Finite Group Databasebeta

← #13323261

Subgroups — #13323261

Order 1792.

730 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2982C211yesmaximal, characteristic, complemented
2C2×C4×#1143C211yesmaximal, characteristic
3#11736731C211yesmaximal, characteristic
4#11729375C211yesmaximal, characteristic, complemented
5#11728691C214yesmaximal, complemented
6C8×#1143C214yesmaximal, complemented
7#11728510C211yesmaximal, characteristic
8#11728499C211yesmaximal, characteristic, complemented
9#11728494C211yesmaximal, characteristic, complemented
10C2×C4×C7×Q8C2211yescharacteristic
11C22×#1143C411yescharacteristic
12C2×C4×D28C2211yescharacteristic
13C7×C8×Q8—22no—
14C7×C8×Q8C2214yescomplemented
15C7×#431—22no—
16C7×#431C2211yescharacteristic
17C7×#431C2211yescharacteristic
18C2×#6409C411yescharacteristic
19C4×#1143C412yes—
20C4×#1143C412yes—
21C4×#1143C2212yes—
22C4×#1143C2212yes—
23C2×#6383C411yescharacteristic
24C2×#6382C411yescharacteristic
25#89697C2211yescharacteristic
26#89697C2211yescharacteristic
27C2×C4×C7⋊C8C2211yescharacteristic
28#89606—22no—
29D7×C22⋊C8—22no—
30C8×D28C2214yes—
31Q8⋊(C7⋊C8)C2214yescomplemented
32#89278C2214yes—
33#89275C2212yescomplemented
34#89274C2212yescomplemented
35#89249C2212yescomplemented
36C8×C7⋊C8C2212yescomplemented
37#17855—71nomaximal, Sylow
38C22×C7×Q8C2×C411yescharacteristic
39C22×D28C2×C411yescharacteristic
40C22×C4×D7—21no—
41C4×C7×Q8—21no—
42C4×C7×Q8—21no—
43C4×C7×Q8C2×C412yes—
44C4×C7×Q8C2312yes—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4C2×C411yescharacteristic
47C2×C42×C7—21no—
48C2×C42×C7C2311yescharacteristic
49C2×#1143C818yescomplemented
50C2×#1143C2×C412yes—
51C2×#1143C2×C412yes—
52C2×(C2×D7)⋊C4—21no—
53C2×C4⋊Dic7C2×C411yescharacteristic
54C22×C7⋊C8C2×C411yescharacteristic
55C22×C7⋊C8C2×C411yescharacteristic
56C2×C8×D7—24no—
57C4×D28—21no—
58C4×D28—21no—
59C4×D28C2×C412yes—
60C4×D28C2312yes—
61C7×C4⋊C8—24no—
62C7×C4⋊C8D412yes—
63C7×C4⋊C8C2312yes—
64C7×C22⋊C8—21no—
65C7×C22⋊C8D412yes—
66C4×C7×C8—24no—
67C4×C7×C8—21no—
68C4×C7×C8C2312yes—
69#6409C2×C412yes—
70#6409C2×C412yes—
71C22⋊(C7⋊C8)—21no—
72C22⋊(C7⋊C8)—21no—
73(C2×D7)⋊C8—24no—
74#6383C2×C412yes—
75#6383C2×C412yes—
76#6382C2×C412yes—
77#6382C2×C412yes—
78C4⋊(C7⋊C8)—21no—
79C4⋊(C7⋊C8)C2312yes—
80C4×C7⋊C8—21no—
81C4×C7⋊C8C2×C411yescharacteristic
82C4×C7⋊C8C2×C411yescharacteristic
83C4×C7⋊C8C2312yes—
84C4×C7⋊C8C2311yescharacteristic
85C4×C7⋊C8C2311yescharacteristic
86#2982D711yescharacteristic, complemented
87#2972—71no—
88C2×C4×SD16—71no—
89SD16⋊C8—74no—
90C8×SD16—74no—
91#1564—71no—
92#1479—71no—
93#1478—71no—
94#1450—71no—
95C23×D7—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—22no—
98C2×C7×Q8C2×C814yescomplemented
99C2×C7×Q8C22×C412yes—
100C22×C4×C7—21no—
101C22×C4×C7—21no—
102C22×C4×C7C22×C411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7#4111yescharacteristic
105C22×Dic7—21no—
106C2×D28—21no—
107C2×D28—22no—
108C2×D28C2×C814yes—
109C2×D28C22×C412yes—
110C2×C4×D7—22no—
111C2×C4×D7—22no—
112C2×C4×D7—82no—
113C2×C7×C8—22no—
114C2×C7×C8C2×D414yes—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—22no—
118C7×C4⋊C4C22×C412yes—
119C42×C7—22no—
