Finite Group Databasebeta

← #13275156

Subgroups — #13275156

Order 1792.

1522 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#389C211yesmaximal, characteristic
2C2×#89753C212yesmaximal
3C2×C4⋊C4×Dic7C212yesmaximal
4#11729318C212yesmaximal
5#11729318C212yesmaximal
6#11729288C211yesmaximal, characteristic
7C4×C7⋊#95C211yesmaximal, characteristic
8#11727161C212yesmaximal
9#11727155C212yesmaximal
10C22×C7×C4⋊C4C2212yes—
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C4⋊C4C2211yescharacteristic
13C22×C42×C7C2211yescharacteristic
14C22×C4⋊Dic7C2212yes—
15C22×C4×Dic7C2212yes—
16C22×C4×Dic7C2212yes—
17C22×C4×Dic7C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19C7×#389C418yescomplemented
20C2×C7×#95C2212yes—
21C2×C7⋊#95C2212yes—
22C2×C7⋊#95C2212yes—
23C2×C7⋊#95C2212yes—
24C2×C7⋊#95C2211yescharacteristic
25C2×C7⋊#95C2212yes—
26C2×C7⋊#95C2211yescharacteristic
27C2×C7⋊#95C2212yes—
28C2×C7⋊#95C2212yes—
29C2×C7⋊#95C2212yes—
30C2×C7⋊#95C2212yes—
31C2×C7⋊#95C2212yes—
32#89753C4116yescomplemented
33C4⋊C4×Dic7—28no—
34#13790—71nomaximal, Sylow
35C23×C4×C7C2312yes—
36C23×C4×C7C2311yescharacteristic
37C23×C4×C7C2312yes—
38C23×C4×C7C2311yescharacteristic
39C23×C4×C7C2311yescharacteristic
40C23×Dic7C2312yes—
41C23×Dic7C2312yes—
42C23×Dic7C2312yes—
43C23×Dic7C2312yes—
44C2×C7×C4⋊C4—24no—
45C2×C7×C4⋊C4—24no—
46C2×C7×C4⋊C4—41no—
47C2×C7×C4⋊C4—41no—
48C2×C7×C4⋊C4C2×C418yes—
49C2×C7×C4⋊C4C2×C418yes—
50C2×C7×C4⋊C4C2×C414yes—
51C2×C7×C4⋊C4C2×C414yes—
52C2×C42×C7—41no—
53C2×C42×C7—41no—
54C2×C42×C7C2×C414yes—
55C2×C4⋊Dic7—24no—
56C2×C4⋊Dic7—24no—
57C2×C4⋊Dic7C2×C418yes—
58C2×C4×Dic7—24no—
59C2×C4×Dic7—24no—
60C2×C4×Dic7—22no—
61C2×C4×Dic7—24no—
62C2×C4×Dic7—41no—
63C2×C4×Dic7—41no—
64C2×C4×Dic7—42no—
65C2×C4×Dic7—42no—
66C2×C4×Dic7—41no—
67C2×C4×Dic7—41no—
68C2×C4×Dic7C2×C418yes—
69C2×C4×Dic7D414yes—
70C7×#95C2×C418yes—
71C7⋊#95—24no—
72C7⋊#95—24no—
73C7⋊#95—24no—
74C7⋊#95—41no—
75C7⋊#95—42no—
76C7⋊#95—42no—
77C7⋊#95—41no—
78C7⋊#95C2×C418yes—
79C7⋊#95C2×C418yes—
80C7⋊#95C2×C418yes—
81C7⋊#95C2×C418yes—
82C2×#389D711yescharacteristic
83C2×#387—72no—
84C2×C4×C4⋊C4—72no—
85#1436—72no—
86#1436—72no—
87#1436—72no—
88#1434—71no—
89#1434—72no—
90C4×#95—71no—
91C24×C7C2411yescharacteristic
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7—22no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—21no—
101C22×C4×C7—22no—
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—22no—
106C22×C4×C7—22no—
107C22×C4×C7—21no—
108C22×C4×C7—21no—
109C22×C4×C7—22no—
110C22×C4×C7—22no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7C22×C412yes—
114C22×C4×C7C22×C414yes—
115C22×C4×C7C22×C414yes—
116C22×C4×C7C22×C414yes—
117C22×C4×C7C22×C414yes—
118C22×C4×C7C22×C414yes—
