Finite Group Databasebeta

← #13326726

Subgroups — #13326726

Order 1792.

1174 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3501C211yesmaximal, characteristic
2C2×#89822C212yesmaximal, complemented
3#11739794C211yesmaximal, characteristic
4#11729658C214yesmaximal, complemented
5#11729647C214yesmaximal, complemented
6#11729444C212yesmaximal, complemented
7#11729354C211yesmaximal, characteristic
8C23×Dic14C2211yescharacteristic
9C7×#554—24no—
10C7×#554—22no—
11C7×D4×Q8—22no—
12C7×#544C2212yes—
13C7×#543—21no—
14C7×#543C2212yes—
15C7×#533—21no—
16C2×C7×C22⋊Q8—21no—
17C2×C7×C22⋊Q8C2212yes—
18C7×#518—21no—
19C2×C4×C7×D4C2211yescharacteristic
20C22×#1144—22no—
21C2×#6487—21no—
22#89992—21no—
23#89984C2211yescharacteristic
24C2×C22⋊(C7⋊C8)C2212yes—
25#89822—24no—
26#89822C2218yes—
27#89751C2212yes—
28C2×#6384—22no—
29#89719C2212yes—
30#89718—21no—
31#89701—21no—
32#89333C2218yes—
33#89323C2212yes—
34#89320C2212yes—
35#18197—71nomaximal, Sylow
36C22×C7×Q8—21no—
37C22×C7×Q8D412yes—
38C22×C7×D4D411yescharacteristic
39C23×C4×C7D411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7—21no—
42C22×Dic14—22no—
43C22×Dic14—21no—
44C22×Dic14—22no—
45C22×Dic14—44no—
46C22×Dic14C2312yes—
47C7×C4⋊Q8—21no—
48C7×C4⋊Q8—21no—
49C7×C4⋊Q8—24no—
50C7×#125—22no—
51C7×#125—24no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—24no—
55C7×C22⋊Q8—24no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8—22no—
58C7×C22⋊Q8—22no—
59C7×C22⋊Q8—22no—
60C7×C22⋊Q8—22no—
61C7×C22⋊Q8—24no—
62C7×C22⋊Q8D414yes—
63C7×C22⋊Q8C2314yes—
64C4×C7×Q8—22no—
65C4×C7×Q8—22no—
66C4×C7×D4—24no—
67C4×C7×D4—24no—
68C4×C7×D4—22no—
69C4×C7×D4—22no—
70C7×#117—22no—
71C2×C7×C4⋊C4—21no—
72C2×C7×C4⋊C4—22no—
73C2×C7×C4⋊C4—22no—
74C2×C7×C4⋊C4D412yes—
75C2×C7×C4⋊C4C2311yescharacteristic
76C2×C7×C22⋊C4—22no—
77C2×C7×C22⋊C4—22no—
78C2×C7×C22⋊C4D411yescharacteristic
79C2×C7×C22⋊C4C2311yescharacteristic
80C2×C42×C7D411yescharacteristic
81C2×C22⋊Dic7—22no—
82C2×C22⋊Dic7—21no—
83C2×#1144—28no—
84C2×#1144—48no—
85C2×C4⋊Dic7—21no—
86#6487—44no—
87C2×Dic7⋊C4—22no—
88C22×C7⋊C8—22no—
89C7⋊#95—22no—
90C7⋊#95—22no—
91C7⋊#95C2312yes—
92C22⋊(C7⋊C8)—22no—
93C22⋊(C7⋊C8)C2314yes—
94#6384—28no—
95#3501D711yescharacteristic
96C2×#451—72no—
97#2409—71no—
98#1605—74no—
99#1602—74no—
100#1520—72no—
101#1459—71no—
102C24×C7C2×D411yescharacteristic
103C24×C7C2×D411yescharacteristic
104C2×C7×Q8—22no—
105C2×C7×Q8—22no—
106C2×C7×Q8—22no—
107C2×C7×Q8—44no—
108C2×C7×Q8—82no—
109C2×C7×Q8C2×D414yes—
110C2×C7×D4—22no—
111C2×C7×D4—22no—
112C2×C7×D4Q1614yescomplemented
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—22no—
116C22×C4×C7—22no—
117C22×C4×C7—21no—
118C22×C4×C7—21no—
119C22×C4×C7—42no—
120C22×C4×C7—42no—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D412yes—
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D411yescharacteristic
125C22×C4×C7C2×D411yescharacteristic
126C22×C4×C7C2×D412yes—
127C22×C4×C7C2411yescharacteristic
128C22×Dic7—22no—
129C22×Dic7—22no—
130C22×Dic7—44no—
131C22×Dic7—42no—
132C2×Dic14—28no—
133C2×Dic14—24no—
134C2×Dic14—24no—
135C2×Dic14—44no—
136C2×Dic14—48no—
137C2×Dic14—44no—
138C2×Dic14—42no—
139C2×Dic14—41no—
140C2×Dic14—41no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—24no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—24no—
