Finite Group Databasebeta

← #13252928

Subgroups — #13252928

Order 1792.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2850C211yesmaximal, characteristic
2C2×#89333C212yesmaximal, complemented
3C2×#89332C212yesmaximal
4C4×C7⋊#95C211yesmaximal, characteristic, complemented
5#11729284C211yesmaximal, characteristic, complemented
6C2×C7×#124C411yescharacteristic, complemented
7C2×C7×C22⋊Q8C2211yescharacteristic
8C2×C7×#121C2211yescharacteristic
9C22×C42×C7C2211yescharacteristic
10C22×C4×Dic7—21no—
11C7×#391C411yescharacteristic, complemented
12C7×#390C411yescharacteristic, complemented
13C7×#388—21no—
14C7×#388—21no—
15C7×#386—21no—
16C7×#386—21no—
17C7×#382C411yescharacteristic, complemented
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2212yes—
20C2×C22⋊(C7⋊C8)C2212yes—
21C2×C4⋊(C7⋊C8)—21no—
22#89333—24no—
23#89332—24no—
24#89319—21no—
25#9713—71nomaximal, Sylow
26C22×C7×Q8C2×C411yescharacteristic
27C22×C7×D4—21no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C23×C4×C7C2311yescharacteristic
32C23×Dic7—22no—
33C7×#124—42no—
34C7×C22⋊Q8—22no—
35C7×C22⋊Q8—22no—
36C7×#121—22no—
37C7×#121—22no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4C2×C411yescharacteristic
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C42×C7—42no—
48C2×C42×C7C2×C411yescharacteristic
49C2×C42×C7C2×C411yescharacteristic
50C2×C42×C7D411yescharacteristic
51C2×C42×C7D411yescharacteristic
52C2×C4×Dic7—24no—
53C2×C4×Dic7—42no—
54C2×C4×Dic7—42no—
55C22×C7⋊C8—22no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7⋊#95—24no—
61C7⋊#95—22no—
62C7⋊#95D414yes—
63C22⋊(C7⋊C8)—22no—
64C22⋊(C7⋊C8)—22no—
65C4⋊(C7⋊C8)—42no—
66#2850D711yescharacteristic
67#1797—71no—
68C2×#328—72no—
69C2×#327—72no—
70C4×#95—71no—
71C24×C7—21no—
72C24×C7C2×D411yescharacteristic
73C2×C7×Q8—22no—
74C2×C7×Q8—41no—
75C2×C7×Q8—41no—
76C2×C7×D4—22no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—42no—
80C2×C7×D4—42no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—22no—
86C22×C4×C7—22no—
87C22×C4×C7—22no—
88C22×C4×C7—22no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7C22⋊C411yescharacteristic
96C22×C4×C7C22⋊C411yescharacteristic
97C22×C4×C7C22⋊C411yescharacteristic
98C22×C4×C7C22⋊C411yescharacteristic
99C22×C4×C7C22×C411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D412yes—
102C22×C4×C7#4112yes—
103C22×Dic7—24no—
104C22×Dic7—24no—
105C22×Dic7—24no—
106C22×Dic7—42no—
107C22×Dic7—42no—
108C22×Dic7—42no—
109C22×Dic7—42no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—41no—
113C7×C4⋊C4—41no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121C42×C7—22no—
122C42×C7—22no—
123C42×C7—41no—
124C42×C7—41no—
125C4×Dic7—48no—
126C2×C7⋊C8—42no—
127C2×C7⋊C8—42no—
128C2×C7⋊C8—42no—
129C2×#124Dic711yescharacteristic, complemented
130C2×C22⋊Q8C2×D711yescharacteristic
131C2×#121C2×D711yescharacteristic
132C22×C42—141no—
133C22×C42C2×D711yescharacteristic
134C2×C4⋊C8—141no—
135C2×C22⋊C8—72no—
136#391Dic711yescharacteristic, complemented
137#390Dic711yescharacteristic, complemented
138#388—21no—
139#388—21no—
140#386—21no—
141#386—21no—
142#382Dic711yescharacteristic, complemented
143C2×#95—72no—
144C2×#95—141no—
145#336—141no—
146#328—144no—
147#327—144no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7C2×C22⋊C411yescharacteristic
160C23×C7#12112yes—
161C23×C7#13512yes—
162C23×C7AGL(1,ℤ/8)11yescharacteristic
163C23×C7#13711yescharacteristic
164C7×Q8—82no—
165C7×D4—82no—
166C7×D4—82no—
167C7×D4—82no—
168C2×C4×C7—24no—
169C2×C4×C7—24no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—21no—
181C2×C4×C7—21no—
182C2×C4×C7—21no—
183C2×C4×C7—42no—
184C2×C4×C7—42no—
185C2×C4×C7—44no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7#9912yes—
196C2×C4×C7#10412yes—
197C2×C4×C7#10412yes—
198C2×C4×C7C4×D412yes—
199C2×C4×C7#12411yescharacteristic
200C2×C4×C7#12711yescharacteristic
201C2×Dic7—28no—
202C2×Dic7—44no—
203C2×Dic7—44no—
204C2×Dic7—44no—
205C2×Dic7—44no—
206C2×Dic7—44no—
207C2×Dic7—44no—
208C7⋊C8—82no—
209C22×Q8C2×Dic711yescharacteristic
210C22×D4—21no—
211C22×D4C2×Dic711yescharacteristic
212C23×C4—142no—
213C23×C4C22⋊D711yescharacteristic
214C23×C4C22⋊D711yescharacteristic
215C23×C4C22×D711yescharacteristic
216C22×C8—142no—