119C22×C4×C7C22×C412yes—
120C22×C4×C7C22×C412yes—
121C22×C4×C7C2×D412yes—
122C22×C4×C7#4112yes—
123C22×Dic7—22no—
124C22×Dic7—22no—
125C22×Dic7—22no—
126C22×Dic7—22no—
127C22×Dic7—22no—
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—22no—
131C22×Dic7—22no—
132C22×Dic7—22no—
133C22×Dic7—22no—
134C22×Dic7—22no—
135C22×Dic7—22no—
136C22×Dic7—22no—
137C22×Dic7—22no—
138C22×Dic7—22no—
139C22×Dic7—22no—
140C22×Dic7—22no—
141C22×Dic7—22no—
142C22×Dic7—22no—
143C22×Dic7—22no—
144C22×Dic7—22no—
145C22×Dic7C22×C414yes—
146C22×Dic7C22×C414yes—
147C22×Dic7C22×C414yes—
148C22×Dic7C22×C414yes—
149C22×Dic7C2×D414yes—
150C22×Dic7#4114yes—
151C7×C4⋊C4—216no—
152C7×C4⋊C4—44no—
153C7×C4⋊C4C42116yescomplemented
154C42×C7—44no—
155C4⋊Dic7—216no—
156C4×Dic7—216no—
157C4×Dic7—44no—
158C4×Dic7—48no—
159C4×Dic7—44no—
160C22×C4⋊C4—72no—
161C22×C4⋊C4C2×D712yes—
162C22×C4⋊C4C2×D711yescharacteristic
163C22×C4⋊C4C2×D711yescharacteristic
164C22×C42—72no—
165C22×C42—71no—
166C22×C42—71no—
167C22×C42—72no—
168C22×C42C2×D711yescharacteristic
169#389Dic718yescomplemented
170#387—716no—
171C4×C4⋊C4—148no—
172C2×#95—72no—
173C2×#95—72no—
174C2×#95—72no—
175C2×#95—72no—
176C2×#95—72no—
177C2×#95—72no—
178C2×#95—72no—
179C2×#95—71no—
180C2×#95—71no—
181C2×#95—72no—
182C2×#95—72no—
183C2×#95C2×D712yes—
184C23×C7C23×C412yes—
185C23×C7C23×C411yescharacteristic
186C23×C7C22×D411yescharacteristic
187C23×C7C2×#4111yescharacteristic
188C23×C7C2×#4112yes—
189C23×C7H(5,𝔽2)11yescharacteristic
190C23×C7H(5,𝔽2)11yescharacteristic
191C23×C7H(5,𝔽2)11yescharacteristic
192C23×C72−1+412yes—
193C23×C72−1+412yes—
194C23×C72−1+411yescharacteristic
195C2×C4×C7—24no—
196C2×C4×C7—24no—
197C2×C4×C7—24no—
198C2×C4×C7—24no—
199C2×C4×C7—24no—
200C2×C4×C7—24no—
201C2×C4×C7—22no—
202C2×C4×C7—22no—
203C2×C4×C7—22no—
204C2×C4×C7—24no—
205C2×C4×C7—24no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—22no—
209C2×C4×C7—22no—
210C2×C4×C7—22no—
211C2×C4×C7—24no—
212C2×C4×C7—22no—
213C2×C4×C7—22no—
214C2×C4×C7—24no—
215C2×C4×C7—22no—
216C2×C4×C7—24no—
217C2×C4×C7—42no—
218C2×C4×C7—42no—
219C2×C4×C7—42no—
220C2×C4×C7—42no—
221C2×C4×C7—42no—
222C2×C4×C7—42no—
223C2×C4×C7C2×C4218yes—
224C2×C4×C7C2×C4214yes—
225C2×C4×C7C4×D414yes—
226C2×Dic7—24no—
227C2×Dic7—24no—
228C2×Dic7—24no—
229C2×Dic7—24no—
230C2×Dic7—24no—
231C2×Dic7—24no—
232C2×Dic7—24no—
233C2×Dic7—24no—
234C2×Dic7—24no—
235C2×Dic7—24no—
236C2×Dic7—24no—
237C2×Dic7—24no—
238C2×Dic7—24no—
239C2×Dic7—24no—
240C2×Dic7—24no—
241C2×Dic7—42no—
242C2×Dic7—42no—
243C2×Dic7—42no—
244C2×Dic7—42no—
245C2×Dic7—42no—
246C2×Dic7—42no—
247C2×Dic7—42no—
248C2×Dic7—42no—
249C2×Dic7—42no—
250C2×Dic7—42no—
251C2×Dic7—42no—
252C2×Dic7—42no—
253C2×Dic7C4×D418yes—