146C7×C4⋊C4—22no—
147C7×C4⋊C4—24no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—24no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4C2×D414yes—
152C7×C22⋊C4—24no—
153C7×C22⋊C4—24no—
154C7×C22⋊C4—22no—
155C7×C22⋊C4—24no—
156C7×C22⋊C4—22no—
157C42×C7—24no—
158C42×C7—21no—
159C42×C7—21no—
160C22⋊Dic7—44no—
161C22⋊Dic7—42no—
162#1144—416no—
163C4⋊Dic7—42no—
164Dic7⋊C4—44no—
165C2×C7⋊C8—24no—
166C2×C7⋊C8—44no—
167C23×Q8—71no—
168C22×Q16—142no—
169#554—24no—
170#554—22no—
171D4×Q8—22no—
172#544C2×D712yes—
173#543—21no—
174#543C2×D712yes—
175#533—21no—
176C2×C22⋊Q8—21no—
177C2×C22⋊Q8—141no—
178C2×C22⋊Q8C2×D712yes—
179#518—21no—
180C2×C4×D4C2×D711yescharacteristic
181#451—78no—
182#451—144no—
183C2×#103—142no—
184C2×C22⋊C8—72no—
185#395—72no—
186#393—72no—
187#391—141no—
188#386—141no—
189#385—141no—
190#380—71no—
191#328—78no—
192#324—72no—
193#323—72no—
194C23×C7—21no—
195C23×C7—21no—
196C23×C7—41no—
197C23×C7—41no—
198C23×C7—41no—
199C23×C7—41no—
200C23×C7#12012yes—
201C23×C7C2×Q1612yes—
202C23×C7C2×Q1612yescomplemented
203C23×C7#13712yes—
204C23×C7C22×D411yescharacteristic
205C7×Q8—44no—
206C7×Q8—44no—
207C7×Q8—82no—
208C7×Q8—82no—
209C7×D4—28no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—24no—
213C2×C4×C7—22no—
214C2×C4×C7—21no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—21no—
218C2×C4×C7—42no—
219C2×C4×C7—41no—
220C2×C4×C7—42no—
221C2×C4×C7—41no—
222C2×C4×C7—41no—
223C2×C4×C7—81no—
224C2×C4×C7—82no—
225C2×C4×C7—82no—
226C2×C4×C7—82no—
227C2×C4×C7—82no—
228C2×C4×C7#12011yescharacteristic
229C2×C4×C7#12011yescharacteristic
230C2×C4×C7C2×Q1612yescomplemented
231C2×C4×C7C22×D412yes—
232C2×Dic7—24no—
233C2×Dic7—48no—
234C2×Dic7—42no—
235C2×Dic7—42no—
236C2×Dic7—42no—
237C2×Dic7—42no—
238C2×Dic7—42no—
239Dic14—28no—
240Dic14—44no—
241Dic14—44no—
242Dic14—44no—
243C7⋊C8—44no—
244C22×Q8—21no—
245C22×Q8—72no—
246C22×Q8—141no—
247C22×Q8—142no—
248C22×Q8—142no—
249C22×Q8—284no—
250C22×Q8C22⋊D712yes—
251C22×D4C22⋊D711yescharacteristic
252C23×C4—141no—
253C23×C4C22⋊D711yescharacteristic
254C23×C4C22×D711yescharacteristic
255C2×Q16—148no—
256C2×Q16—288no—
257C22×C8—142no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—24no—
261#125—22no—
262#125—24no—
263C22⋊Q8—22no—
264C22⋊Q8—22no—
265C22⋊Q8—24no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—22no—
271C22⋊Q8—22no—
272C22⋊Q8—24no—
273C22⋊Q8—284no—
274C22⋊Q8C22⋊D714yes—
275C22⋊Q8C22×D714yes—
276C4×Q8—22no—
277C4×Q8—22no—
278C4×D4—24no—
279C4×D4—24no—
280C4×D4—22no—
281C4×D4—22no—
282#117—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—142no—
287C2×C4⋊C4—141no—
288C2×C4⋊C4C22⋊D712yes—
289C2×C4⋊C4C22×D711yescharacteristic
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—142no—
293C2×C22⋊C4—141no—
294C2×C22⋊C4C22⋊D711yescharacteristic
295C2×C22⋊C4C22×D711yescharacteristic
296C2×C42C22⋊D711yescharacteristic
297#103—148no—
298C22⋊C8—74no—
299C22⋊C8—142no—
300#95—72no—
301#95—142no—
302#95—142no—
303C22×C7—21no—
304C22×C7—22no—
305C22×C7—21no—
306C22×C7—22no—
307C22×C7—22no—
308C22×C7—42no—
309C22×C7—42no—
310C22×C7—42no—
311C22×C7—41no—
312C22×C7—41no—
313C22×C7—81no—
314C22×C7#45114yes—
315C22×C7C2×#12011yescharacteristic
316C22×C7C22×Q1611yescharacteristic
