Finite Group Databasebeta

← #13337348

Subgroups — #13337348

Order 1792.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3227C211yesmaximal, characteristic, complemented
2#11740755C211yesmaximal, characteristic
3#11740722C211yesmaximal, characteristic, complemented
4#11740644C211yesmaximal, characteristic, complemented
5C2×#89341C211yesmaximal, characteristic, complemented
6#11728893C212yesmaximal, complemented
7#11728892C212yesmaximal, complemented
8#11728889C212yesmaximal, complemented
9#11728888C212yesmaximal, complemented
10#11728526C211yesmaximal, characteristic, complemented
11#11728500C211yesmaximal, characteristic, complemented
12C7×#532C2211yescharacteristic
13C2×C7×C4⋊Q8C2211yescharacteristic
14C2×#6511—21no—
15#90168C2211yescharacteristic
16C7×#498—21no—
17C7×#486—21no—
18C7×#481—22no—
19C7×#476C2212yes—
20C7×#476C2212yes—
21C7×#431C2211yescharacteristic
22#90006—21no—
23#89865C2212yes—
24#89861—22no—
25#89860C2212yes—
26#89859—21no—
27#89844—21no—
28#89810—22no—
29#89787C2212yes—
30#89784C2212yes—
31#89783—22no—
32D4×C7⋊C8C2212yes—
33#89779C2212yes—
34#89738—21no—
35#89738—21no—
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C2×C4⋊(C7⋊C8)C2211yescharacteristic
38C4×#1136C2211yescharacteristic
39#89341C2212yes—
40#89341C2212yes—
41#89281C2211yescharacteristic
42#89281C2211yescharacteristic
43#89280C2211yescharacteristic
44#89280C2211yescharacteristic
45#89275C2212yes—
46#89274C2212yes—
47#19236—71nomaximal, Sylow
48C2×C7×#41—21no—
49C22×C7×Q8—21no—
50C2×#1157—21no—
51C7×C4⋊Q8—21no—
52C7×C4⋊Q8—21no—
53C7×C4⋊Q8—21no—
54C7×C4⋊Q8D411yescharacteristic
55C7×C4⋊Q8C2312yes—
56C7×#127D411yescharacteristic
57C7×#124—21no—
58C7×#121—21no—
59C7×#121D412yes—
60C4×C7×D4—21no—
61C4×C7×D4C2312yes—
62C2×C7×C4⋊C4—21no—
63C2×C7×C4⋊C4—21no—
64C2×C42×C7C2311yescharacteristic
65#6511—44no—
66C2×#1141—22no—
67#6493—21no—
68#6493—21no—
69C2×#1136—22no—
70C2×#1136D411yescharacteristic
71C2×#1136D411yescharacteristic
72C22×C7⋊C8—21no—
73C22×C7⋊C8—21no—
74C22×C7⋊C8—22no—
75#6438—21no—
76#6437C2311yescharacteristic
77C4×D28—21no—
78C4×D28—21no—
79#6432C2311yescharacteristic
80C7×#106—22no—
81C7×#106—22no—
82C7×C4⋊C8C2311yescharacteristic
83C7×C4⋊C8C2311yescharacteristic
84C7×#102—22no—
85C7×#102—22no—
86C7×C22⋊C8—21no—
87C4×C7×C8—21no—
88#6406—22no—
89C22⋊(C7⋊C8)—22no—
90C22⋊(C7⋊C8)—22no—
91#6384—22no—
92#6384—22no—
93#6383—22no—
94#6383—22no—
95#6383—22no—
96#6381—22no—
97C4⋊(C7⋊C8)—21no—
98C4⋊(C7⋊C8)—21no—
99C4⋊(C7⋊C8)C2312yes—
100C4⋊(C7⋊C8)C2311yescharacteristic
101C4⋊(C7⋊C8)C2311yescharacteristic
102#6377D412yes—
103C4×C7⋊C8C2312yes—
104#3245—71no—
105#3227D711yescharacteristic, complemented
106#3141—71no—
107#2935—71no—
108#1703—72no—
109#1697—72no—
110#1683—72no—
111#1677—72no—
112#1565—71no—
113C2×#333—71no—
114#1540—71no—
115C7×#41—42no—
116C7×#41—42no—
117C2×C7×Q8—21no—
118C2×C7×Q8—22no—
119C2×C7×Q8—41no—
120C2×C7×Q8—82no—
121C2×C7×D4—22no—
122C2×C7×D4—21no—
123C2×C7×D4—21no—
124C2×C7×D4C2×D412yes—
125C22×C4×C7—22no—
126C22×C4×C7—41no—
127C22×C4×C7C2×D411yescharacteristic
128C22×C4×C7C2×D411yescharacteristic
129C22×C4×C7#4111yescharacteristic
130C2×C22⋊D7—41no—
131#1157—42no—
132#1157—42no—
133C2×D28—21no—
134C2×D28—21no—
135C2×D28—21no—
136C2×C4×D7—41no—
137C2×Dic14—21no—