254C23×C4—72no—
255C23×C4—72no—
256C23×C4—72no—
257C23×C4—72no—
258C23×C4C22×D712yes—
259C23×C4C22×D711yescharacteristic
260C23×C4C22×D712yes—
261C23×C4C22×D711yescharacteristic
262C23×C4C22×D711yescharacteristic
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—78no—
268C2×C4⋊C4—144no—
269C2×C4⋊C4—144no—
270C2×C4⋊C4C4×D718yes—
271C2×C4⋊C4C2×Dic718yes—
272C2×C4⋊C4C2×Dic714yes—
273C2×C4⋊C4C2×Dic714yes—
274C2×C42—41no—
275C2×C42—41no—
276C2×C42—78no—
277C2×C42—74no—
278C2×C42—144no—
279C2×C42—144no—
280C2×C42—144no—
281C2×C42—142no—
282C2×C42—281no—
283C2×C42—281no—
284C2×C42—281no—
285C2×C42—281no—
286C2×C42—282no—
287C2×C42—282no—
288C2×C42C2×Dic714yes—
289#95—78no—
290#95—78no—
291#95—78no—
292#95—78no—
293#95—144no—
294#95—144no—
295#95—144no—
296#95—281no—
297#95—281no—
298#95—282no—
299#95—282no—
300#95C2×Dic718yes—
301C22×C7C22×C4211yescharacteristic
302C22×C7C2×C4×D411yescharacteristic
303C22×C7C2×C4×D412yes—
304C22×C7C4×#4112yes—
305C22×C7C4×#4111yescharacteristic
306C22×C7#51811yescharacteristic
307C22×C7#51911yescharacteristic
308C22×C7#51911yescharacteristic
309C22×C7#51912yes—
310C22×C7#52012yes—
311C22×C7#52012yes—
312C22×C7#52011yescharacteristic
313C22×C7#52012yes—
314C22×C7#53411yescharacteristic
315C22×C7#53612yes—
316C22×C7#53711yescharacteristic
317C22×C7#53811yescharacteristic
318C22×C7#53912yes—
319C22×C7#53911yescharacteristic
320C22×C7#53912yes—
321C22×C7#54112yes—
322C22×C7#54112yes—
323C22×C7#54212yes—
324C4×C7—24no—
325C4×C7—28no—
326C4×C7—24no—
327C4×C7—44no—
328C4×C7—42no—
329Dic7—28no—
330Dic7—44no—
331Dic7—44no—
332Dic7—44no—
333C24C23×D711yescharacteristic
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—74no—
356C22×C4—74no—
357C22×C4—74no—
358C22×C4—74no—
359C22×C4—74no—
360C22×C4—74no—
361C22×C4—142no—
362C22×C4—142no—
363C22×C4—142no—
364C22×C4—142no—
365C22×C4—142no—
366C22×C4—142no—
367C22×C4—142no—
368C22×C4—142no—
369C22×C4—142no—
370C22×C4—142no—
371C22×C4—142no—
372C22×C4—142no—
373C22×C4—142no—
374C22×C4—142no—
375C22×C4—142no—
376C22×C4—142no—
377C22×C4—142no—
378C22×C4—142no—
379C22×C4—142no—
380C22×C4—142no—
381C22×C4—142no—
382C22×C4—142no—
383C22×C4C2×C4×D714yes—
384C22×C4C2×C4×D714yes—
385C22×C4C2×C4×D714yes—
386C22×C4D4×D712yes—
387C22×C4#115912yes—
388C22×C4C22×Dic712yes—
389C22×C4C22×Dic714yes—
390C22×C4C22×Dic714yes—
391C22×C4C22×Dic712yes—
392C22×C4C22×Dic712yes—
393C4⋊C4—216no—
394C4⋊C4—44no—
395C4⋊C4—1416no—
396C4⋊C4C4×Dic7116yescomplemented
397C42—44no—
398C42—1416no—
399C42—284no—
400C42—284no—
401C42—288no—
402C2×C7C42×D411yescharacteristic
403C2×C7#227411yescharacteristic
404C2×C7#227611yescharacteristic
405C2×C7#232212yes—
406C2×C7#232312yes—
407C2×C7#232511yescharacteristic
