Finite Group Databasebeta

← #13516516

Subgroups — #13516516

Order 1792.

1800 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7⋊#564C211yesmaximal, characteristic
2C7×#2382C211yesmaximal, characteristic
3Dic7×#125C211yesmaximal, characteristic
4#11733438C214yesmaximal
5#11733411C214yesmaximal
6#11732424C214yesmaximal
7#11732423C214yesmaximal
8#11732388C214yesmaximal
9#11732388C214yesmaximal
10#11731606C212yesmaximal
11#11731595C212yesmaximal
12C2×C7×#125C2211yescharacteristic
13C7⋊#564C4116yescomplemented
14C2×#6453C2214yes—
15C2×#6452C2214yes—
16C2×#6452C2214yes—
17C2×#6433C2212yes—
18C7×#389C2214yes—
19C7×#387C2214yes—
20C7×#387C2214yes—
21C7×#382C2212yes—
22#89757C2212yes—
23#89753C2214yes—
24#89750C2214yes—
25#89750C2214yes—
26#89749C2214yes—
27C4⋊C4×Dic7C2214yes—
28C4⋊C4×Dic7C2212yes—
29C4⋊C4×Dic7C2212yes—
30C4⋊C4×Dic7C2212yes—
31C4⋊C4×Dic7C2214yes—
32#89745C2212yes—
33#89745C2212yes—
34#89744C2214yes—
35#89744C2214yes—
36#89708C2211yescharacteristic
37#89705C2214yes—
38#89705C2214yes—
39C42×Dic7C2211yescharacteristic
40C42×Dic7C2214yes—
41#89700C2214yes—
42#89427C2214yes—
43#89427C2214yes—
44#89426C2214yes—
45#89426C2214yes—
46#89425C2214yes—
47#89425C2214yes—
48#89424C2214yes—
49#89424C2214yes—
50#89422C2214yes—
51#89422C2214yes—
52#89421C2214yes—
53#89421C2214yes—
54#89421C2214yes—
55#89421C2214yes—
56#89419C2214yes—
57#89419C2214yes—
58C4×#564—71nomaximal, Sylow
59C7×#125C2×C418yes—
60C2×C7×C4⋊C4C2312yes—
61C2×C7×C4⋊C4C2314yes—
62C2×C7×C4⋊C4C2312yes—
63C2×C7×C4⋊C4C2312yes—
64C2×C7×C4⋊C4C2314yes—
65C2×C42×C7C2314yes—
66C2×C42×C7C2311yescharacteristic
67C2×C4⋊Dic7C2312yes—
68C2×C4⋊Dic7C2314yes—
69C2×C4⋊Dic7C2312yes—
70C2×C4⋊Dic7C2312yes—
71C2×C4⋊Dic7C2314yes—
72C2×Dic7⋊C4C2314yes—
73C2×Dic7⋊C4C2314yes—
74C2×Dic7⋊C4C2314yes—
75C2×Dic7⋊C4C2314yes—
76C2×Dic7⋊C4C2314yes—
77C2×Dic7⋊C4C2314yes—
78C2×Dic7⋊C4C2314yes—
79C2×Dic7⋊C4C2314yes—
80C2×C4×Dic7C2314yes—
81C2×C4×Dic7C2311yescharacteristic
82C2×C4×Dic7C2314yes—
83C2×C4×Dic7C2312yes—
84C2×C4×Dic7C2314yes—
85C2×C4×Dic7C2314yes—
86C2×C4×Dic7C2314yes—
87C2×C4×Dic7C2311yescharacteristic
88C2×C4×Dic7C2312yes—
89C2×C4×Dic7C2314yes—
90#6453C2×C4132yes—
91#6452C2×C4132yes—
92#6452C2×C4132yes—
93#6433C2×C4116yes—
94C7×#95C2314yes—
95C7×#95C2314yes—
96C7×#95C2314yes—
97C7×#95C2314yes—
98C7⋊#95C2314yes—
99C7⋊#95C2314yes—
100C7⋊#95C2314yes—
101C7⋊#95C2314yes—
102C7⋊#95C2314yes—
103C7⋊#95C2314yes—
104C7⋊#95C2314yes—
105C7⋊#95C2314yes—
106C7⋊#95C2314yes—
107C7⋊#95C2314yes—
108C7⋊#95C2314yes—
109C2×#564—71no—
110#2382—74no—
111#2382—74no—
112#2382—74no—
113#2382—72no—
114#2382D711yescharacteristic
115C4×#125—74no—
116C4×#125—74no—
117C4×#125—74no—
118C4×#125—72no—
119C4×#125—71no—
120C22×C4×C7C2414yes—
121C22×C4×C7C2414yes—
122C22×C4×C7C2412yes—
123C22×C4×C7C2414yes—
124C22×C4×C7C2411yescharacteristic
