Finite Group Databasebeta

← #13249533

Subgroups — #13249533

Order 1792.

800 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740144C211yesmaximal, characteristic
2C2×#89421C211yesmaximal, characteristic, complemented
3C2×C7×#327C211yesmaximal, characteristic
4C2×#89332C211yesmaximal, characteristic
5#11727664C211yesmaximal, characteristic, complemented
6#11725481C211yesmaximal, characteristic, complemented
7#11725418C211yesmaximal, characteristic, complemented
8C2×C7×#121C2211yescharacteristic
9C2×#6498—21no—
10C22×Dic7⋊C4—21no—
11C22×C4×Dic7—21no—
12C22×C4×Dic7C2211yescharacteristic
13C2×C7×C22⋊C8C2211yescharacteristic
14C2×C7×#95C2211yescharacteristic
15#89993C412yescomplemented
16#89992—21no—
17#89984C2211yescharacteristic
18C2×C7⋊#95C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89745C412yescomplemented
21#89716—22no—
22#89421—22no—
23#89421—22no—
24C7×#327—22no—
25#89345—22no—
26#89332—22no—
27#9148—71nomaximal, Sylow
28C22×C7×D4—21no—
29C22×C7×D4C2×C411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7C2311yescharacteristic
32C23×Dic7—21no—
33C23×Dic7D411yescharacteristic
34C23×Dic7D411yescharacteristic
35C22×C7×C8—21no—
36C7×#121—21no—
37C7×#121—21no—
38C7×#121—21no—
39C7×#121—21no—
40C2×C7×C4⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C22⋊Dic7—21no—
43C2×C22⋊Dic7—22no—
44C2×C22⋊Dic7—22no—
45#6498—42no—
46#6498—42no—
47C2×C4⋊Dic7C2×C412yes—
48C2×Dic7⋊C4—24no—
49C2×Dic7⋊C4—28no—
50C2×Dic7⋊C4—22no—
51C2×C4×Dic7—21no—
52C2×C4×Dic7—21no—
53C2×C4×Dic7—21no—
54C2×C4×Dic7—21no—
55C2×C4×Dic7—24no—
56C2×C4×Dic7—42no—
57C2×C4×Dic7—42no—
58C2×C4×Dic7C2×C412yes—
59C2×C4×Dic7D412yes—
60C22×C7⋊C8—21no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×#95—21no—
64C7×#95D412yes—
65C7⋊#95—21no—
66C7⋊#95—21no—
67C7⋊#95—22no—
68C7⋊#95D412yes—
69C22⋊(C7⋊C8)—21no—
70C22⋊(C7⋊C8)—21no—
71#2854—71no—
72C2×C4×C4⋊C4—71no—
73#1817—71no—
74C2×#327—71no—
75C2×#327D711yescharacteristic
76#1294—71no—
77#1294—71no—
78C24×C7—21no—
79C24×C7C2×D411yescharacteristic
80C2×C7×D4—21no—
81C2×C7×D4—21no—
82C2×C7×D4—42no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C22×C4×C7—24no—
90C22×C4×C7—22no—
91C22×C4×C7—21no—
92C22×C4×C7—42no—
93C22×C4×C7—42no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C22×C411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7#4111yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C22×Dic7—21no—
103C22×Dic7—21no—
104C22×Dic7—21no—
105C22×Dic7—21no—
106C22×Dic7—21no—
107C22×Dic7—21no—
108C22×Dic7—24no—
109C22×Dic7—24no—
110C22×Dic7—22no—
111C22×Dic7—22no—
112C22×Dic7—42no—
113C22×Dic7—41no—
114C22×Dic7—41no—
115C22×Dic7C22⋊C412yes—
116C22×Dic7C22⋊C412yes—
117C22×Dic7D812yes—
118C22×Dic7SD1612yes—
119C2×C7×C8—42no—
120C2×C7×C8—41no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—41no—
125C22⋊Dic7—41no—
126C22⋊Dic7—41no—
127C22⋊Dic7—44no—
128C22⋊Dic7—44no—
129C4⋊Dic7—42no—
130Dic7⋊C4—216no—
131Dic7⋊C4—44no—
132C4×Dic7—22no—
133C4×Dic7—22no—
134C4×Dic7—22no—
135C4×Dic7—22no—
136C4×Dic7—48no—
137C2×C7⋊C8—42no—
138C2×C7⋊C8—41no—
139C2×#123—141no—
140C2×#121C2×D711yescharacteristic
