Finite Group Databasebeta

← #13326787

Subgroups — #13326787

Order 1792.

1002 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3501C211yesmaximal, characteristic
2C2×#89815C212yesmaximal, complemented
3#11739818C211yesmaximal, characteristic
4#11729652C214yesmaximal, complemented
5#11729644C214yesmaximal, complemented
6#11729444C212yesmaximal, complemented
7#11729354C211yesmaximal, characteristic
8C7×#554—24no—
9C7×#554—22no—
10C7×D4×Q8—22no—
11C7×#544C2212yes—
12C7×#543—21no—
13C7×#543C2212yes—
14C7×#533—21no—
15C2×C7×C22⋊Q8—21no—
16C2×C7×C22⋊Q8C2212yes—
17C7×#518—21no—
18C2×C4×C7×D4C2211yescharacteristic
19C2×#6498—21no—
20C22×C4⋊Dic7C2211yescharacteristic
21C2×#6409—22no—
22#89984—21no—
23#89984C2211yescharacteristic
24C2×C22⋊(C7⋊C8)C2212yes—
25#89815—24no—
26#89815C2218yes—
27#89755C2212yes—
28C2×#6382—22no—
29#89725C2212yes—
30#89715—21no—
31#89700—21no—
32#89333C2218yes—
33#89323C2212yes—
34#89320C2212yes—
35#18204—71nomaximal, Sylow
36C22×C7×Q8—21no—
37C22×C7×Q8D412yes—
38C22×C7×D4D411yescharacteristic
39C23×C4×C7D411yescharacteristic
40C23×C4×C7C2311yescharacteristic
41C23×Dic7—21no—
42C7×C4⋊Q8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—24no—
45C7×#125—22no—
46C7×#125—24no—
47C7×C22⋊Q8—22no—
48C7×C22⋊Q8—22no—
49C7×C22⋊Q8—24no—
50C7×C22⋊Q8—24no—
51C7×C22⋊Q8—24no—
52C7×C22⋊Q8—22no—
53C7×C22⋊Q8—22no—
54C7×C22⋊Q8—22no—
55C7×C22⋊Q8—22no—
56C7×C22⋊Q8—24no—
57C7×C22⋊Q8D414yes—
58C7×C22⋊Q8C2314yes—
59C4×C7×Q8—22no—
60C4×C7×Q8—22no—
61C4×C7×D4—24no—
62C4×C7×D4—24no—
63C4×C7×D4—22no—
64C4×C7×D4—22no—
65C7×#117—22no—
66C2×C7×C4⋊C4—21no—
67C2×C7×C4⋊C4—22no—
68C2×C7×C4⋊C4—22no—
69C2×C7×C4⋊C4D412yes—
70C2×C7×C4⋊C4C2311yescharacteristic
71C2×C7×C22⋊C4—22no—
72C2×C7×C22⋊C4—22no—
73C2×C7×C22⋊C4D411yescharacteristic
74C2×C7×C22⋊C4C2311yescharacteristic
75C2×C42×C7D411yescharacteristic
76C2×C22⋊Dic7—21no—
77C2×C22⋊Dic7—22no—
78#6498—44no—
79C2×C4⋊Dic7—22no—
80C2×C4⋊Dic7—22no—
81C2×C4⋊Dic7—21no—
82C2×C4⋊Dic7—21no—
83C2×C4⋊Dic7C2312yes—
84C2×Dic7⋊C4—22no—
85C22×C7⋊C8—22no—
86#6409—28no—
87C7⋊#95—22no—
88C7⋊#95—22no—
89C7⋊#95C2312yes—
90C22⋊(C7⋊C8)—22no—
91C22⋊(C7⋊C8)C2314yes—
92#6382—28no—
93#3501D711yescharacteristic
94C2×#483—72no—
95#2486—71no—
96#1611—74no—
97#1608—74no—
98#1520—72no—
99#1459—71no—
100C24×C7C2×D411yescharacteristic
101C24×C7C2×D411yescharacteristic
102C2×C7×Q8—22no—
103C2×C7×Q8—22no—
104C2×C7×Q8—22no—
105C2×C7×Q8—44no—
106C2×C7×Q8—82no—
107C2×C7×Q8C2×D414yes—
108C2×C7×D4—22no—
109C2×C7×D4—22no—
110C2×C7×D4SD1614yes—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—22no—
114C22×C4×C7—22no—
115C22×C4×C7—21no—
116C22×C4×C7—21no—
117C22×C4×C7—42no—
118C22×C4×C7—42no—
119C22×C4×C7C2×D411yescharacteristic
120C22×C4×C7C2×D412yes—
121C22×C4×C7C2×D411yescharacteristic
122C22×C4×C7C2×D411yescharacteristic
123C22×C4×C7C2×D411yescharacteristic
124C22×C4×C7C2×D412yes—
125C22×C4×C7C2411yescharacteristic
126C22×Dic7—22no—
127C22×Dic7—22no—
128C22×Dic7—42no—
129C22×Dic7—42no—
130C22×Dic7—42no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—24no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—24no—