125C22×Dic7C2411yescharacteristic
126C22×Dic7C2412yes—
127C22×Dic7C2414yes—
128C22×Dic7C2414yes—
129C22×Dic7C2414yes—
130C22×Dic7C2411yescharacteristic
131C7×C4⋊C4—42no—
132C7×C4⋊C4—42no—
133C7×C4⋊C4—44no—
134C7×C4⋊C4C22×C418yes—
135C7×C4⋊C4C22×C4116yes—
136C42×C7—44no—
137C42×C7C22×C414yes—
138C4⋊Dic7—42no—
139C4⋊Dic7—44no—
140C4⋊Dic7C22×C418yes—
141C4⋊Dic7C22×C418yes—
142C4⋊Dic7C22×C4116yes—
143Dic7⋊C4—44no—
144Dic7⋊C4—44no—
145Dic7⋊C4—44no—
146Dic7⋊C4—44no—
147Dic7⋊C4C22×C4116yes—
148Dic7⋊C4C22×C4116yes—
149Dic7⋊C4C22×C4116yes—
150Dic7⋊C4C22×C4116yes—
151C4×Dic7—28no—
152C4×Dic7—28no—
153C4×Dic7—22no—
154C4×Dic7—24no—
155C4×Dic7—28no—
156C4×Dic7—44no—
157C4×Dic7—44no—
158C4×Dic7—41no—
159C4×Dic7—42no—
160C4×Dic7C22×C4116yes—
161#564—716no—
162C2×#125—74no—
163C2×#125—74no—
164C2×#125—74no—
165C2×#125—72no—
166C2×#125C2×D711yescharacteristic
167#389—74no—
168#389—74no—
169#389—72no—
170#389—72no—
171#389—72no—
172#389—74no—
173#389—74no—
174#389—74no—
175#389C2×D714yes—
176#387—74no—
177#387—74no—
178#387—74no—
179#387—74no—
180#387—74no—
181#387—74no—
182#387—74no—
183#387—74no—
184#387—74no—
185#387—74no—
186#387—74no—
187#387—74no—
188#387—74no—
189#387C2×D714yes—
190#387C2×D714yes—
191#382—74no—
192#382—71no—
193#382—74no—
194#382—74no—
195#382C2×D712yes—
196C4×C4⋊C4—74no—
197C4×C4⋊C4—72no—
198C4×C4⋊C4—72no—
199C4×C4⋊C4—74no—
200C4×C4⋊C4—72no—
201C4×C4⋊C4—74no—
202C4×C4⋊C4—74no—
203C4×C4⋊C4—74no—
204C4×C4⋊C4—74no—
205C43—74no—
206C43—71no—
207C23×C7C2511yescharacteristic
208C2×C4×C7—22no—
209C2×C4×C7—24no—
210C2×C4×C7—21no—
211C2×C4×C7—21no—
212C2×C4×C7—22no—
213C2×C4×C7—24no—
214C2×C4×C7—24no—
215C2×C4×C7—24no—
216C2×C4×C7—24no—
217C2×C4×C7—24no—
218C2×C4×C7—24no—
219C2×C4×C7—24no—
220C2×C4×C7C23×C418yes—
221C2×C4×C7C23×C412yes—
222C2×C4×C7C23×C414yes—
223C2×Dic7—24no—
224C2×Dic7—24no—
225C2×Dic7—24no—
226C2×Dic7—24no—
227C2×Dic7—21no—
228C2×Dic7—22no—
229C2×Dic7—24no—
230C2×Dic7—22no—
231C2×Dic7—22no—
232C2×Dic7—24no—
233C2×Dic7—24no—
234C2×Dic7—22no—
235C2×Dic7C23×C418yes—
236C2×Dic7C23×C418yes—
237C2×Dic72−1+412yes—
238C2×Dic72−1+414yes—
239#125—732no—
240#125—732no—
241#125—732no—
242#125—716no—
243#125C4×D718yes—
244C2×C4⋊C4—74no—
245C2×C4⋊C4—74no—
246C2×C4⋊C4—74no—
247C2×C4⋊C4—74no—
248C2×C4⋊C4—72no—
249C2×C4⋊C4—74no—
250C2×C4⋊C4—72no—
251C2×C4⋊C4—74no—
252C2×C4⋊C4—74no—
253C2×C4⋊C4—74no—
254C2×C4⋊C4—74no—
255C2×C4⋊C4—74no—
256C2×C4⋊C4—72no—
257C2×C4⋊C4C22×D714yes—
258C2×C4⋊C4C22×D712yes—
259C2×C4⋊C4C22×D712yes—
260C2×C4⋊C4C22×D714yes—
261C2×C4⋊C4C22×D712yes—
262C2×C42—74no—
263C2×C42—74no—
264C2×C42—74no—
265C2×C42—74no—
266C2×C42—72no—
267C2×C42—71no—
268C2×C42—74no—
269C2×C42—74no—
270C2×C42—72no—
271C2×C42—71no—
272C2×C42C22×D714yes—
273C2×C42C22×D711yescharacteristic
274#95—74no—
275#95—74no—
276#95—74no—
277#95—74no—
278#95—74no—