141C22×C4⋊C4—141no—
142C22×C42—71no—
143C22×C42—141no—
144C2×C22⋊C8—71no—
145C2×C22⋊C8C2×D711yescharacteristic
146#390—72no—
147#389—72no—
148#388—142no—
149#386—141no—
150#380—71no—
151C4×C4⋊C4—142no—
152C4×C4⋊C4—142no—
153C2×#95—71no—
154C2×#95C2×D711yescharacteristic
155#336—142no—
156#327—22no—
157#327—142no—
158C23×C7—22no—
159C23×C7—22no—
160C23×C7—41no—
161C23×C7—42no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—42no—
165C23×C7C2×C22⋊C411yescharacteristic
166C23×C7#12011yescharacteristic
167C23×C7#12311yescharacteristic
168C23×C7C2×D811yescharacteristic
169C23×C7C2×SD1611yescharacteristic
170C23×C7AGL(1,ℤ/8)11yescharacteristic
171C23×C7#13711yescharacteristic
172C7×D4—81no—
173C7×D4—81no—
174C7×D4—81no—
175C7×D4—81no—
176C7×D4—81no—
177C7×D4—81no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—24no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—44no—
187C2×C4×C7—44no—
188C2×C4×C7—42no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7C4×D411yescharacteristic
193C2×C4×C7C4×D411yescharacteristic
194C2×C4×C7#12111yescharacteristic
195C2×C4×C7#12111yescharacteristic
196C2×Dic7—24no—
197C2×Dic7—24no—
198C2×Dic7—22no—
199C2×Dic7—22no—
200C2×Dic7—22no—
201C2×Dic7—22no—
202C2×Dic7—22no—
203C2×Dic7—22no—
204C2×Dic7—22no—
205C2×Dic7—22no—
206C2×Dic7—24no—
207C2×Dic7—42no—
208C2×Dic7—44no—
209C2×Dic7—44no—
210C2×Dic7—42no—
211C2×Dic7—42no—
212C2×Dic7—42no—
213C2×Dic7—42no—
214C2×Dic7#10214yes—
215C2×Dic7#10414yes—
216C7×C8—81no—
217C7⋊C8—81no—
218C22×D4—21no—
219C22×D4C4×D711yescharacteristic
220C23×C4—21no—
221C23×C4—71no—
222C23×C4—71no—
223C23×C4—141no—
224C23×C4C22×D711yescharacteristic
225C22×C8—21no—
226C22×C8—141no—
227#123—282no—
228#123—282no—
229#121—21no—
230#121—21no—
231#121—21no—
232#121—21no—
233C2×C4⋊C4—72no—
234C2×C4⋊C4—148no—
235C2×C4⋊C4—144no—
236C2×C4⋊C4—142no—
237C2×C4⋊C4C4×D711yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—142no—
240C2×C22⋊C4—142no—
241C2×C22⋊C4—141no—
242C2×C42—72no—
243C2×C42—72no—
244C2×C42—141no—
245C2×C42—141no—
246C2×C42—141no—
247C2×C42—141no—
248C2×C42—144no—
249C2×C42—282no—
250C2×C42—282no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—141no—
254C22⋊C8—141no—
255#95—21no—
256#95—72no—
257#95—142no—
258#95—141no—
259#95—141no—
260#95C22⋊D712yes—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—21no—
264C22×C7—21no—
265C22×C7—22no—
266C22×C7—22no—
267C22×C7—22no—
268C22×C7—22no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7—81no—
272C22×C7—82no—
273C22×C7—82no—
274C22×C7#38611yescharacteristic
275C22×C7#41012yes—
276C22×C7C2×#10211yescharacteristic
277C22×C7#41711yescharacteristic
278C22×C7C2×#10412yes—
279C22×C7#44711yescharacteristic
280C22×C7#44811yescharacteristic
281C22×C7#48011yescharacteristic
282C22×C7#48111yescharacteristic
283C4×C7—21no—
284C4×C7—21no—
285C4×C7—21no—
286C4×C7—21no—
287C4×C7—44no—
288C4×C7—81no—
289Dic7—24no—
290Dic7—24no—
291Dic7—44no—
292Dic7—82no—
293C24—21no—
294C24D4×D711yescharacteristic
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—72no—
312C22×C4—72no—
313C22×C4—72no—
314C22×C4—72no—
315C22×C4—144no—
