Finite Group Databasebeta

← #13382895

Subgroups — #13382895

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#428C211yesmaximal, characteristic
2C2×#89891C211yesmaximal, characteristic
3#11739677C211yesmaximal, characteristic
4#11739602C211yesmaximal, characteristic
5#11729739C211yesmaximal, characteristic
6#11729334C211yesmaximal, characteristic
7C2×#89324C211yesmaximal, characteristic
8#11728376C212yesmaximal, complemented
9#11728375C212yesmaximal, complemented
10#11728322C212yesmaximal, complemented
11#11728322C212yesmaximal, complemented
12C22×C7×M16C2211yescharacteristic
13C2×C7×#117C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C2×#6489C2211yescharacteristic
16C2×#6489C2211yescharacteristic
17C22×C4⋊Dic7C2211yescharacteristic
18C23×C7⋊C8C2211yescharacteristic
19C7×#428—24no—
20C2×C7×#107C2212yes—
21C2×C7×#106C2212yes—
22#89982—22no—
23#89891—24no—
24C2×Dic7⋊C8C2212yes—
25C2×Dic7⋊C8C2212yes—
26#89749—21no—
27#89748—21no—
28#89745C2212yes—
29#89745C2212yes—
30#89745C2212yes—
31#89745C2212yes—
32#89714—22no—
33#89700—21no—
34#89700—21no—
35#89346C2212yes—
36#89346C2212yes—
37#89324C2212yes—
38#89324C2212yes—
39#89324C2211yescharacteristic
40#89324C2211yescharacteristic
41#89324C2211yescharacteristic
42#89324C2211yescharacteristic
43#23671—71nomaximal, Sylow
44C23×C4×C7—21no—
45C23×C4×C7C2311yescharacteristic
46C23×Dic7—21no—
47C2×C7×M16—22no—
48C2×C7×M16—22no—
49C2×C7×M16D414yes—
50C22×C7×C8C2312yes—
51C7×#117—22no—
52C7×#117—22no—
53C2×C7×C4⋊C4—22no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4C2312yes—
57C2×C7×C4⋊C4C2311yescharacteristic
58C2×C7×C4⋊C4C2311yescharacteristic
59C2×C7×C22⋊C4—21no—
60C2×C42×C7—21no—
61C2×C22⋊Dic7—21no—
62C2×C22⋊Dic7D412yes—
63#6489—22no—
64#6489—22no—
65#6489—22no—
66#6489Q814yes—
67C2×C4⋊Dic7—21no—
68C2×C4⋊Dic7—22no—
69C2×C4⋊Dic7D411yescharacteristic
70C2×C4⋊Dic7D411yescharacteristic
71C2×C4⋊Dic7Q814yes—
72C2×C4⋊Dic7C2312yes—
73C2×C4⋊Dic7C2311yescharacteristic
74C2×C4⋊Dic7C2311yescharacteristic
75C2×Dic7⋊C4—22no—
76C2×Dic7⋊C4—22no—
77C2×Dic7⋊C4—22no—
78C2×Dic7⋊C4—22no—
79C2×C4×Dic7—21no—
80C2×C4×Dic7C2312yes—
81C22×C7⋊C8—22no—
82C22×C7⋊C8—22no—
83C22×C7⋊C8—22no—
84C22×C7⋊C8C2311yescharacteristic
85C22×C7⋊C8C2311yescharacteristic
86C7×#107—24no—
87C7×#106—24no—
88C7⋊#95—22no—
89C7⋊#95—22no—
90C7⋊#95—22no—
91C7⋊#95—22no—
92Dic7⋊C8—24no—
93Dic7⋊C8—24no—
94C2×#428D711yescharacteristic
95C2×#424—71no—
96#2317—71no—
97#2316—71no—
98#2088—72no—
99#2088—72no—
100#2087—72no—
101#2086—72no—
102#1736—71no—
103#1735—71no—
104C2×#340—71no—
105C24×C7C2×D411yescharacteristic
