Finite Group Databasebeta

← #13512490

Subgroups — #13512490

Order 1792.

2250 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6450C211yesmaximal, characteristic
2C7×#2313C211yesmaximal, characteristic
3#11739600C211yesmaximal, characteristic
4#11732391C218yesmaximal, complemented
5#11732383C218yesmaximal, complemented
6#11731498C212yesmaximal
7#11731496C212yesmaximal
8#11731314C212yesmaximal
9#11731308C212yesmaximal
10#11730890C212yesmaximal
11C4×#6440C212yesmaximal
12C23×Dic14C2211yescharacteristic
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×Dic7⋊C4C2211yescharacteristic
15C22×Dic7⋊C4C2211yescharacteristic
16C22×C4×Dic7C2211yescharacteristic
17C22×C4×Dic7C2211yescharacteristic
18C22×C4×Dic7C2211yescharacteristic
19C2×#6450C4116yescomplemented
20C2×#6450C2214yes—
21C2×#6450C2214yes—
22C2×#6440C2214yes—
23C7×#387C2218yes—
24C7×#381C2212yes—
25C7×#380C2212yes—
26C4×C7×C22⋊C4C2212yes—
27#89984C2212yes—
28#89983C2212yes—
29C4×C22⋊Dic7C2212yes—
30#89753C2218yes—
31#89749C2214yes—
32C4⋊C4×Dic7C2214yes—
33#89744C2218yes—
34#89715C2214yes—
35#89714C2214yes—
36C22⋊C4×Dic7C2214yes—
37#89705C2218yes—
38C4×Dic7⋊C4C2214yes—
39C42×Dic7C2214yes—
40#89424C2218yes—
41#89423C2214yes—
42#89423C2214yes—
43#89422C2214yes—
44#89421C2214yes—
45#89421C2214yes—
46#89420C2214yes—
47#89420C2214yes—
48#89420C2214yes—
49#89419C2218yes—
50#89418C2218yes—
51#89418C2218yes—
52#34059—71nomaximal, Sylow
53C23×C4×C7C2311yescharacteristic
54C23×C4×C7C2311yescharacteristic
55C23×C4×C7C2311yescharacteristic
56C23×Dic7C2311yescharacteristic
57C23×Dic7C2311yescharacteristic
58C23×Dic7C2311yescharacteristic
59C23×Dic7C2311yescharacteristic
60C22×Dic14—28no—
61C22×Dic14C2×C418yes—
62C22×Dic14C2314yes—
63C2×C7×C4⋊C4C2×C418yes—
64C2×C7×C4⋊C4C2314yes—
65C2×C7×C4⋊C4C2314yes—
66C2×C7×C22⋊C4C2314yes—
67C2×C42×C7C2314yes—
68C2×C22⋊Dic7C2314yes—
69C2×C4⋊Dic7C2314yes—
70C2×Dic7⋊C4C2×C418yes—
71C2×Dic7⋊C4C2×C418yes—
72C2×Dic7⋊C4C2318yes—
73C2×Dic7⋊C4C2318yes—
74C2×Dic7⋊C4C2314yes—
75C2×Dic7⋊C4C2314yes—
76C2×C4×Dic7C2×C418yes—
77C2×C4×Dic7C2×C418yes—
78C2×C4×Dic7C2×C418yes—
79C2×C4×Dic7C2318yes—
80C2×C4×Dic7C2318yes—
81C2×C4×Dic7C2314yes—
82C2×C4×Dic7C2312yes—
83C2×C4×Dic7C2312yes—
84C2×C4×Dic7C2312yes—
85C2×C4×Dic7C2312yes—
86C2×C4×Dic7C2312yes—
87C2×C4×Dic7C2312yes—
88#6450C2×C4132yes—
89#6450C2×C4132yes—
90#6440—216no—
91C7×#95C2314yes—
92C7×#95C2314yes—
93C7×#95C2314yes—
94C7×#95C2314yes—
95C7⋊#95C2314yes—
96C7⋊#95C2318yes—
97C7⋊#95C2314yes—
98C7⋊#95C2318yes—
99C7⋊#95C2318yes—
100C7⋊#95C2314yes—
101C7⋊#95C2318yes—
102C7⋊#95C2314yes—
103C22×C4×Q8—71no—
104#2320—78no—
105#2316—72no—
106#2313—72no—
107#2313—71no—
108#2313D711yescharacteristic
109#2310—72no—
110#2295—78no—
111#2283—72no—
112#2283—72no—
113C4×#117—72no—
114C24×C7C2411yescharacteristic
115C22×C4×C7—24no—
116C22×C4×C7—42no—
117C22×C4×C7—42no—
118C22×C4×C7C22×C414yes—
119C22×C4×C7C22×C414yes—
120C22×C4×C7C22×C414yes—
121C22×C4×C7C2418yes—
122C22×C4×C7C2412yes—
123C22×C4×C7C2412yes—
124C22×C4×C7C2412yes—
