Finite Group Databasebeta

← #13323735

Subgroups — #13323735

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#11736742C211yesmaximal, characteristic
2#11735903C211yesmaximal, characteristic, complemented
3C2×C8×D28C211yesmaximal, characteristic, complemented
4C7×#1452C211yesmaximal, characteristic, complemented
5#11728516C211yesmaximal, characteristic, complemented
6#11728512C211yesmaximal, characteristic, complemented
7#11728503C211yesmaximal, characteristic, complemented
8#11725835C214yesmaximal, complemented
9#11725831C214yesmaximal, complemented
10C2×#6464—21no—
11C22×C8×D7—21no—
12#90168C2211yescharacteristic
13C2×C4×D28C2211yescharacteristic
14C7×#431C2211yescharacteristic
15C7×#431C2211yescharacteristic
16C2×C4×C7×C8C2211yescharacteristic
17#89882—21no—
18C2×(C2×D7)⋊C8—21no—
19#89696C2211yescharacteristic
20C2×C4⋊(C7⋊C8)C2211yescharacteristic
21#89618C412yes—
22#89615C412yes—
23#89508C412yes—
24#89504C412yes—
25C4×D56C2214yes—
26#89460—21no—
27#89460C2212yescomplemented
28C8×D28—21no—
29C8×D28—21no—
30C8×D28—21no—
31C8×D28C2212yes—
32C8×D28C2212yes—
33C8×D28C2212yescomplemented
34C7×#322C2212yescomplemented
35C7×#322C2212yescomplemented
36#89254C2214yes—
37#89254C2214yescomplemented
38#89252C2214yescomplemented
39#17907—71nomaximal, Sylow
40C2×#1157—21no—
41C22×D28C2×C411yescharacteristic
42C22×C4×D7—21no—
43C22×C7×C8D411yescharacteristic
44C22×C7×C8D411yescharacteristic
45C2×C42×C7C2311yescharacteristic
46#6493—21no—
47#6493C2×C412yes—
48C2×(C2×D7)⋊C4—21no—
49C2×C4⋊Dic7C2×C411yescharacteristic
50C2×#1136—21no—
51C22×C7⋊C8—21no—
52C2×D56C2×C414yes—
53#6464—44no—
54C2×#1131—22no—
55C2×C8×D7—22no—
56C2×C8×D7—22no—
57C2×C8×D7—22no—
58C2×C8×D7—44no—
59#6438—21no—
60#6437C2×C411yescharacteristic
61C4×D28—21no—
62C4×D28—21no—
63C4×D28—21no—
64C4×D28C2×C412yes—
65C4×D28C2312yes—
66C4×D28C2312yes—
67#6432C2×C411yescharacteristic
68C7×C4⋊C8—21no—
69C7×C4⋊C8C2×C412yes—
70C7×C22⋊C8—21no—
71C7×C22⋊C8C2×C412yes—
72C4×C7×C8D411yescharacteristic
73C4×C7×C8D411yescharacteristic
74C4×C7×C8C2312yes—
75C4×C7×C8C2312yes—
76C4×C7×C8C2311yescharacteristic
77C4×C7×C8C2311yescharacteristic
78#6395C2×C414yes—
79#6395C2×C414yes—
80(C2×D7)⋊C8—22no—
81(C2×D7)⋊C8—22no—
82(C2×D7)⋊C8—22no—
83(C2×D7)⋊C8—22no—
84#6392C2×C414yes—
85C4⋊(C7⋊C8)—21no—
86C4⋊(C7⋊C8)—21no—
87C4⋊(C7⋊C8)C2312yes—
88C4⋊(C7⋊C8)C2312yes—
89#3100—71no—
90#2931—71no—
91C2×C8×D4—71no—
92#1586—74no—
93#1580—74no—
94#1570—71no—
95#1486—71no—
96#1480—71no—
97#1452D711yescharacteristic, complemented
98C23×D7—21no—
99C22×C4×C7C22×C411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7#4111yescharacteristic
102C2×C22⋊D7—22no—
103C22×Dic7—21no—
104#1157—42no—
105#1157—42no—
106C2×D28—21no—
107C2×D28—24no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28C22×C412yes—
111C2×D28C22×C412yes—
112C2×C4×D7—22no—
113C2×C4×D7—22no—
114C2×C4×D7—22no—
115C2×C4×D7—42no—
116C2×C4×D7—42no—
117C2×Dic14—21no—
118C2×C7×C8—22no—
119C2×C7×C8—22no—
120C2×C7×C8—22no—
121C2×C7×C8C22×C414yes—
122C2×C7×C8C2×D411yescharacteristic
123C2×C7×C8C2×D411yescharacteristic
124C2×C7×C8C2×D411yescharacteristic
125C2×C7×C8C2×D411yescharacteristic