279#95—74no—
280#95—74no—
281#95—74no—
282#95—74no—
283#95—74no—
284#95—74no—
285#95C22×D714yes—
286#95C22×D714yes—
287#95C22×D714yes—
288#95C22×D714yes—
289C22×C7C24×C411yescharacteristic
290C22×C7C2×2−1+411yescharacteristic
291C22×C7C2×2−1+412yes—
292C22×C7#58512yes—
293C22×C7#58511yescharacteristic
294C4×C7—22no—
295C4×C7—24no—
296C4×C7—28no—
297C4×C7—44no—
298C4×C7—44no—
299Dic7—28no—
300Dic7—28no—
301Dic7—41no—
302Dic7—42no—
303Dic7—44no—
304Dic7#56414yescomplemented
305C22×C4—71no—
306C22×C4—74no—
307C22×C4—74no—
308C22×C4—71no—
309C22×C4—74no—
310C22×C4—72no—
311C22×C4C23×D714yes—
312C22×C4C23×D714yes—
313C22×C4C23×D714yes—
314C22×C4C23×D712yes—
315C22×C4C23×D711yescharacteristic
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—716no—
320C4⋊C4—716no—
321C4⋊C4—716no—
322C4⋊C4—716no—
323C4⋊C4—716no—
324C4⋊C4—78no—
325C4⋊C4—78no—
326C4⋊C4—284no—
327C4⋊C4—284no—
328C4⋊C4—284no—
329C4⋊C4—284no—
330C4⋊C4—284no—
331C4⋊C4—282no—
332C4⋊C4C2×C4×D7116yes—
333C4⋊C4C2×C4×D718yes—
334C42—44no—
335C42—716no—
336C42—142no—
337C42—148no—
338C42—144no—
339C42—148no—
340C42—148no—
341C42—282no—
342C42—281no—
343C42—284no—
344C42—284no—
345C42C2×C4×D714yes—
346C2×C7C4×2−1+411yescharacteristic
347C2×C7C4×2−1+412yes—
348C2×C7C2×#56411yescharacteristic
349C2×C7#353111yescharacteristic
350C2×C7#353112yes—
351C23C24×D711yescharacteristic
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—24no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—72no—
365C2×C4—74no—
366C2×C4—78no—
367C2×C4—78no—
368C2×C4—144no—
369C2×C4—144no—
370C2×C4—144no—
371C2×C4—144no—
372C2×C4—142no—
373C2×C4—144no—
374C2×C4—142no—
375C2×C4—141no—
376C2×C4—144no—
377C2×C4—142no—
378C2×C4—142no—
379C2×C4—144no—
380C2×C4C22×C4×D718yes—
381C2×C4C22×C4×D712yes—
382C2×C4C22×C4×D714yes—
383C7C4×#56411yescharacteristic, complemented, Sylow
384C22C23×C4×D711yescharacteristic
385C22#9061812yes—
386C22D7×2−1+411yescharacteristic
387C22D7×2−1+412yes—
388C22#9062011yescharacteristic
389C4—22no—
390C4—28no—
391C4—24no—
392C4—44no—
393C4—44no—
394C4—74no—
395C4—148no—
396C4—148no—
397C4—281no—
398C4—284no—
399C4—282no—
400C2#1174294111yescharacteristic
401C2#1174295512yes—
402C2#1174416211yescharacteristic
403C2#1174418412yes—
404C2D7×#56411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC7×#2382C2
socle, prime radical, cosocleC23×C7C25
hypercenter#2382D7
hypocenterC7C4×#564
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #2382

Complements

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

NH ≅ G/Ntype
13. C7⋊#564C4semidirect
304. Dic7#564semidirect
383. C7C4×#564semidirect

Maximal subgroups

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

subgroupindex
C2×C7⋊#5642
C7×#23822
Dic7×#1252
#117334382
#117334112
#117324242
#117324232
#117323882
#117316062
#117315952
C4×#5647