Finite Group Databasebeta

← #13339347

Subgroups — #13339347

Order 1792.

684 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3036C211yesmaximal, characteristic
2#11740762C211yesmaximal, characteristic, complemented
3#11739749C211yesmaximal, characteristic, complemented
4C2×#89779C211yesmaximal, characteristic, complemented
5#11729558C211yesmaximal, characteristic, complemented
6#11728796C214yesmaximal, complemented
7#11728762C214yesmaximal, complemented
8#11728262C211yesmaximal, characteristic, complemented
9#11728248C211yesmaximal, characteristic, complemented
10C2×C7×AGL(1,ℤ/8)—21no—
11C7×#532C2211yescharacteristic
12C2×C4×C7×D4C2211yescharacteristic
13C2×#6432C2211yescharacteristic
14C7×#463—22no—
15C7×#461C2212yes—
16C7×#459C2212yes—
17C7×#423C2211yescharacteristic
18C7×#417—21no—
19C2×#6406—21no—
20#89979—22no—
21#89865C2212yescomplemented
22#89848C2212yescomplemented
23#89812—22no—
24#89791—22no—
25#89784—22no—
26#89784C2214yes—
27#89779—22no—
28#89779C2214yes—
29C2×#6381—21no—
30#89697C2211yescharacteristic
31#89697C2211yescharacteristic
32C2×C4⋊(C7⋊C8)C2211yescharacteristic
33#89342C2212yescomplemented
34#89339C2212yescomplemented
35#89281C2214yescomplemented
36#89274C2214yescomplemented
37#19421—71nomaximal, Sylow
38C2×C7×#41—21no—
39C22×C7×D4D411yescharacteristic
40C23×C4×C7—21no—
41C22×Dic14—21no—
42C7×AGL(1,ℤ/8)—44no—
43C2×C7×SD16—22no—
44C2×C7×D8—22no—
45C2×C7×M16—21no—
46C7×C4⋊Q8C2311yescharacteristic
47C7×#127C2311yescharacteristic
48C7×#124—21no—
49C7×#121—21no—
50C7×#121—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C4×C7×D4—21no—
55C4×C7×D4D412yes—
56C4×C7×D4C2312yes—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C42×C7C2311yescharacteristic
60C2×#1142—24no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—21no—
63C2×#1136—24no—
64C22×C7⋊C8—21no—
65#6432—21no—
66#6432—21no—
67#6432—21no—
68#6432C2312yes—
69C7×C4⋊C8—21no—
70C7×C4⋊C8C2312yes—
71C7×#103—22no—
72C7×#102—22no—
73#6406—24no—
74C22⋊(C7⋊C8)—21no—
75C22⋊(C7⋊C8)—24no—
76C22⋊(C7⋊C8)D412yes—
77#6384—24no—
78#6381—24no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)D412yes—
81C4⋊(C7⋊C8)C2312yes—
82C4⋊(C7⋊C8)C2312yes—
83C4×C7⋊C8—21no—
84C4×C7⋊C8C2312yes—
85#3206—71no—
86C2×#474—71no—
87#3036D711yescharacteristic
88#2975—71no—
89#1699—74no—
90#1685—74no—
91#1571—71no—
92#1549—71no—
93#1540—71no—
94C24×C7—21no—
95C7×#41—42no—
96C7×#41—42no—
97C2×C7×Q8—21no—
98C2×C7×D4—21no—
99C2×C7×D4—22no—
100C2×C7×D4—22no—
101C2×C7×D4—21no—
102C2×C7×D4—21no—
103C2×C7×D4—21no—
104C2×C7×D4—41no—
105C2×C7×D4C2×D412yes—
106C22×C4×C7—21no—
107C22×C4×C7—22no—
108C22×C4×C7—22no—
109C22×C4×C7—41no—
110C22×C4×C7—82no—
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2×D411yescharacteristic
113C22×C4×C7#4111yescharacteristic
114C22×Dic7—41no—
115C2×Dic14—22no—
116C2×Dic14—41no—
117C2×Dic14—82no—
118C7×SD16—44no—
119C7×D8—44no—
120C7×M16—42no—