120C42×C7—22no—
121C42×C7C22×C411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7#4111yescharacteristic
124C42×C7C2411yescharacteristic
125#1143C2×C818yescomplemented
126#1143C2×C818yescomplemented
127(C2×D7)⋊C4—22no—
128(C2×D7)⋊C4—22no—
129C4⋊Dic7—21no—
130C4⋊Dic7C22×C412yes—
131C2×C7⋊C8—22no—
132C2×C7⋊C8—22no—
133C2×C7⋊C8—22no—
134C2×C7⋊C8C2×C814yescomplemented
135C2×C7⋊C8C22×C411yescharacteristic
136C2×C7⋊C8C22×C411yescharacteristic
137C2×C7⋊C8C22×C411yescharacteristic
138C2×C7⋊C8C22×C411yescharacteristic
139C8×D7—84no—
140C22×SD16—71no—
141C2×C4×Q8C2×D711yescharacteristic
142C2×C4×D4—71no—
143C8×Q8—22no—
144C8×Q8C2×D714yescomplemented
145C4×SD16—72no—
146C4×SD16—72no—
147C4×SD16—72no—
148C4×SD16—72no—
149C8×D4—74no—
150C8×D4—142no—
151#431—22no—
152#431—71no—
153#431—71no—
154#431C2×D711yescharacteristic
155#431C2×D711yescharacteristic
156C2×#106—71no—
157C2×#103—71no—
158C2×#102—71no—
159C2×C22⋊C8—142no—
160C2×C4×C8—71no—
161#335—72no—
162#334—72no—
163Q8⋊C8—74no—
164D4⋊C8—74no—
165#322—72no—
166C82—72no—
167C23×C7C4×D411yescharacteristic
168C22×D7—22no—
169C22×D7—24no—
170C22×D7—81no—
171C22×D7—81no—
172C7×Q8—22no—
173C7×Q8—24no—
174C7×Q8C22×C814yes—
175C2×C4×C7—22no—
176C2×C4×C7—24no—
177C2×C4×C7—22no—
178C2×C4×C7—22no—
179C2×C4×C7—41no—
180C2×C4×C7—81no—
181C2×C4×C7C4×D411yescharacteristic
182C2×C4×C7C4×D411yescharacteristic
183C2×C4×C7C4×D411yescharacteristic
184C2×C4×C7C22×C812yes—
185C2×C4×C7#13112yes—
186C2×C4×C7C23×C411yescharacteristic
187C2×C4×C7C22×D411yescharacteristic
188C2×C4×C7C2×#4111yescharacteristic
189C2×Dic7—22no—
190C2×Dic7—81no—
191D28—22no—
192D28—24no—
193D28C22×C814yes—
194C4×D7—44no—
195C4×D7—82no—
196C7×C8—24no—
197C7×C8—42no—
198C7⋊C8—21no—
199C7⋊C8—21no—
200C7⋊C8—42no—
201C7⋊C8—42no—
202C7⋊C8C22×C812yes—
203C7⋊C8C22×C812yes—
204C22×Q8C4×D711yescharacteristic
205C22×D4—71no—
206C23×C4—141no—
207C2×SD16—78no—
208C2×SD16—72no—
209C2×SD16—72no—
210C22×C8—71no—
211C22×C8—71no—
212C22×C8—144no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8C4×D712yes—
216C4×Q8C22×D712yes—
217C4×D4—72no—
218C4×D4—72no—
219C4×D4—141no—
220C4×D4—141no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—71no—
223C2×C4⋊C4C4×D711yescharacteristic
224C2×C22⋊C4—141no—
225C2×C42—21no—
226C2×C42C22×D711yescharacteristic
227#106—72no—
228#106—72no—
229C4⋊C8—24no—
230C4⋊C8—72no—
231C4⋊C8—141no—
232C4⋊C8C22⋊D712yes—
233C4⋊C8C22×D712yes—
234#103—72no—
235#103—72no—
236#102—72no—
237#102—72no—
238C22⋊C8—21no—
239C22⋊C8—141no—
240C22⋊C8—141no—
241C22⋊C8—144no—
242C22⋊C8C22⋊D712yes—
243C4×C8—24no—
244C4×C8—21no—
245C4×C8—71no—
246C4×C8—71no—
247C4×C8—71no—
248C4×C8—71no—
249C4×C8—72no—
250C4×C8—141no—
251C4×C8C22×D712yes—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7C8×D412yes—
255C22×C7C2×C4×D411yescharacteristic
256C2×D7—24no—
257C2×D7—42no—
258C2×D7—42no—
259C2×D7—84no—
260C4×C7—24no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—42no—
265C4×C7C8×D412yes—
266C4×C7C23×C811yescharacteristic
267C4×C7C2×#13111yescharacteristic
268C4×C7#57712yes—
269Dic7—42no—
270C24—141no—
271C2×Q8—22no—
272C2×Q8—21no—
273C2×Q8C8×D714yescomplemented
274C2×Q8C2×C4×D712yes—