408C2×C7#232711yescharacteristic
409C2×C7#233211yescharacteristic
410C2×C7#233412yes—
411C2×C7#233611yescharacteristic
412C2×C7#233612yes—
413C23C22×C4×D712yes—
414C23C2×D4×D711yescharacteristic
415C23C2×#115911yescharacteristic
416C23#654811yescharacteristic
417C23#654811yescharacteristic
418C23#655112yes—
419C23#655112yes—
420C23D7×#4112yes—
421C23#655311yescharacteristic
422C23#655411yescharacteristic
423C23C23×Dic711yescharacteristic
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—78no—
453C2×C4—144no—
454C2×C4—144no—
455C2×C4—144no—
456C2×C4—144no—
457C2×C4—144no—
458C2×C4—144no—
459C2×C4—144no—
460C2×C4—144no—
461C2×C4—144no—
462C2×C4—144no—
463C2×C4—144no—
464C2×C4—144no—
465C2×C4—144no—
466C2×C4—144no—
467C2×C4—144no—
468C2×C4—282no—
469C2×C4—282no—
470C2×C4—282no—
471C2×C4—282no—
472C2×C4—282no—
473C2×C4—282no—
474C2×C4—282no—
475C2×C4—282no—
476C2×C4—282no—
477C2×C4—282no—
478C2×C4—282no—
479C2×C4—282no—
480C2×C4C2×C4×Dic718yes—
481C2×C4C2×C4×Dic714yes—
482C2×C4D4×Dic714yes—
483C7#1379011yescharacteristic, complemented, Sylow
484C22#9019512yes—
485C22#9019512yes—
486C22C4×#115912yes—
487C22C4×D4×D712yes—
488C22#9026312yes—
489C22#9026512yes—
490C22#9028811yescharacteristic
491C22#9029311yescharacteristic
492C22#9029611yescharacteristic
493C22#9030811yescharacteristic
494C22#9032912yes—
495C22#9033112yes—
496C22#9038312yes—
497C22#9038712yes—
498C22#9041512yes—
499C22#9042212yes—
500C22C22×C4×Dic711yescharacteristic
501C22C2×D4×Dic711yescharacteristic
502C22#9048911yescharacteristic
503C22#9050311yescharacteristic
504C22#9050311yescharacteristic
505C22#41×Dic711yescharacteristic
506C22#9051811yescharacteristic
507C4—28no—
508C4—24no—
509C4—24no—
510C4—42no—
511C4—44no—
512C4—148no—
513C4—284no—
514C4—284no—
515C4—284no—
516C2#1173341612yes—
517C2#1173342012yes—
518C2#1173342312yes—
519C2#1173364112yes—
520C2#1173958911yescharacteristic
521C2C4×D4×Dic711yescharacteristic
522C2#1173989511yescharacteristic
523C2#1174007911yescharacteristic
524C2#1174019111yescharacteristic
525C2#1174046911yescharacteristic
526C2#1174067211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D7
commutatorC22×C7C22×C42
FittingC2×C7×#389C2
socle, prime radical, cosocleC24×C7C24
hypercenterC2×#389D7
hypocenterC7#13790
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C2×#389

Complements

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

NH ≅ G/Ntype
19. C7×#389C4semidirect
32. #89753C4semidirect
153. C7×C4⋊C4C42semidirect
169. #389Dic7semidirect
396. C4⋊C4C4×Dic7semidirect
483. C7#13790semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3892
C2×#897532
C2×C4⋊C4×Dic72
#117293182
#117292882
C4×C7⋊#952
#117271612
#117271552
#137907