121C2×C7×C8—22no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4C2×D412yes—
126C7×C22⋊C4—22no—
127C7×C22⋊C4—22no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C42×C7C2×D411yescharacteristic
131C42×C7C2×D411yescharacteristic
132C42×C7#4111yescharacteristic
133C42×C7C2411yescharacteristic
134#1142—84no—
135C4⋊Dic7—22no—
136C4⋊Dic7—22no—
137C4⋊Dic7—41no—
138C4⋊Dic7—41no—
139#1136—84no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—22no—
142C2×C7⋊C8—42no—
143C2×C7⋊C8C2×D414yes—
144C2×AGL(1,ℤ/8)—21no—
145#532C2×D711yescharacteristic
146C2×C4⋊Q8—71no—
147C2×C4×D4C2×D711yescharacteristic
148#492—72no—
149#487—72no—
150#474—74no—
151#474—142no—
152#463—22no—
153#461—142no—
154#461C2×D712yes—
155#459C2×D712yes—
156#448—142no—
157#436—74no—
158#436—142no—
159#431—71no—
160#431—71no—
161C2×#107—141no—
162#423C2×D711yescharacteristic
163C2×C4⋊C8—71no—
164#417—21no—
165C2×#102—141no—
166#407—142no—
167#335—74no—
168#332—72no—
169#332—74no—
170#331—72no—
171C23×C7—22no—
172C23×C7—42no—
173C23×C7—41no—
174C23×C7—81no—
175C23×C7—81no—
176C23×C7#12111yescharacteristic
177C7×Q8—41no—
178C7×Q8—41no—
179C7×D4—22no—
180C7×D4—22no—
181C7×D4—41no—
182C7×D4—41no—
183C7×D4—42no—
184C7×D4—42no—
185C7×D4—81no—
186C7×D4—82no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—41no—
190C2×C4×C7—44no—
191C2×C4×C7—41no—
192C2×C4×C7—82no—
193C2×C4×C7—81no—
194C2×C4×C7—82no—
195C2×C4×C7#12111yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7AGL(1,ℤ/8)12yes—
199C2×C4×C7#13712yes—
200C2×C4×C7C22×D411yescharacteristic
201C2×C4×C7C22×D411yescharacteristic
202C2×C4×C7C2×#4111yescharacteristic
203C2×Dic7—42no—
204C2×Dic7—82no—
205Dic14—44no—
206Dic14—82no—
207C7×C8—42no—
208C7⋊C8—24no—
209C7⋊C8—42no—
210C7⋊C8—42no—
211C2×#41—21no—
212C22×Q8—141no—
213C22×D4C22⋊D711yescharacteristic
214C23×C4—21no—
215AGL(1,ℤ/8)—44no—
216C2×SD16—22no—
217C2×SD16—144no—
218C2×D8—22no—
219C2×M16—21no—
220C2×M16—144no—
221C22×C8—141no—
222C4⋊Q8—72no—
223C4⋊Q8—141no—
224C4⋊Q8—141no—
225C4⋊Q8—141no—
226C4⋊Q8C22×D711yescharacteristic
227#127C22×D711yescharacteristic
228#124—21no—
229#121—21no—
230#121—21no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4C22⋊D712yes—
236C4×D4C22×D712yes—
237C2×C4⋊C4—141no—
238C2×C4⋊C4—141no—
239C2×C4⋊C4C22⋊D711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C42C22×D711yescharacteristic
242#107—144no—
243C4⋊C8—21no—
244C4⋊C8—72no—
245C4⋊C8—72no—
246C4⋊C8—72no—
247C4⋊C8—141no—
248C4⋊C8C22×D712yes—
249#103—22no—
250#103—144no—
251#102—22no—
252#102—144no—
253C22⋊C8—72no—
254C22⋊C8—144no—
255C22⋊C8—141no—
256C4×C8—72no—
257C4×C8—141no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—42no—
261C22×C7—42no—
262C22×C7—42no—
263C22×C7—84no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—81no—
267C22×C7#46812yes—
268C22×C7C2×#12111yescharacteristic
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—42no—
273C4×C7—81no—
