Finite Group Databasebeta

← #13323736

Subgroups — #13323736

Order 1792.

766 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
1#11740743C211yesmaximal, characteristic
2#11740720C211yesmaximal, characteristic, complemented
3C2×#89606C211yesmaximal, characteristic, complemented
4C7×#1480C211yesmaximal, characteristic, complemented
5#11728516C211yesmaximal, characteristic, complemented
6#11728496C211yesmaximal, characteristic, complemented
7#11728175C211yesmaximal, characteristic, complemented
8#11725912C214yesmaximal, complemented
9#11725905C214yesmaximal, complemented
10C7×#532C2211yescharacteristic
11C2×#6511—21no—
12C22×C8×D7—21no—
13C2×C4×D28C2211yescharacteristic
14C7×#431C2211yescharacteristic
15C2×C7×C4⋊C8C2211yescharacteristic
16#90006—21no—
17C2×(C2×D7)⋊C8—21no—
18#89859C412yes—
19#89844C412yes—
20#89808C412yes—
21#89805C412yes—
22C4×#1141C2214yes—
23#89784—21no—
24#89784C2212yescomplemented
25D4×C7⋊C8—21no—
26D4×C7⋊C8C2212yescomplemented
27#89697C2211yescharacteristic
28#89696C2211yescharacteristic
29C2×C4×C7⋊C8C2211yescharacteristic
30#89606—21no—
31#89606—21no—
32#89606C2212yes—
33#89606C2212yes—
34C7×D4⋊C8C2214yescomplemented
35#89275C2214yescomplemented
36#89254C2214yes—
37#89250C2212yescomplemented
38#89249C2212yescomplemented
39#17907—71nomaximal, Sylow
40C2×C7×#41—21no—
41C22×D28C2×C411yescharacteristic
42C22×C4×D7—21no—
43C22×C7×C8—21no—
44C7×C4⋊Q8C2×C411yescharacteristic
45C7×#127C2×C411yescharacteristic
46C7×#124—21no—
47C7×#121—21no—
48C7×#121C2×C412yes—
49C4×C7×D4—21no—
50C4×C7×D4C2312yes—
51C2×C42×C7C2311yescharacteristic
52#6511—44no—
53C2×#1141C2×C414yes—
54C2×(C2×D7)⋊C4—21no—
55C2×C4⋊Dic7C2×C411yescharacteristic
56C2×#1136—21no—
57C2×#1136—22no—
58C22×C7⋊C8—22no—
59C22×C7⋊C8D411yescharacteristic
60C22×C7⋊C8D411yescharacteristic
61C2×C8×D7—22no—
62C2×C8×D7—22no—
63C2×C8×D7—44no—
64C4×D28—21no—
65C4×D28—21no—
66C4×D28C2×C412yes—
67C4×D28C2312yes—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8C2312yes—
71C7×C22⋊C8—21no—
72C4×C7×C8C2312yes—
73#6406C2×C414yes—
74C22⋊(C7⋊C8)—22no—
75C22⋊(C7⋊C8)—22no—
76C22⋊(C7⋊C8)C2×C412yes—
77(C2×D7)⋊C8—22no—
78(C2×D7)⋊C8—22no—
79#6383C2×C414yes—
80#6381C2×C414yes—
81C4⋊(C7⋊C8)—21no—
82C4⋊(C7⋊C8)C2×C412yes—
83C4⋊(C7⋊C8)C2312yes—
84C4×C7⋊C8D411yescharacteristic
85C4×C7⋊C8D411yescharacteristic
86C4×C7⋊C8C2312yes—
87C4×C7⋊C8C2311yescharacteristic
88C4×C7⋊C8C2311yescharacteristic
89#3100—71no—
90#2931—71no—
91C2×C8×D4—71no—
92#1586—74no—
93#1580—74no—
94#1570—71no—
95#1486—71no—
96#1480D711yescharacteristic, complemented
97#1452—71no—
98C23×D7—21no—
99C7×#41—42no—
100C7×#41—42no—
101C2×C7×Q8—21no—
102C2×C7×D4—22no—
103C2×C7×D4—21no—
104C2×C7×D4—21no—
105C2×C7×D4C22×C412yes—
106C22×C4×C7—22no—
107C22×C4×C7C22×C411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7#4111yescharacteristic
110C22×Dic7—21no—
111C2×D28—21no—
112C2×D28—24no—
113C2×D28C22×C412yes—
114C2×C4×D7—22no—
115C2×C4×D7—22no—
116C2×C4×D7—42no—
117C2×C4×D7—42no—
118C2×C7×C8—22no—
119C2×C7×C8—22no—
120C2×C7×C8—42no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4C22×C412yes—
123C7×C22⋊C4—22no—
124C7×C22⋊C4—22no—
