Finite Group Databasebeta

← #13252781

Subgroups — #13252781

Order 1792.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739522C211yesmaximal, characteristic
2C7×#1434C211yesmaximal, characteristic, complemented
3C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4C2×#89247C212yesmaximal, complemented
5C2×#89246C212yesmaximal
6C22×C42×C7C2211yescharacteristic
7C2×#6493C2211yescharacteristic
8C2×#6487C2211yescharacteristic
9C2×#6438C411yescharacteristic, complemented
10C2×C7×#95—21no—
11C2×C7×#95—21no—
12C2×C7×#95C2212yes—
13#89985—21no—
14#89985—21no—
15C2×C22⋊(C7⋊C8)C2212yes—
16#89712—21no—
17#89712—21no—
18#89711C411yescharacteristic, complemented
19#89708C411yescharacteristic, complemented
20#89701C411yescharacteristic, complemented
21C2×C4×C7⋊C8—21no—
22#89319—21no—
23#89247—24no—
24#89246—24no—
25#9681—71nomaximal, Sylow
26C23×C4×C7—22no—
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C23×C4×C7C2311yescharacteristic
30C22×C22⋊D7—21no—
31C22×D28C2×C411yescharacteristic
32C22×Dic14C2×C411yescharacteristic
33C2×C42×C7—42no—
34C2×C42×C7C2×C411yescharacteristic
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7D411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C22⋊Dic7—21no—
39#6493—22no—
40#6493—22no—
41C2×(C2×D7)⋊C4—21no—
42C2×(C2×D7)⋊C4—21no—
43C2×(C2×D7)⋊C4—21no—
44C2×(C2×D7)⋊C4—21no—
45C2×(C2×D7)⋊C4—21no—
46C2×C4⋊Dic7C2×C411yescharacteristic
47C2×C4⋊Dic7C2×C411yescharacteristic
48#6487—22no—
49#6487—22no—
50C2×Dic7⋊C4—21no—
51C22×C7⋊C8—22no—
52#6438—42no—
53C7×#95—24no—
54C7×#95—22no—
55C7×#95—22no—
56C7×#95—42no—
57C7⋊#95—21no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C22⋊(C7⋊C8)—22no—
62C22⋊(C7⋊C8)D414yes—
63C4×C7⋊C8—42no—
64#2850—71no—
65C4×C22⋊C8—71no—
66C2×#328—72no—
67C2×#327—72no—
68#1434D711yescharacteristic, complemented
69C24×C7C2×D411yescharacteristic
70C23×D7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—22no—
75C22×C4×C7—24no—
76C22×C4×C7—41no—
77C22×C4×C7—41no—
78C22×C4×C7—41no—
79C22×C4×C7—41no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22×C411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D412yes—
93C22×C4×C7#4112yes—
94C2×C22⋊D7—41no—
95C2×C22⋊D7—41no—
96C2×C22⋊D7—42no—
97C2×C22⋊D7—42no—
98C22×Dic7—21no—
99C22×Dic7—21no—
100C22×Dic7—21no—
101C2×D28—22no—
102C2×D28—41no—
103C2×D28—41no—
104C2×Dic14—22no—
105C2×Dic14—41no—
106C2×Dic14—41no—
107C42×C7—22no—
108C42×C7—22no—
109C42×C7—41no—
110C42×C7—41no—
111C22⋊Dic7—41no—
112C22⋊Dic7—41no—
113(C2×D7)⋊C4—42no—
114(C2×D7)⋊C4—42no—
115(C2×D7)⋊C4—42no—
116(C2×D7)⋊C4—42no—
117(C2×D7)⋊C4—42no—
118C4⋊Dic7—22no—
119C4⋊Dic7—22no—
120Dic7⋊C4—42no—
121C2×C7⋊C8—24no—
122C2×C7⋊C8—42no—
123C2×C7⋊C8—42no—
124C2×#124—71no—
125C2×C22⋊Q8—71no—
126C2×#121—71no—
127C22×C42C2×D711yescharacteristic
128C2×C22⋊C8—72no—
