Finite Group Databasebeta

← C22×#89247 #13320877

Subgroups — C22×#89247 #13320877

Order 1792.

1846 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6487C211yesmaximal, characteristic
2C22×C7×#95C211yesmaximal, characteristic
3C22×C22⋊(C7⋊C8)C211yesmaximal, characteristic
4C2×#89247C2112yesmaximal, direct product factor
5C24×C4×C7—21no—
6C24×C4×C7C2211yescharacteristic
7C23×Dic14C411yescharacteristic, complemented
8C22×C22⋊Dic7—21no—
9C22×C4⋊Dic7C411yescharacteristic, complemented
10C2×#6487—23no—
11C2×#6487—23no—
12C2×#6487C416yescomplemented
13C2×#6487C2216yes—
14C22×Dic7⋊C4—21no—
15C23×C7⋊C8—21no—
16C2×C7×#95—23no—
17C2×C7×#95C2216yes—
18C2×C22⋊(C7⋊C8)—23no—
19C2×C22⋊(C7⋊C8)C2216yes—
20#89247C22116yesdirect product factor
21C22×#328—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C23×C4×C7—28no—
24C23×C4×C7—26no—
25C23×C4×C7—44no—
26C23×C4×C7—44no—
27C23×C4×C7—44no—
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7C2×C413yes—
30C23×C4×C7D411yescharacteristic
31C23×C4×C7D413yes—
32C23×C4×C7D413yes—
33C23×C4×C7C2313yes—
34C23×Dic7—21no—
35C23×Dic7—21no—
36C22×Dic14—23no—
37C22×Dic14—44no—
38C22×Dic14C2×C416yescomplemented
39C2×C22⋊Dic7—26no—
40C2×C22⋊Dic7—26no—
41C2×C4⋊Dic7—23no—
42C2×C4⋊Dic7C2×C416yescomplemented
43#6487—212no—
44#6487—26no—
45#6487—26no—
46#6487C2×C4112yescomplemented
47#6487C2314yes—
48C2×Dic7⋊C4—26no—
49C2×Dic7⋊C4—26no—
50C22×C7⋊C8—26no—
51C22×C7⋊C8—44no—
52C7×#95—26no—
53C7×#95C2314yes—
54C22⋊(C7⋊C8)—26no—
55C22⋊(C7⋊C8)C2314yes—
56C22×C22⋊Q8—71no—
57C22×C22⋊C8—71no—
58C22×#95D711yescharacteristic
59C2×#328—712no—
60C24×C7—24no—
61C24×C7—24no—
62C24×C7C22⋊C413yes—
63C24×C7C22⋊C411yescharacteristic
64C24×C7SD1614yes—
65C24×C7Q1614yes—
66C24×C7C2×D413yes—
67C22×C4×C7—24no—
68C22×C4×C7—24no—
69C22×C4×C7—224no—
70C22×C4×C7—24no—
71C22×C4×C7—46no—
72C22×C4×C7—46no—
73C22×C4×C7—46no—
74C22×C4×C7—46no—
75C22×C4×C7—432no—
76C22×C4×C7—412no—
77C22×C4×C7—412no—
78C22×C4×C7C22⋊C413yes—
79C22×C4×C7C22⋊C413yes—
80C22×C4×C7C22⋊C416yes—
81C22×C4×C7C22×C413yes—
82C22×C4×C7C22×C413yes—
83C22×C4×C7C2×D413yes—
84C22×C4×C7C2×D413yes—
85C22×C4×C7C2×D413yes—
86C22×C4×C7C2411yescharacteristic
87C22×Dic7—26no—
88C22×Dic7—26no—
89C22×Dic7—44no—
90C22×Dic7—44no—
91C2×Dic14—26no—
92C2×Dic14—412no—
93C2×Dic14—412no—
94C2×Dic14C22×C414yescomplemented
95C22⋊Dic7—212no—
96C22⋊Dic7—24no—
97C4⋊Dic7—26no—
98C4⋊Dic7C22×C414yescomplemented
99Dic7⋊C4—212no—
100Dic7⋊C4—24no—
101C2×C7⋊C8—24no—
102C2×C7⋊C8—412no—
103C23×Q8—71no—
104C24×C4—21no—
105C24×C4C2×D711yescharacteristic
106C23×C8—141no—
107C2×C22⋊Q8—76no—
108C2×C22⋊Q8—76no—
109C2×C22⋊Q8—143no—
110C2×C22⋊Q8—143no—
111C22×C4⋊C4—71no—
