Finite Group Databasebeta

← #13247178

Subgroups — #13247178

Order 1792.

544 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2283C211yesmaximal, characteristic
2#11729288C211yesmaximal, characteristic
3#11729285C211yesmaximal, characteristic
4#11725430C212yesmaximal
5#11725430C212yesmaximal
6C22×C42×C7C2211yescharacteristic
7C7×#383—21no—
8C7×#382C411yescharacteristic, complemented
9C7×#382C411yescharacteristic, complemented
10C7×#381—21no—
11C7×#380C2211yescharacteristic
12C7×#380C2211yescharacteristic
13C4×C7×C4⋊C4C411yescharacteristic, complemented
14C4×C7×C4⋊C4C411yescharacteristic, complemented
15C4×C7×C22⋊C4—21no—
16C7×#376—21no—
17C2×C7⋊#95—21no—
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2212yes—
20C2×C22⋊(C7⋊C8)C2212yes—
21#89319—21no—
22#89319—21no—
23#8714—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7—22no—
28C2×C7×C4⋊C4C2×C411yescharacteristic
29C2×C7×C4⋊C4C2×C411yescharacteristic
30C2×C7×C4⋊C4C2×C411yescharacteristic
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4—21no—
33C2×C7×C22⋊C4—21no—
34C2×C42×C7—24no—
35C2×C42×C7—21no—
36C2×C42×C7—21no—
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7C2×C411yescharacteristic
39C2×C42×C7D411yescharacteristic
40C2×C42×C7D411yescharacteristic
41C22×C7⋊C8—22no—
42C7×#95—21no—
43C7×#95—21no—
44C7×#95—21no—
45C7×#95—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7⋊#95—24no—
51C7⋊#95—24no—
52C7⋊#95—42no—
53C22⋊(C7⋊C8)—24no—
54#2283D711yescharacteristic
55#1817—71no—
56#1434—71no—
57#1294—72no—
58#1294—72no—
59C24×C7C2×D411yescharacteristic
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—41no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7C22⋊C411yescharacteristic
74C22×C4×C7C22⋊C411yescharacteristic
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22×C411yescharacteristic
78C22×C4×C7C2×D411yescharacteristic
79C22×C4×C7#4114yes—
80C22×Dic7—24no—
81C22×Dic7—42no—
82C22×Dic7—42no—
83C22×Dic7—42no—
84C22×Dic7—42no—
85C22×Dic7—42no—
86C22×Dic7—42no—
87C7×C4⋊C4—41no—
88C7×C4⋊C4—41no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C22⋊C4—42no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C42×C7—22no—
95C42×C7—22no—
96C42×C7—22no—
97C42×C7—22no—
98C42×C7—42no—
99C42×C7—42no—
100C2×C7⋊C8—42no—
101C2×C7⋊C8—42no—
102C2×C7⋊C8—42no—
103C22×C42C2×D711yescharacteristic
104C2×C22⋊C8—72no—
105#383—21no—
106#382Dic711yescharacteristic, complemented
107#382Dic711yescharacteristic, complemented
108#381—21no—
109#380C2×D711yescharacteristic
110#380C2×D711yescharacteristic
111C4×C4⋊C4Dic711yescharacteristic, complemented
112C4×C4⋊C4Dic711yescharacteristic, complemented
113C4×C22⋊C4—21no—
114#376—21no—
115C2×#95—72no—
116C2×#95—141no—
117C2×#95—141no—
118#336—141no—
119#336—141no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7C2×C22⋊C411yescharacteristic
125C23×C7#12312yes—
126C23×C7#13512yes—
127C23×C7AGL(1,ℤ/8)11yescharacteristic
128C23×C7#13711yescharacteristic
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
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—44no—
145C2×C4×C7—44no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—41no—
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#10412yes—
161C2×C4×C7#11712yes—
162C2×C4×C7#12312yes—
163C2×Dic7—44no—
164C2×Dic7—44no—
165C2×Dic7—44no—
166C2×Dic7—82no—
167C2×Dic7—82no—
168C2×Dic7—82no—
169C2×Dic7—82no—
170C7⋊C8—82no—
171C23×C4—142no—
172C23×C4C22⋊D711yescharacteristic
173C23×C4C22⋊D711yescharacteristic
174C23×C4C22×D711yescharacteristic
175C22×C8—142no—
176C2×C4⋊C4C2×Dic711yescharacteristic
177C2×C4⋊C4C2×Dic711yescharacteristic
178C2×C4⋊C4C2×Dic711yescharacteristic
