Finite Group Databasebeta

← #13245409

Subgroups — #13245409

Order 1792.

782 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#556C211yesmaximal, characteristic
2#11729442C212yesmaximal, complemented
3C4×#6408C211yesmaximal, characteristic
4#11729356C211yesmaximal, characteristic
5#11729344C212yesmaximal, complemented
6C7×#556—24no—
7C7×#556—24no—
8C2×C7×C4⋊Q8C411yescharacteristic, complemented
9C2×C7×#126—21no—
10C2×C7×#126—21no—
11C2×C7×#124—21no—
12C2×C7×C22⋊Q8C411yescharacteristic, complemented
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8—21no—
15C2×C4×C7×Q8C411yescharacteristic, complemented
16C2×C4×C7×D4C2211yescharacteristic
17C2×C7×#117C411yescharacteristic, complemented
18C2×C7×#117C2211yescharacteristic
19C2×#6408—21no—
20C4×C22⋊Dic7C2212yes—
21#89979C2212yes—
22#89333—24no—
23#89327—24no—
24#89323—21no—
25#89321—21no—
26#89320—21no—
27#8365—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C22×C7×Q8C2×C411yescharacteristic
30C22×C7×D4D411yescharacteristic
31C23×C4×C7C2×C411yescharacteristic
32C23×C4×C7C2311yescharacteristic
33C7×C4⋊Q8—22no—
34C7×C4⋊Q8—22no—
35C7×#126—28no—
36C7×#126—28no—
37C7×#124—28no—
38C7×C22⋊Q8—22no—
39C7×C22⋊Q8—22no—
40C7×C22⋊Q8—22no—
41C7×C22⋊Q8D414yes—
42C4×C7×Q8—28no—
43C4×C7×Q8—22no—
44C4×C7×Q8—22no—
45C4×C7×D4—22no—
46C4×C7×D4—22no—
47C7×#117—22no—
48C7×#117—22no—
49C7×#117—22no—
50C7×#117D414yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2×C411yescharacteristic
56C2×C7×C4⋊C4C2×C411yescharacteristic
57C2×C7×C4⋊C4C2×C411yescharacteristic
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—21no—
63C2×C7×C22⋊C4D411yescharacteristic
64C2×C7×C22⋊C4D411yescharacteristic
65C2×C42×C7—21no—
66C2×C42×C7—21no—
67C2×C42×C7—21no—
68C2×C42×C7C2×C411yescharacteristic
69C2×C22⋊Dic7—22no—
70C2×C4×Dic7—22no—
71C2×#1136—22no—
72#6408—24no—
73C7⋊#95—22no—
74C22⋊(C7⋊C8)—22no—
75C22⋊(C7⋊C8)—22no—
76C2×#556D711yescharacteristic
77#1826—72no—
78C4×#99—71no—
79#1508—72no—
80#1465—71no—
81C24×C7C22⋊C411yescharacteristic
82C24×C7C2×D411yescharacteristic
83C2×C7×Q8—24no—
84C2×C7×Q8—21no—
85C2×C7×Q8—42no—
86C2×C7×Q8—42no—
87C2×C7×Q8—42no—
88C2×C7×Q8C22⋊C412yes—
89C2×C7×D4—22no—
90C2×C7×D4—22no—
91C2×C7×D4—41no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—42no—
99C22×C4×C7—42no—
100C22×C4×C7C22⋊C411yescharacteristic
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C22×C411yescharacteristic
104C22×C4×C7C2×D412yes—
105C22×C4×C7C2×D412yes—
106C22×C4×C7C2×D411yescharacteristic
107C22×Dic7—42no—
108C7×C4⋊C4—24no—
109C7×C4⋊C4—24no—
110C7×C4⋊C4—24no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—24no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4C22⋊C412yes—
117C7×C4⋊C4C22⋊C412yes—
118C7×C4⋊C4C22⋊C412yes—
119C7×C22⋊C4—24no—
120C7×C22⋊C4—24no—
121C7×C22⋊C4—24no—
122C7×C22⋊C4—24no—
123C7×C22⋊C4—22no—
124C7×C22⋊C4—22no—
125C42×C7—24no—
126C42×C7—24no—
127C42×C7—21no—
128C42×C7—21no—
129C42×C7—24no—
130C22⋊Dic7—24no—
131C22⋊Dic7—42no—
132C4×Dic7—44no—
133#1136—44no—
134C2×C7⋊C8—42no—
135#556—24no—
136#556—24no—
137C2×C4⋊Q8Dic711yescharacteristic, complemented
138C2×#126—21no—
139C2×#126—21no—
140C2×#124—21no—
141C2×C22⋊Q8Dic711yescharacteristic, complemented
142C2×C22⋊Q8C2×D711yescharacteristic
143C2×C4×Q8—21no—
144C2×C4×Q8Dic711yescharacteristic, complemented
145C2×C4×D4C2×D711yescharacteristic
146C2×#117Dic711yescharacteristic, complemented
147C2×#117C2×D711yescharacteristic
148C2×#99—141no—
149#407—72no—
150C4×C22⋊C4—72no—
151#342—141no—
152#339—144no—
153#328—144no—
154#324—141no—
155#323—141no—
156C23×C7—21no—
157C23×C7—21no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—21no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7C2×C22⋊C411yescharacteristic
166C23×C7#12012yes—
167C7×Q8—42no—
168C7×Q8—48no—
169C7×Q8—42no—
170C7×D4—44no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—24no—
174C2×C4×C7—22no—
175C2×C4×C7—22no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—42no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7#10412yes—
194C2×C4×C7C2×C22⋊C411yescharacteristic
195C2×C4×C7C2×C22⋊C411yescharacteristic