129C2×C4×C8—141no—
130#391—71no—
131#390—71no—
132#388—141no—
133#388—141no—
134#386—141no—
135#386—141no—
136#382—71no—
137C2×#95—21no—
138C2×#95—21no—
139C2×#95C2×D712yes—
140#336—141no—
141#328—144no—
142#327—144no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7C2×C22⋊C411yescharacteristic
148C23×C7#12411yescharacteristic
149C23×C7#12711yescharacteristic
150C23×C7#13512yes—
151C23×C7#13512yes—
152C22×D7—41no—
153C22×D7—41no—
154C22×D7—41no—
155C22×D7—41no—
156C22×D7—41no—
157C22×D7—41no—
158C22×D7—41no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—44no—
176C2×C4×C7—44no—
177C2×C4×C7—44no—
178C2×C4×C7—44no—
179C2×C4×C7—44no—
180C2×C4×C7—82no—
181C2×C4×C7—82no—
182C2×C4×C7—82no—
183C2×C4×C7—82no—
184C2×C4×C7#9912yes—
185C2×C4×C7#10412yes—
186C2×C4×C7C4×D412yes—
187C2×C4×C7#12112yes—
188C22⋊D7—82no—
189C22⋊D7—82no—
190C2×Dic7—41no—
191C2×Dic7—41no—
192C2×Dic7—41no—
193C2×Dic7—41no—
194C2×Dic7—41no—
195C2×Dic7—41no—
196C2×Dic7—41no—
197C2×Dic7—41no—
198C2×Dic7—41no—
199D28—82no—
200Dic14—82no—
201C7⋊C8—44no—
202C22×Q8—71no—
203C22×D4—71no—
204C22×D4—141no—
205C23×C4—22no—
206C23×C4C22⋊D711yescharacteristic
207C23×C4C22⋊D711yescharacteristic
208C23×C4C22×D711yescharacteristic
209C22×C8—142no—
210#124—282no—
211C22⋊Q8—142no—
212C22⋊Q8—142no—
213#121—142no—
214#121—142no—
215C2×C4⋊C4—71no—
216C2×C4⋊C4—71no—
217C2×C4⋊C4—141no—
218C2×C22⋊C4—141no—
219C2×C22⋊C4—141no—
220C2×C22⋊C4—141no—
221C2×C22⋊C4—141no—
222C2×C22⋊C4—141no—
223C2×C22⋊C4—141no—
224C2×C42—42no—
225C2×C42C4×D711yescharacteristic
226C2×C42C4×D711yescharacteristic
227C2×C42D2811yescharacteristic
228C2×C42D2811yescharacteristic
229C22⋊C8—74no—
230C22⋊C8—142no—
231C4×C8—282no—
232#95—24no—
233#95—22no—
234#95—22no—
235#95—42no—
236#95—141no—
237#95—141no—
238#95—141no—
239#95—141no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—22no—
245C22×C7—22no—
246C22×C7—22no—
247C22×C7—22no—
248C22×C7#39011yescharacteristic
249C22×C7C2×#9911yescharacteristic
250C22×C7#41011yescharacteristic
251C22×C7#41611yescharacteristic
252C22×C7#41611yescharacteristic
253C22×C7C2×#10411yescharacteristic
254C22×C7#42111yescharacteristic
255C22×C7#48812yes—
256C22×C7#49512yes—
257C2×D7—81no—
258C2×D7—81no—
259C2×D7—81no—
260C2×D7—81no—
261C2×D7—81no—
262C2×D7—81no—
263C2×D7—81no—
264C4×C7—22no—
265C4×C7—22no—
266C4×C7—22no—
267C4×C7—22no—
268C4×C7—41no—
269C4×C7—41no—
270C4×C7—84no—
271Dic7—81no—
272Dic7—81no—
273Dic7—81no—
274C24—141no—
275C24C2×C22⋊D711yescharacteristic
276C2×Q8—142no—
277C2×Q8—281no—
278C2×Q8—281no—
279C2×D4—142no—
280C2×D4—281no—
281C2×D4—281no—
282C2×D4—281no—
283C2×D4—281no—
284C2×D4—282no—
285C2×D4—282no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—141no—
302C22×C4—141no—
303C22×C4—141no—
304C22×C4(C2×D7)⋊C411yescharacteristic