275C2×D4—74no—
276C2×D4—72no—
277C2×D4—142no—
278C2×D4—141no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—141no—
282C22×C4—142no—
283C22×C4—142no—
284C22×C4—562no—
285C22×C4C2×C4×D711yescharacteristic
286C22×C4#115711yescharacteristic
287C22×C4C2×C22⋊D711yescharacteristic
288SD16—78no—
289SD16—78no—
290C2×C8—22no—
291C2×C8—74no—
292C2×C8—71no—
293C2×C8—71no—
294C2×C8—71no—
295C2×C8—71no—
296C2×C8—142no—
297C2×C8—142no—
298C2×C8—142no—
299C2×C8—564no—
300C2×C8C2×C22⋊D714yes—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—72no—
305C4⋊C4—141no—
306C4⋊C4C2×C4×D712yes—
307C22⋊C4—142no—
308C22⋊C4—142no—
309C42—22no—
310C42—22no—
311C42C2×C4×D711yescharacteristic
312C42#115711yescharacteristic
313C42C2×C22⋊D711yescharacteristic
314C42C23×D711yescharacteristic
315C2×C7—22no—
316C2×C7C2×C8×D411yescharacteristic
317C2×C7#294911yescharacteristic
318C2×C7#295711yescharacteristic
319D7—44no—
320C23—142no—
321C23—144no—
322C23—561no—
323C23—561no—
324C23C4×C22⋊D711yescharacteristic
325Q8—24no—
326Q8—22no—
327Q8C2×C8×D714yes—
328D4—74no—
329D4—144no—
330D4—142no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—81no—
337C2×C4—142no—
338C2×C4—284no—
339C2×C4—562no—
340C2×C4—561no—
341C2×C4C2×C8×D712yes—
342C2×C4#646412yes—
343C2×C4C4×C22⋊D711yescharacteristic
344C2×C4C4×C22⋊D711yescharacteristic
345C2×C4C4×C22⋊D711yescharacteristic
346C2×C4C22×C4×D711yescharacteristic
347C2×C4C2×#115711yescharacteristic
348C2×C4C22×C22⋊D711yescharacteristic
349C8—24no—
350C8—42no—
351C8—72no—
352C8—72no—
353C8—141no—
354C8—141no—
355C8—282no—
356C8—282no—
357C7#1785511yescharacteristic, complemented, Sylow
358C22—21no—
359C22—21no—
360C22—144no—
361C22—282no—
362C22—282no—
363C22—564no—
364C22C8×C22⋊D712yes—
365C22C2×C4×C22⋊D711yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—24no—
370C4—42no—
371C4—282no—
372C4C8×C22⋊D712yes—
373C4C22×C8×D711yescharacteristic
374C4C2×#646411yescharacteristic
375C4#9052812yes—
376C2—22no—
377C2—284no—
378C2#1173999311yescharacteristic
379C2C2×C8×C22⋊D711yescharacteristic
380C2#1174125711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D7
commutatorC4×C7C23×C8
FrattiniC42C23×D7
FittingC7×#2982C2
socleC22×C7C2×C4×D4
prime radicalC22×D28C2×C4
cosocleC42×C7C24
hypercenter#2982D7
hypocenterC7#17855
LCSG ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2982

Complements

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

NH ≅ G/Ntype
1. C7×#2982C2semidirect
4. #11729375C2semidirect
5. #11728691C2semidirect
6. C8×#1143C2semidirect
8. #11728499C2semidirect
9. #11728494C2semidirect
14. C7×C8×Q8C22semidirect
31. Q8⋊(C7⋊C8)C22semidirect
33. #89275C22semidirect
34. #89274C22semidirect
35. #89249C22semidirect
36. C8×C7⋊C8C22semidirect
49. C2×#1143C8semidirect
86. #2982D7semidirect
98. C2×C7×Q8C2×C8semidirect
125. #1143C2×C8semidirect
126. #1143C2×C8semidirect
134. C2×C7⋊C8C2×C8semidirect
144. C8×Q8C2×D7semidirect
273. C2×Q8C8×D7semidirect
357. C7#17855semidirect

Maximal subgroups

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

subgroupindex
C7×#29822
C2×C4×#11432
#117367312
#117293752
#117286912
C8×#11432
#117285102
#117284992
#117284942
#178557