Finite Group Databasebeta

← #13248796

Subgroups — #13248796

Order 1792.

800 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#395C211yesmaximal, characteristic
2#11739601C211yesmaximal, characteristic
3#11729886C212yesmaximal
4C2×#89323C211yesmaximal, characteristic
5#11725418C212yesmaximal
6C23×C7×Q8C2211yescharacteristic
7C22×C7×C4⋊C4—21no—
8C22×C7×C4⋊C4C2211yescharacteristic
9C2×#6489—21no—
10C7×#395—22no—
11C7×#395C414yescomplemented
12C2×C7×#95—21no—
13C2×C7×#95C2211yescharacteristic
14C2×#6409—24no—
15#89984—21no—
16#89984C2212yes—
17C2×C22⋊(C7⋊C8)C2212yes—
18#89757—24no—
19#89323—22no—
20#9015—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C22×C7×Q8—22no—
23C22×C7×Q8—22no—
24C22×C7×Q8—44no—
25C22×C7×Q8C2×C412yes—
26C23×C4×C7—21no—
27C23×C4×C7—21no—
28C23×C4×C7D411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—22no—
33C2×C7×C4⋊C4—24no—
34C2×C7×C4⋊C4—44no—
35C2×C7×C4⋊C4C2×C412yes—
36C2×C7×C4⋊C4D412yes—
37C2×C22⋊Dic7—22no—
38#6489—42no—
39#6489—42no—
40C2×C4⋊Dic7—22no—
41C2×C4⋊Dic7—22no—
42C2×C4⋊Dic7D414yes—
43C2×C4×Dic7—22no—
44C22×C7⋊C8—22no—
45C7×#95—24no—
46C7×#95—21no—
47C7×#95C2×C412yes—
48#6409—48no—
49C7⋊#95—22no—
50C7⋊#95—22no—
51C22⋊(C7⋊C8)—22no—
52C22⋊(C7⋊C8)—22no—
53C2×#395D711yescharacteristic
54#2316—71no—
55#1787—72no—
56C2×#324—71no—
57#1294—72no—
58C24×C7C2×D411yescharacteristic
59C2×C7×Q8—28no—
60C2×C7×Q8—24no—
61C2×C7×Q8—44no—
62C2×C7×Q8—44no—
63C2×C7×Q8—44no—
64C2×C7×Q8—48no—
65C2×C7×Q8—42no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—24no—
71C22×C4×C7—24no—
72C22×C4×C7—22no—
73C22×C4×C7—22no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—42no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7—41no—
80C22×C4×C7—41no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—42no—
84C22×C4×C7C22⋊C412yes—
85C22×C4×C7C22⋊C412yes—
86C22×C4×C7SD1612yes—
87C22×C4×C7Q1612yes—
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D412yes—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7#4112yes—
92C22×Dic7—24no—
93C7×C4⋊C4—24no—
94C7×C4⋊C4—24no—
95C7×C4⋊C4—48no—
96C22⋊Dic7—42no—
97C22⋊Dic7—42no—
98C4⋊Dic7—44no—
99C4⋊Dic7—44no—
100C4⋊Dic7—82no—
101C4×Dic7—44no—
102C2×C7⋊C8—44no—
103C2×C7⋊C8—42no—
104C23×Q8C2×D711yescharacteristic
105C2×#117—141no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4C2×D711yescharacteristic
108C2×#103—144no—
109C2×C22⋊C8—72no—
110#395—22no—
111#395Dic714yescomplemented
112#389—144no—
113#380—72no—
114#380—141no—
115C2×#95—21no—
116C2×#95C2×D711yescharacteristic
117#324—142no—
118C23×C7—22no—
119C23×C7—22no—
120C23×C7C2×C22⋊C411yescharacteristic
121C23×C7#12011yescharacteristic
122C23×C7#12311yescharacteristic
123C23×C7C2×SD1611yescharacteristic
124C23×C7C2×Q1611yescharacteristic
125C23×C7AGL(1,ℤ/8)11yescharacteristic
126C23×C7#13711yescharacteristic
127C7×Q8—28no—
128C7×Q8—48no—
129C7×Q8—44no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—24no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—24no—
138C2×C4×C7—24no—
139C2×C4×C7—22no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7—44no—
145C2×C4×C7—44no—
146C2×C4×C7—42no—
147C2×C4×C7—44no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—44no—
152C2×C4×C7—42no—
153C2×C4×C7—81no—
154C2×C4×C7#10314yescomplemented
155C2×C4×C7C4×D412yes—
156C2×C4×C7#12112yes—
157C2×Dic7—44no—
158C2×Dic7—44no—
159C2×Dic7—44no—
160C7⋊C8—82no—
161C22×Q8—21no—
162C22×Q8—22no—
163C22×Q8—22no—
164C22×Q8—44no—
