Finite Group Databasebeta

← #13249752

Subgroups — #13249752

Order 1792.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2356C211yesmaximal, characteristic, complemented
2#11739886C211yesmaximal, characteristic
3#11729375C211yesmaximal, characteristic, complemented
4#11727299C211yesmaximal, characteristic
5#11727223C211yesmaximal, characteristic
6#11725437C211yesmaximal, characteristic, complemented
7#11725426C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C2×#6505—21no—
10#90168C2211yescharacteristic
11C7×#391—21no—
12C7×#387—22no—
13C7×#383—21no—
14C7×#382C2211yescharacteristic
15C4×C7×C4⋊C4—21no—
16C4×C7×C4⋊C4C2211yescharacteristic
17C7×#376—21no—
18#89803C412yescomplemented
19#89800—22no—
20#89799C412yescomplemented
21#89738—21no—
22#89697C2211yescharacteristic
23#89697C2211yescharacteristic
24#89696C2211yescharacteristic
25Q8⋊(C7⋊C8)—22no—
26#89246—21no—
27#89246—21no—
28#89246—22no—
29#9182—71nomaximal, Sylow
30C22×C7×Q8D411yescharacteristic
31C2×#1157—21no—
32C4×C7×Q8—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8C2×C412yes—
35C4×C7×Q8D412yes—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4D411yescharacteristic
40C2×C42×C7—21no—
41C2×C42×C7—22no—
42C2×C42×C7—21no—
43C2×C42×C7C2311yescharacteristic
44C2×#1144—22no—
45#6505—42no—
46#6505—42no—
47C2×#1143—22no—
48#6493—21no—
49#6493—21no—
50C2×#1136—21no—
51#6438—21no—
52#6437C2×C411yescharacteristic
53C4×D28—21no—
54C4×D28—21no—
55#6432C2×C411yescharacteristic
56C7×#95—22no—
57C7×#95—22no—
58C7×#95—22no—
59C7×#95—21no—
60C7×#95D411yescharacteristic
61C7×#95D411yescharacteristic
62C7×#95D412yes—
63C22⋊(C7⋊C8)—21no—
64C22⋊(C7⋊C8)—21no—
65#6384—22no—
66#6383—22no—
67C4⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)—21no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)C2×C412yes—
71C4×C7⋊C8—21no—
72C4×C7⋊C8—21no—
73#3037—71no—
74#2356D711yescharacteristic, complemented
75#1517—71no—
76#1502—71no—
77#1479—71no—
78#1303—71no—
79#1286—71no—
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8—21no—
83C2×C7×Q8—21no—
84C2×C7×Q8—21no—
85C2×C7×Q8C22⋊C412yes—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—22no—
90C22×C4×C7—21no—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C22×C4×C7C2×D411yescharacteristic
94C2×C22⋊D7—41no—
95#1157—41no—
96#1157—41no—
97#1157—41no—
98#1157—41no—
99C2×D28—21no—
100C2×D28—21no—
101C2×D28—21no—
102C2×C4×D7—41no—
103C2×Dic14—21no—
104C7×C4⋊C4—22no—
105C7×C4⋊C4—22no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—42no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4C22⋊C412yes—
111C42×C7—22no—
112C42×C7—22no—
113C42×C7—44no—
114C42×C7—41no—
115C42×C7—41no—
116C42×C7C22×C411yescharacteristic
117C42×C7C2×D411yescharacteristic
118C42×C7#4111yescharacteristic
119C42×C7#4111yescharacteristic
120#1144—44no—
121#1143—44no—
122(C2×D7)⋊C4—41no—
123(C2×D7)⋊C4—41no—
124C4⋊Dic7—21no—
125C4⋊Dic7—21no—
126#1136—41no—
127#1136—41no—
128C2×C7⋊C8—22no—
129C2×C7⋊C8—41no—
130C2×C7⋊C8—41no—
131C2×#137—141no—
132#532—71no—
133C2×C4×Q8C2×D711yescharacteristic
134#464—142no—
135#462—72no—
136#460—72no—
137#431—71no—
138#431—71no—
139#423—71no—
140#417—141no—
141#391—21no—
142#387—22no—
143#383—21no—
144#382C2×D711yescharacteristic
145C4×C4⋊C4—21no—
146C4×C4⋊C4C2×D711yescharacteristic
147#376—21no—
148#327—141no—
149#327—141no—
150#327—142no—
151Q8⋊C8—142no—
152C23×C7#12011yescharacteristic
153C22×D7—41no—
154C7×Q8—22no—
155C7×Q8—22no—
156C7×Q8—41no—
157C7×Q8—41no—
158C2×C4×C7—22no—
159C2×C4×C7—22no—
160C2×C4×C7—22no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—42no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7C2×C22⋊C411yescharacteristic
175C2×C4×C7C4×D411yescharacteristic
176C2×C4×C7C4×D411yescharacteristic
177C2×C4×C7#12111yescharacteristic
178C2×C4×C7#12111yescharacteristic
179C2×C4×C7#12311yescharacteristic
180C2×C4×C7#13511yescharacteristic
181C2×C4×C7#13511yescharacteristic
182C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
184C22⋊D7—81no—
185C22⋊D7—81no—
186C2×Dic7—41no—
187D28—41no—
188D28—41no—
189D28—81no—
190D28—81no—
191C4×D7—81no—
192C4×D7—81no—
193Dic14—41no—
