Finite Group Databasebeta

← #13339333

Subgroups — #13339333

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#459C211yesmaximal, characteristic
2#11740614C211yesmaximal, characteristic, complemented
3#11739749C211yesmaximal, characteristic, complemented
4#11739741C211yesmaximal, characteristic, complemented
5#11729513C211yesmaximal, characteristic, complemented
6#11728764C214yesmaximal, complemented
7#11728762C214yesmaximal, complemented
8#11728191C211yesmaximal, characteristic, complemented
9#11728176C211yesmaximal, characteristic, complemented
10C22×C7×D8—21no—
11C2×C7×#127C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13#90161C2211yescharacteristic
14C7×#459—22no—
15C7×#459C2214yes—
16C2×C7×C4⋊C8C2211yescharacteristic
17C2×C7×#102—21no—
18#89990—22no—
19#89987—21no—
20#89979—22no—
21#89848C2212yescomplemented
22#89847—22no—
23#89842C2212yescomplemented
24#89784—22no—
25#89784C2214yescomplemented
26#89781—22no—
27#89780C2212yescomplemented
28#89779C2212yescomplemented
29#89736—21no—
30#89697C2211yescharacteristic
31#89697C2211yescharacteristic
32#89696C2211yescharacteristic
33#89339C2214yescomplemented
34#89281C2212yescomplemented
35#89276C2212yescomplemented
36#89274C2214yescomplemented
37#19420—71nomaximal, Sylow
38C22×C7×D4—21no—
39C22×C7×D4—21no—
40C22×C7×D4—21no—
41C22×C7×D4D411yescharacteristic
42C23×C4×C7—21no—
43C2×C7×D8—24no—
44C2×C7×D8—44no—
45C22×C7×C8—21no—
46C7×#127—21no—
47C7×#127—21no—
48C7×#127—21no—
49C7×#127C2312yes—
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4D412yes—
53C4×C7×D4C2312yes—
54C2×C7×C4⋊C4D411yescharacteristic
55C2×C7×C22⋊C4—21no—
56C2×C42×C7C2311yescharacteristic
57C2×#1142—24no—
58#6489—21no—
59#6487—21no—
60#6487—21no—
61C2×#1136—21no—
62C2×#1136—24no—
63#6433—21no—
64#6432C2311yescharacteristic
65#6432C2311yescharacteristic
66C4×Dic14—21no—
67C4×Dic14—21no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8C2312yes—
70C7×#102—24no—
71#6406—22no—
72#6406—22no—
73#6406—24no—
74C22⋊(C7⋊C8)—21no—
75C22⋊(C7⋊C8)—24no—
76C22⋊(C7⋊C8)D412yes—
77#6382—22no—
78#6381—22no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)D412yes—
81C4⋊(C7⋊C8)C2312yes—
82C4⋊(C7⋊C8)C2312yes—
83C4×C7⋊C8—21no—
84C4×C7⋊C8C2312yes—
85#3208—71no—
86#3077—71no—
87C2×#459D711yescharacteristic
88#2975—71no—
89#1697—74no—
90#1685—74no—
91#1571—71no—
92#1545—71no—
93#1542—71no—
94C24×C7—21no—
95C24×C7—41no—
96C2×C7×D4—21no—
97C2×C7×D4—22no—
98C2×C7×D4—22no—
99C2×C7×D4—22no—
100C2×C7×D4—22no—
101C2×C7×D4—22no—
102C2×C7×D4—41no—
103C2×C7×D4—41no—
104C2×C7×D4—41no—
105C2×C7×D4—42no—
106C2×C7×D4—42no—
107C2×C7×D4—82no—
108C2×C7×D4—82no—
109C2×C7×D4C2×D412yes—
110C22×C4×C7—21no—
111C22×C4×C7—22no—
112C22×C4×C7—22no—
113C22×C4×C7—82no—
114C22×C4×C7C2×D411yescharacteristic
115C22×C4×C7C2×D411yescharacteristic
116C22×C4×C7#4111yescharacteristic
117C2×Dic14—21no—
118C2×Dic14—21no—
119C7×D8—48no—
120C2×C7×C8—22no—
121C2×C7×C8—42no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4C2×D412yes—
