Finite Group Databasebeta

← #13252786

Subgroups — #13252786

Order 1792.

700 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739522C211yesmaximal, characteristic
2C7×C4⋊#95C211yesmaximal, characteristic, complemented
3#11729283C211yesmaximal, characteristic, complemented
4C2×#89247C212yesmaximal, complemented
5C2×#89246C212yesmaximal
6C22×C7×C4⋊C4—21no—
7C22×C42×C7C2211yescharacteristic
8C2×#6493C2211yescharacteristic
9C2×#6487C2211yescharacteristic
10C2×#6438C411yescharacteristic, complemented
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×#6377—21no—
22#89319—21no—
23#89247—24no—
24#89246—24no—
25#9682—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×C7×C4⋊C4—24no—
34C2×C7×C4⋊C4—42no—
35C2×C7×C4⋊C4—42no—
36C2×C42×C7—42no—
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C2×C22⋊Dic7—21no—
42#6493—22no—
43#6493—22no—
44C2×(C2×D7)⋊C4—21no—
45C2×(C2×D7)⋊C4—21no—
46C2×(C2×D7)⋊C4—21no—
47C2×(C2×D7)⋊C4—21no—
48C2×(C2×D7)⋊C4—21no—
49C2×C4⋊Dic7C2×C411yescharacteristic
50C2×C4⋊Dic7C2×C411yescharacteristic
51#6487—22no—
52#6487—22no—
53C2×Dic7⋊C4—21no—
54C22×C7⋊C8—22no—
55#6438—42no—
56C7×#95—22no—
57C7×#95—22no—
58C7×#95—42no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95—21no—
62C7⋊#95—21no—
63C22⋊(C7⋊C8)—22no—
64C22⋊(C7⋊C8)D414yes—
65#6377—42no—
66#2850—71no—
67#1749—71no—
68C2×#328—72no—
69C2×#327—72no—
70C4⋊#95D711yescharacteristic, complemented
71C24×C7C2×D411yescharacteristic
72C23×D7—21no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—22no—
76C22×C4×C7—22no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—24no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—41no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7—42no—
87C22×C4×C7—42no—
88C22×C4×C7—42no—
89C22×C4×C7—42no—
90C22×C4×C7C22⋊C411yescharacteristic
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C22×C411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D412yes—
97C22×C4×C7#4112yes—
98C2×C22⋊D7—41no—
99C2×C22⋊D7—41no—
100C2×C22⋊D7—41no—
101C2×C22⋊D7—41no—
102C2×C22⋊D7—41no—
103C2×C22⋊D7—41no—
104C22×Dic7—21no—
105C22×Dic7—21no—
106C22×Dic7—21no—
107C2×D28—22no—
108C2×D28—41no—
109C2×D28—41no—
110C2×Dic14—22no—
111C2×Dic14—41no—
112C2×Dic14—41no—
113C7×C4⋊C4—48no—
114C42×C7—22no—
115C42×C7—22no—
116C42×C7—41no—
117C42×C7—41no—
118C22⋊Dic7—41no—
119C22⋊Dic7—41no—
120(C2×D7)⋊C4—42no—
121(C2×D7)⋊C4—42no—
122(C2×D7)⋊C4—42no—
123(C2×D7)⋊C4—42no—
124(C2×D7)⋊C4—42no—
125C4⋊Dic7—22no—
126C4⋊Dic7—22no—
127Dic7⋊C4—42no—
128C2×C7⋊C8—24no—
129C2×C7⋊C8—42no—
130C2×C7⋊C8—42no—
131C2×#124—71no—
132C2×C22⋊Q8—71no—
133C2×#121—71no—
134C22×C4⋊C4—21no—
135C22×C42C2×D711yescharacteristic
136C2×C22⋊C8—72no—
137C2×#97—141no—
138#391—71no—
139#390—71no—
140#388—141no—
141#388—141no—
142#386—141no—
143#386—141no—
144#382—71no—
145C2×#95—21no—
146C2×#95C2×D712yes—
147#336—141no—
148#328—144no—
149#327—144no—
150C23×C7—21no—
151C23×C7—21no—
152C23×C7—21no—
153C23×C7—21no—
154C23×C7C2×C22⋊C411yescharacteristic