112C22×C4⋊C4—141no—
113C22×C22⋊C4—141no—
114C2×C22⋊C8—76no—
115C2×C22⋊C8—143no—
116C2×#95—23no—
117C2×#95C2×D716yes—
118#328—716no—
119C23×C7—216no—
120C23×C7—216no—
121C23×C7—26no—
122C23×C7—26no—
123C23×C7—26no—
124C23×C7—26no—
125C23×C7#9914yes—
126C23×C7#103112yes—
127C23×C7#10314yes—
128C23×C7#10414yes—
129C23×C7C2×C22⋊C413yes—
130C23×C7C2×C22⋊C413yes—
131C23×C7C2×SD1616yes—
132C23×C7C2×Q1616yes—
133C23×C7C22×D411yescharacteristic
134C2×C4×C7—26no—
135C2×C4×C7—26no—
136C2×C4×C7—26no—
137C2×C4×C7—26no—
138C2×C4×C7—28no—
139C2×C4×C7—46no—
140C2×C4×C7—43no—
141C2×C4×C7—43no—
142C2×C4×C7—43no—
143C2×C4×C7—41no—
144C2×C4×C7—44no—
145C2×C4×C7—448no—
146C2×C4×C7—44no—
147C2×C4×C7C2×C22⋊C413yes—
148C2×C4×C7C2×C22⋊C413yes—
149C2×C4×C7C23×C411yescharacteristic
150C2×C4×C7C22×D411yescharacteristic
151C2×Dic7—24no—
152C2×Dic7—24no—
153C2×Dic7—412no—
154C2×Dic7—412no—
155Dic14—412no—
156Dic14—44no—
157C7⋊C8—44no—
158C25C22⋊D711yescharacteristic
159C22×Q8—76no—
160C22×Q8—143no—
161C22×Q8—284no—
162C23×C4—28no—
163C23×C4—26no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4—44no—
167C23×C4—141no—
168C23×C4—141no—
169C23×C4C4×D711yescharacteristic
170C23×C4C4×D713yes—
171C23×C4D2811yescharacteristic
172C23×C4D2813yes—
173C23×C4C22⋊D713yes—
174C23×C4C22×D713yes—
175C22×C8—146no—
176C22×C8—284no—
177C22⋊Q8—712no—
178C22⋊Q8—74no—
179C22⋊Q8—1412no—
180C22⋊Q8—146no—
181C22⋊Q8—146no—
182C2×C4⋊C4—76no—
183C2×C4⋊C4—143no—
184C2×C4⋊C4—146no—
185C2×C4⋊C4—146no—
186C2×C22⋊C4—146no—
187C2×C22⋊C4—146no—
188C22⋊C8—74no—
189C22⋊C8—146no—
190#95—26no—
191#95C22×D714yes—
192C22×C7—21no—
193C22×C7—23no—
194C22×C7—21no—
195C22×C7—23no—
196C22×C7—212no—
197C22×C7—224no—
198C22×C7—212no—
199C22×C7#328116yescomplemented
200C22×C7C2×#9916yes—
201C22×C7C2×#10316yes—
202C22×C7C2×#10316yes—
203C22×C7C2×#10416yes—
204C22×C7C22×C22⋊C411yescharacteristic
205C22×C7C22×SD1611yescharacteristic
206C22×C7C22×Q1611yescharacteristic
207C4×C7—21no—
208C4×C7—23no—
209C4×C7—21no—
210C4×C7—23no—
211C4×C7—48no—
212Dic7—44no—
213Dic7—44no—
214C24—24no—
215C24—24no—
216C24(C2×D7)⋊C413yes—
217C24(C2×D7)⋊C411yescharacteristic
218C24#114214yes—
219C24#114414yes—
220C24C2×C22⋊D713yes—
221C2×Q8—74no—
222C2×Q8—146no—
223C2×Q8—2812no—
224C2×Q8—2812no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—224no—
228C22×C4—24no—
229C22×C4—46no—
230C22×C4—46no—
231C22×C4—46no—
232C22×C4—46no—
233C22×C4—412no—
234C22×C4—432no—
235C22×C4—412no—
236C22×C4—146no—
237C22×C4—146no—
238C22×C4—284no—
239C22×C4—284no—
240C22×C4(C2×D7)⋊C413yes—
241C22×C4(C2×D7)⋊C413yes—
242C22×C4(C2×D7)⋊C416yes—
243C22×C4C2×C4×D713yes—
244C22×C4C2×C4×D713yes—