126C42×C7C22×C411yescharacteristic
127C42×C7C2×D411yescharacteristic
128C42×C7#4111yescharacteristic
129C42×C7C2411yescharacteristic
130(C2×D7)⋊C4—22no—
131(C2×D7)⋊C4—22no—
132(C2×D7)⋊C4—22no—
133(C2×D7)⋊C4—22no—
134C4⋊Dic7—21no—
135C4⋊Dic7—21no—
136C4⋊Dic7C22×C412yes—
137C4⋊Dic7C22×C412yes—
138#1136—42no—
139C2×C7⋊C8—22no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—42no—
142D56—28no—
143#1131—44no—
144C8×D7—44no—
145C8×D7—44no—
146C8×D7—44no—
147C2×#131—141no—
148C23×C8—141no—
149#532—71no—
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#431C2×D711yescharacteristic
165#431C2×D711yescharacteristic
166#423—71no—
167C2×C4⋊C8—71no—
168#408—141no—
169C2×C22⋊C8—141no—
170C2×C4×C8C2×D711yescharacteristic
171#334—74no—
172D4⋊C8—74no—
173D4⋊C8—74no—
174#322C2×D712yescomplemented
175#322C2×D712yescomplemented
176C23×C7C4×D411yescharacteristic
177C22×D7—24no—
178C22×D7—22no—
179C22×D7—22no—
180C22×D7—42no—
181C22×D7—42no—
182C2×C4×C7—22no—
183C2×C4×C7—41no—
184C2×C4×C7C4×D411yescharacteristic
185C2×C4×C7C4×D411yescharacteristic
186C2×C4×C7C4×D411yescharacteristic
187C2×C4×C7AGL(1,ℤ/8)12yes—
188C2×C4×C7AGL(1,ℤ/8)12yes—
189C2×C4×C7C23×C411yescharacteristic
190C2×C4×C7C22×D411yescharacteristic
191C2×C4×C7C2×#4111yescharacteristic
192C22⋊D7—44no—
193C2×Dic7—22no—
194C2×Dic7—22no—
195C2×Dic7—42no—
196D28—24no—
197D28—22no—
198D28—24no—
199D28—42no—
200D28—42no—
201D28#13114yescomplemented
202C4×D7—42no—
203C4×D7—42no—
204C4×D7—42no—
205C4×D7—42no—
206C4×D7—44no—
207Dic14—42no—
208C7×C8—24no—
209C7×C8—21no—
210C7×C8—21no—
211C7×C8—21no—
212C7×C8—21no—
213C7×C8—42no—
214C7⋊C8—42no—
215C7⋊C8—42no—
216C2×#41—141no—
217C22×D4—71no—
218C23×C4—141no—
219C2×D8—74no—
220#131—284no—
221C2×M16—141no—
222C2×M16—142no—
223C22×C8—141no—
224C22×C8—142no—
225C22×C8—142no—
226C22×C8—142no—
227C22×C8—284no—
228C22×C8D2811yescharacteristic
229C22×C8D2811yescharacteristic
230C4⋊Q8—71no—
231#127—71no—
232#124—141no—
233#121—72no—
234#121—141no—
235C4×D4—72no—
236C4×D4—72no—
237C4×D4—72no—
238C4×D4—141no—
239C4×D4—141no—
240C4×D4—141no—
241C2×C4⋊C4—71no—
242C2×C22⋊C4—141no—
243C2×C42C22×D711yescharacteristic
244#107—74no—
245C4⋊C8—21no—
246C4⋊C8—72no—
247C4⋊C8—72no—
248C4⋊C8—141no—
249C4⋊C8—141no—
250C4⋊C8C4×D712yes—
251#102—74no—
252#102—74no—
253C22⋊C8—21no—
254C22⋊C8—142no—
255C22⋊C8—142no—
256C22⋊C8—142no—
257C22⋊C8—142no—
258C22⋊C8C4×D712yes—
259C4×C8D2811yescharacteristic
260C4×C8D2811yescharacteristic
261C4×C8C22×D712yes—
262C4×C8C22×D712yes—
263C4×C8C22×D711yescharacteristic
264C4×C8C22×D711yescharacteristic
265C22×C7—21no—
266C22×C7—21no—
267C22×C7#44212yes—
268C22×C7C2×C4×D411yescharacteristic
269C2×D7—24no—
270C2×D7—48no—
271C2×D7—42no—
272C2×D7—42no—
273C2×D7—42no—
274C2×D7—42no—
275C2×D7—42no—
276C4×C7—21no—
277C4×C7—21no—
278C4×C7—21no—
279C4×C7#44212yes—
280C4×C7C2×#13112yes—
281C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
282C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
283Dic7—42no—
284Dic7—42no—
285C24—141no—
286#41—282no—
287#41—282no—
288C2×Q8—141no—
289C2×D4—72no—
290C2×D4—72no—
291C2×D4—144no—
292C2×D4—142no—
293C2×D4—141no—