165C22×Q8C2×Dic712yes—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C22⋊D711yescharacteristic
169C23×C4C22⋊D711yescharacteristic
170C23×C4C22×D711yescharacteristic
171C22×C8—142no—
172#117—282no—
173#117—282no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—74no—
179C2×C4⋊C4—142no—
180C2×C4⋊C4—142no—
181C2×C4⋊C4C2×Dic712yes—
182C2×C4⋊C4C22⋊D712yes—
183C2×C22⋊C4—142no—
184C2×C42—142no—
185#103—288no—
186C22⋊C8—142no—
187C22⋊C8—142no—
188#95—24no—
189#95—21no—
190#95—142no—
191#95—142no—
192#95C2×Dic712yes—
193C22×C7—22no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—22no—
197C22×C7—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—21no—
201C22×C7#38611yescharacteristic
202C22×C7C2×#10311yescharacteristic
203C22×C7#41711yescharacteristic
204C22×C7#45011yescharacteristic
205C22×C7#45111yescharacteristic
206C22×C7#45612yes—
207C22×C7B(4,𝔽2)12yes—
208C22×C7#48311yescharacteristic
209C22×C7#48411yescharacteristic
210C4×C7—24no—
211C4×C7—22no—
212C4×C7—22no—
213C4×C7—44no—
214C4×C7—44no—
215C4×C7—42no—
216Dic7—84no—
217C24C2×C22⋊D711yescharacteristic
218C2×Q8—28no—
219C2×Q8—24no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—48no—
224C2×Q8—42no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—144no—
244C22×C4#114312yes—
245C22×C4#114412yes—
246C22×C4C22⋊Dic712yes—
247C22×C4C22⋊Dic712yes—
248C22×C4D4×D712yes—
249C22×C4#115912yes—
250C22×C4C22×Dic711yescharacteristic
251C22×C4C2×C22⋊D711yescharacteristic
252C2×C8—284no—
253C2×C8—282no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—48no—
257C4⋊C4—284no—
258C4⋊C4—284no—
259C4⋊C4—562no—
260C22⋊C4—282no—
261C22⋊C4—282no—
262C42—284no—
263C2×C7—21no—
264C2×C7—21no—
265C2×C7—21no—
266C2×C7—21no—
267C2×C7#160412yes—
268C2×C7#160512yes—
269C2×C7#187211yescharacteristic
270C2×C7#188611yescharacteristic
271C2×C7#188911yescharacteristic
272C23—22no—
273C23—22no—
274C23#649811yescharacteristic
275C23#650011yescharacteristic
276C23C2×#114311yescharacteristic
277C23C2×#114411yescharacteristic
278C23#651211yescharacteristic
279C23#651411yescharacteristic
280C23C2×C22⋊Dic711yescharacteristic
281Q8—28no—
282Q8—48no—
283Q8—44no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4—284no—
309C2×C4—284no—
310C2×C4—284no—
311C2×C4#640914yescomplemented
312C2×C4D4×Dic712yes—
313C2×C4#650212yes—
314C8—562no—
315C7#901511yescharacteristic, complemented, Sylow
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—21no—
323C22—21no—
324C22#8980412yes—
325C22#8981511yescharacteristic
326C22#8981611yescharacteristic
327C22#8981811yescharacteristic
328C22#8982211yescharacteristic
329C22#8986712yes—
330C22#8999211yescharacteristic
331C22C2×#640911yescharacteristic
332C22#9000411yescharacteristic
333C4—22no—
334C4—22no—
335C4—24no—
336C4—44no—
337C4—44no—
338C4—42no—
339C4—564no—
340C2—21no—
341C2—21no—
342C2—21no—
343C2—21no—
344C2#1172746012yes—
345C2#1172746112yes—
346C2#1172942311yescharacteristic
347C2#1172944611yescharacteristic
348C2#1172953811yescharacteristic
subgroupG/H
centerC23#6500
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#395C2
socleC23×C7#120
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenterC2×#395D7
hypocenterC7#9015
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
11. C7×#395C4semidirect
111. #395Dic7semidirect
154. C2×C4×C7#103semidirect
311. C2×C4#6409semidirect
315. C7#9015semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3952
#117396012
#117298862
C2×#893232
#117254182
#90157