
232C2×#41—141no—
233C22×Q8C22⋊D711yescharacteristic
234C2×SD16—22no—
235C2×SD16—142no—
236C2×SD16—142no—
237C2×D8—142no—
238C4⋊Q8—71no—
239C4⋊Q8C22×D711yescharacteristic
240#127—71no—
241#127C22×D711yescharacteristic
242#124—21no—
243#124—141no—
244#121—21no—
245#121—21no—
246#121—141no—
247#121—141no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8C22⋊D712yes—
251C4×Q8C22×D712yes—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—141no—
255C4×D4—141no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22⋊D711yescharacteristic
258C2×C42—21no—
259C2×C42C22×D711yescharacteristic
260#107—142no—
261#106—22no—
262C4⋊C8—72no—
263C4⋊C8—72no—
264C4⋊C8—72no—
265C4⋊C8—142no—
266C4⋊C8—142no—
267C4⋊C8C22×D712yes—
268#103—22no—
269#103—142no—
270#103—142no—
271#102—22no—
272#102—142no—
273#102—142no—
274C22⋊C8—21no—
275C22⋊C8—72no—
276C22⋊C8—141no—
277C22⋊C8—141no—
278#97—141no—
279C4×C8—21no—
280C4×C8—71no—
281C4×C8—71no—
282C4×C8—142no—
283C4×C8—142no—
284C4×C8—141no—
285C22×C7—21no—
286C22×C7—21no—
287C22×C7—21no—
288C22×C7—81no—
289C22×C7—81no—
290C22×C7—81no—
291C22×C7C2×#12111yescharacteristic
292C2×D7—81no—
293C2×D7—81no—
294C2×D7—81no—
295C4×C7—21no—
296C4×C7—24no—
297C4×C7—21no—
298C4×C7—21no—
299C4×C7—21no—
300C4×C7—42no—
301C4×C7—81no—
302C4×C7C2×#13512yes—
303C4×C7#57712yes—
304Dic7—81no—
305#41—82no—
306#41—562no—
307C2×Q8—24no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—141no—
311C2×Q8C2×C22⋊D712yes—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—41no—
316C2×D4—141no—
317C2×D4—141no—
318C2×D4—141no—
319C2×D4—281no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—281no—
324C22×C4D4×D711yescharacteristic
325C22×C4#115911yescharacteristic
326C22×C4C2×C22⋊D711yescharacteristic
327SD16—44no—
328SD16—284no—
329SD16—562no—
330D8—562no—
331C2×C8—22no—
332C2×C8—72no—
333C2×C8—72no—
334C2×C8—142no—
335C2×C8—142no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—141no—
342C4⋊C4—141no—
343C4⋊C4C2×C22⋊D712yes—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—281no—
347C22⋊C4—281no—
348C42—22no—
349C42—22no—
350C42D4×D711yescharacteristic
351C42#115911yescharacteristic
352C42C2×C22⋊D711yescharacteristic
353C42C23×D711yescharacteristic
354C2×C7—41no—
355C2×C7—161no—
356C2×C7#304311yescharacteristic
357C2×C7#305011yescharacteristic
358C2×C7#306811yescharacteristic
359D7—161no—
360C23—41no—
361C23—281no—
362C23#650211yescharacteristic
363Q8—24no—
364Q8—22no—
365Q8—42no—
366Q8—561no—
367Q8#651314yes—
368D4—42no—
369D4—42no—
370D4—81no—
371D4—82no—
372D4—282no—
373D4—282no—
374D4—561no—
375D4—562no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—281no—
386C2×C4—562no—
387C2×C4#650211yescharacteristic
388C2×C4#650211yescharacteristic
389C2×C4#650211yescharacteristic
390C2×C4C2×D4×D711yescharacteristic
391C2×C4C2×#115911yescharacteristic
392C2×C4C22×C22⋊D711yescharacteristic
393C8—42no—
394C8—142no—
395C8—142no—
396C8—282no—
397C8—282no—
398C7#1940111yescharacteristic, complemented, Sylow
399C22—21no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—561no—
406C22—561no—
407C22—561no—
408C22C2×#650211yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—24no—
413C4—21no—
414C4—42no—
415C4—81no—
416C4—561no—
417C4#9045312yes—
418C4C2×#651312yes—
419C2—41no—
420C2—161no—
421C2—1121no—
422C2#1174036711yescharacteristic
423C2#1174129411yescharacteristic
424C2#1174144711yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3103C2
socleC22×C7C2×#121
prime radical#11740748C2
hypercenter#3103D7
hypocenterC7#19401
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. C7×#3103C2semidirect
3. #11739889C2semidirect
4. #11739880C2semidirect
5. #11729562C2semidirect
6. #11728873C2semidirect
7. #11728870C2semidirect
8. #11728776C2semidirect
9. #11728775C2semidirect
10. #11728527C2semidirect
11. #11728501C2semidirect
102. #3103D7semidirect
398. C7#19401semidirect

Maximal subgroups

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

subgroupindex
C7×#31032
#117407482
#117398892
#117398802
#117295622
#117288732
#117288702
#117287762
#117287752
#117285272
#117285012
#194017