106C22×C4×C7—21no—
107C22×C4×C7—21no—
108C22×C4×C7—22no—
109C22×C4×C7—42no—
110C22×C4×C7—81no—
111C22×C4×C7—81no—
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7C2×D412yes—
114C22×C4×C7C2×Q812yes—
115C22×C4×C7C2×Q812yes—
116C22×C4×C7#4112yes—
117C22×C4×C7#4112yes—
118C22×C4×C7#4112yes—
119C22×C4×C7C2411yescharacteristic
120C22×Dic7—21no—
121C22×Dic7—21no—
122C22×Dic7—22no—
123C22×Dic7—24no—
124C22×Dic7—81no—
125C22×Dic7—81no—
126C22×Dic7C2×D412yes—
127C22×Dic7C2×D412yes—
128C7×M16—28no—
129C2×C7×C8—22no—
130C2×C7×C8—22no—
131C2×C7×C8C2×D414yes—
132C7×C4⋊C4—21no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—21no—
135C7×C4⋊C4—22no—
136C7×C4⋊C4—21no—
137C7×C4⋊C4—21no—
138C7×C4⋊C4—42no—
139C7×C22⋊C4—42no—
140C42×C7—42no—
141C22⋊Dic7—24no—
142C22⋊Dic7—24no—
143C4⋊Dic7—22no—
144C4⋊Dic7—22no—
145C4⋊Dic7—24no—
146C4⋊Dic7—22no—
147C4⋊Dic7—21no—
148C4⋊Dic7—21no—
149C4⋊Dic7C2×Q814yes—
150C4⋊Dic7C2×Q812yes—
151C4⋊Dic7C2×Q812yes—
152Dic7⋊C4—82no—
153Dic7⋊C4—82no—
154Dic7⋊C4—82no—
155Dic7⋊C4—82no—
156C4×Dic7—24no—
157C4×Dic7—22no—
158C4×Dic7—22no—
159C2×C7⋊C8—21no—
160C2×C7⋊C8—21no—
161C2×C7⋊C8—21no—
162C2×C7⋊C8—21no—
163C2×C7⋊C8—21no—
164C2×C7⋊C8—21no—
165C2×C7⋊C8—44no—
166C22×M16C2×D711yescharacteristic
167C23×C8—71no—
168C2×#117—71no—
169C2×#117—71no—
170C2×#117C2×D711yescharacteristic
171C22×C4⋊C4—71no—
172C22×C4⋊C4C2×D711yescharacteristic
173#428—24no—
174C2×#107C2×D712yes—
175C2×#106C2×D712yes—
176#424—144no—
177C2×C4⋊C8—72no—
178C2×C4⋊C8—72no—
179#389—72no—
180#389—72no—
181#389—72no—
182#389—72no—
183#389—141no—
184#389—141no—
185#385—142no—
186#384—141no—
187#382—141no—
188#380—142no—
189#340—72no—
190#340—72no—
191#340—71no—
192#340—72no—
193#340—71no—
194#340—72no—
195#340—71no—
196#340—71no—
197C23×C7—21no—
198C23×C7—21no—
199C23×C7—21no—
200C23×C7—21no—
201C23×C7#12112yes—
202C23×C7C22⋊Q812yes—
203C23×C7C22⋊Q812yes—
204C23×C7C22×D411yescharacteristic
205C2×C4×C7—41no—
206C2×C4×C7—41no—
207C2×C4×C7—42no—
208C2×C4×C7—41no—
209C2×C4×C7—42no—
210C2×C4×C7—41no—
211C2×C4×C7—42no—
212C2×C4×C7—41no—
213C2×C4×C7—41no—
214C2×C4×C7—84no—
215C2×C4×C7#12112yes—
216C2×C4×C7C22⋊Q812yes—
217C2×C4×C7C22⋊Q812yes—
218C2×C4×C7#12314yes—
219C2×C4×C7#12514yes—
220C2×C4×C7#12514yes—
221C2×C4×C7C4⋊Q814yes—
222C2×C4×C7C22×D411yescharacteristic
223C2×C4×C7C22×Q811yescharacteristic
224C2×C4×C7C22×Q811yescharacteristic
225C2×C4×C7C2×#4111yescharacteristic
226C2×C4×C7C2×#4111yescharacteristic
227C2×C4×C7C2×#4111yescharacteristic
228C2×Dic7—22no—
229C2×Dic7—22no—
230C2×Dic7—22no—
231C2×Dic7—22no—
232C2×Dic7—24no—
233C2×Dic7—22no—
234C2×Dic7—22no—
