Finite Group Databasebeta

← #13364188

Subgroups — #13364188

Order 1792.

830 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89609C211yesmaximal, characteristic
2#11731763C211yesmaximal, characteristic
3#11731197C211yesmaximal, characteristic
4C7×#1947C211yesmaximal, characteristic, complemented
5#11728385C212yesmaximal
6#11728370C212yesmaximal, complemented
7#11728208C211yesmaximal, characteristic, complemented
8#11727322C211yesmaximal, characteristic, complemented
9#11727220C211yesmaximal, characteristic, complemented
10#11726624C212yesmaximal
11#11726551C212yesmaximal, complemented
12C22×#1131C2212yes—
13C2×#6455—21no—
14C2×C42×D7C2211yescharacteristic
15C2×C7×#107C2212yes—
16C2×C7×C4⋊C8C2211yescharacteristic
17C7×#384C2211yescharacteristic
18C4×C7×C4⋊C4C2211yescharacteristic
19C2×#6391C2212yes—
20C2×Dic7⋊C8C2212yes—
21#89759—21no—
22#89759C2211yescharacteristic
23#89759C2212yes—
24#89759C2211yescharacteristic
25#89749—21no—
26#89749C2211yescharacteristic
27#89749C2211yescharacteristic
28C4⋊C4×Dic7—21no—
29C4⋊C4×Dic7C2212yes—
30C2×#6382C2212yes—
31C2×#6381C2212yes—
32C4×(C2×D7)⋊C4—21no—
33#89707C2211yescharacteristic
34C4×C4⋊Dic7C2211yescharacteristic
35#89704—21no—
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37#89609C418yescomplemented
38#89436—21no—
39#89423—21no—
40C7×#340C2212yes—
41#89346C2212yes—
42#89324C2212yes—
43#89324C2212yes—
44#21963—71nomaximal, Sylow
45C22×C4×D7D411yescharacteristic
46C22×C4×D7D411yescharacteristic
47C22×C4×D7C2311yescharacteristic
48C22×C7×C8C2312yes—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4C2311yescharacteristic
52C2×C7×C4⋊C4C2312yes—
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C42×C7—21no—
55C2×C42×C7C2311yescharacteristic
56C2×(C2×D7)⋊C4—22no—
57C2×(C2×D7)⋊C4—21no—
58C2×(C2×D7)⋊C4—21no—
59C2×C4⋊Dic7—21no—
60C2×C4⋊Dic7—21no—
61C2×C4⋊Dic7C2311yescharacteristic
62C2×C4⋊Dic7C2312yes—
63C2×C4⋊Dic7C2311yescharacteristic
64C2×C4×Dic7—21no—
65C2×C4×Dic7—22no—
66C2×C4×Dic7D411yescharacteristic
67C2×C4×Dic7D411yescharacteristic
68C2×C4×Dic7C2311yescharacteristic
69C22×C7⋊C8C2312yes—
70C2×#1131—42no—
71C2×#1131—42no—
72C2×#1131C2×C418yes—
73#6455—44no—
74C42×D7C2×C414yes—
75C42×D7Q814yes—
76C7×#107—42no—
77C7×C4⋊C8C2×C414yes—
78C7×#95—21no—
79C7⋊#95—21no—
80C7⋊#95—21no—
81C7⋊#95—21no—
82C7⋊#95—21no—
83C7⋊#95—21no—
84C7⋊#95—21no—
85C7⋊#95—21no—
86C7⋊#95—22no—
87C7⋊#95—22no—
88#6391—42no—
89Dic7⋊C8C2×C418yes—
90#6382—42no—
91#6381—42no—
92C4⋊(C7⋊C8)C2×C414yes—
93C2×#423—71no—
94#2285—71no—
95#2283—71no—
96#1949—72no—
97#1949—72no—
98#1948—71no—
99#1948—71no—
100#1947—71no—
101#1947D711yescharacteristic, complemented
102#1809—72no—
103#1809—72no—
