Finite Group Databasebeta

← #13245484

Subgroups — #13245484

Order 1792.

818 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#554C211yesmaximal, characteristic
2#11729444C212yesmaximal, complemented
3#11729373C211yesmaximal, characteristic
4#11729354C211yesmaximal, characteristic
5#11729308C212yesmaximal, complemented
6C7×#554—24no—
7C7×#554—24no—
8C2×C7×C4⋊Q8C411yescharacteristic, complemented
9C2×C7×#125—21no—
10C2×C7×C22⋊Q8—21no—
11C2×C7×C22⋊Q8—21no—
12C2×C7×C22⋊Q8C411yescharacteristic, complemented
13C2×C7×C22⋊Q8C2211yescharacteristic
14C2×C4×C7×Q8C411yescharacteristic, complemented
15C2×C4×C7×D4—21no—
16C2×C4×C7×D4C2211yescharacteristic
17C22×C7×C4⋊C4C411yescharacteristic, complemented
18C22×C7×C4⋊C4C2211yescharacteristic
19C2×#6408—21no—
20#89984C2212yes—
21C2×C22⋊(C7⋊C8)C2212yes—
22#89333—24no—
23#89324—24no—
24#89323—21no—
25#89321—21no—
26#89320—21no—
27#8380—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C22×C7×Q8C2×C411yescharacteristic
30C22×C7×D4D411yescharacteristic
31C23×C4×C7—21no—
32C23×C4×C7—21no—
33C23×C4×C7C2×C411yescharacteristic
34C23×C4×C7C2311yescharacteristic
35C7×C4⋊Q8—22no—
36C7×C4⋊Q8—22no—
37C7×#125—28no—
38C7×C22⋊Q8—28no—
39C7×C22⋊Q8—28no—
40C7×C22⋊Q8—22no—
41C7×C22⋊Q8—22no—
42C7×C22⋊Q8—22no—
43C7×C22⋊Q8D414yes—
44C4×C7×Q8—22no—
45C4×C7×Q8—22no—
46C4×C7×D4—28no—
47C4×C7×D4—22no—
48C4×C7×D4—22no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—22no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4C2×C411yescharacteristic
59C2×C7×C4⋊C4C2×C411yescharacteristic
60C2×C7×C4⋊C4C2×C411yescharacteristic
61C2×C7×C4⋊C4D411yescharacteristic
62C2×C7×C4⋊C4D414yes—
63C2×C7×C22⋊C4—21no—
64C2×C7×C22⋊C4—21no—
65C2×C7×C22⋊C4D411yescharacteristic
66C2×C7×C22⋊C4D411yescharacteristic
67C2×C42×C7—21no—
68C2×C42×C7C2×C411yescharacteristic
69C2×C22⋊Dic7—22no—
70C2×C4⋊Dic7—22no—
71C22×C7⋊C8—22no—
72#6408—42no—
73C7⋊#95—22no—
74C22⋊(C7⋊C8)—22no—
75C22⋊(C7⋊C8)—22no—
76C2×#554D711yescharacteristic
77#1824—72no—
78#1799—71no—
79#1520—72no—
80#1459—71no—
81C24×C7C22⋊C411yescharacteristic
82C24×C7C2×D411yescharacteristic
83C2×C7×Q8—24no—
84C2×C7×Q8—21no—
85C2×C7×Q8—42no—
86C2×C7×Q8—82no—
87C2×C7×Q8C22⋊C412yes—
88C2×C7×D4—22no—
89C2×C7×D4—22no—
90C2×C7×D4SD1612yes—
91C2×C7×D4Q1612yes—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—24no—
97C22×C4×C7—24no—
98C22×C4×C7—21no—
99C22×C4×C7—21no—
100C22×C4×C7—42no—
101C22×C4×C7—42no—
102C22×C4×C7—82no—
103C22×C4×C7—82no—
104C22×C4×C7C22⋊C411yescharacteristic
105C22×C4×C7C22⋊C411yescharacteristic
106C22×C4×C7C22⋊C411yescharacteristic
107C22×C4×C7C22×C411yescharacteristic
108C22×C4×C7C2×D412yes—
109C22×C4×C7C2×D412yes—
110C22×C4×C7C2×D411yescharacteristic
111C22×Dic7—42no—
112C7×C4⋊C4—24no—
113C7×C4⋊C4—24no—
114C7×C4⋊C4—24no—
115C7×C4⋊C4—24no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—24no—
120C7×C4⋊C4—24no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4C22⋊C412yes—