155C23×C7#12411yescharacteristic
156C23×C7#12711yescharacteristic
157C23×C7AGL(1,ℤ/8)11yescharacteristic
158C23×C7AGL(1,ℤ/8)11yescharacteristic
159C23×C7#13711yescharacteristic
160C23×C7#13711yescharacteristic
161C22×D7—41no—
162C22×D7—41no—
163C22×D7—41no—
164C22×D7—41no—
165C22×D7—41no—
166C22×D7—41no—
167C22×D7—41no—
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—21no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—44no—
189C2×C4×C7—44no—
190C2×C4×C7—44no—
191C2×C4×C7—82no—
192C2×C4×C7—82no—
193C2×C4×C7—82no—
194C2×C4×C7—82no—
195C2×C4×C7#9912yes—
196C2×C4×C7#10412yes—
197C2×C4×C7C4×D412yes—
198C2×C4×C7#12112yes—
199C22⋊D7—81no—
200C22⋊D7—81no—
201C22⋊D7—81no—
202C22⋊D7—81no—
203C2×Dic7—41no—
204C2×Dic7—41no—
205C2×Dic7—41no—
206C2×Dic7—41no—
207C2×Dic7—41no—
208C2×Dic7—41no—
209C2×Dic7—41no—
210C2×Dic7—41no—
211C2×Dic7—41no—
212D28—82no—
213Dic14—82no—
214C7⋊C8—44no—
215C22×Q8—71no—
216C22×D4—71no—
217C22×D4—141no—
218C23×C4—22no—
219C23×C4C22⋊D711yescharacteristic
220C23×C4C22⋊D711yescharacteristic
221C23×C4C22×D711yescharacteristic
222C22×C8—142no—
223#124—282no—
224C22⋊Q8—142no—
225C22⋊Q8—142no—
226#121—142no—
227#121—142no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—71no—
232C2×C4⋊C4—71no—
233C2×C4⋊C4—141no—
234C2×C22⋊C4—141no—
235C2×C22⋊C4—141no—
236C2×C22⋊C4—141no—
237C2×C22⋊C4—141no—
238C2×C22⋊C4—141no—
239C2×C22⋊C4—141no—
240C2×C42—42no—
241C2×C42C4×D711yescharacteristic
242C2×C42C4×D711yescharacteristic
243C2×C42D2811yescharacteristic
244C2×C42D2811yescharacteristic
245C22⋊C8—74no—
246C22⋊C8—142no—
247#97—282no—
248#95—22no—
249#95—22no—
250#95—42no—
251#95—141no—
252#95—141no—
253#95—141no—
254#95—141no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7#39011yescharacteristic
268C22×C7C2×#9911yescharacteristic
269C22×C7#41011yescharacteristic
270C22×C7#41711yescharacteristic
271C22×C7#41711yescharacteristic
272C22×C7C2×#10411yescharacteristic
273C22×C7#42111yescharacteristic
274C22×C7#48911yescharacteristic
275C22×C7#48911yescharacteristic
276C22×C7#49611yescharacteristic
277C22×C7#49711yescharacteristic
278C2×D7—81no—
279C2×D7—81no—
280C2×D7—81no—
281C2×D7—81no—
282C2×D7—81no—
283C2×D7—81no—
284C2×D7—81no—
285C4×C7—22no—
286C4×C7—22no—
287C4×C7—22no—
288C4×C7—22no—
289C4×C7—41no—
290C4×C7—41no—
291C4×C7—84no—
292Dic7—81no—
293Dic7—81no—
294Dic7—81no—
295C24—141no—
296C24C2×C22⋊D711yescharacteristic
297C2×Q8—142no—
298C2×Q8—281no—
299C2×Q8—281no—
300C2×D4—142no—
301C2×D4—281no—
302C2×D4—281no—
303C2×D4—281no—
304C2×D4—281no—
305C2×D4—281no—
306C2×D4—281no—
307C2×D4—281no—
308C2×D4—281no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—141no—
327C22×C4—141no—
328C22×C4—141no—
329C22×C4(C2×D7)⋊C411yescharacteristic
330C22×C4(C2×D7)⋊C411yescharacteristic
331C22×C4(C2×D7)⋊C411yescharacteristic
332C22×C4(C2×D7)⋊C411yescharacteristic
333C22×C4C2×C4×D711yescharacteristic
334C22×C4C2×D2811yescharacteristic
335C22×C4D4×D712yes—
336C22×C4#116112yes—