104C23×D7C2×D411yescharacteristic
105C22×C4×C7—21no—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2411yescharacteristic
111C22×Dic7—21no—
112C22×Dic7—21no—
113C22×Dic7—22no—
114C22×Dic7C2×D411yescharacteristic
115C22×Dic7C2×D411yescharacteristic
116C22×Dic7C2×D411yescharacteristic
117C2×C4×D7—22no—
118C2×C4×D7—22no—
119C2×C4×D7—21no—
120C2×C4×D7—21no—
121C2×C4×D7—21no—
122C2×C4×D7—21no—
123C2×C4×D7—41no—
124C2×C4×D7—41no—
125C2×C4×D7C4⋊C414yes—
126C2×C4×D7C4⋊C414yes—
127C2×C4×D7C22×C412yes—
128C2×C4×D7C2×Q812yes—
129C2×C7×C8—22no—
130C2×C7×C8—22no—
131C2×C7×C8C22×C414yes—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—21no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—42no—
139C7×C4⋊C4—42no—
140C42×C7—42no—
141C42×C7C22×C412yes—
142C42×C7C2×Q812yes—
143(C2×D7)⋊C4—44no—
144(C2×D7)⋊C4—42no—
145(C2×D7)⋊C4—42no—
146C4⋊Dic7—22no—
147C4⋊Dic7—21no—
148C4⋊Dic7—21no—
149C4⋊Dic7—22no—
150C4⋊Dic7—21no—
151C4⋊Dic7—21no—
152C4⋊Dic7—42no—
153C4⋊Dic7—42no—
154C4×Dic7—44no—
155C4×Dic7—42no—
156C4×Dic7C4⋊C414yes—
157C4×Dic7C4⋊C414yes—
158C4×Dic7C22×C412yes—
159C4×Dic7C2×Q812yes—
160C2×C7⋊C8—22no—
161C2×C7⋊C8—22no—
162C2×C7⋊C8C22×C414yes—
163#1131—48no—
164C22×M16—72no—
165C2×#117—141no—
166C22×C42—71no—
167C2×#107—72no—
168C2×#107C2×D712yes—
169C2×#106—72no—
170C2×#106—72no—
171#423—78no—
172C2×C4⋊C8—72no—
173C2×C4⋊C8—71no—
174C2×C4⋊C8C2×D711yescharacteristic
175#384—71no—
176#384C2×D711yescharacteristic
177#383—141no—
178#382—71no—
179#382—71no—
180#382—141no—
181#381—141no—
182#380—71no—
183#380—71no—
184#380—72no—
185#380—141no—
186C4×C4⋊C4—72no—
187C4×C4⋊C4—71no—
188C4×C4⋊C4—141no—
189C4×C4⋊C4C2×D711yescharacteristic
190C4×C22⋊C4—141no—
191#376—141no—
192#340—72no—
193#340—72no—
194#340—72no—
195#340C2×D712yes—
196C23×C7C22×D411yescharacteristic
197C22×D7—21no—
198C22×D7—21no—
199C22×D7—21no—
200C22×D7—21no—
201C22×D7—21no—
202C22×D7—21no—
203C22×D7C2×C4⋊C412yes—
204C2×C4×C7—21no—
205C2×C4×C7—21no—
206C2×C4×C7—41no—
207C2×C4×C7—42no—
208C2×C4×C7—41no—
209C2×C4×C7—42no—
210C2×C4×C7—41no—
211C2×C4×C7—41no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—42no—
215C2×C4×C7C2×C4⋊C412yes—
216C2×C4×C7C2×C4⋊C412yes—
217C2×C4×C7AGL(1,ℤ/8)12yes—
218C2×C4×C7#13712yes—
219C2×C4×C7C23×C411yescharacteristic
220C2×C4×C7C22×Q811yescharacteristic
221C2×C4×C7H(5,𝔽2)11yescharacteristic
222C2×C4×C7H(5,𝔽2)11yescharacteristic
223C2×C4×C72−1+411yescharacteristic
224C2×C4×C72−1+411yescharacteristic
225C2×Dic7—21no—
226C2×Dic7—21no—
227C2×Dic7—21no—
228C2×Dic7—21no—
229C2×Dic7—21no—
230C2×Dic7—21no—
231C2×Dic7—41no—
232C2×Dic7—42no—
233C2×Dic7—41no—