123C7×C4⋊C4C22⋊C412yes—
124C7×C4⋊C4C22⋊C412yes—
125C7×C22⋊C4—24no—
126C7×C22⋊C4—24no—
127C7×C22⋊C4—22no—
128C7×C22⋊C4—22no—
129C42×C7—24no—
130C42×C7—21no—
131C42×C7—21no—
132C22⋊Dic7—42no—
133C22⋊Dic7—42no—
134C4⋊Dic7—44no—
135C2×C7⋊C8—44no—
136C2×C7⋊C8—42no—
137#554—24no—
138#554—24no—
139C2×C4⋊Q8Dic711yescharacteristic, complemented
140C2×#125—21no—
141C2×C22⋊Q8—21no—
142C2×C22⋊Q8—21no—
143C2×C22⋊Q8Dic711yescharacteristic, complemented
144C2×C22⋊Q8C2×D711yescharacteristic
145C2×C4×Q8Dic711yescharacteristic, complemented
146C2×C4×D4—21no—
147C2×C4×D4C2×D711yescharacteristic
148C22×C4⋊C4Dic711yescharacteristic, complemented
149C22×C4⋊C4C2×D711yescharacteristic
150C2×#99—141no—
151C2×C22⋊C8—72no—
152#380—72no—
153#342—141no—
154#340—144no—
155#328—144no—
156#324—141no—
157#323—141no—
158C23×C7—21no—
159C23×C7—21no—
160C23×C7—41no—
161C23×C7—41no—
162C23×C7—41no—
163C23×C7—41no—
164C23×C7#10312yes—
165C23×C7C2×C22⋊C411yescharacteristic
166C23×C7#12012yes—
167C23×C7C2×SD1611yescharacteristic
168C23×C7C2×Q1611yescharacteristic
169C23×C7#13712yes—
170C7×Q8—42no—
171C7×Q8—42no—
172C7×Q8—84no—
173C7×D4—24no—
174C7×D4—24no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—24no—
178C2×C4×C7—22no—
179C2×C4×C7—22no—
180C2×C4×C7—21no—
181C2×C4×C7—22no—
182C2×C4×C7—21no—
183C2×C4×C7—22no—
184C2×C4×C7—22no—
185C2×C4×C7—41no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—84no—
196C2×C4×C7—84no—
197C2×C4×C7#10312yes—
198C2×C4×C7C2×C22⋊C411yescharacteristic
199C2×C4×C7C2×C22⋊C411yescharacteristic
200C2×C4×C7#12012yes—
201C2×Dic7—82no—
202C2×Dic7—82no—
203C2×Dic7—82no—
204C7⋊C8—82no—
205C22×Q8—21no—
206C22×Q8C2×Dic711yescharacteristic
207C22×D4C22⋊D711yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C2×Dic711yescharacteristic
211C23×C4C22×D711yescharacteristic
212C22×C8—142no—
213C4⋊Q8—22no—
214C4⋊Q8—22no—
215#125—28no—
216C22⋊Q8—28no—
217C22⋊Q8—28no—
218C22⋊Q8—22no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8C22⋊D714yes—
222C4×Q8—22no—
223C4×Q8—22no—
224C4×D4—28no—
225C4×D4—22no—
226C4×D4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—142no—
237C2×C4⋊C4C2×Dic711yescharacteristic
238C2×C4⋊C4C2×Dic711yescharacteristic
239C2×C4⋊C4C2×Dic711yescharacteristic
240C2×C4⋊C4C22⋊D711yescharacteristic
241C2×C4⋊C4C22⋊D714yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—142no—
245C2×C22⋊C4C22⋊D711yescharacteristic
246C2×C22⋊C4C22⋊D711yescharacteristic
247C2×C42—21no—
248C2×C42C2×Dic711yescharacteristic
249#99—282no—
250C22⋊C8—142no—
251C22⋊C8—142no—
252#95—142no—
253C22×C7—22no—
254C22×C7—21no—
255C22×C7—22no—
256C22×C7—21no—
257C22×C7—22no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—42no—
261C22×C7—42no—
262C22×C7—42no—
263C22×C7—41no—
264C22×C7—41no—
265C22×C7#37911yescharacteristic
266C22×C7C2×#10311yescharacteristic
267C22×C7#41911yescharacteristic
268C22×C7#44812yes—
269C22×C7#45112yes—
270C4×C7—22no—
271C4×C7—21no—
272C4×C7—22no—