125C22×Dic7—24no—
126C22×Dic7—24no—
127C22×Dic7—42no—
128C22×Dic7C22×C414yes—
129C22×Dic7C22×C414yes—
130C22×Dic7C22×C414yes—
131C22×Dic7C22×C414yes—
132C22×Dic7#4118yes—
133C22×Dic7C2418yes—
134C22×Dic7C2411yescharacteristic
135C22×Dic7C2412yes—
136C22×Dic7C2412yes—
137C22×Dic7C2412yes—
138C22×Dic7C2411yescharacteristic
139C2×Dic14—232no—
140C2×Dic14—216no—
141C2×Dic14C22×C4116yes—
142C7×C4⋊C4—44no—
143C7×C4⋊C4C22×C4116yes—
144C7×C22⋊C4—28no—
145C42×C7—44no—
146C22⋊Dic7—28no—
147C4⋊Dic7—44no—
148Dic7⋊C4—216no—
149Dic7⋊C4—48no—
150Dic7⋊C4C22×C4116yes—
151Dic7⋊C4C22×C4116yes—
152C4×Dic7—216no—
153C4×Dic7—216no—
154C4×Dic7—44no—
155C4×Dic7C22×C418yes—
156C4×Dic7C22×C418yes—
157C4×Dic7C22×C418yes—
158C4×Dic7C22×C418yes—
159C4×Dic7C22×C418yes—
160C4×Dic7C22×C418yes—
161C23×Q8—71no—
162C2×C4×Q8—716no—
163C2×C4×Q8—74no—
164C2×C4×Q8—74no—
165C2×#117—74no—
166C22×C4⋊C4—71no—
167C22×C4⋊C4—71no—
168C22×C4⋊C4C2×D711yescharacteristic
169C22×C42—71no—
170C22×C42—71no—
171C22×C42—71no—
172#391—78no—
173#391—78no—
174#389—78no—
175#389—78no—
176#387—78no—
177#387—78no—
178#387—78no—
179#387C2×D718yes—
180#383—74no—
181#383—74no—
182#382—74no—
183#382—74no—
184#381—72no—
185#381—74no—
186#381C2×D712yes—
187#380—72no—
188#380—74no—
189#380C2×D712yes—
190C4×C4⋊C4—74no—
191C4×C4⋊C4—74no—
192C4×C4⋊C4—74no—
193C4×C4⋊C4—74no—
194C4×C22⋊C4—72no—
195C4×C22⋊C4—74no—
196C4×C22⋊C4C2×D712yes—
197#376—74no—
198#376—74no—
199#376—74no—
200C43—74no—
201C23×C7—22no—
202C23×C7—22no—
203C23×C7C23×C412yes—
204C23×C7C2×#4114yes—
205C23×C7C2511yescharacteristic
206C2×C4×C7—28no—
207C2×C4×C7—24no—
208C2×C4×C7—24no—
209C2×C4×C7—24no—
210C2×C4×C7—28no—
211C2×C4×C7—28no—
212C2×C4×C7—28no—
213C2×C4×C7—44no—
214C2×C4×C7—44no—
215C2×C4×C7C23×C414yes—
216C2×C4×C7C23×C414yes—
217C2×C4×C7C23×C414yes—
218C2×Dic7—28no—
219C2×Dic7—24no—
220C2×Dic7—22no—
221C2×Dic7—24no—
222C2×Dic7—28no—
223C2×Dic7—28no—
224C2×Dic7—28no—
225C2×Dic7—28no—
226C2×Dic7—44no—
227C2×Dic7#117116yes—
228C2×Dic7C23×C412yes—
229C2×Dic7C23×C414yes—
230C2×Dic7C23×C414yes—
231C2×Dic7C23×C414yes—
232C2×Dic7C23×C412yes—
233C2×Dic7C2×#4118yes—
234C2×Dic72−1+414yes—
235Dic14—232no—
236C22×Q8—78no—
237C22×Q8—74no—
238C22×Q8—148no—
239C23×C4—71no—
240C23×C4—71no—
241C23×C4—71no—
242C23×C4—71no—
243C23×C4C22×D711yescharacteristic
244C23×C4C22×D711yescharacteristic
245C23×C4C22×D711yescharacteristic
246C4×Q8—732no—
247C4×Q8—732no—
248#117—1416no—
249C2×C4⋊C4—78no—
250C2×C4⋊C4—78no—
251C2×C4⋊C4—74no—
252C2×C4⋊C4—74no—
253C2×C4⋊C4—78no—
254C2×C4⋊C4—78no—
255C2×C4⋊C4—74no—
256C2×C4⋊C4C4×D718yes—
257C2×C4⋊C4C22×D714yes—
258C2×C4⋊C4C22×D714yes—
259C2×C22⋊C4—74no—
260C2×C22⋊C4C22×D714yes—
261C2×C42—78no—
262C2×C42—78no—
263C2×C42—78no—
264C2×C42—72no—
265C2×C42—72no—
266C2×C42—72no—
267C2×C42—72no—
268C2×C42—72no—
269C2×C42—72no—
270C2×C42—78no—
271C2×C42—78no—
272C2×C42—74no—
273C2×C42C22×D714yes—
274#95—78no—
275#95—74no—
276#95—78no—
277#95—74no—
278#95—78no—
279#95—74no—
280#95—78no—
281#95—74no—
282#95C22×D714yes—