234C2×Dic7—42no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—42no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7C2×C4⋊C412yes—
241C2×Dic7C2×C4⋊C412yes—
242C2×Dic7C2×C4⋊C412yes—
243C4×D7—22no—
244C4×D7—22no—
245C4×D7—22no—
246C4×D7—22no—
247C4×D7—28no—
248C4×D7—44no—
249C7×C8—24no—
250C7⋊C8—24no—
251C23×C4—71no—
252C23×C4—71no—
253C23×C4—71no—
254C2×M16—78no—
255C2×M16—282no—
256C2×M16—282no—
257C22×C8—72no—
258C22×C8C22×D712yes—
259#117—284no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—71no—
263C2×C4⋊C4—71no—
264C2×C4⋊C4—72no—
265C2×C4⋊C4—141no—
266C2×C4⋊C4—141no—
267C2×C4⋊C4C22×D711yescharacteristic
268C2×C4⋊C4C22×D712yes—
269C2×C4⋊C4C22×D711yescharacteristic
270C2×C22⋊C4—142no—
271C2×C22⋊C4—141no—
272C2×C22⋊C4—141no—
273C2×C42—21no—
274C2×C42—71no—
275C2×C42—71no—
276C2×C42—71no—
277C2×C42—74no—
278C2×C42—74no—
279C2×C42—142no—
280C2×C42—141no—
281C2×C42C22×D711yescharacteristic
282#107—42no—
283#107—282no—
284#106—282no—
285#106—282no—
286C4⋊C8—78no—
287C4⋊C8—74no—
288C4⋊C8C4×D714yes—
289#95—21no—
290#95—141no—
291#95—141no—
292#95—141no—
293#95—141no—
294#95—141no—
295#95—141no—
296#95—142no—
297#95—142no—
298#95—141no—
299C22×C7C22×C4⋊C411yescharacteristic
300C22×C7#53711yescharacteristic
301C22×C7#53711yescharacteristic
302C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
303C22×C7C2×#13711yescharacteristic
304C22×C7#57511yescharacteristic
305C22×C7#57511yescharacteristic
306C2×D7—22no—
307C2×D7—22no—
308C2×D7—22no—
309C2×D7—41no—
310C2×D7—41no—
311C2×D7—41no—
312C2×D7—41no—
313C4×C7—22no—
314C4×C7—81no—
315C4×C7—82no—
316C4×C7—81no—
317C4×C7#42814yes—
318C4×C7#51811yescharacteristic
319C4×C7#51911yescharacteristic
320C4×C7#54411yescharacteristic
321C4×C7#54411yescharacteristic
322Dic7—22no—
323Dic7—22no—
324Dic7—22no—
325Dic7—82no—
326Dic7—81no—
327Dic7—81no—
328C24—71no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—71no—
333C22×C4—71no—
334C22×C4—71no—
335C22×C4—74no—
336C22×C4—72no—
337C22×C4—72no—
338C22×C4—74no—
339C22×C4—142no—
340C22×C4—142no—
341C22×C4—141no—
342C22×C4—141no—
343C22×C4—141no—
344C22×C4—141no—
345C22×C4—141no—
346C22×C4—141no—
347C22×C4—142no—
348C22×C4—281no—
349C22×C4—281no—
350C22×C4D4×D711yescharacteristic
351C22×C4D4×D711yescharacteristic
352C22×C4C23×D711yescharacteristic
353M16—288no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—74no—
357C2×C8—142no—
358C2×C8—142no—
359C2×C8C2×C4×D714yes—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—141no—
369C4⋊C4—141no—
370C4⋊C4—142no—
371C4⋊C4—141no—
372C4⋊C4—142no—
373C4⋊C4—141no—
374C4⋊C4—282no—
375C4⋊C4—282no—
376C22⋊C4—282no—
377C22⋊C4—282no—
378C22⋊C4—284no—
379C42—42no—
380C42—74no—
381C42—74no—
382C42—72no—
383C42—72no—
384C42—284no—