124C7×C22⋊C4—22no—
125C7×C22⋊C4—22no—
126C42×C7C2×D411yescharacteristic
127C42×C7C2×D411yescharacteristic
128C42×C7#4111yescharacteristic
129C42×C7C2411yescharacteristic
130C22⋊Dic7—41no—
131#1142—84no—
132C4⋊Dic7—21no—
133C4⋊Dic7—21no—
134C4⋊Dic7—21no—
135C4⋊Dic7—21no—
136Dic7⋊C4—41no—
137Dic7⋊C4—41no—
138C4×Dic7—41no—
139#1136—42no—
140#1136—84no—
141C2×C7⋊C8—22no—
142C2×C7⋊C8—22no—
143C2×C7⋊C8C2×D414yes—
144C22×D8—21no—
145#533—71no—
146C2×#127C2×D711yescharacteristic
147C2×C4×D4C2×D711yescharacteristic
148#492—72no—
149#486—72no—
150#478—142no—
151#476—72no—
152#474—72no—
153#460—142no—
154#459—22no—
155#459C2×D714yes—
156#450—142no—
157#436—74no—
158#436—142no—
159#431—71no—
160#431—71no—
161#428—141no—
162#423—71no—
163C2×C4⋊C8C2×D711yescharacteristic
164#418—141no—
165C2×#102—21no—
166#407—142no—
167#335—74no—
168#333—72no—
169#332—72no—
170#331—74no—
171C23×C7—22no—
172C23×C7—42no—
173C23×C7—42no—
174C23×C7—81no—
175C23×C7—81no—
176C23×C7—82no—
177C23×C7—82no—
178C23×C7—82no—
179C23×C7#12111yescharacteristic
180C7×D4—22no—
181C7×D4—22no—
182C7×D4—42no—
183C7×D4—42no—
184C7×D4—44no—
185C7×D4—42no—
186C7×D4—82no—
187C7×D4—82no—
188C7×D4—82no—
189C7×D4—82no—
190C2×C4×C7—22no—
191C2×C4×C7—22no—
192C2×C4×C7—41no—
193C2×C4×C7—44no—
194C2×C4×C7—82no—
195C2×C4×C7—81no—
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7AGL(1,ℤ/8)12yes—
200C2×C4×C7#13712yes—
201C2×C4×C7C22×D411yescharacteristic
202C2×C4×C7C22×D411yescharacteristic
203C2×C4×C7C2×#4111yescharacteristic
204C2×Dic7—41no—
205C2×Dic7—41no—
206Dic14—42no—
207Dic14—81no—
208C7×C8—42no—
209C7⋊C8—24no—
210C7⋊C8—42no—
211C7⋊C8—42no—
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C22⋊D711yescharacteristic
216C23×C4—21no—
217C2×SD16—144no—
218C2×D8—24no—
219C2×D8—44no—
220C2×M16—144no—
221C2×M16—141no—
222C22×C8—21no—
223C4⋊Q8—71no—
224C4⋊Q8—71no—
225#127—21no—
226#127—21no—
227#127—21no—
228#127C22×D712yes—
229#125—141no—
230C22⋊Q8—141no—
231C22⋊Q8—141no—
232C4×Q8—141no—
233C4×Q8—141no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4C22⋊D712yes—
237C4×D4C22×D712yes—
238#117—141no—
239C2×C4⋊C4C22⋊D711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C42C22×D711yescharacteristic
242#107—142no—
243#106—142no—
244C4⋊C8—21no—
245C4⋊C8—72no—
246C4⋊C8—72no—
247C4⋊C8—72no—
248C4⋊C8—141no—
249C4⋊C8C22×D712yes—
250#102—24no—
251#102—144no—
252#102—142no—
253#102—142no—
254C22⋊C8—72no—
255C22⋊C8—144no—
256C22⋊C8—141no—
257C4×C8—72no—
258C4×C8—141no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—42no—
262C22×C7—42no—
263C22×C7—42no—
264C22×C7—84no—
265C22×C7—82no—
266C22×C7—82no—
267C22×C7—82no—
268C22×C7—164no—
269C22×C7#46812yes—
270C22×C7C2×#12111yescharacteristic
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—42no—
275C4×C7#46812yes—
276C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
277C4×C7C2×#13711yescharacteristic
278C4×C7#57612yes—
279Dic7—81no—
280Dic7—81no—
281C24—21no—
