Finite Group Databasebeta

← #13251556

Subgroups — #13251556

Order 1792.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2409C211yesmaximal, characteristic
2#11739746C211yesmaximal, characteristic
3#11729886C212yesmaximal, complemented
4#11729373C211yesmaximal, characteristic
5#11725431C212yesmaximal, complemented
6C23×C7×Q8C2211yescharacteristic
7C2×C7×C22⋊Q8—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#395C412yescomplemented
10C7×#393C412yescomplemented
11C7×#391—21no—
12C7×#386—21no—
13C7×#385—21no—
14C7×#380C2211yescharacteristic
15C2×#6409—24no—
16C2×#6408—21no—
17#89984C2212yes—
18C2×C22⋊(C7⋊C8)—21no—
19C2×C22⋊(C7⋊C8)C2212yes—
20D4×C7⋊C8—24no—
21#89697—21no—
22#89321—21no—
23#9467—71nomaximal, Sylow
24C22×C7×Q8—21no—
25C22×C7×Q8—22no—
26C22×C7×Q8—22no—
27C22×C7×Q8—44no—
28C22×C7×Q8C2×C412yes—
29C22×C7×D4D411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7D411yescharacteristic
32C23×C4×C7C2311yescharacteristic
33C7×C22⋊Q8—44no—
34C4×C7×D4—22no—
35C4×C7×D4—22no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4C2×C411yescharacteristic
42C2×C7×C22⋊C4D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C2×C22⋊Dic7—22no—
45C2×C4⋊Dic7—22no—
46C22×C7⋊C8—22no—
47C22×C7⋊C8D414yes—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95C2×C412yes—
51#6409—48no—
52#6408—42no—
53C7⋊#95—22no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—22no—
57C4⋊(C7⋊C8)—22no—
58C4×C7⋊C8—22no—
59#2972—71no—
60#2409D711yescharacteristic
61#1799—71no—
62#1787—72no—
63#1289—72no—
64C24×C7C22⋊C411yescharacteristic
65C24×C7C2×D411yescharacteristic
66C2×C7×Q8—24no—
67C2×C7×Q8—28no—
68C2×C7×Q8—24no—
69C2×C7×Q8—44no—
70C2×C7×Q8—44no—
71C2×C7×Q8—42no—
72C2×C7×Q8—48no—
73C2×C7×Q8—41no—
74C2×C7×Q8—41no—
75C2×C7×D4—22no—
76C2×C7×D4—22no—
77C2×C7×D4SD1612yes—
78C2×C7×D4Q1612yes—
79C22×C4×C7—22no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—22no—
83C22×C4×C7—42no—
84C22×C4×C7—44no—
85C22×C4×C7—42no—
86C22×C4×C7—42no—
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22×C411yescharacteristic
91C22×C4×C7C2×D412yes—
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7#4112yes—
94C22×Dic7—42no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—44no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—44no—
101C7×C22⋊C4—42no—
102C42×C7—21no—
103C42×C7—21no—
104C22⋊Dic7—42no—
105C22⋊Dic7—42no—
106C4⋊Dic7—44no—
107C2×C7⋊C8—24no—
108C2×C7⋊C8—44no—
109C2×C7⋊C8—82no—
110C23×Q8C2×D711yescharacteristic
111C2×C22⋊Q8—21no—
112C2×C4×D4C2×D711yescharacteristic
113C8×D4—144no—
114#431—141no—
115C2×#103—144no—
116C2×#99—141no—
117C2×C22⋊C8—72no—
118C2×C22⋊C8—141no—
119#395Dic712yescomplemented
120#393Dic712yescomplemented
121#391—21no—
122#386—21no—
123#385—21no—
124#380—72no—
125#380C2×D711yescharacteristic
126#342—141no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7#10312yescomplemented
134C23×C7C2×C22⋊C411yescharacteristic
135C23×C7#12112yes—
136C23×C7C2×SD1611yescharacteristic
137C23×C7C2×Q1611yescharacteristic
138C23×C7#13512yes—
139C7×Q8—28no—
140C7×Q8—44no—
141C7×Q8—44no—
142C7×Q8—44no—
143C7×D4—24no—
144C7×D4—24no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—24no—
148C2×C4×C7—22no—
149C2×C4×C7—24no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—41no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—48no—
156C2×C4×C7—42no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—81no—
164C2×C4×C7#10312yescomplemented
165C2×C4×C7C4×D412yes—
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12311yescharacteristic
168C2×Dic7—82no—
169C2×Dic7—82no—
170C2×Dic7—82no—
171C7⋊C8—44no—
172C22×Q8—21no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—44no—
176C22×Q8C2×Dic712yes—
177C22×D4C22⋊D711yescharacteristic
178C23×C4—21no—
179C23×C4C22⋊D711yescharacteristic
180C23×C4C22×D711yescharacteristic
