Finite Group Databasebeta

← #13253855

Subgroups — #13253855

Order 1792.

600 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×#89419C211yesmaximal, characteristic
2#11727662C211yesmaximal, characteristic
3#11727660C211yesmaximal, characteristic
4C7×#1280C211yesmaximal, characteristic
5#11725480C211yesmaximal, characteristic
6#11725480C211yesmaximal, characteristic
7#11725419C211yesmaximal, characteristic
8C22×C4⋊Dic7C411yescharacteristic
9C22×Dic7⋊C4C411yescharacteristic
10C22×Dic7⋊C4C411yescharacteristic
11C22×Dic7⋊C4C411yescharacteristic
12C22×C4×Dic7C2211yescharacteristic
13C2×C7×C22⋊C8C2211yescharacteristic
14C2×C7×C22⋊C8C2211yescharacteristic
15C2×C7×#95C2211yescharacteristic
16C2×C7⋊#95C2211yescharacteristic
17C2×C22⋊(C7⋊C8)C2211yescharacteristic
18C2×C22⋊(C7⋊C8)C2211yescharacteristic
19C2×#6389—21no—
20C2×Dic7⋊C8—21no—
21#89419—21no—
22#89419—21no—
23#89419—21no—
24#89419—21no—
25#89345—21no—
26#89345—21no—
27#9883—71nomaximal, Sylow
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C23×Dic7C2×C411yescharacteristic
32C23×Dic7C2×C411yescharacteristic
33C23×Dic7C2×C411yescharacteristic
34C23×Dic7C2×C411yescharacteristic
35C22×C7×C8—21no—
36C22×C7×C8—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7—21no—
39C2×C4⋊Dic7—42no—
40C2×Dic7⋊C4—22no—
41C2×Dic7⋊C4—21no—
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—22no—
44C2×Dic7⋊C4—21no—
45C2×Dic7⋊C4—21no—
46C2×Dic7⋊C4—21no—
47C2×Dic7⋊C4—21no—
48C2×Dic7⋊C4—42no—
49C2×Dic7⋊C4—42no—
50C2×C4×Dic7—42no—
51C2×C4×Dic7D411yescharacteristic
52C2×C4×Dic7D411yescharacteristic
53C2×C4×Dic7D411yescharacteristic
54C2×C4×Dic7D411yescharacteristic
55C22×C7⋊C8—21no—
56C22×C7⋊C8—21no—
57C7×C22⋊C8—21no—
58C7×C22⋊C8—21no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8Q812yes—
61C7×#95—21no—
62C7×#95—21no—
63C7⋊#95—21no—
64C7⋊#95—21no—
65C22⋊(C7⋊C8)—21no—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)Q812yes—
69#6389—42no—
70Dic7⋊C8—42no—
71C2×#389—71no—
72#1797—71no—
73#1749—71no—
74#1280—71no—
75#1280—71no—
76#1280—71no—
77#1280D711yescharacteristic
78C24×C7C22×C411yescharacteristic
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—41no—
85C22×C4×C7—41no—
86C22×C4×C7—41no—
87C22×C4×C7—41no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7M1612yes—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×Q811yescharacteristic
94C22×C4×C7C2×Q811yescharacteristic
95C22×C4×C7#4111yescharacteristic
96C22×C4×C7#4111yescharacteristic
97C22×Dic7—21no—
98C22×Dic7—21no—
99C22×Dic7—22no—
100C22×Dic7—22no—
101C22×Dic7—21no—
102C22×Dic7—21no—
103C22×Dic7—21no—
104C22×Dic7—21no—
105C22×Dic7—22no—
106C22×Dic7—21no—
107C22×Dic7—21no—
108C22×Dic7—21no—
109C22×Dic7—41no—
110C22×Dic7—41no—
111C22×Dic7—41no—
112C22×Dic7—41no—
113C22×Dic7—41no—
114C22×Dic7C22⋊C411yescharacteristic
115C22×Dic7C22⋊C411yescharacteristic
116C22×Dic7C22⋊C411yescharacteristic
117C22×Dic7C22⋊C411yescharacteristic
118C22×Dic7M1612yes—
119C2×C7×C8—22no—
120C2×C7×C8—41no—
121C2×C7×C8—41no—
122C2×C7×C8—41no—
123C2×C7×C8—41no—
124C2×C7×C8—41no—
125C4⋊Dic7—42no—
126C4⋊Dic7—42no—
127Dic7⋊C4—24no—
128Dic7⋊C4—24no—
129Dic7⋊C4—42no—
130Dic7⋊C4—42no—
131Dic7⋊C4—42no—
132Dic7⋊C4—42no—
133C4×Dic7—22no—
134C4×Dic7—22no—
135C4×Dic7—41no—
136C4×Dic7—41no—
137C2×C7⋊C8—22no—
138C2×C7⋊C8—41no—
139C2×C7⋊C8—41no—
140C2×C7⋊C8—41no—