235C2×Dic7—22no—
236C2×Dic7—22no—
237C2×Dic7—41no—
238C2×Dic7—42no—
239C2×Dic7—41no—
240C2×Dic7—42no—
241C2×Dic7—41no—
242C2×Dic7—41no—
243C2×Dic7—84no—
244C7×C8—24no—
245C7⋊C8—41no—
246C7⋊C8—41no—
247C23×C4—21no—
248C23×C4—141no—
249C23×C4C22×D711yescharacteristic
250C2×M16—22no—
251C2×M16—22no—
252C2×M16C22⋊D714yes—
253C22×C8—71no—
254C22×C8—71no—
255C22×C8—142no—
256C22×C8—142no—
257C22×C8—142no—
258C22×C8C22×D712yes—
259#117—22no—
260#117—22no—
261#117—74no—
262#117—142no—
263#117—142no—
264#117—142no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—74no—
269C2×C4⋊C4—72no—
270C2×C4⋊C4—71no—
271C2×C4⋊C4—71no—
272C2×C4⋊C4—71no—
273C2×C4⋊C4—71no—
274C2×C4⋊C4—142no—
275C2×C4⋊C4—141no—
276C2×C4⋊C4—142no—
277C2×C4⋊C4—142no—
278C2×C4⋊C4—142no—
279C2×C4⋊C4—142no—
280C2×C4⋊C4C22×D712yes—
281C2×C4⋊C4C22×D711yescharacteristic
282C2×C4⋊C4C22×D711yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—72no—
285C2×C22⋊C4—141no—
286C2×C42—21no—
287C2×C42—72no—
288C2×C42—141no—
289#107—24no—
290#106—24no—
291C4⋊C8—144no—
292C4⋊C8—144no—
293#95—142no—
294#95—142no—
295#95—142no—
296#95—142no—
297C22×C7—22no—
298C22×C7—22no—
299C22×C7—22no—
300C22×C7—22no—
301C22×C7—22no—
302C22×C7—22no—
303C22×C7#39814yes—
304C22×C7C2×#12111yescharacteristic
305C22×C7C2×C22⋊Q811yescharacteristic
306C22×C7C2×C22⋊Q811yescharacteristic
307C22×C7#57611yescharacteristic
308C22×C7#57711yescharacteristic
309C22×C7#57711yescharacteristic
310C22×C7#57811yescharacteristic
311C4×C7—81no—
312C4×C7—82no—
313C4×C7—81no—
314C4×C7#39814yes—
315C4×C7C2×#12311yescharacteristic
316C4×C7C2×#12511yescharacteristic
317C4×C7C2×#12511yescharacteristic
318C4×C7C2×C4⋊Q811yescharacteristic
319Dic7—44no—
320Dic7—42no—
321Dic7—42no—
322Dic7—42no—
323Dic7—42no—
324C24D4×D711yescharacteristic
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—72no—
330C22×C4—72no—
331C22×C4—81no—
332C22×C4—81no—
333C22×C4—141no—
334C22×C4—141no—
335C22×C4—142no—
336C22×C4—144no—
337C22×C4—561no—
338C22×C4—561no—
339C22×C4#115712yes—
340C22×C4#115712yes—
341C22×C4D4×D711yescharacteristic
342C22×C4#115912yes—
343C22×C4Q8×D712yes—
344C22×C4Q8×D712yes—
345C22×C4C2×C22⋊D712yes—
346C22×C4C23×D711yescharacteristic
347M16—28no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—141no—
351C2×C8—141no—
352C2×C8—141no—
353C2×C8—141no—
354C2×C8—141no—
355C2×C8—141no—
356C2×C8—284no—
357C2×C8C2×C22⋊D714yes—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—42no—
365C4⋊C4—74no—
366C4⋊C4—72no—
367C4⋊C4—72no—
368C4⋊C4—144no—
369C4⋊C4—142no—
370C4⋊C4—142no—
371C4⋊C4—142no—
372C4⋊C4—141no—
373C4⋊C4—141no—
