Finite Group Databasebeta

← #13337375

Subgroups — #13337375

Order 1792.

658 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3229C211yesmaximal, characteristic, complemented
2#11740808C211yesmaximal, characteristic
3#11740775C211yesmaximal, characteristic, complemented
4#11740648C211yesmaximal, characteristic
5C2×#89342C211yesmaximal, characteristic, complemented
6#11728875C212yesmaximal, complemented
7#11728874C212yesmaximal, complemented
8#11728871C212yesmaximal, complemented
9#11728870C212yesmaximal, complemented
10#11728526C211yesmaximal, characteristic
11#11728500C211yesmaximal, characteristic, complemented
12C7×#533C2211yescharacteristic
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×#6505—21no—
15C2×#6505—21no—
16#90168C2211yescharacteristic
17C7×#498—21no—
18C7×#487—21no—
19C7×#483—22no—
20C7×#475C2212yes—
21C7×#475C2212yes—
22C7×#431C2211yescharacteristic
23#89864C2212yes—
24#89858—22no—
25#89857C2212yes—
26#89856—21no—
27#89855—21no—
28#89820—22no—
29#89799C2212yes—
30#89796—22no—
31#89796C2212yes—
32Q8×C7⋊C8C2212yes—
33#89792C2212yes—
34#89771—21no—
35#89769—21no—
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C4×#1136C2211yescharacteristic
39#89342C2212yes—
40#89342C2212yes—
41#89281C2211yescharacteristic
42#89281C2211yescharacteristic
43#89280C2211yescharacteristic
44#89280C2211yescharacteristic
45#89275C2212yes—
46#89274C2212yes—
47#19239—71nomaximal, Sylow
48C22×C7×Q8—21no—
49C2×#1157—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8D411yescharacteristic
54C7×C4⋊Q8D411yescharacteristic
55C7×C4⋊Q8C2312yes—
56C7×#125—21no—
57C7×C22⋊Q8—21no—
58C7×C22⋊Q8D412yes—
59C4×C7×Q8—21no—
60C4×C7×Q8C2312yes—
61C7×#117—21no—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C42×C7C2311yescharacteristic
65C2×#1144—22no—
66C2×#1144—22no—
67#6505—44no—
68#6505—44no—
69C2×#1143—22no—
70C2×#1143—22no—
71C2×#1143—22no—
72#6493—21no—
73#6493—21no—
74C2×#1136D411yescharacteristic
75C2×#1136D411yescharacteristic
76C22×C7⋊C8—21no—
77C22×C7⋊C8—21no—
78#6438—21no—
79#6437C2311yescharacteristic
80C4×D28—21no—
81C4×D28—21no—
82#6432C2311yescharacteristic
83C7×#106—22no—
84C7×#106—22no—
85C7×C4⋊C8C2311yescharacteristic
86C7×C4⋊C8C2311yescharacteristic
87C7×#103—22no—
88C7×#103—22no—
89C7×C22⋊C8—21no—
90C4×C7×C8—21no—
91#6409—22no—
92#6383—22no—
93#6382—22no—
94C4⋊(C7⋊C8)—21no—
95C4⋊(C7⋊C8)—21no—
96C4⋊(C7⋊C8)—22no—
97C4⋊(C7⋊C8)—22no—
98C4⋊(C7⋊C8)C2312yes—
99C4⋊(C7⋊C8)C2311yescharacteristic
100C4⋊(C7⋊C8)C2311yescharacteristic
101#6377—22no—
102#6377D412yes—
103C4×C7⋊C8—22no—
104C4×C7⋊C8C2312yes—
105#3246—71no—
106#3229D711yescharacteristic, complemented
107#3152—71no—
108#2962—71no—
109#1690—72no—
110#1686—72no—
111#1670—72no—
112#1666—72no—
113#1565—71no—
114#1540—71no—
115C2×#332—71no—
116C2×C7×Q8—21no—
117C2×C7×Q8—22no—
118C2×C7×Q8—41no—
119C2×C7×Q8—42no—
120C2×C7×Q8—42no—
121C2×C7×Q8C2×D412yes—
122C22×C4×C7—41no—
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7#4111yescharacteristic
126C2×C22⋊D7—41no—
127#1157—42no—
128#1157—42no—
129C2×D28—21no—
130C2×D28—21no—
131C2×D28—21no—
132C2×C4×D7—41no—
133C2×Dic14—21no—
134C2×C7×C8—21no—