274C4×C7#46812yes—
275C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
276C4×C7C2×#13711yescharacteristic
277C4×C7#57612yes—
278Dic7—82no—
279C24—21no—
280#41—42no—
281#41—42no—
282C2×Q8—21no—
283C2×Q8—142no—
284C2×Q8—281no—
285C2×Q8—562no—
286C2×D4—22no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4C2×C22⋊D712yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—82no—
299C22×C4—281no—
300C22×C4D4×D711yescharacteristic
301C22×C4#115911yescharacteristic
302C22×C4C2×C22⋊D711yescharacteristic
303SD16—44no—
304SD16—564no—
305D8—44no—
306M16—42no—
307M16—564no—
308C2×C8—22no—
309C2×C8—74no—
310C2×C8—142no—
311C2×C8—142no—
312C2×C8—282no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—142no—
317C4⋊C4—142no—
318C4⋊C4—281no—
319C4⋊C4—281no—
320C4⋊C4C2×C22⋊D712yes—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42D4×D711yescharacteristic
326C42#115911yescharacteristic
327C42C2×C22⋊D711yescharacteristic
328C42C23×D711yescharacteristic
329C2×C7—22no—
330C2×C7—82no—
331C2×C7—161no—
332C2×C7#304111yescharacteristic
333C2×C7C2×#46811yescharacteristic
334C2×C7#306911yescharacteristic
335C23—22no—
336C23—42no—
337C23—41no—
338C23—81no—
339C23—81no—
340C23#650211yescharacteristic
341Q8—41no—
342Q8—41no—
343Q8—284no—
344Q8—562no—
345D4—22no—
346D4—22no—
347D4—42no—
348D4—41no—
349D4—41no—
350D4—42no—
351D4—81no—
352D4—82no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—282no—
362C2×C4—562no—
363C2×C4#649512yes—
364C2×C4#650211yescharacteristic
365C2×C4#650211yescharacteristic
366C2×C4#650211yescharacteristic
367C2×C4#651412yes—
368C2×C4C2×D4×D711yescharacteristic
369C2×C4C2×#115911yescharacteristic
370C2×C4C22×C22⋊D711yescharacteristic
371C8—42no—
372C8—144no—
373C8—282no—
374C8—282no—
375C7#1942111yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#8981412yes—
386C22C2×#650211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—81no—
392C4—562no—
393C4#8981412yes—
394C4#9046512yes—
395C4C2×#649511yescharacteristic
396C4C2×#651411yescharacteristic
397C2—22no—
398C2—82no—
399C2—161no—
400C2C2×#8981411yescharacteristic
401C2#1174137911yescharacteristic
402C2#1174144811yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
Fitting, prime radicalC7×#3036C2
socleC22×C7C2×#121
hypercenter#3036D7
hypocenterC7#19421
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #11740762C2semidirect
3. #11739749C2semidirect
4. C2×#89779C2semidirect
5. #11729558C2semidirect
6. #11728796C2semidirect
7. #11728762C2semidirect
8. #11728262C2semidirect
9. #11728248C2semidirect
21. #89865C22semidirect
22. #89848C22semidirect
33. #89342C22semidirect
34. #89339C22semidirect
35. #89281C22semidirect
36. #89274C22semidirect
375. C7#19421semidirect

Maximal subgroups

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

subgroupindex
C7×#30362
#117407622
#117397492
C2×#897792
#117295582
#117287962
#117287622
#117282622
#117282482
#194217