181C22×C8—74no—
182C22×C8—142no—
183C22⋊Q8—44no—
184C4×D4—22no—
185C4×D4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—142no—
189C2×C4⋊C4C2×Dic711yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—142no—
193C2×C22⋊C4C2×Dic711yescharacteristic
194C2×C22⋊C4C22⋊D711yescharacteristic
195C2×C42C22⋊D711yescharacteristic
196C4⋊C8—142no—
197#103—288no—
198#99—282no—
199C22⋊C8—142no—
200C22⋊C8—142no—
201C22⋊C8—142no—
202C4×C8—142no—
203#95—22no—
204#95—22no—
205#95—142no—
206#95C2×Dic712yes—
207C22×C7—22no—
208C22×C7—21no—
209C22×C7—22no—
210C22×C7—21no—
211C22×C7—22no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—42no—
215C22×C7—42no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—81no—
219C22×C7#38611yescharacteristic
220C22×C7C2×#10311yescharacteristic
221C22×C7#41611yescharacteristic
222C22×C7#46512yes—
223C22×C7#46712yes—
224C4×C7—21no—
225C4×C7—22no—
226C4×C7—22no—
227C4×C7—44no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7—82no—
231C4×C7B(4,𝔽2)12yes—
232Dic7—162no—
233C24C22⋊Dic711yescharacteristic
234C24C2×C22⋊D711yescharacteristic
235C2×Q8—24no—
236C2×Q8—28no—
237C2×Q8—24no—
238C2×Q8—44no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—48no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4#114312yes—
247C2×D4#114412yes—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—282no—
257C22×C4C22⋊Dic711yescharacteristic
258C22×C4C22⋊Dic711yescharacteristic
259C22×C4C22⋊Dic711yescharacteristic
260C22×C4D4×D712yes—
261C22×C4#115912yes—
262C22×C4C22×Dic711yescharacteristic
263C22×C4C2×C22⋊D711yescharacteristic
264C2×C8—144no—
265C2×C8—284no—
266C2×C8—562no—
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4—44no—
270C4⋊C4—284no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—44no—
274C22⋊C4—42no—
275C22⋊C4—282no—
276C22⋊C4—282no—
277C42—21no—
278C42—21no—
279C2×C7—22no—
280C2×C7—21no—
281C2×C7—21no—
282C2×C7—22no—
283C2×C7—42no—
284C2×C7#188911yescharacteristic
285C2×C7#189211yescharacteristic
286C2×C7#190911yescharacteristic
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#640912yescomplemented
294C23#650212yes—
295C23C2×#114311yescharacteristic
296C23C2×#114411yescharacteristic
297C23#651312yes—
298C23C2×C22⋊Dic711yescharacteristic
299Q8—28no—
300Q8—44no—
301Q8—44no—
302Q8—44no—
303D4—24no—
304D4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—48no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—81no—
324C2×C4—562no—
325C2×C4—562no—
326C2×C4—562no—
327C2×C4#640912yescomplemented
328C2×C4D4×Dic712yes—
329C2×C4#649811yescharacteristic
330C2×C4#650011yescharacteristic
331C8—284no—
332C7#946711yescharacteristic, complemented, Sylow
333C22—21no—
334C22—22no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—42no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—81no—
345C22#8982012yes—
346C22#8982312yes—
347C22#8999211yescharacteristic
348C22C2×#640911yescharacteristic
349C22#9000511yescharacteristic
350C4—21no—
351C4—22no—
352C4—22no—
353C4—44no—
354C4—41no—
355C4—41no—
356C4—82no—
357C4—1122no—
358C4#8980412yes—
359C2—21no—
360C2—22no—
361C2—21no—
362C2—22no—
363C2—42no—
364C2#1172942311yescharacteristic
365C2#1172945011yescharacteristic
366C2#1172961911yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2409C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#2409D7
hypocenterC7#9467
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #11729886C2semidirect
5. #11725431C2semidirect
9. C7×#395C4semidirect
10. C7×#393C4semidirect
119. #395Dic7semidirect
120. #393Dic7semidirect
133. C23×C7#103semidirect
164. C2×C4×C7#103semidirect
293. C23#6409semidirect
327. C2×C4#6409semidirect
332. C7#9467semidirect

Maximal subgroups

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

subgroupindex
C7×#24092
#117397462
#117298862
#117293732
#117254312
#94677