337C2×C8—144no—
338C2×C8—282no—
339C2×C8—282no—
340C4⋊C4—48no—
341C4⋊C4—142no—
342C4⋊C4—142no—
343C4⋊C4—282no—
344C22⋊C4—282no—
345C22⋊C4—282no—
346C22⋊C4—281no—
347C22⋊C4—281no—
348C22⋊C4—282no—
349C22⋊C4—282no—
350C22⋊C4—282no—
351C42—22no—
352C42—22no—
353C42—41no—
354C42—41no—
355C2×C7—21no—
356C2×C7—21no—
357C2×C7—21no—
358C2×C7—21no—
359C2×C7#150411yescharacteristic
360C2×C7#150911yescharacteristic
361C2×C7#151211yescharacteristic
362C2×C7#151311yescharacteristic
363C2×C7#195911yescharacteristic
364C2×C7#196411yescharacteristic
365C2×C7#198011yescharacteristic
366D7—161no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—281no—
372C23—281no—
373C23—281no—
374C23—281no—
375C23—281no—
376C23—281no—
377C23—281no—
378C23C2×(C2×D7)⋊C411yescharacteristic
379C23#649511yescharacteristic
380C23#649511yescharacteristic
381C23#650311yescharacteristic
382C23#650511yescharacteristic
383C23#650511yescharacteristic
384C23#651011yescharacteristic
385Q8—562no—
386D4—562no—
387D4—561no—
388D4—561no—
389D4—561no—
390D4—561no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—281no—
419C2×C4—281no—
420C2×C4—281no—
421C2×C4—281no—
422C2×C4—281no—
423C2×C4—281no—
424C2×C4—281no—
425C2×C4—281no—
426C2×C4—281no—
427C2×C4#637912yes—
428C2×C4#638012yes—
429C2×C4#645612yes—
430C2×C4#645812yes—
431C8—284no—
432C7#968211yescharacteristic, complemented, Sylow
433C22—21no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—561no—
446C22—561no—
447C22—561no—
448C22—561no—
449C22—561no—
450C22—561no—
451C22—561no—
452C22C2×#637911yescharacteristic
453C22C2×#638011yescharacteristic
454C22#8973811yescharacteristic
455C22#8973811yescharacteristic
456C22#8976011yescharacteristic
457C22#8977711yescharacteristic
458C22#8977811yescharacteristic
459C22#8984611yescharacteristic
460C22#8985811yescharacteristic
461C22#8986111yescharacteristic
462C22#8986111yescharacteristic
463C4—22no—
464C4—22no—
465C4—22no—
466C4—22no—
467C4—41no—
468C4—41no—
469C4—84no—
470C4—561no—
471C4—561no—
472C4—561no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—21no—
477C2—1121no—
478C2#1172715211yescharacteristic
479C2#1172722411yescharacteristic
480C2#1172724611yescharacteristic
481C2#1172726211yescharacteristic
482C2#1172728211yescharacteristic
483C2#1172729811yescharacteristic
484C2#1172733511yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×C4⋊#95C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenterC4⋊#95D7
hypocenterC7#9682
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4⋊#95

Complements

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

NH ≅ G/Ntype
2. C7×C4⋊#95C2semidirect
3. #11729283C2semidirect
4. C2×#89247C2semidirect
10. C2×#6438C4semidirect
18. #89711C4semidirect
19. #89708C4semidirect
20. #89701C4semidirect
70. C4⋊#95D7semidirect
432. C7#9682semidirect

Maximal subgroups

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

subgroupindex
#117395222
C7×C4⋊#952
#117292832
C2×#892472
C2×#892462
#96827