217#124—42no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220#121—22no—
221#121—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4C2×Dic711yescharacteristic
224C2×C4⋊C4C2×Dic711yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—42no—
232C2×C42—144no—
233C2×C42—282no—
234C2×C42—282no—
235C2×C42C2×Dic711yescharacteristic
236C2×C42C2×Dic711yescharacteristic
237C2×C42C22⋊D711yescharacteristic
238C2×C42C22⋊D711yescharacteristic
239C4⋊C8—282no—
240C22⋊C8—142no—
241C22⋊C8—142no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—74no—
247#95—144no—
248#95—142no—
249C22×C7—22no—
250C22×C7—22no—
251C22×C7—22no—
252C22×C7—22no—
253C22×C7—21no—
254C22×C7—21no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7—81no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7#39011yescharacteristic
265C22×C7C2×#9911yescharacteristic
266C22×C7#41011yescharacteristic
267C22×C7#41611yescharacteristic
268C22×C7#41711yescharacteristic
269C22×C7C2×#10411yescharacteristic
270C22×C7C2×#10411yescharacteristic
271C22×C7#46312yes—
272C22×C7#46412yes—
273C4×C7—21no—
274C4×C7—22no—
275C4×C7—21no—
276C4×C7—22no—
277C4×C7—22no—
278C4×C7—22no—
279C4×C7—41no—
280C4×C7—81no—
281C4×C7—81no—
282C4×C7—81no—
283Dic7—48no—
284C24—21no—
285C24C2×C22⋊D711yescharacteristic
286C2×Q8—22no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×D4—22no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—41no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—144no—
309C22×C4—144no—
310C22×C4—144no—
311C22×C4—282no—
312C22×C4—282no—
313C22×C4—282no—
314C22×C4—282no—
315C22×C4C22⋊Dic711yescharacteristic
316C22×C4C22⋊Dic711yescharacteristic
317C22×C4C22⋊Dic711yescharacteristic
318C22×C4C22⋊Dic711yescharacteristic
319C22×C4D4×D712yes—
320C22×C4#115912yes—
321C22×C4C22×Dic711yescharacteristic
322C22×C4C2×C22⋊D711yescharacteristic
323C2×C8—282no—
324C2×C8—282no—
325C2×C8—282no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C42—22no—
338C42—22no—
339C42—41no—
340C42—41no—
341C42—288no—
342C2×C7—22no—
343C2×C7—22no—
344C2×C7—161no—
345C2×C7#150011yescharacteristic
346C2×C7#150111yescharacteristic
347C2×C7#150411yescharacteristic
348C2×C7#150911yescharacteristic
349C2×C7#196211yescharacteristic
350C2×C7#196411yescharacteristic
351C2×C7#198311yescharacteristic
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23#650112yes—
364C23#651211yescharacteristic
365C23#651312yes—
366C23#651411yescharacteristic
367C23C2×C22⋊Dic711yescharacteristic
368Q8—82no—
369D4—82no—
370D4—82no—
371D4—82no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—148no—
400C2×C4—284no—
401C2×C4—284no—
402C2×C4—284no—
403C2×C4—284no—
404C2×C4—284no—
405C2×C4—284no—
406C2×C4#640812yes—
407C2×C4#641112yes—
408C2×C4#641112yes—
409C2×C4D4×Dic712yes—
410C2×C4#649911yescharacteristic
411C2×C4#650311yescharacteristic
412C8—562no—
413C7#971311yescharacteristic, complemented, Sylow
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—22no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#8982612yes—
430C22#8982912yes—
431C22C2×#640811yescharacteristic
432C22#8999311yescharacteristic
433C22#9000411yescharacteristic
434C22#9000511yescharacteristic
435C22C2×#641111yescharacteristic
436C22C2×#641111yescharacteristic
437C22#9001211yescharacteristic
438C4—21no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—288no—
449C2—22no—
450C2—22no—
451C2—161no—
452C2#1172944011yescharacteristic
453C2#1172946111yescharacteristic
454C2#1172946211yescharacteristic
455C2#1172947511yescharacteristic
456C2#1172950811yescharacteristic
457C2#1172950911yescharacteristic
458C2#1172956911yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2850C2
socleC23×C7C2×C22⋊C4
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenter#2850D7
hypocenterC7#9713
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
2. C2×#89333C2semidirect
4. C4×C7⋊#95C2semidirect
5. #11729284C2semidirect
6. C2×C7×#124C4semidirect
11. C7×#391C4semidirect
12. C7×#390C4semidirect
17. C7×#382C4semidirect
129. C2×#124Dic7semidirect
136. #391Dic7semidirect
137. #390Dic7semidirect
142. #382Dic7semidirect
413. C7#9713semidirect

Maximal subgroups

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

subgroupindex
C7×#28502
C2×#893332
C2×#893322
C4×C7⋊#952
#117292842
#97137