141C2×C7⋊C8—41no—
142C2×C7⋊C8—41no—
143C22×C4⋊C4—71no—
144C22×C4⋊C4—71no—
145C22×C4⋊C4—71no—
146C22×C4⋊C4—71no—
147C22×C42—71no—
148C2×C4⋊C8—141no—
149C2×C22⋊C8—71no—
150C2×C22⋊C8—71no—
151C2×C22⋊C8C2×D711yescharacteristic
152C2×C22⋊C8C2×D711yescharacteristic
153C2×#97—141no—
154#389—141no—
155#389—141no—
156#389—141no—
157#389—141no—
158C2×#95—71no—
159C2×#95C2×D711yescharacteristic
160#336—141no—
161#336—141no—
162C23×C7—21no—
163C23×C7—21no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7C2×C22⋊C411yescharacteristic
167C23×C7C4×Q811yescharacteristic
168C23×C7C4×Q811yescharacteristic
169C23×C7C2×M1611yescharacteristic
170C23×C7C2×M1611yescharacteristic
171C23×C7#13111yescharacteristic
172C23×C7#13111yescharacteristic
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—22no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—81no—
190C2×C4×C7—81no—
191C2×C4×C7C22⋊Q811yescharacteristic
192C2×C4×C7C22⋊Q811yescharacteristic
193C2×C4×C7#12411yescharacteristic
194C2×C4×C7C4⋊Q811yescharacteristic
195C2×Dic7—24no—
196C2×Dic7—24no—
197C2×Dic7—21no—
198C2×Dic7—21no—
199C2×Dic7—21no—
200C2×Dic7—21no—
201C2×Dic7—21no—
202C2×Dic7—21no—
203C2×Dic7—22no—
204C2×Dic7—21no—
205C2×Dic7—21no—
206C2×Dic7—21no—
207C2×Dic7—42no—
208C2×Dic7—42no—
209C2×Dic7—42no—
210C2×Dic7—41no—
211C2×Dic7—41no—
212C2×Dic7—42no—
213C2×Dic7—41no—
214C2×Dic7—41no—
215C2×Dic7—41no—
216C2×Dic7—81no—
217C2×Dic7—81no—
218C2×Dic7#9912yes—
219C2×Dic7#10012yes—
220C2×Dic7#10112yes—
221C7×C8—42no—
222C7×C8—81no—
223C7⋊C8—42no—
224C7⋊C8—81no—
225C23×C4—71no—
226C23×C4—71no—
227C23×C4—71no—
228C23×C4—71no—
229C23×C4C4×D711yescharacteristic
230C23×C4C4×D711yescharacteristic
231C23×C4C22×D711yescharacteristic
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—141no—
235C22×C8—141no—
236C2×C4⋊C4—142no—
237C2×C4⋊C4—141no—
238C2×C4⋊C4—141no—
239C2×C4⋊C4—142no—
240C2×C4⋊C4—141no—
241C2×C4⋊C4—141no—
242C2×C4⋊C4—141no—
243C2×C4⋊C4—141no—
244C2×C4⋊C4—141no—
245C2×C4⋊C4—141no—
246C2×C4⋊C4—282no—
247C2×C4⋊C4—282no—
248C2×C4⋊C4—282no—
249C2×C42—71no—
250C2×C42—71no—
251C2×C42—71no—
252C2×C42—71no—
253C2×C42—282no—
254C4⋊C8—282no—
255C22⋊C8—21no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—72no—
259C22⋊C8—141no—
260C22⋊C8—141no—
261C22⋊C8—141no—
262C22⋊C8Dic1412yes—
263#97—282no—
264#95—21no—
265#95—21no—
266#95—141no—
267#95—141no—
268C22×C7—21no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—21no—
277C22×C7—21no—
278C22×C7—21no—
279C22×C7—21no—
280C22×C7#39111yescharacteristic
281C22×C7#40711yescharacteristic
282C22×C7#40811yescharacteristic
283C22×C7C2×#9911yescharacteristic
284C22×C7#41011yescharacteristic
285C22×C7C2×#10011yescharacteristic
286C22×C7C2×#10111yescharacteristic
287C22×C7#44611yescharacteristic
288C22×C7#44611yescharacteristic
289C22×C7#44611yescharacteristic
290C22×C7#44611yescharacteristic
291C4×C7—21no—
292C4×C7—21no—
293C4×C7—21no—
294C4×C7—21no—
295C4×C7—42no—
296C4×C7—81no—
297Dic7—22no—
298Dic7—22no—
299Dic7—22no—
300Dic7—42no—
301Dic7—41no—
302Dic7—81no—
303C24C2×C4×D711yescharacteristic
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—71no—
316C22×C4—71no—
317C22×C4—71no—
318C22×C4—71no—
319C22×C4—72no—
320C22×C4—142no—
321C22×C4—142no—
322C22×C4—142no—
323C22×C4—141no—
324C22×C4—141no—
325C22×C4—141no—
326C22×C4—141no—
327C22×C4—141no—