245C22×C4C2×D2813yes—
246C22×C4C2×D2813yes—
247C22×C4C2×C22⋊D713yes—
248C22×C4C23×D711yescharacteristic
249C2×C8—144no—
250C2×C8—2812no—
251C4⋊C4—74no—
252C4⋊C4—1412no—
253C4⋊C4—146no—
254C4⋊C4—144no—
255C22⋊C4—1412no—
256C22⋊C4—144no—
257C2×C7—21no—
258C2×C7—21no—
259C2×C7—26no—
260C2×C7C2×#328112yescomplemented
261C2×C7C22×#9911yescharacteristic
262C2×C7C22×#10311yescharacteristic
263C2×C7C22×#10411yescharacteristic
264C23—26no—
265C23—26no—
266C23—26no—
267C23—26no—
268C23—216no—
269C23—216no—
270C23#637914yes—
271C23#638014yes—
272C23#6384112yes—
273C23#638414yes—
274C23C2×(C2×D7)⋊C413yes—
275C23C2×(C2×D7)⋊C413yes—
276C23C2×#114216yes—
277C23C2×#114416yes—
278C23C22×C22⋊D711yescharacteristic
279Q8—284no—
280Q8—2812no—
281C2×C4—26no—
282C2×C4—26no—
283C2×C4—26no—
284C2×C4—26no—
285C2×C4—28no—
286C2×C4—46no—
287C2×C4—43no—
288C2×C4—43no—
289C2×C4—43no—
290C2×C4—41no—
291C2×C4—448no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—144no—
295C2×C4—144no—
296C2×C4—2812no—
297C2×C4—2812no—
298C2×C4C2×(C2×D7)⋊C413yes—
299C2×C4C2×(C2×D7)⋊C413yes—
300C2×C4C22×C4×D711yescharacteristic
301C2×C4C22×D2811yescharacteristic
302C8—284no—
303C7C22×#32811yescharacteristic, complemented, Sylow
304C22—212no—
305C22—212no—
306C22—224no—
307C22—21no—
308C22—23no—
309C22—21no—
310C22—23no—
311C22#89247116yesdirect product factor
312C22C2×#637916yes—
313C22C2×#638016yes—
314C22C2×#638416yes—
315C22C2×#638416yes—
316C22C22×(C2×D7)⋊C411yescharacteristic
317C22C22×#114211yescharacteristic
318C22C22×#114411yescharacteristic
319C4—21no—
320C4—23no—
321C4—23no—
322C4—21no—
323C4—48no—
324C4—284no—
325C4—284no—
326C2—21no—
327C2—21no—
328C2—26no—
329C2C2×#89247112yesdirect product factor
330C2C22×#637911yescharacteristic
331C2C22×#638011yescharacteristic
332C2C22×#638411yescharacteristic
subgroupG/H
centerC24(C2×D7)⋊C4
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC22×C7×#95C2
socleC24×C7C22⋊C4
prime radicalC25×C7D4
cosocleC22×C4×C7C24
hypercenterC22×#95D7
hypocenterC7C22×#328
LCSG ▷ C2×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C24 ◁ C24×C4 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
4. C2×#89247C2direct (N × H)
7. C23×Dic14C4semidirect
9. C22×C4⋊Dic7C4semidirect
12. C2×#6487C4semidirect
20. #89247C22direct (N × H)
38. C22×Dic14C2×C4semidirect
42. C2×C4⋊Dic7C2×C4semidirect
46. #6487C2×C4semidirect
94. C2×Dic14C22×C4semidirect
98. C4⋊Dic7C22×C4semidirect
199. C22×C7#328semidirect
260. C2×C7C2×#328semidirect
303. C7C22×#328semidirect
311. C22#89247direct (N × H)
329. C2C2×#89247direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#64872
C22×C7×#952
C22×C22⋊(C7⋊C8)2
C2×#892472
C22×#3287