374C4⋊C4—562no—
375C4⋊C4—562no—
376C4⋊C4—562no—
377C4⋊C4—562no—
378C22⋊C4—42no—
379C22⋊C4—144no—
380C22⋊C4—144no—
381C42—42no—
382C42—144no—
383C42—142no—
384C42—142no—
385C2×C7—24no—
386C2×C7C2×#39811yescharacteristic
387C2×C7#306211yescharacteristic
388C2×C7#306311yescharacteristic
389C2×C7#308111yescharacteristic
390C2×C7#308211yescharacteristic
391C2×C7#308311yescharacteristic
392C2×C7#308411yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23#645912yes—
398C23#645912yes—
399C23#650112yes—
400C23C2×D4×D711yescharacteristic
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—84no—
411C2×C4—142no—
412C2×C4—142no—
413C2×C4—144no—
414C2×C4—142no—
415C2×C4—142no—
416C2×C4—142no—
417C2×C4—142no—
418C2×C4—142no—
419C2×C4—142no—
420C2×C4—281no—
421C2×C4—282no—
422C2×C4—281no—
423C2×C4—282no—
424C2×C4—281no—
425C2×C4—281no—
426C2×C4—564no—
427C2×C4#645214yes—
428C2×C4#645214yes—
429C2×C4#645912yes—
430C2×C4#645912yes—
431C2×C4#649214yes—
432C2×C4#650112yes—
433C2×C4#650714yes—
434C2×C4C2×#115711yescharacteristic
435C2×C4C2×#115711yescharacteristic
436C2×C4C2×#115911yescharacteristic
437C2×C4C2×Q8×D711yescharacteristic
438C2×C4C2×Q8×D711yescharacteristic
439C2×C4C22×C22⋊D711yescharacteristic
440C8—24no—
441C8—281no—
442C8—281no—
443C7#2367111yescharacteristic, complemented, Sylow
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—22no—
450C22#8975214yes—
451C22C2×#645911yescharacteristic
452C22C2×#645911yescharacteristic
453C22#9046511yescharacteristic
454C22#9046611yescharacteristic
455C22#9046911yescharacteristic
456C22#9047011yescharacteristic
457C22C2×#650111yescharacteristic
458C4—82no—
459C4—81no—
460C4—81no—
461C4—284no—
462C4—282no—
463C4—282no—
464C4—282no—
465C4—282no—
466C4#8975214yes—
467C4C2×#645211yescharacteristic
468C4C2×#645211yescharacteristic
469C4C2×#649211yescharacteristic
470C4C2×#650711yescharacteristic
471C2—24no—
472C2#1173668711yescharacteristic
473C2#1173668811yescharacteristic
474C2#1173668911yescharacteristic
475C2#1173669111yescharacteristic
476C2C2×#8975211yescharacteristic
477C2#1174139611yescharacteristic
478C2#1174143111yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC2×C7×#428C2
socleC23×C7C22×D4
prime radicalC24×C7C2×D4
hypercenterC2×#428D7
hypocenterC7#23671
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
8. #11728376C2semidirect
9. #11728375C2semidirect
10. #11728322C2semidirect
11. #11728322C2semidirect
443. C7#23671semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4282
C2×#898912
#117396772
#117396022
#117297392
#117293342
C2×#893242
#117283762
#117283752
#117283222
#236717