283#95C22×D714yes—
284#95C22×D714yes—
285#95C22×D714yes—
286C22×C7—22no—
287C22×C7—22no—
288C22×C7—28no—
289C22×C7C2×#11714yes—
290C22×C7C24×C411yescharacteristic
291C22×C7C22×#4112yes—
292C22×C7C2×2−1+412yes—
293C22×C7#58512yes—
294C4×C7—24no—
295C4×C7—24no—
296C4×C7—24no—
297C4×C7—48no—
298Dic7—24no—
299Dic7—22no—
300Dic7—24no—
301Dic7—48no—
302Dic7C2×#11718yes—
303Dic7#51914yes—
304C24C23×D711yescharacteristic
305C2×Q8—716no—
306C2×Q8—1432no—
307C2×Q8—1416no—
308C22×C4—24no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—74no—
312C22×C4—78no—
313C22×C4—71no—
314C22×C4—72no—
315C22×C4—74no—
316C22×C4—74no—
317C22×C4—71no—
318C22×C4—72no—
319C22×C4—72no—
320C22×C4—74no—
321C22×C4—78no—
322C22×C4—144no—
323C22×C4—144no—
324C22×C4—282no—
325C22×C4C2×C4×D714yes—
326C22×C4C2×C4×D714yes—
327C22×C4C2×C4×D714yes—
328C22×C4C23×D718yes—
329C22×C4C23×D712yes—
330C22×C4C23×D712yes—
331C22×C4C23×D712yes—
332C4⋊C4—44no—
333C4⋊C4—716no—
334C4⋊C4—716no—
335C4⋊C4—1416no—
336C4⋊C4—288no—
337C4⋊C4—284no—
338C4⋊C4C2×C4×D7116yes—
339C22⋊C4—28no—
340C22⋊C4—148no—
341C42—44no—
342C42—78no—
343C42—78no—
344C42—78no—
345C42—78no—
346C42—78no—
347C42—78no—
348C42—1416no—
349C42—1416no—
350C42—284no—
351C2×C7—24no—
352C2×C7C22×#11711yescharacteristic
353C2×C7C2×#51911yescharacteristic
354C2×C7#342811yescharacteristic
355C2×C7#346114yes—
356C23—22no—
357C23—22no—
358C23C22×C4×D712yes—
359C23D7×#4114yes—
360C23C24×D711yescharacteristic
361Q8—1432no—
362C2×C4—28no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—28no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—716no—
372C2×C4—72no—
373C2×C4—74no—
374C2×C4—78no—
375C2×C4—74no—
376C2×C4—72no—
377C2×C4—74no—
378C2×C4—74no—
379C2×C4—148no—
380C2×C4—148no—
381C2×C4—144no—
382C2×C4—144no—
383C2×C4—142no—
384C2×C4—148no—
385C2×C4—148no—
386C2×C4—148no—
387C2×C4—284no—
388C2×C4C22×C4×D714yes—
389C2×C4C22×C4×D714yes—
390C2×C4C22×C4×D714yes—
391C7#3405911yescharacteristic, complemented, Sylow
392C22—28no—
393C22—22no—
394C22—22no—
395C22D7×#11714yes—
396C22C23×C4×D711yescharacteristic
397C22C2×D7×#4112yes—
398C22#9061812yes—
399C22D7×2−1+412yes—
400C4—24no—
401C4—24no—
402C4—24no—
403C4—48no—
404C4—78no—
405C4—74no—
406C4—144no—
407C4—144no—
408C4—142no—
409C4—288no—
410C2—24no—
411C2C2×D7×#11711yescharacteristic
412C2#1174292411yescharacteristic
413C2D7×#51911yescharacteristic
414C2#1174349714yes—
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2313C2
socle, cosocleC23×C7C25
prime radicalC24×C7C24
hypercenter#2313D7
hypocenterC7#34059
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2313

Complements

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

NH ≅ G/Ntype
4. #11732391C2semidirect
5. #11732383C2semidirect
19. C2×#6450C4semidirect
391. C7#34059semidirect

Maximal subgroups

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

subgroupindex
C22×#64502
C7×#23132
#117396002
#117323912
#117323832
#117314982
#117314962
#117313142
#117313082
#117308902
C4×#64402
#340597