125C42×C7C22×C411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7#4111yescharacteristic
128C42×C7C2411yescharacteristic
129#1141—28no—
130(C2×D7)⋊C4—22no—
131(C2×D7)⋊C4—22no—
132C4⋊Dic7—21no—
133C4⋊Dic7C22×C412yes—
134#1136—44no—
135#1136—42no—
136C2×C7⋊C8—22no—
137C2×C7⋊C8—22no—
138C2×C7⋊C8—22no—
139C2×C7⋊C8—44no—
140C2×C7⋊C8C22×C414yes—
141C2×C7⋊C8C2×D411yescharacteristic
142C2×C7⋊C8C2×D411yescharacteristic
143C2×C7⋊C8C2×D411yescharacteristic
144C2×C7⋊C8C2×D411yescharacteristic
145C8×D7—44no—
146C8×D7—44no—
147C2×#131—141no—
148C23×C8—141no—
149#532C2×D711yescharacteristic
150C2×C4×D4—71no—
151#480—72no—
152#474—72no—
153#459—72no—
154#447—72no—
155C4×D8—74no—
156#436—72no—
157#436—141no—
158C8×D4—72no—
159C8×D4—72no—
160C8×D4—72no—
161C8×D4—141no—
162C8×D4—141no—
163C8×D4—141no—
164#431—71no—
165#431C2×D711yescharacteristic
166#423—71no—
167C2×C4⋊C8C2×D711yescharacteristic
168#408—141no—
169C2×C22⋊C8—141no—
170C2×C4×C8—71no—
171#334—74no—
172D4⋊C8—74no—
173D4⋊C8C2×D714yescomplemented
174#322—72no—
175#322—72no—
176C23×C7—22no—
177C23×C7C4×D411yescharacteristic
178C22×D7—24no—
179C22×D7—22no—
180C22×D7—42no—
181C22×D7—42no—
182C7×Q8—42no—
183C7×D4—24no—
184C7×D4—42no—
185C7×D4—44no—
186C7×D4—42no—
187C2×C4×C7—22no—
188C2×C4×C7—22no—
189C2×C4×C7—41no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7C4×D411yescharacteristic
193C2×C4×C7C4×D411yescharacteristic
194C2×C4×C7C4×D411yescharacteristic
195C2×C4×C7AGL(1,ℤ/8)12yes—
196C2×C4×C7AGL(1,ℤ/8)12yes—
197C2×C4×C7C23×C411yescharacteristic
198C2×C4×C7C22×D411yescharacteristic
199C2×C4×C7C2×#4111yescharacteristic
200C2×Dic7—22no—
201C2×Dic7—42no—
202D28—24no—
203D28—22no—
204D28#13114yescomplemented
205C4×D7—42no—
206C4×D7—42no—
207C4×D7—44no—
208C7×C8—42no—
209C7×C8—42no—
210C7⋊C8—24no—
211C7⋊C8—21no—
212C7⋊C8—21no—
213C7⋊C8—21no—
214C7⋊C8—21no—
215C7⋊C8—42no—
216C2×#41—21no—
217C22×D4—71no—
218C23×C4—141no—
219C2×D8—74no—
220#131—284no—
221C2×M16—142no—
222C2×M16—141no—
223C22×C8—21no—
224C22×C8—71no—
225C22×C8—71no—
226C22×C8—142no—
227C22×C8—142no—
228C22×C8—142no—
229C22×C8—284no—
230C4⋊Q8C4×D711yescharacteristic
231#127C4×D711yescharacteristic
232#124—21no—
233#121—21no—
234#121C4×D712yes—
235C4×D4—21no—
236C4×D4—72no—
237C4×D4—72no—
238C4×D4—141no—
239C4×D4—141no—
240C4×D4C22×D712yes—
241C2×C4⋊C4—71no—
242C2×C22⋊C4—141no—
243C2×C42C22×D711yescharacteristic
244#107—74no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—72no—
248C4⋊C8—72no—
249C4⋊C8—141no—
250C4⋊C8C22×D712yes—
251#102—74no—
252#102—74no—
253C22⋊C8—21no—
254C22⋊C8—72no—
255C22⋊C8—142no—
256C22⋊C8—142no—
257C22⋊C8—142no—
258C22⋊C8—142no—
259C4×C8—71no—
260C4×C8—71no—
261C4×C8—72no—
262C4×C8—71no—
263C4×C8—71no—
264C4×C8C22×D712yes—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—42no—
268C22×C7—42no—
269C22×C7—42no—
270C22×C7#44212yes—
271C22×C7C2×C4×D411yescharacteristic
272C2×D7—24no—
273C2×D7—48no—
274C2×D7—42no—
275C2×D7—42no—
276C4×C7—21no—
277C4×C7—21no—
278C4×C7—21no—
279C4×C7—42no—
280C4×C7#44212yes—
281C4×C7C2×#13112yes—
282C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
283C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
284Dic7—42no—