328C22×C4—141no—
329C22×C4—141no—
330C22×C4—141no—
331C22×C4—141no—
332C22×C4—281no—
333C22×C4—281no—
334C22×C4—281no—
335C22×C4—281no—
336C22×C4—281no—
337C22×C4#113112yes—
338C22×C4C2×Dic1411yescharacteristic
339C22×C4#115711yescharacteristic
340C22×C4D4×D711yescharacteristic
341C22×C4D4×D711yescharacteristic
342C22×C4#115911yescharacteristic
343C22×C4Q8×D711yescharacteristic
344C2×C8—22no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—142no—
351C2×C8—281no—
352C2×C8—281no—
353C2×C8—281no—
354C2×C8—281no—
355C2×C8—281no—
356C4⋊C4—144no—
357C4⋊C4—144no—
358C4⋊C4—282no—
359C4⋊C4—282no—
360C4⋊C4—282no—
361C4⋊C4—282no—
362C4⋊C4—282no—
363C4⋊C4—282no—
364C42—142no—
365C42—142no—
366C42—281no—
367C42—281no—
368C2×C7—21no—
369C2×C7—21no—
370C2×C7—21no—
371C2×C7—21no—
372C2×C7#146311yescharacteristic
373C2×C7#146611yescharacteristic
374C2×C7#147011yescharacteristic
375C2×C7#147111yescharacteristic
376C2×C7#198511yescharacteristic
377C2×C7#199611yescharacteristic
378C2×C7#199711yescharacteristic
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23C4×Dic1411yescharacteristic
384C23D7×C22⋊C411yescharacteristic
385C23#645011yescharacteristic
386C23C2×#113111yescharacteristic
387C23#646411yescharacteristic
388C23D7×M1611yescharacteristic
389C23#647011yescharacteristic
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—72no—
407C2×C4—72no—
408C2×C4—72no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—144no—
412C2×C4—144no—
413C2×C4—141no—
414C2×C4—141no—
415C2×C4—141no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—142no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—282no—
424C2×C4—282no—
425C2×C4—282no—
426C2×C4—281no—
427C2×C4—281no—
428C2×C4—282no—
429C2×C4—281no—
430C2×C4—281no—
431C2×C4—281no—
432C2×C4—561no—
433C2×C4—561no—
434C2×C4#644111yescharacteristic
435C2×C4#644811yescharacteristic
436C2×C4#645111yescharacteristic
437C2×C4#645911yescharacteristic
438C8—42no—
439C8—81no—
440C8—282no—
441C8—561no—
442C7#988311yescharacteristic, complemented, Sylow
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22#8941811yescharacteristic
456C22#8945111yescharacteristic
457C22#8947311yescharacteristic
458C22#8949811yescharacteristic
459C22#8949911yescharacteristic
460C22#8951211yescharacteristic
461C22D7×#9911yescharacteristic
462C22D7×#10011yescharacteristic
463C22D7×#10111yescharacteristic
464C22#8960011yescharacteristic
465C22#8960311yescharacteristic
466C4—21no—
467C4—21no—
468C4—21no—
469C4—21no—
470C4—42no—
471C4—81no—
472C4—142no—
473C4—142no—
474C4—142no—
475C4—282no—
476C4—281no—
477C4—561no—
478C2—21no—
479C2—21no—
480C2—21no—
481C2—21no—
482C2#1172585711yescharacteristic
483C2#1172587711yescharacteristic
484C2#1172588211yescharacteristic
485C2#1172589811yescharacteristic
486C2#1172637411yescharacteristic
487C2#1172670911yescharacteristic
488C2#1172672811yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutator, prime radicalC24×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1280C2
socleC23×C7C2×C22⋊C4
cosocleC23×C4×C7C23
hypercenter#1280D7
hypocenterC7#9883
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
442. C7#9883semidirect

Maximal subgroups

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

subgroupindex
C2×#894192
#117276622
#117276602
C7×#12802
#117254802
#117254192
#98837