196C2×C4×C7#12012yes—
197C2×Dic7—44no—
198C2×Dic7—82no—
199C2×Dic7—82no—
200C7⋊C8—82no—
201C22×Q8—21no—
202C22×Q8C2×Dic711yescharacteristic
203C22×D4C22⋊D711yescharacteristic
204C23×C4C2×Dic711yescharacteristic
205C23×C4C22×D711yescharacteristic
206C2×M16—142no—
207C4⋊Q8—22no—
208C4⋊Q8—22no—
209#126—28no—
210#126—28no—
211#124—28no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8—22no—
215C22⋊Q8C22⋊D714yes—
216C4×Q8—28no—
217C4×Q8—22no—
218C4×Q8—22no—
219C4×D4—22no—
220C4×D4—22no—
221#117—22no—
222#117—22no—
223#117—22no—
224#117C22⋊D714yes—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4C2×Dic711yescharacteristic
230C2×C4⋊C4C2×Dic711yescharacteristic
231C2×C4⋊C4C2×Dic711yescharacteristic
232C2×C4⋊C4C22⋊D711yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—142no—
238C2×C22⋊C4C22⋊D711yescharacteristic
239C2×C22⋊C4C22⋊D711yescharacteristic
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—142no—
244C2×C42C2×Dic711yescharacteristic
245#99—144no—
246C22⋊C8—142no—
247C22⋊C8—142no—
248#95—142no—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—22no—
252C22×C7—22no—
253C22×C7—22no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—81no—
258C22×C7—81no—
259C22×C7—81no—
260C22×C7—81no—
261C22×C7#37911yescharacteristic
262C22×C7C2×#10411yescharacteristic
263C22×C7#42111yescharacteristic
264C4×C7—22no—
265C4×C7—21no—
266C4×C7—22no—
267C4×C7—41no—
268C4×C7—42no—
269C4×C7—41no—
270C4×C7—42no—
271C4×C7—42no—
272C4×C7—42no—
273C4×C7#41012yes—
274Dic7—84no—
275C24C22⋊Dic711yescharacteristic
276C24C2×C22⋊D711yescharacteristic
277C2×Q8—24no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8C22⋊Dic712yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—41no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—282no—
295C22×C4C22⋊Dic711yescharacteristic
296C22×C4C22⋊Dic711yescharacteristic
297C22×C4C22⋊Dic711yescharacteristic
298C22×C4C22×Dic711yescharacteristic
299C22×C4C2×C22⋊D712yes—
300C22×C4C2×C22⋊D712yes—
301C22×C4C2×C22⋊D711yescharacteristic
302M16—284no—
303C2×C8—282no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4C22⋊Dic712yes—
313C4⋊C4C22⋊Dic712yes—
314C4⋊C4C22⋊Dic712yes—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—144no—
322C22⋊C4—282no—
323C42—24no—
324C42—24no—
325C42—21no—
326C42—21no—
327C42—24no—
328C42—284no—
329C2×C7—21no—
330C2×C7—21no—
331C2×C7—42no—
332C2×C7—81no—
333C2×C7#178211yescharacteristic
334C2×C7#178911yescharacteristic
335C2×C7#179611yescharacteristic
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23C2×C22⋊Dic711yescharacteristic
346C23#651612yes—
347Q8—42no—
348Q8—48no—
349Q8—42no—
350D4—44no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—24no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—284no—
374C2×C4—562no—
375C2×C4—562no—
376C2×C4#641112yes—
377C2×C4C2×C22⋊Dic711yescharacteristic
378C2×C4C2×C22⋊Dic711yescharacteristic
379C2×C4#651612yes—
380C8—562no—
381C7#836511yescharacteristic, complemented, Sylow
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22C2×#641111yescharacteristic
395C22#9000811yescharacteristic
396C22#9001911yescharacteristic
397C4—22no—
398C4—22no—
399C4—21no—
400C4—42no—
401C4—42no—
402C4—41no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—564no—
407C4#9001212yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—81no—
412C2#1172989211yescharacteristic
413C2#1172989311yescharacteristic
414C2#1172989811yescharacteristic
subgroupG/H
centerC22#90019
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#556C2
socleC22×C7#379
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenterC2×#556D7
hypocenterC7#8365
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
2. #11729442C2semidirect
5. #11729344C2semidirect
8. C2×C7×C4⋊Q8C4semidirect
12. C2×C7×C22⋊Q8C4semidirect
15. C2×C4×C7×Q8C4semidirect
17. C2×C7×#117C4semidirect
137. C2×C4⋊Q8Dic7semidirect
141. C2×C22⋊Q8Dic7semidirect
144. C2×C4×Q8Dic7semidirect
146. C2×#117Dic7semidirect
381. C7#8365semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5562
#117294422
C4×#64082
#117293562
#117293442
#83657