136C7×C4⋊C4—22no—
137C7×C4⋊C4—24no—
138C7×C4⋊C4—22no—
139C7×C4⋊C4—24no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4C2×D414yes—
142C7×C22⋊C4—24no—
143C7×C22⋊C4—24no—
144C7×C22⋊C4—22no—
145C7×C22⋊C4—24no—
146C7×C22⋊C4—22no—
147C42×C7—24no—
148C42×C7—21no—
149C42×C7—21no—
150C22⋊Dic7—42no—
151C22⋊Dic7—44no—
152C4⋊Dic7—24no—
153C4⋊Dic7—42no—
154C4⋊Dic7—41no—
155C4⋊Dic7—41no—
156Dic7⋊C4—44no—
157C2×C7⋊C8—24no—
158C2×C7⋊C8—44no—
159#554—24no—
160#554—22no—
161D4×Q8—22no—
162#544C2×D712yes—
163#543—21no—
164#543C2×D712yes—
165#533—21no—
166C2×#123—141no—
167C2×C22⋊Q8—21no—
168C2×C22⋊Q8C2×D712yes—
169#518—21no—
170C2×C4×D4C2×D711yescharacteristic
171C22×C4⋊C4—71no—
172#483—78no—
173#483—144no—
174C2×#106—142no—
175C2×#103—142no—
176C2×C22⋊C8—72no—
177#398—72no—
178#397—72no—
179#389—141no—
180#381—141no—
181#380—71no—
182#380—141no—
183#328—78no—
184#324—72no—
185#323—72no—
186C23×C7—21no—
187C23×C7—21no—
188C23×C7—41no—
189C23×C7—41no—
190C23×C7—41no—
191C23×C7—41no—
192C23×C7#12012yes—
193C23×C7C2×SD1612yes—
194C23×C7C2×SD1612yes—
195C23×C7#13712yes—
196C23×C7C22×D411yescharacteristic
197C7×Q8—44no—
198C7×Q8—44no—
199C7×Q8—82no—
200C7×Q8—82no—
201C7×D4—28no—
202C2×C4×C7—21no—
203C2×C4×C7—21no—
204C2×C4×C7—24no—
205C2×C4×C7—22no—
206C2×C4×C7—21no—
207C2×C4×C7—24no—
208C2×C4×C7—24no—
209C2×C4×C7—21no—
210C2×C4×C7—42no—
211C2×C4×C7—41no—
212C2×C4×C7—42no—
213C2×C4×C7—41no—
214C2×C4×C7—41no—
215C2×C4×C7—81no—
216C2×C4×C7—82no—
217C2×C4×C7—82no—
218C2×C4×C7—82no—
219C2×C4×C7—82no—
220C2×C4×C7#12011yescharacteristic
221C2×C4×C7#12011yescharacteristic
222C2×C4×C7C2×SD1612yes—
223C2×C4×C7C22×D412yes—
224C2×Dic7—42no—
225C2×Dic7—42no—
226C2×Dic7—42no—
227C2×Dic7—42no—
228C2×Dic7—42no—
229C2×Dic7—42no—
230C2×Dic7—84no—
231C7⋊C8—44no—
232C22×Q8—21no—
233C22×Q8C22⋊D712yes—
234C22×D4C22⋊D711yescharacteristic
235C23×C4—141no—
236C23×C4C22⋊D711yescharacteristic
237C23×C4C22×D711yescharacteristic
238C22×C8—142no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—24no—
242#125—22no—
243#125—24no—
244#123—284no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—24no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—22no—
251C22⋊Q8—22no—
252C22⋊Q8—22no—
253C22⋊Q8—22no—
254C22⋊Q8—24no—
255C22⋊Q8C22⋊D714yes—
256C22⋊Q8C22×D714yes—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—22no—
262C4×D4—22no—
263#117—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—72no—
268C2×C4⋊C4—142no—
269C2×C4⋊C4—142no—
270C2×C4⋊C4—141no—
271C2×C4⋊C4—142no—
272C2×C4⋊C4—141no—
273C2×C4⋊C4C22⋊D712yes—
274C2×C4⋊C4C22×D711yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—141no—
278C2×C22⋊C4—142no—
279C2×C22⋊C4C22⋊D711yescharacteristic
280C2×C22⋊C4C22×D711yescharacteristic
281C2×C42C22⋊D711yescharacteristic
282#106—148no—
283#103—148no—
284C22⋊C8—74no—
285C22⋊C8—142no—
286#95—72no—
287#95—142no—
288#95—142no—
289C22×C7—21no—
290C22×C7—22no—
291C22×C7—21no—
292C22×C7—22no—
293C22×C7—22no—
294C22×C7—42no—
295C22×C7—42no—
296C22×C7—42no—
297C22×C7—41no—
298C22×C7—41no—
299C22×C7—81no—
300C22×C7#44814yes—