285C24—141no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×D4—22no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—72no—
293C2×D4—144no—
294C2×D4—141no—
295C2×D4C2×C4×D712yes—
296C22×C4—22no—
297C22×C4—141no—
298C22×C4—142no—
299C22×C4—142no—
300C22×C4—282no—
301C22×C4—282no—
302C22×C4C2×C4×D711yescharacteristic
303C22×C4D4×D711yescharacteristic
304C22×C4#115911yescharacteristic
305D8—148no—
306M16—284no—
307M16—282no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—42no—
311C2×C8—74no—
312C2×C8—71no—
313C2×C8—71no—
314C2×C8—71no—
315C2×C8—71no—
316C2×C8—142no—
317C2×C8—142no—
318C2×C8—142no—
319C2×C8—284no—
320C2×C8—284no—
321C2×C8—284no—
322C4⋊C4—21no—
323C4⋊C4—72no—
324C4⋊C4—141no—
325C4⋊C4C2×C4×D712yes—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C42C2×C4×D711yescharacteristic
331C42D4×D711yescharacteristic
332C42#115911yescharacteristic
333C42C23×D711yescharacteristic
334C2×C7—22no—
335C2×C7—82no—
336C2×C7#292911yescharacteristic
337C2×C7C2×#44211yescharacteristic
338C2×C7#295511yescharacteristic
339D7—44no—
340C23—22no—
341C23—144no—
342C23—142no—
343C23—282no—
344C23—282no—
345C23(C22⋊D7)⋊C411yescharacteristic
346Q8—42no—
347D4—24no—
348D4—42no—
349D4—44no—
350D4—42no—
351D4—74no—
352D4—144no—
353D4—142no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—142no—
360C2×C4—282no—
361C2×C4—282no—
362C2×C4—282no—
363C2×C4—284no—
364C2×C4(C22⋊D7)⋊C411yescharacteristic
365C2×C4(C22⋊D7)⋊C411yescharacteristic
366C2×C4(C22⋊D7)⋊C411yescharacteristic
367C2×C4#647412yes—
368C2×C4#647712yes—
369C2×C4C22×C4×D711yescharacteristic
370C2×C4C2×D4×D711yescharacteristic
371C2×C4C2×#115911yescharacteristic
372C8—42no—
373C8—42no—
374C8—144no—
375C8—141no—
376C8—141no—
377C8—141no—
378C8—141no—
379C8—282no—
380C7#1790711yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—144no—
387C22—288no—
388C22—282no—
389C22—282no—
390C22#8955712yes—
391C22C2×(C22⋊D7)⋊C411yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—42no—
396C4—282no—
397C4#8955712yes—
398C4D7×#13112yes—
399C4C2×#647411yescharacteristic
400C4C2×#647711yescharacteristic
401C2—22no—
402C2—82no—
403C2—284no—
404C2#1173610011yescharacteristic
405C2C2×#8955711yescharacteristic
406C2#1173643411yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1480C2
socleC22×C7C2×C4×D4
prime radical#11740743C2
cosocleC42×C7C24
hypercenter#1480D7
hypocenterC7#17907
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1480

Complements

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

NH ≅ G/Ntype
2. #11740720C2semidirect
3. C2×#89606C2semidirect
4. C7×#1480C2semidirect
5. #11728516C2semidirect
6. #11728496C2semidirect
7. #11728175C2semidirect
8. #11725912C2semidirect
9. #11725905C2semidirect
24. #89784C22semidirect
26. D4×C7⋊C8C22semidirect
34. C7×D4⋊C8C22semidirect
35. #89275C22semidirect
37. #89250C22semidirect
38. #89249C22semidirect
96. #1480D7semidirect
173. D4⋊C8C2×D7semidirect
204. D28#131semidirect
380. C7#17907semidirect

Maximal subgroups

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

subgroupindex
#117407432
#117407202
C2×#896062
C7×#14802
#117285162
#117284962
#117281752
#117259122
#117259052
#179077