138C2×C7×C8—21no—
139C2×C7×C8—21no—
140C7×C4⋊C4—21no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—41no—
144C7×C4⋊C4—41no—
145C7×C4⋊C4C2×D412yes—
146C7×C22⋊C4—22no—
147C7×C22⋊C4—22no—
148C42×C7C2×D411yescharacteristic
149C42×C7C2×D411yescharacteristic
150C42×C7#4111yescharacteristic
151C42×C7C2411yescharacteristic
152#1141—82no—
153(C2×D7)⋊C4—41no—
154(C2×D7)⋊C4—41no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157#1136—22no—
158#1136—22no—
159#1136—44no—
160C2×C7⋊C8—21no—
161C2×C7⋊C8—21no—
162C2×C7⋊C8—22no—
163C2×C7⋊C8—44no—
164C2×C7⋊C8—42no—
165C2×C7⋊C8—81no—
166C2×C7⋊C8C2×D412yes—
167C2×C7⋊C8C2×D412yes—
168C2×#131—141no—
169#532—71no—
170#532C2×D711yescharacteristic
171C2×C4⋊Q8C2×D711yescharacteristic
172#498—21no—
173#489—142no—
174#487—72no—
175#486—21no—
176#486—72no—
177#481—22no—
178#476C2×D712yes—
179#476C2×D712yes—
180#474—72no—
181#474—141no—
182#466—142no—
183#459—72no—
184#459—141no—
185#436—72no—
186#435—142no—
187C8×D4—72no—
188#431C2×D711yescharacteristic
189C2×C4⋊C8—71no—
190C2×C4⋊C8—71no—
191#417—141no—
192#417—141no—
193#408—141no—
194C4×M16—71no—
195#335—72no—
196#334—72no—
197#333—72no—
198#333—72no—
199#332—71no—
200#332—71no—
201#331—71no—
202#331—71no—
203C23×C7—22no—
204C23×C7#12111yescharacteristic
205C22×D7—41no—
206C7×Q8—42no—
207C7×Q8—82no—
208C7×Q8—82no—
209C7×D4—22no—
210C7×D4—22no—
211C7×D4—44no—
212C7×D4—42no—
213C7×D4—81no—
214C2×C4×C7—21no—
215C2×C4×C7—21no—
216C2×C4×C7—22no—
217C2×C4×C7—41no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—82no—
222C2×C4×C7#12111yescharacteristic
223C2×C4×C7#12111yescharacteristic
224C2×C4×C7#12111yescharacteristic
225C2×C4×C7#13512yes—
226C2×C4×C7AGL(1,ℤ/8)12yes—
227C2×C4×C7C22×D411yescharacteristic
228C2×C4×C7C22×D411yescharacteristic
229C2×C4×C7C2×#4111yescharacteristic
230C22⋊D7—82no—
231C2×Dic7—41no—
232D28—41no—
233D28—41no—
234D28—41no—
235D28—41no—
236D28—81no—
237C4×D7—82no—
238Dic14—41no—
239Dic14—41no—
240C7×C8—41no—
241C7×C8—41no—
242C7⋊C8—22no—
243C7⋊C8—22no—
244C7⋊C8—41no—
245C7⋊C8—41no—
246C7⋊C8—42no—
247C2×#41—21no—
248C2×#41—141no—
249C22×Q8—21no—
250C2×D8—142no—
251#131—284no—
252C2×M16—71no—
253C2×M16—71no—
254C2×M16—142no—
255C22×C8—142no—
256C22×C8—141no—
257C22×C8—141no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—71no—
262C4⋊Q8C22⋊D711yescharacteristic
263C4⋊Q8C22×D712yes—
264#127—71no—
265#127C22⋊D711yescharacteristic
266#124—21no—
267#124—141no—
268#121—21no—
269#121—141no—
270#121—141no—
271#121C22⋊D712yes—
272C4×D4—21no—
273C4×D4—141no—
274C4×D4—141no—
275C4×D4C22×D712yes—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C42C22×D711yescharacteristic
279#107—142no—
280#106—22no—
281#106—22no—
282C4⋊C8—71no—
283C4⋊C8—71no—
284C4⋊C8—72no—
285C4⋊C8—141no—
286C4⋊C8—141no—
287C4⋊C8C22×D711yescharacteristic
288C4⋊C8C22×D711yescharacteristic
289#103—142no—
290#103—142no—
291#102—22no—
292#102—22no—
293#102—142no—
294#102—142no—
295#102—142no—
296#102—142no—
297C22⋊C8—21no—
298C22⋊C8—142no—
299C22⋊C8—142no—
300#97—72no—
301C4×C8—21no—
302C4×C8—72no—
303C22×C7—21no—
304C22×C7—21no—
305C22×C7—42no—
306C22×C7—42no—
307C22×C7—42no—
308C22×C7#46312yes—
309C22×C7C2×#12111yescharacteristic
310C2×D7—81no—
311C2×D7—81no—
312C2×D7—81no—
313C4×C7—21no—
314C4×C7—21no—