301C22×C7C2×#12011yescharacteristic
302C22×C7C22×SD1611yescharacteristic
303C22×C7C2×#13711yescharacteristic
304C4×C7—22no—
305C4×C7—22no—
306C4×C7—21no—
307C4×C7—41no—
308C4×C7—44no—
309C4×C7—41no—
310C4×C7—44no—
311C4×C7#45812yes—
312Dic7—82no—
313Dic7—82no—
314C24C2×C22⋊D711yescharacteristic
315C24C2×C22⋊D711yescharacteristic
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—22no—
319C2×Q8—44no—
320C2×Q8—82no—
321C2×Q8C2×C22⋊D714yes—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4#114314yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—142no—
334C22×C4—142no—
335C22×C4—282no—
336C22×C4—282no—
337C22×C4—282no—
338C22×C4C2×C22⋊D711yescharacteristic
339C22×C4C2×C22⋊D711yescharacteristic
340C22×C4C2×C22⋊D712yes—
341C22×C4C2×C22⋊D711yescharacteristic
342C22×C4C2×C22⋊D711yescharacteristic
343C22×C4C2×C22⋊D712yes—
344C22×C4C23×D711yescharacteristic
345C2×C8—144no—
346C2×C8—284no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—144no—
358C4⋊C4—282no—
359C4⋊C4—284no—
360C4⋊C4—281no—
361C4⋊C4—281no—
362C4⋊C4C2×C22⋊D714yes—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—22no—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—282no—
369C22⋊C4—284no—
370C42—24no—
371C42—21no—
372C42—21no—
373C2×C7—21no—
374C2×C7—21no—
375C2×C7—24no—
376C2×C7—42no—
377C2×C7C2×#44811yescharacteristic
378C2×C7#302211yescharacteristic
379C2×C7C2×#45811yescharacteristic
380C23—21no—
381C23—21no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23C2×#114312yes—
387C23C2×#114312yes—
388C23#650512yes—
389C23#651612yes—
390C23C22×C22⋊D711yescharacteristic
391Q8—44no—
392Q8—44no—
393Q8—82no—
394Q8—82no—
395D4—28no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—21no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—81no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—282no—
415C2×C4—282no—
416C2×C4—282no—
417C2×C4—282no—
418C2×C4—282no—
419C2×C4—282no—
420C2×C4—564no—
421C2×C4C2×#114312yes—
422C2×C4#651611yescharacteristic
423C2×C4#651611yescharacteristic
424C2×C4C22×C22⋊D712yes—
425C8—284no—
426C7#1820411yescharacteristic, complemented, Sylow
427C22—21no—
428C22—22no—
429C22—21no—
430C22—22no—
431C22—22no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22#8999914yes—
439C22C22×#114311yescharacteristic
440C22C2×#650511yescharacteristic
441C22C2×#651611yescharacteristic
442C4—22no—
443C4—22no—
444C4—21no—
445C4—41no—
446C4—44no—
447C4—41no—
448C4—44no—
449C4—562no—
450C4—562no—
451C4#9001512yes—
452C2—21no—
453C2—21no—
454C2—24no—
455C2—42no—
456C2C2×#8999911yescharacteristic
457C2C2×#9001511yescharacteristic
458C2#1174156811yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#3501C2
socleC22×C7C2×#120
prime radicalC22×C7×D4D4
hypercenter#3501D7
hypocenterC7#18204
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×#89815C2semidirect
4. #11729652C2semidirect
5. #11729644C2semidirect
6. #11729444C2semidirect
426. C7#18204semidirect

Maximal subgroups

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

subgroupindex
C7×#35012
C2×#898152
#117398182
#117296522
#117296442
#117294442
#117293542
#182047