294C2×D4—141no—
295C2×D4—141no—
296C22×C4—141no—
297C22×C4—142no—
298C22×C4—142no—
299C22×C4—142no—
300C22×C4—282no—
301C22×C4—282no—
302C22×C4C2×C4×D711yescharacteristic
303C22×C4C2×D2811yescharacteristic
304C22×C4#115711yescharacteristic
305D8—148no—
306M16—284no—
307M16—282no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—142no—
312C2×C8—142no—
313C2×C8—284no—
314C2×C8—284no—
315C2×C8—284no—
316C2×C8—282no—
317C2×C8C2×C4×D714yes—
318C2×C8C2×D2811yescharacteristic
319C2×C8C2×D2811yescharacteristic
320C2×C8C2×D2811yescharacteristic
321C2×C8C2×D2811yescharacteristic
322C4⋊C4—72no—
323C4⋊C4—72no—
324C4⋊C4—141no—
325C4⋊C4—141no—
326C22⋊C4—142no—
327C22⋊C4—142no—
328C22⋊C4—142no—
329C22⋊C4—142no—
330C42C2×C4×D711yescharacteristic
331C42C2×D2811yescharacteristic
332C42#115711yescharacteristic
333C42C23×D711yescharacteristic
334C2×C7—22no—
335C2×C7#292911yescharacteristic
336C2×C7C2×#44211yescharacteristic
337C2×C7#295511yescharacteristic
338D7—44no—
339D7—82no—
340C23—144no—
341C23—142no—
342C23—142no—
343C23—282no—
344C23—282no—
345C23C4×D2811yescharacteristic
346Q8—282no—
347D4—74no—
348D4—144no—
349D4—142no—
350D4—144no—
351D4—282no—
352D4—284no—
353D4—282no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—142no—
357C2×C4—142no—
358C2×C4—282no—
359C2×C4—282no—
360C2×C4—282no—
361C2×C4—282no—
362C2×C4—282no—
363C2×C4—284no—
364C2×C4C4×D2811yescharacteristic
365C2×C4C4×D2811yescharacteristic
366C2×C4C4×D2811yescharacteristic
367C2×C4#647112yes—
368C2×C4#647112yes—
369C2×C4C22×C4×D711yescharacteristic
370C2×C4C22×D2811yescharacteristic
371C2×C4C2×#115711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8—21no—
376C8—24no—
377C8—42no—
378C8—282no—
379C8—282no—
380C7#1790711yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—144no—
384C22—288no—
385C22—282no—
386C22—282no—
387C22—282no—
388C22—282no—
389C22—282no—
390C22#8948312yes—
391C22C2×C4×D2811yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—282no—
396C4—282no—
397C4#8948312yes—
398C4C2×#647111yescharacteristic
399C4C2×#647111yescharacteristic
400C4D7×#13112yes—
401C2—22no—
402C2—284no—
403C2—562no—
404C2C2×#8948311yescharacteristic
405C2#1173599811yescharacteristic
406C2#1173602411yescharacteristic
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC42C23×D7
FittingC7×#1452C2
socleC22×C7C2×C4×D4
prime radical#11736742C2
cosocleC42×C7C24
hypercenter#1452D7
hypocenterC7#17907
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #1452

Complements

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

NH ≅ G/Ntype
2. #11735903C2semidirect
3. C2×C8×D28C2semidirect
4. C7×#1452C2semidirect
5. #11728516C2semidirect
6. #11728512C2semidirect
7. #11728503C2semidirect
8. #11725835C2semidirect
9. #11725831C2semidirect
27. #89460C22semidirect
33. C8×D28C22semidirect
34. C7×#322C22semidirect
35. C7×#322C22semidirect
37. #89254C22semidirect
38. #89252C22semidirect
97. #1452D7semidirect
174. #322C2×D7semidirect
175. #322C2×D7semidirect
201. 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
#117367422
#117359032
C2×C8×D282
C7×#14522
#117285162
#117285122
#117285032
#117258352
#117258312
#179077