Finite Group Databasebeta

← #13253871

Subgroups — #13253871

Order 1792.

566 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2851C211yesmaximal, characteristic
2#11729288C211yesmaximal, characteristic
3C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic
4#11725430C212yesmaximal
5#11725430C212yesmaximal
6C2×C7×#125C411yescharacteristic, complemented
7C22×C42×C7C2211yescharacteristic
8C7×#389C411yescharacteristic, complemented
9C7×#389C411yescharacteristic, complemented
10C7×#387—21no—
11C7×#387—21no—
12C7×#385—21no—
13C7×#385—21no—
14C7×#382C411yescharacteristic, complemented
15C7×#380C2211yescharacteristic
16C7×#380C2211yescharacteristic
17C2×C7⋊#95—21no—
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2212yes—
20C2×C22⋊(C7⋊C8)C2212yes—
21C2×C4×C7⋊C8—21no—
22#89319—21no—
23#9886—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7—22no—
28C7×#125—42no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C42×C7—42no—
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C22×C7⋊C8—22no—
45C7×#95—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7⋊#95—24no—
52C7⋊#95—24no—
53C7⋊#95—42no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)Q814yes—
56C4×C7⋊C8—42no—
57#2851D711yescharacteristic
58C4×C22⋊C8—71no—
59#1434—71no—
60#1294—72no—
61#1294—72no—
62C24×C7C2×D411yescharacteristic
63C22×C4×C7—22no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22×C411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7C2×Q812yes—
82C22×C4×C7#4112yes—
83C22×Dic7—24no—
84C22×Dic7—42no—
85C22×Dic7—42no—
86C22×Dic7—42no—
87C22×Dic7—42no—
88C22×Dic7—42no—
89C22×Dic7—42no—
90C7×C4⋊C4—41no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—42no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—42no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C42×C7—22no—
101C42×C7—22no—
102C42×C7—41no—
103C42×C7—41no—
104C2×C7⋊C8—24no—
105C2×C7⋊C8—42no—
106C2×C7⋊C8—42no—
107C2×#125Dic711yescharacteristic, complemented
108C22×C42C2×D711yescharacteristic
109C2×C22⋊C8—72no—
110C2×C4×C8—141no—
111#389Dic711yescharacteristic, complemented
112#389Dic711yescharacteristic, complemented
113#387—21no—
114#387—21no—
115#385—21no—
116#385—21no—
117#382Dic711yescharacteristic, complemented
118#380C2×D711yescharacteristic
119#380C2×D711yescharacteristic
120C2×#95—72no—
121C2×#95—141no—
122C2×#95—141no—
123#336—141no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7C2×C22⋊C411yescharacteristic
129C23×C7#12411yescharacteristic
130C23×C7C4⋊Q811yescharacteristic
131C23×C7#13512yes—
132C23×C7#13512yes—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—44no—
150C2×C4×C7—44no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7#9912yes—
164C2×C4×C7#10412yes—
165C2×C4×C7C4×Q812yes—
166C2×C4×C7C22⋊Q812yes—
167C2×Dic7—44no—
168C2×Dic7—44no—
169C2×Dic7—44no—
170C2×Dic7—82no—
171C2×Dic7—82no—
172C2×Dic7—82no—
173C2×Dic7—82no—
174C7⋊C8—44no—
175C23×C4—142no—
176C23×C4C22⋊D711yescharacteristic
177C23×C4C22⋊D711yescharacteristic
178C23×C4C22×D711yescharacteristic
179C22×C8—142no—
180#125—42no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4C2×Dic711yescharacteristic
186C2×C4⋊C4C2×Dic711yescharacteristic
187C2×C4⋊C4C2×Dic711yescharacteristic
188C2×C4⋊C4C2×Dic711yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C42—42no—
192C2×C42C2×Dic711yescharacteristic
193C2×C42C2×Dic711yescharacteristic
194C2×C42C22⋊D711yescharacteristic
195C2×C42C22⋊D711yescharacteristic
196C22⋊C8—74no—
197C22⋊C8—142no—
198C4×C8—282no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—144no—
206#95—144no—
207#95—282no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—22no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—22no—
216C22×C7#39111yescharacteristic
217C22×C7C2×#9911yescharacteristic
218C22×C7#41011yescharacteristic
219C22×C7#41611yescharacteristic
220C22×C7#41611yescharacteristic
221C22×C7C2×#10411yescharacteristic
222C22×C7#42111yescharacteristic
223C22×C7#48812yes—
224C22×C7#49912yes—
225C4×C7—22no—
226C4×C7—22no—
227C4×C7—22no—
228C4×C7—22no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—81no—
232C4×C7—81no—
233C4×C7—81no—
234C4×C7—81no—
235Dic7—84no—
236C24C2×C22⋊D711yescharacteristic
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—144no—
250C22×C4—282no—
251C22×C4—282no—
252C22×C4—282no—
253C22×C4—282no—
254C22×C4—282no—
255C22×C4—282no—
256C22×C4C22⋊Dic711yescharacteristic
257C22×C4C22⋊Dic711yescharacteristic
258C22×C4C22⋊Dic711yescharacteristic
259C22×C4C22⋊Dic711yescharacteristic
260C22×C4Q8×D712yes—
261C22×C4#116112yes—
262C22×C4C22×Dic711yescharacteristic
263C22×C4C2×C22⋊D711yescharacteristic
264C2×C8—144no—
265C2×C8—282no—
266C2×C8—282no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C42—22no—
278C42—22no—
279C42—41no—
280C42—41no—
281C2×C7—22no—
282C2×C7—22no—
283C2×C7#150011yescharacteristic
284C2×C7#150111yescharacteristic
285C2×C7#151611yescharacteristic
286C2×C7#152111yescharacteristic
287C2×C7#198611yescharacteristic
288C2×C7#199111yescharacteristic
289C2×C7#199611yescharacteristic
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23#650711yescharacteristic
295C23#651011yescharacteristic
296C23#651312yes—
297C23#651312yes—
298C23C2×C22⋊Dic711yescharacteristic
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—284no—
330C2×C4—284no—
331C2×C4—284no—
332C2×C4—562no—
333C2×C4—562no—
334C2×C4—562no—
335C2×C4—562no—
336C2×C4#640812yes—
337C2×C4#641112yes—
338C2×C4Q8×Dic712yes—
339C2×C4#650912yes—
340C8—284no—
341C7#988611yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22#8983612yes—
351C22#8984012yes—
352C22C2×#640811yescharacteristic
353C22#9000211yescharacteristic
354C22#9000511yescharacteristic
355C22#9000511yescharacteristic
356C22C2×#641111yescharacteristic
357C22#9000811yescharacteristic
358C22#9001211yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—41no—
364C4—41no—
365C4—81no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—564no—
370C2—22no—
371C2—22no—
372C2#1172945511yescharacteristic
373C2#1172946111yescharacteristic
374C2#1172946411yescharacteristic
375C2#1172949411yescharacteristic
376C2#1172950811yescharacteristic
377C2#1172951111yescharacteristic
378C2#1172959511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2851C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2851D7
hypocenterC7#9886
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
6. C2×C7×#125C4semidirect
8. C7×#389C4semidirect
9. C7×#389C4semidirect
14. C7×#382C4semidirect
107. C2×#125Dic7semidirect
111. #389Dic7semidirect
112. #389Dic7semidirect
117. #382Dic7semidirect
341. C7#9886semidirect

Maximal subgroups

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

subgroupindex
C7×#28512
#117292882
C4×C22⋊(C7⋊C8)2
#117254302
#98867