282C24—41no—
283C2×Q8—141no—
284C2×Q8—141no—
285C2×D4—22no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—82no—
297C2×D4—82no—
298C2×D4C2×C22⋊D712yes—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—82no—
303C22×C4D4×D711yescharacteristic
304C22×C4#115911yescharacteristic
305C22×C4C2×C22⋊D711yescharacteristic
306SD16—564no—
307D8—48no—
308M16—282no—
309M16—564no—
310C2×C8—22no—
311C2×C8—42no—
312C2×C8—74no—
313C2×C8—142no—
314C2×C8—142no—
315C4⋊C4—21no—
316C4⋊C4—141no—
317C4⋊C4—141no—
318C4⋊C4—141no—
319C4⋊C4—141no—
320C4⋊C4—281no—
321C4⋊C4—281no—
322C4⋊C4C2×C22⋊D712yes—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—281no—
326C42—281no—
327C42D4×D711yescharacteristic
328C42#115911yescharacteristic
329C42C2×C22⋊D711yescharacteristic
330C42C23×D711yescharacteristic
331C2×C7—22no—
332C2×C7—82no—
333C2×C7—162no—
334C2×C7#304111yescharacteristic
335C2×C7C2×#46811yescharacteristic
336C2×C7#306811yescharacteristic
337C23—22no—
338C23—42no—
339C23—42no—
340C23—81no—
341C23—81no—
342C23—82no—
343C23—82no—
344C23—82no—
345C23#650211yescharacteristic
346Q8—282no—
347Q8—561no—
348D4—22no—
349D4—22no—
350D4—44no—
351D4—42no—
352D4—42no—
353D4—42no—
354D4—82no—
355D4—82no—
356D4—82no—
357D4—82no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—44no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—281no—
365C2×C4—281no—
366C2×C4#649512yes—
367C2×C4#650211yescharacteristic
368C2×C4#650211yescharacteristic
369C2×C4#650211yescharacteristic
370C2×C4#651412yes—
371C2×C4C2×D4×D711yescharacteristic
372C2×C4C2×#115911yescharacteristic
373C2×C4C22×C22⋊D711yescharacteristic
374C8—42no—
375C8—144no—
376C8—282no—
377C8—282no—
378C7#1942011yescharacteristic, complemented, Sylow
379C22—21no—
380C22—21no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—84no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—164no—
389C22#8981412yes—
390C22C2×#650211yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—561no—
396C4—561no—
397C4#8981412yes—
398C4#9044812yes—
399C4C2×#649511yescharacteristic
400C4C2×#651411yescharacteristic
401C2—22no—
402C2—82no—
403C2—162no—
404C2C2×#8981411yescharacteristic
405C2#1174127111yescharacteristic
406C2#1174138811yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC2×C7×#459C2
socleC22×C7C2×#121
hypercenterC2×#459D7
hypocenterC7#19420
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
2. #11740614C2semidirect
3. #11739749C2semidirect
4. #11739741C2semidirect
5. #11729513C2semidirect
6. #11728764C2semidirect
7. #11728762C2semidirect
8. #11728191C2semidirect
9. #11728176C2semidirect
21. #89848C22semidirect
23. #89842C22semidirect
25. #89784C22semidirect
27. #89780C22semidirect
28. #89779C22semidirect
33. #89339C22semidirect
34. #89281C22semidirect
35. #89276C22semidirect
36. #89274C22semidirect
378. C7#19420semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4592
#117406142
#117397492
#117397412
#117295132
#117287642
#117287622
#117281912
#117281762
#194207