305C22×C4(C2×D7)⋊C411yescharacteristic
306C22×C4(C2×D7)⋊C411yescharacteristic
307C22×C4(C2×D7)⋊C411yescharacteristic
308C22×C4C2×C4×D711yescharacteristic
309C22×C4C2×D2811yescharacteristic
310C22×C4D4×D712yes—
311C22×C4#116112yes—
312C2×C8—144no—
313C2×C8—282no—
314C2×C8—282no—
315C4⋊C4—142no—
316C4⋊C4—142no—
317C4⋊C4—282no—
318C22⋊C4—282no—
319C22⋊C4—282no—
320C22⋊C4—281no—
321C22⋊C4—281no—
322C22⋊C4—282no—
323C22⋊C4—282no—
324C22⋊C4—282no—
325C42—22no—
326C42—22no—
327C42—41no—
328C42—41no—
329C2×C7—22no—
330C2×C7—22no—
331C2×C7#150011yescharacteristic
332C2×C7#150111yescharacteristic
333C2×C7#151611yescharacteristic
334C2×C7#152111yescharacteristic
335C2×C7#195811yescharacteristic
336C2×C7#196411yescharacteristic
337C2×C7#198011yescharacteristic
338D7—161no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—281no—
344C23—281no—
345C23—281no—
346C23—281no—
347C23—281no—
348C23—281no—
349C23—281no—
350C23C2×(C2×D7)⋊C411yescharacteristic
351C23#650311yescharacteristic
352C23#651011yescharacteristic
353C23#651312yes—
354C23#651312yes—
355Q8—562no—
356D4—562no—
357D4—562no—
358D4—562no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—281no—
385C2×C4—281no—
386C2×C4—281no—
387C2×C4—281no—
388C2×C4—281no—
389C2×C4—281no—
390C2×C4—281no—
391C2×C4—281no—
392C2×C4—281no—
393C2×C4#637912yes—
394C2×C4#638012yes—
395C2×C4#645612yes—
396C2×C4#645812yes—
397C8—284no—
398C7#968111yescharacteristic, complemented, Sylow
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—22no—
406C22—22no—
407C22—561no—
408C22—561no—
409C22—561no—
410C22—561no—
411C22—561no—
412C22—561no—
413C22—561no—
414C22C2×#637911yescharacteristic
415C22C2×#638011yescharacteristic
416C22#8976011yescharacteristic
417C22#8977411yescharacteristic
418C22#8977411yescharacteristic
419C22#8977711yescharacteristic
420C22#8977811yescharacteristic
421C22#8983112yes—
422C22#8984012yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—84no—
430C4—561no—
431C4—561no—
432C4—561no—
433C2—22no—
434C2—22no—
435C2—1121no—
436C2#1172718511yescharacteristic
437C2#1172719711yescharacteristic
438C2#1172724611yescharacteristic
439C2#1172728211yescharacteristic
440C2#1172728611yescharacteristic
441C2#1172730511yescharacteristic
442C2#1172731911yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1434C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenter#1434D7
hypocenterC7#9681
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C7×#1434C2semidirect
3. C4×C22⋊(C7⋊C8)C2semidirect
4. C2×#89247C2semidirect
9. C2×#6438C4semidirect
18. #89711C4semidirect
19. #89708C4semidirect
20. #89701C4semidirect
68. #1434D7semidirect
398. C7#9681semidirect

Maximal subgroups

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

subgroupindex
#117395222
C7×#14342
C4×C22⋊(C7⋊C8)2
C2×#892472
C2×#892462
#96817