135C2×C7×C8—21no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—21no—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—41no—
144C7×C4⋊C4—41no—
145C7×C4⋊C4C2×D412yes—
146C7×C22⋊C4—22no—
147C42×C7—22no—
148C42×C7C2×D411yescharacteristic
149C42×C7C2×D411yescharacteristic
150C42×C7#4111yescharacteristic
151C42×C7C2411yescharacteristic
152#1144—44no—
153#1144—44no—
154#1143—44no—
155#1143—44no—
156#1143—82no—
157(C2×D7)⋊C4—41no—
158(C2×D7)⋊C4—41no—
159C4⋊Dic7—21no—
160C4⋊Dic7—21no—
161#1136—22no—
162#1136SD1614yescomplemented
163C2×C7⋊C8—22no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8—21no—
166C2×C7⋊C8—42no—
167C2×C7⋊C8—81no—
168C2×C7⋊C8C2×D412yes—
169C2×C7⋊C8C2×D412yes—
170C2×#137—141no—
171C2×#137—141no—
172#533C2×D711yescharacteristic
173#532—71no—
174C2×C4⋊Q8C2×D711yescharacteristic
175#498—21no—
176#497—142no—
177#494—72no—
178#492—72no—
179#487—21no—
180#483—22no—
181#476—141no—
182#475—72no—
183#475C2×D712yes—
184#475C2×D712yes—
185#465—142no—
186#461—141no—
187#460—72no—
188#446—72no—
189#446—142no—
190C8×Q8—72no—
191#431C2×D711yescharacteristic
192#424—141no—
193C2×C4⋊C8—71no—
194C2×C4⋊C8—71no—
195#405—141no—
196C4×M16—71no—
197#335—72no—
198#334—72no—
199#332—72no—
200#332—72no—
201#332—71no—
202#332—71no—
203#331—71no—
204#331—71no—
205C23×C7#12111yescharacteristic
206C22×D7—41no—
207C7×Q8—22no—
208C7×Q8—22no—
209C7×Q8—42no—
210C7×Q8—42no—
211C7×Q8—81no—
212C7×Q8—82no—
213C2×C4×C7—21no—
214C2×C4×C7—21no—
215C2×C4×C7—22no—
216C2×C4×C7—22no—
217C2×C4×C7—41no—
218C2×C4×C7—42no—
219C2×C4×C7—82no—
220C2×C4×C7#12111yescharacteristic
221C2×C4×C7#12111yescharacteristic
222C2×C4×C7#12111yescharacteristic
223C2×C4×C7C2×SD1612yes—
224C2×C4×C7AGL(1,ℤ/8)12yes—
225C2×C4×C7C22×D411yescharacteristic
226C2×C4×C7C22×D411yescharacteristic
227C2×C4×C7C2×#4111yescharacteristic
228C22⋊D7—82no—
229C2×Dic7—41no—
230D28—41no—
231D28—41no—
232D28—41no—
233D28—41no—
234D28—81no—
235C4×D7—82no—
236Dic14—41no—
237Dic14—41no—
238C7×C8—41no—
239C7×C8—41no—
240C7⋊C8—22no—
241C7⋊C8—41no—
242C7⋊C8—41no—
243C7⋊C8—42no—
244C7⋊C8C2×SD1614yes—
245C2×#41—141no—
246C22×Q8—21no—
247#137—284no—
248#137—284no—
249C2×Q16—142no—
250C2×Q16—142no—
251C2×SD16—142no—
252C2×SD16—142no—
253C2×SD16—142no—
254C2×M16—71no—
255C2×M16—71no—
256C22×C8—141no—
257C22×C8—141no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—71no—
262C4⋊Q8C22⋊D711yescharacteristic
263C4⋊Q8C22⋊D711yescharacteristic
264C4⋊Q8C22×D712yes—
265#127—71no—
266#125—21no—
267#124—141no—
268C22⋊Q8—21no—
269C22⋊Q8C22⋊D712yes—
270#121—141no—
271#121—141no—
272C4×Q8—21no—
273C4×Q8C22×D712yes—
274C4×D4—141no—
275C4×D4—141no—
276#117—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C42C22×D711yescharacteristic
280#106—22no—
281#106—22no—
282#106—142no—
283C4⋊C8—71no—
284C4⋊C8—71no—
285C4⋊C8—72no—
286C4⋊C8—142no—
287C4⋊C8—142no—
288C4⋊C8—141no—
289C4⋊C8—141no—
290C4⋊C8C22×D711yescharacteristic
291C4⋊C8C22×D711yescharacteristic
292#103—22no—
293#103—22no—
294#103—142no—
295#102—142no—
296C22⋊C8—21no—
297#97—72no—
298#97—142no—
299C4×C8—21no—
300C4×C8—72no—
301C4×C8—142no—
302C22×C7—21no—
303C22×C7—21no—
304C22×C7#46012yes—
305C22×C7C2×#12111yescharacteristic
306C2×D7—81no—