179C2×C4⋊C4C2×Dic711yescharacteristic
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C42—24no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42C2×Dic711yescharacteristic
186C2×C42C2×Dic711yescharacteristic
187C2×C42C22⋊D711yescharacteristic
188C2×C42C22⋊D711yescharacteristic
189C22⋊C8—144no—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—21no—
197#95—21no—
198#95—144no—
199#95—144no—
200#95—282no—
201C22×C7—22no—
202C22×C7—22no—
203C22×C7—22no—
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—22no—
209C22×C7#38111yescharacteristic
210C22×C7#41011yescharacteristic
211C22×C7#41011yescharacteristic
212C22×C7#41611yescharacteristic
213C22×C7#41711yescharacteristic
214C22×C7C2×#10411yescharacteristic
215C22×C7#42111yescharacteristic
216C22×C7#48212yes—
217C22×C7#48512yes—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—22no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—81no—
225C4×C7—81no—
226C4×C7—81no—
227C4×C7—81no—
228Dic7—84no—
229C24C2×C22⋊D711yescharacteristic
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—144no—
244C22×C4—282no—
245C22×C4—282no—
246C22×C4—282no—
247C22×C4—282no—
248C22×C4—282no—
249C22×C4—282no—
250C22×C4C22⋊Dic711yescharacteristic
251C22×C4C22⋊Dic711yescharacteristic
252C22×C4C22⋊Dic711yescharacteristic
253C22×C4C22⋊Dic711yescharacteristic
254C22×C4#115714yes—
255C22×C4C22×Dic711yescharacteristic
256C22×C4C2×C22⋊D711yescharacteristic
257C2×C8—282no—
258C2×C8—282no—
259C2×C8—282no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C42—22no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—42no—
272C42—42no—
273C2×C7—22no—
274C2×C7—22no—
275C2×C7#150011yescharacteristic
276C2×C7#150911yescharacteristic
277C2×C7#151211yescharacteristic
278C2×C7#152111yescharacteristic
279C2×C7#182011yescharacteristic
280C2×C7#182511yescharacteristic
281C2×C7#182611yescharacteristic
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23#649212yes—
287C23#651211yescharacteristic
288C23#651312yes—
289C23#651411yescharacteristic
290C23C2×C22⋊Dic711yescharacteristic
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—284no—
323C2×C4—284no—
324C2×C4—284no—
325C2×C4—562no—
326C2×C4—562no—
327C2×C4—562no—
328C2×C4—562no—
329C2×C4#641112yes—
330C2×C4#648912yes—
331C2×C4#649212yes—
332C8—562no—
333C7#871411yescharacteristic, complemented, Sylow
334C22—22no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22#8976812yes—
343C22#8977512yes—
344C22#8998311yescharacteristic
345C22#9000411yescharacteristic
346C22#9000511yescharacteristic
347C22C2×#641111yescharacteristic
348C22#9000811yescharacteristic
349C22#9001211yescharacteristic
350C22#9001211yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—564no—
362C2—22no—
363C2—22no—
364C2#1172932911yescharacteristic
365C2#1172934411yescharacteristic
366C2#1172934811yescharacteristic
367C2#1172950811yescharacteristic
368C2#1172950911yescharacteristic
369C2#1172951011yescharacteristic
370C2#1172951111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2283C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2283D7
hypocenterC7#8714
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2283

Complements

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

NH ≅ G/Ntype
8. C7×#382C4semidirect
9. C7×#382C4semidirect
13. C4×C7×C4⋊C4C4semidirect
14. C4×C7×C4⋊C4C4semidirect
106. #382Dic7semidirect
107. #382Dic7semidirect
111. C4×C4⋊C4Dic7semidirect
112. C4×C4⋊C4Dic7semidirect
333. C7#8714semidirect

Maximal subgroups

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

subgroupindex
C7×#22832
#117292882
#117292852
#117254302
#87147