385C42—282no—
386C42C2×C4×D712yes—
387C42Q8×D712yes—
388C2×C7#231611yescharacteristic
389C2×C7C2×#42811yescharacteristic
390C2×C7#291111yescharacteristic
391C2×C7#325111yescharacteristic
392C2×C7#325111yescharacteristic
393C2×C7#325211yescharacteristic
394C2×C7#325211yescharacteristic
395D7—42no—
396C23—72no—
397C23—141no—
398C23—141no—
399C23—141no—
400C23—141no—
401C23—141no—
402C23—141no—
403C23C2×D4×D711yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—72no—
416C2×C4—72no—
417C2×C4—72no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—141no—
424C2×C4—142no—
425C2×C4—142no—
426C2×C4—148no—
427C2×C4—142no—
428C2×C4—142no—
429C2×C4—284no—
430C2×C4—282no—
431C2×C4—281no—
432C2×C4—281no—
433C2×C4—281no—
434C2×C4—281no—
435C2×C4—282no—
436C2×C4—281no—
437C2×C4—282no—
438C2×C4—281no—
439C2×C4D7×C4⋊C412yes—
440C2×C4D7×C4⋊C412yes—
441C2×C4#647412yes—
442C2×C4#648112yes—
443C2×C4C22×C4×D711yescharacteristic
444C2×C4#654811yescharacteristic
445C2×C4C2×Q8×D711yescharacteristic
446C2×C4#655111yescharacteristic
447C2×C4#655311yescharacteristic
448C2×C4#655411yescharacteristic
449C8—24no—
450C8—144no—
451C7#2196311yescharacteristic, complemented, Sylow
452C22—142no—
453C22—142no—
454C22—142no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22—281no—
459C22C2×D7×C4⋊C411yescharacteristic
460C22#9029911yescharacteristic
461C22#9032011yescharacteristic
462C22C2×#647411yescharacteristic
463C22C2×#648111yescharacteristic
464C22#9046411yescharacteristic
465C22#9046811yescharacteristic
466C4—22no—
467C4—81no—
468C4—81no—
469C4—82no—
470C4—142no—
471C4—142no—
472C4—142no—
473C4—562no—
474C4—561no—
475C4—561no—
476C4#8965314yes—
477C4#9023611yescharacteristic
478C4#9026311yescharacteristic
479C4#9038011yescharacteristic
480C4#9042011yescharacteristic
481C2—282no—
482C2#1173364011yescharacteristic
483C2C2×#8965311yescharacteristic
484C2#1173665311yescharacteristic
485C2#1173778911yescharacteristic
486C2#1173794311yescharacteristic
487C2#1173842211yescharacteristic
488C2#1173857711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1947C2
socleC23×C7C22×D4
prime radicalC23×D7C2×D4
cosocleC22×C4×C7C24
hypercenter#1947D7
hypocenterC7#21963
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
4. C7×#1947C2semidirect
6. #11728370C2semidirect
7. #11728208C2semidirect
8. #11727322C2semidirect
9. #11727220C2semidirect
11. #11726551C2semidirect
37. #89609C4semidirect
101. #1947D7semidirect
451. C7#21963semidirect

Maximal subgroups

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

subgroupindex
C2×#896092
#117317632
#117311972
C7×#19472
#117283852
#117283702
#117282082
#117273222
#117272202
#117266242
#117265512
#219637