307C2×D7—81no—
308C2×D7—81no—
309C4×C7—21no—
310C4×C7—21no—
311C4×C7—21no—
312C4×C7—42no—
313C4×C7—42no—
314C4×C7—82no—
315C4×C7#46012yes—
316C4×C7C22×SD1611yescharacteristic
317C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
318C4×C7#57711yescharacteristic
319C4×C7#57811yescharacteristic
320Dic7—81no—
321#41—282no—
322#41—282no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—141no—
329C2×Q8C2×C22⋊D712yes—
330C2×D4—141no—
331C2×D4—141no—
332C2×D4—141no—
333C2×D4—281no—
334C22×C4—41no—
335C22×C4—281no—
336C22×C4D4×D711yescharacteristic
337C22×C4#115911yescharacteristic
338C22×C4C2×C22⋊D711yescharacteristic
339Q16—284no—
340Q16—284no—
341SD16—284no—
342SD16—284no—
343SD16—562no—
344M16—74no—
345M16—142no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—72no—
349C2×C8—72no—
350C2×C8—141no—
351C2×C8—141no—
352C2×C8—142no—
353C2×C8—282no—
354C2×C8—561no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—141no—
365C4⋊C4—141no—
366C4⋊C4C2×C22⋊D712yes—
367C22⋊C4—22no—
368C22⋊C4—281no—
369C22⋊C4—281no—
370C42—22no—
371C42D4×D711yescharacteristic
372C42#115911yescharacteristic
373C42C2×C22⋊D711yescharacteristic
374C42C23×D711yescharacteristic
375C2×C7—22no—
376C2×C7C2×#46011yescharacteristic
377C2×C7#306311yescharacteristic
378C2×C7#306911yescharacteristic
379D7—161no—
380C23—281no—
381C23#650211yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—42no—
385Q8—42no—
386Q8—81no—
387Q8—82no—
388Q8—281no—
389Q8—281no—
390D4—281no—
391D4—281no—
392D4—281no—
393D4—281no—
394D4—561no—
395D4—562no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—82no—
403C2×C4—281no—
404C2×C4—562no—
405C2×C4D7×SD1612yes—
406C2×C4#647712yes—
407C2×C4#650211yescharacteristic
408C2×C4#650211yescharacteristic
409C2×C4#650211yescharacteristic
410C2×C4C2×D4×D711yescharacteristic
411C2×C4C2×#115911yescharacteristic
412C2×C4C22×C22⋊D711yescharacteristic
413C8—41no—
414C8—41no—
415C8—74no—
416C8—142no—
417C8—281no—
418C8—281no—
419C8—282no—
420C7#1923911yescharacteristic, complemented, Sylow
421C22—21no—
422C22—21no—
423C22—561no—
424C22—561no—
425C22—561no—
426C22#8993512yes—
427C22C2×#650211yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—42no—
432C4—42no—
433C4—82no—
434C4—561no—
435C4#8993512yes—
436C4C2×D7×SD1611yescharacteristic
437C4C2×#647711yescharacteristic
438C4#9052811yescharacteristic
439C4#9052911yescharacteristic
440C2—22no—
441C2—1121no—
442C2#1174084511yescharacteristic
443C2C2×#8993511yescharacteristic
444C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3229C2
socleC22×C7C2×#121
prime radical#90168C22
hypercenter#3229D7
hypocenterC7#19239
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
1. C7×#3229C2semidirect
3. #11740775C2semidirect
5. C2×#89342C2semidirect
6. #11728875C2semidirect
7. #11728874C2semidirect
8. #11728871C2semidirect
9. #11728870C2semidirect
11. #11728500C2semidirect
106. #3229D7semidirect
162. #1136SD16semidirect
420. C7#19239semidirect

Maximal subgroups

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

subgroupindex
C7×#32292
#117408082
#117407752
#117406482
C2×#893422
#117288752
#117288742
#117288712
#117288702
#117285262
#117285002
#192397