273C4×C7—42no—
274C4×C7—42no—
275C4×C7—42no—
276C4×C7—41no—
277C4×C7—41no—
278C4×C7—42no—
279C4×C7#41012yes—
280Dic7—162no—
281C24C22⋊Dic711yescharacteristic
282C24C2×C22⋊D711yescharacteristic
283C2×Q8—24no—
284C2×Q8—21no—
285C2×Q8—42no—
286C2×Q8—82no—
287C2×Q8C22⋊Dic712yes—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4#114312yes—
291C2×D4#114412yes—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—282no—
305C22×C4C22⋊Dic711yescharacteristic
306C22×C4C22⋊Dic711yescharacteristic
307C22×C4C22⋊Dic711yescharacteristic
308C22×C4C22×Dic711yescharacteristic
309C22×C4C2×C22⋊D712yes—
310C22×C4C2×C22⋊D712yes—
311C22×C4C2×C22⋊D711yescharacteristic
312C2×C8—284no—
313C2×C8—282no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—22no—
324C4⋊C4—284no—
325C4⋊C4C22⋊Dic712yes—
326C4⋊C4C22⋊Dic712yes—
327C4⋊C4C22⋊Dic712yes—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—282no—
333C22⋊C4—282no—
334C42—24no—
335C42—21no—
336C42—21no—
337C2×C7—22no—
338C2×C7—21no—
339C2×C7—21no—
340C2×C7—22no—
341C2×C7—42no—
342C2×C7#178211yescharacteristic
343C2×C7#178711yescharacteristic
344C2×C7#179511yescharacteristic
345C23—21no—
346C23—21no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23#640912yes—
352C23C2×#114311yescharacteristic
353C23#650512yes—
354C23C2×#114411yescharacteristic
355C23C2×C22⋊Dic711yescharacteristic
356C23#651612yes—
357Q8—42no—
358Q8—42no—
359Q8—84no—
360D4—24no—
361D4—24no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—562no—
385C2×C4—562no—
386C2×C4—562no—
387C2×C4#640912yes—
388C2×C4C2×C22⋊Dic711yescharacteristic
389C2×C4C2×C22⋊Dic711yescharacteristic
390C2×C4#651612yes—
391C8—562no—
392C7#838011yescharacteristic, complemented, Sylow
393C22—21no—
394C22—22no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—41no—
399C22—42no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22C2×#640911yescharacteristic
406C22#8999711yescharacteristic
407C22#8999912yes—
408C22#9000012yes—
409C22#9001911yescharacteristic
410C4—22no—
411C4—22no—
412C4—21no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—41no—
419C4—1122no—
420C4#9001212yes—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—22no—
425C2—42no—
426C2#1172988611yescharacteristic
427C2#1172989311yescharacteristic
428C2#1172989511yescharacteristic
subgroupG/H
centerC22#90019
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#554C2
socleC22×C7#379
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenterC2×#554D7
hypocenterC7#8380
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
2. #11729444C2semidirect
5. #11729308C2semidirect
8. C2×C7×C4⋊Q8C4semidirect
12. C2×C7×C22⋊Q8C4semidirect
14. C2×C4×C7×Q8C4semidirect
17. C22×C7×C4⋊C4C4semidirect
139. C2×C4⋊Q8Dic7semidirect
143. C2×C22⋊Q8Dic7semidirect
145. C2×C4×Q8Dic7semidirect
148. C22×C4⋊C4Dic7semidirect
392. C7#8380semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5542
#117294442
#117293732
#117293542
#117293082
#83807