315C4×C7—21no—
316C4×C7—42no—
317C4×C7—82no—
318C4×C7#46312yes—
319C4×C7C2×#13511yescharacteristic
320C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
321C4×C7#57711yescharacteristic
322C4×C7#57811yescharacteristic
323Dic7—81no—
324#41—42no—
325#41—42no—
326#41—282no—
327#41—282no—
328C2×Q8—21no—
329C2×Q8—22no—
330C2×Q8—41no—
331C2×Q8—82no—
332C2×Q8—141no—
333C2×D4—22no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—141no—
337C2×D4—141no—
338C2×D4—141no—
339C2×D4—281no—
340C2×D4C2×C22⋊D712yes—
341C22×C4—22no—
342C22×C4—41no—
343C22×C4—281no—
344C22×C4D4×D711yescharacteristic
345C22×C4#115911yescharacteristic
346C22×C4C2×C22⋊D711yescharacteristic
347D8—562no—
348M16—142no—
349M16—142no—
350M16—284no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—72no—
354C2×C8—72no—
355C2×C8—141no—
356C2×C8—141no—
357C2×C8—142no—
358C2×C8—284no—
359C2×C8—282no—
360C2×C8—561no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—141no—
367C4⋊C4—141no—
368C4⋊C4C2×C22⋊D712yes—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—281no—
372C22⋊C4—281no—
373C42D4×D711yescharacteristic
374C42#115911yescharacteristic
375C42C2×C22⋊D711yescharacteristic
376C42C23×D711yescharacteristic
377C2×C7—22no—
378C2×C7—82no—
379C2×C7C2×#46311yescharacteristic
380C2×C7#306311yescharacteristic
381C2×C7#306811yescharacteristic
382D7—161no—
383C23—22no—
384C23—281no—
385C23#650211yescharacteristic
386Q8—42no—
387Q8—82no—
388Q8—82no—
389Q8—281no—
390Q8—281no—
391D4—22no—
392D4—22no—
393D4—44no—
394D4—42no—
395D4—81no—
396D4—281no—
397D4—281no—
398D4—281no—
399D4—281no—
400D4—561no—
401D4—562no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4—281no—
411C2×C4—562no—
412C2×C4#647712yes—
413C2×C4#647912yes—
414C2×C4#650211yescharacteristic
415C2×C4#650211yescharacteristic
416C2×C4#650211yescharacteristic
417C2×C4C2×D4×D711yescharacteristic
418C2×C4C2×#115911yescharacteristic
419C2×C4C22×C22⋊D711yescharacteristic
420C8—41no—
421C8—41no—
422C8—142no—
423C8—142no—
424C8—281no—
425C8—281no—
426C8—282no—
427C7#1923611yescharacteristic, complemented, Sylow
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—561no—
434C22—561no—
435C22—561no—
436C22#8993712yes—
437C22C2×#650211yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—82no—
443C4—561no—
444C4#8993712yes—
445C4C2×#647711yescharacteristic
446C4C2×#647911yescharacteristic
447C4#9052811yescharacteristic
448C4#9052911yescharacteristic
449C2—22no—
450C2—82no—
451C2—1121no—
452C2#1174084511yescharacteristic
453C2C2×#8993711yescharacteristic
454C2#1174138811yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC7×#3227C2
socleC22×C7C2×#121
prime radical#11740755C2
hypercenter#3227D7
hypocenterC7#19236
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
1. C7×#3227C2semidirect
3. #11740722C2semidirect
4. #11740644C2semidirect
5. C2×#89341C2semidirect
6. #11728893C2semidirect
7. #11728892C2semidirect
8. #11728889C2semidirect
9. #11728888C2semidirect
10. #11728526C2semidirect
11. #11728500C2semidirect
105. #3227D7semidirect
427. C7#19236semidirect

Maximal subgroups

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

subgroupindex
C7×#32272
#117407552
#117407222
#117406442
C2×#893412
#117288932
#117288922
#117288892
#117288882
#117285262
#117285002
#192367