316C22×C4—141no—
317C22×C4—141no—
318C22×C4—141no—
319C22×C4—141no—
320C22×C4—141no—
321C22×C4—141no—
322C22×C4—144no—
323C22×C4—142no—
324C22×C4—142no—
325C22×C4—282no—
326C22×C4—281no—
327C22×C4—281no—
328C22×C4C2×C4×D711yescharacteristic
329C22×C4#115711yescharacteristic
330C22×C4D4×D711yescharacteristic
331C22×C4D4×D711yescharacteristic
332C22×C4#115911yescharacteristic
333C22×C4C2×C22⋊D711yescharacteristic
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—282no—
337C2×C8—281no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—1416no—
341C4⋊C4—282no—
342C4⋊C4—284no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—284no—
346C22⋊C4—284no—
347C22⋊C4—281no—
348C22⋊C4—281no—
349C42—142no—
350C42—142no—
351C42—142no—
352C42—142no—
353C42—288no—
354C2×C7—21no—
355C2×C7—21no—
356C2×C7—21no—
357C2×C7—21no—
358C2×C7—161no—
359C2×C7#150212yes—
360C2×C7#150912yes—
361C2×C7#187411yescharacteristic
362C2×C7#189611yescharacteristic
363C2×C7#189911yescharacteristic
364C23—22no—
365C23—22no—
366C23—41no—
367C23—42no—
368C23—41no—
369C23—41no—
370C23—42no—
371C23D7×C22⋊C411yescharacteristic
372C23#644611yescharacteristic
373C23D7×D811yescharacteristic
374C23#647411yescharacteristic
375C23D7×SD1611yescharacteristic
376C23#647811yescharacteristic
377C23#650011yescharacteristic
378D4—81no—
379D4—81no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—74no—
399C2×C4—74no—
400C2×C4—144no—
401C2×C4—144no—
402C2×C4—142no—
403C2×C4—142no—
404C2×C4—142no—
405C2×C4—142no—
406C2×C4—142no—
407C2×C4—142no—
408C2×C4—142no—
409C2×C4—142no—
410C2×C4—144no—
411C2×C4—282no—
412C2×C4—284no—
413C2×C4—284no—
414C2×C4—282no—
415C2×C4—282no—
416C2×C4—282no—
417C2×C4—282no—
418C2×C4(C22⋊D7)⋊C411yescharacteristic
419C2×C4#644711yescharacteristic
420C2×C4C4×C22⋊D711yescharacteristic
421C2×C4#650111yescharacteristic
422C8—81no—
423C8—561no—
424C7#914811yescharacteristic, complemented, Sylow
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—82no—
437C22—82no—
438C22#8951212yes—
439C22D7×#10211yescharacteristic
440C22#8954311yescharacteristic
441C22#8954611yescharacteristic
442C22#8954911yescharacteristic
443C22D7×#10412yes—
444C22#8972811yescharacteristic
445C22#8992811yescharacteristic
446C22#8994211yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—44no—
452C4—81no—
453C4—144no—
454C4—144no—
455C4—284no—
456C4—562no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—161no—
462C2#1172603112yes—
463C2#1172603312yes—
464C2#1172778311yescharacteristic
465C2#1172795411yescharacteristic
466C2#1172813611yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#327C2
socleC23×C7C2×C22⋊C4
prime radicalC2×C7×#121C22
cosocleC23×C4×C7C23
hypercenterC2×#327D7
hypocenterC7#9148
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. C2×#89421C2semidirect
5. #11727664C2semidirect
6. #11725481C2semidirect
7. #11725418C2semidirect
15. #89993C4semidirect
20. #89745C4semidirect
424. C7#9148semidirect

Maximal subgroups

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

subgroupindex
#117401442
C2×#894212
C2×C7×#3272
C2×#893322
#117276642
#117254812
#117254182
#91487