194Dic14—41no—
195C7⋊C8—42no—
196C7⋊C8—81no—
197C7⋊C8—81no—
198C2×#41—141no—
199C22×Q8C22⋊D711yescharacteristic
200#137—282no—
201#137—282no—
202C2×Q16—142no—
203C2×SD16—142no—
204C2×M16—141no—
205C4⋊Q8—71no—
206#127—71no—
207#124—141no—
208#121—141no—
209#121—141no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8C4×D712yes—
213C4×Q8D2812yes—
214C4×D4—141no—
215C4×D4—141no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊D711yescharacteristic
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—21no—
223C2×C42C22×D711yescharacteristic
224C4⋊C8—72no—
225C4⋊C8—141no—
226C4⋊C8—141no—
227C4⋊C8—141no—
228#103—142no—
229#102—142no—
230C22⋊C8—141no—
231C22⋊C8—141no—
232C4×C8—141no—
233C4×C8—141no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—21no—
238#95C22⋊D711yescharacteristic
239#95C22⋊D711yescharacteristic
240#95C22⋊D712yes—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7#38611yescharacteristic
244C22×C7#44911yescharacteristic
245C22×C7#44911yescharacteristic
246C2×D7—81no—
247C2×D7—81no—
248C2×D7—81no—
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—42no—
254C4×C7—42no—
255C4×C7—42no—
256C4×C7—81no—
257C4×C7—82no—
258C4×C7#41611yescharacteristic
259C4×C7#41811yescharacteristic
260C4×C7#48211yescharacteristic
261C4×C7#48211yescharacteristic
262Dic7—81no—
263#41—281no—
264#41—281no—
265#41—281no—
266#41—281no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—141no—
273C2×Q8(C2×D7)⋊C412yes—
274C2×D4—141no—
275C2×D4—141no—
276C2×D4—141no—
277C2×D4—281no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—281no—
284C22×C4C2×C22⋊D711yescharacteristic
285C22×C4C2×C22⋊D711yescharacteristic
286C22×C4C2×C22⋊D711yescharacteristic
287Q16—284no—
288SD16—284no—
289M16—281no—
290M16—281no—
291C2×C8—142no—
292C2×C8—281no—
293C2×C8—281no—
294C4⋊C4—22no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—141no—
301C4⋊C4—141no—
302C4⋊C4(C2×D7)⋊C412yes—
303C22⋊C4—281no—
304C22⋊C4—281no—
305C42—22no—
306C42—22no—
307C42—44no—
308C42—41no—
309C42—41no—
310C42C2×C4×D711yescharacteristic
311C42C2×D2811yescharacteristic
312C42#115711yescharacteristic
313C42#115711yescharacteristic
314C2×C7—21no—
315C2×C7—21no—
316C2×C7#187511yescharacteristic
317C2×C7#190511yescharacteristic
318C2×C7#190611yescharacteristic
319D7—161no—
320C23—281no—
321C23#651611yescharacteristic
322Q8—22no—
323Q8—22no—
324Q8—41no—
325Q8—41no—
326Q8—281no—
327Q8—281no—
328D4—281no—
329D4—281no—
330D4—561no—
331D4—561no—
332D4—561no—
333D4—561no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—281no—
351C2×C4—561no—
352C2×C4—561no—
353C2×C4C2×(C2×D7)⋊C411yescharacteristic
354C2×C4C4×C22⋊D711yescharacteristic
355C2×C4C4×C22⋊D711yescharacteristic
356C2×C4#649211yescharacteristic
357C2×C4#649311yescharacteristic
358C2×C4#649311yescharacteristic
359C2×C4#651211yescharacteristic
360C2×C4#651211yescharacteristic
361C2×C4#651311yescharacteristic
362C2×C4#651311yescharacteristic
363C8—282no—
364C8—561no—
365C8—561no—
366C7#918211yescharacteristic, complemented, Sylow
367C22—21no—
368C22—21no—
369C22—561no—
370C22—561no—
371C22—561no—
372C22#8998511yescharacteristic
373C22#9001011yescharacteristic
374C22#9001011yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—81no—
383C4—82no—
384C4—561no—
385C4#8977311yescharacteristic
386C4#8977411yescharacteristic
387C4#8977511yescharacteristic
388C4#8977511yescharacteristic
389C2—21no—
390C2—21no—
391C2—1121no—
392C2#1172934011yescharacteristic
393C2#1172939911yescharacteristic
394C2#1172941611yescharacteristic
subgroupG/H
centerC22#89985
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2356C2
socleC22×C7#386
prime radical#90168C22
cosocleC2×C42×C7C23
hypercenter#2356D7
hypocenterC7#9182
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
1. C7×#2356C2semidirect
3. #11729375C2semidirect
6. #11725437C2semidirect
7. #11725426C2semidirect
18. #89803C4semidirect
20. #89799C4semidirect
74. #2356D7semidirect
366. C7#9182semidirect

Maximal subgroups

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

subgroupindex
C7×#23562
#117398862
#117293752
#117272992
#117272232
#117254372
#117254262
#91827