317C22×C7C2×#13711yescharacteristic
318C4×C7—22no—
319C4×C7—22no—
320C4×C7—21no—
321C4×C7—41no—
322C4×C7—44no—
323C4×C7—41no—
324C4×C7—44no—
325C4×C7B(4,𝔽2)12yes—
326Dic7—44no—
327Dic7—82no—
328C24C2×C22⋊D711yescharacteristic
329C24C2×C22⋊D711yescharacteristic
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—22no—
333C2×Q8—44no—
334C2×Q8—82no—
335C2×Q8—144no—
336C2×Q8—148no—
337C2×Q8—144no—
338C2×Q8—282no—
339C2×Q8—284no—
340C2×Q8—284no—
341C2×Q8—288no—
342C2×Q8—281no—
343C2×Q8—281no—
344C2×Q8C2×C22⋊D714yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4#114414yescomplemented
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—142no—
357C22×C4—142no—
358C22×C4—284no—
359C22×C4—282no—
360C22×C4C2×C22⋊D711yescharacteristic
361C22×C4C2×C22⋊D711yescharacteristic
362C22×C4C2×C22⋊D712yes—
363C22×C4C2×C22⋊D711yescharacteristic
364C22×C4C2×C22⋊D711yescharacteristic
365C22×C4C2×C22⋊D712yes—
366C22×C4C23×D711yescharacteristic
367Q16—2816no—
368C2×C8—144no—
369C2×C8—284no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—282no—
381C4⋊C4—284no—
382C4⋊C4C2×C22⋊D714yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—22no—
386C22⋊C4—24no—
387C22⋊C4—22no—
388C22⋊C4—284no—
389C22⋊C4—282no—
390C42—24no—
391C42—21no—
392C42—21no—
393C2×C7—21no—
394C2×C7—21no—
395C2×C7—24no—
396C2×C7—42no—
397C2×C7C2×#45111yescharacteristic
398C2×C7#302211yescharacteristic
399C2×C7C2×B(4,𝔽2)11yescharacteristic
400C23—21no—
401C23—21no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23#650512yes—
407C23C2×#114412yes—
408C23C2×#114412yescomplemented
409C23#651612yes—
410C23C22×C22⋊D711yescharacteristic
411Q8—44no—
412Q8—44no—
413Q8—82no—
414Q8—82no—
415Q8—148no—
416Q8—284no—
417Q8—284no—
418Q8—284no—
419D4—28no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—21no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—21no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—144no—
439C2×C4—282no—
440C2×C4—288no—
441C2×C4—282no—
442C2×C4—282no—
443C2×C4—282no—
444C2×C4—282no—
445C2×C4C2×#114412yescomplemented
446C2×C4#651611yescharacteristic
447C2×C4#651611yescharacteristic
448C2×C4C22×C22⋊D712yes—
449C8—284no—
450C7#1819711yescharacteristic, complemented, Sylow
451C22—21no—
452C22—22no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—41no—
460C22—41no—
461C22—81no—
462C22#9000014yes—
463C22C2×#650511yescharacteristic
464C22C22×#114411yescharacteristic
465C22C2×#651611yescharacteristic
466C4—22no—
467C4—22no—
468C4—21no—
469C4—41no—
470C4—44no—
471C4—41no—
472C4—44no—
473C4—284no—
474C4—562no—
475C4#9001612yes—
476C2—21no—
477C2—21no—
478C2—24no—
479C2—42no—
480C2C2×#9000011yescharacteristic
481C2C2×#9001611yescharacteristic
482C2#1174156811yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#3501D7
hypocenterC7#18197
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
2. C2×#89822C2semidirect
4. #11729658C2semidirect
5. #11729647C2semidirect
6. #11729444C2semidirect
112. C2×C7×D4Q16semidirect
202. C23×C7C2×Q16semidirect
230. C2×C4×C7C2×Q16semidirect
347. C2×D4#1144semidirect
408. C23C2×#1144semidirect
445. C2×C4C2×#1144semidirect
450. C7#18197semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
C2×#898222
#117397942
#117296582
#117296472
#117294442
#117293542
#181977