Finite Group Databasebeta

← #13352009

Subgroups — #13352009

Order 1792.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C7×#117C211yesmaximal, characteristic
2C2×#89697C211yesmaximal, characteristic
3C2×C4×#1136C211yesmaximal, characteristic
4#11727232C214yesmaximal, complemented
5#11727202C214yesmaximal
6#11727142C211yesmaximal, characteristic, complemented
7#11727142C211yesmaximal, characteristic, complemented
8C4×#6377C211yesmaximal, characteristic, complemented
9C42×C7⋊C8C211yesmaximal, characteristic, complemented
10C2×C7×#117C412yes—
11C22×C42×C7C2211yescharacteristic
12C22×#1136C2212yes—
13C4×C7×C4⋊C4C412yes—
14C4×C7×C4⋊C4C2212yes—
15C4×C7×C22⋊C4C412yes—
16C4×C7×C22⋊C4C2212yes—
17C7×#376C412yes—
18C43×C7C2211yescharacteristic
19C43×C7C2211yescharacteristic
20C2×C22⋊(C7⋊C8)C2212yes—
21#89697C418yescomplemented
22C2×C4⋊(C7⋊C8)C2212yes—
23C4×#1136C418yescomplemented
24C2×#6377C2214yes—
25C2×#6377C2212yes—
26C2×C4×C7⋊C8C2214yes—
27C2×C4×C7⋊C8C2211yescharacteristic
28C2×C4×C7⋊C8C2211yescharacteristic
29C2×C4×C7⋊C8C2211yescharacteristic
30C2×C4×C7⋊C8C2211yescharacteristic
31#89319C2214yes—
32#89319C2214yes—
33#20472—71nomaximal, Sylow
34C23×C4×C7C2×C411yescharacteristic
35C23×C4×C7C2×C411yescharacteristic
36C23×C4×C7C2311yescharacteristic
37C7×#117—28no—
38C2×C7×C4⋊C4C2×C414yes—
39C2×C7×C22⋊C4C2×C414yes—
40C2×C42×C7C2×C414yes—
41C2×C42×C7C2×C414yes—
42C2×C42×C7C2×C412yes—
43C2×C42×C7C2×C412yes—
44C2×C42×C7C2×C412yes—
45C2×C42×C7C2×C411yescharacteristic
46C2×C42×C7C2×C411yescharacteristic
47C2×C42×C7C2×C412yes—
48C2×C42×C7C2312yes—
49C2×C42×C7C2312yes—
50C2×C42×C7C2311yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C2×#1136—28no—
53C2×#1136C2×C418yes—
54C22×C7⋊C8C2312yes—
55C22×C7⋊C8C2312yes—
56C22×C7⋊C8C2312yes—
57C22×C7⋊C8C2312yes—
58C7×#95C2×C414yes—
59C7×#95C2×C414yes—
60C22⋊(C7⋊C8)C2×C418yes—
61C4⋊(C7⋊C8)C2×C418yes—
62#6377—28no—
63#6377C2×C418yes—
64C4×C7⋊C8—28no—
65C4×C7⋊C8C2×C414yes—
66C4×C7⋊C8C2×C414yes—
67C4×C7⋊C8C2×C414yes—
68C4×C7⋊C8C2×C414yes—
69C2×#431—71no—
70C2×C4×M16—71no—
71C4×#117D711yescharacteristic
72#1771—74no—
73#1753—74no—
74#1744—71no—
75#1744—71no—
76C4×#97—71no—
77C42×C8—71no—
78C24×C7C22×C411yescharacteristic
79C22×C4×C7—24no—
80C22×C4×C7—24no—
81C22×C4×C7—24no—
82C22×C4×C7C4214yes—
83C22×C4×C7C4214yes—
84C22×C4×C7C22×C414yes—
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C22×C412yes—
89C22×C4×C7C22×C412yes—
90C22×C4×C7C22×C414yes—
91C22×C4×C7C22×C411yescharacteristic
92C22×C4×C7C22×C411yescharacteristic
93C22×C4×C7C2411yescharacteristic
94C7×C4⋊C4—28no—
95C7×C22⋊C4—28no—
96C42×C7—24no—
97C42×C7—24no—
98C42×C7—24no—
99C42×C7—24no—
100C42×C7—24no—
101C42×C7C4214yes—
102C42×C7C4214yes—
103C42×C7M1618yescomplemented
104C42×C7C22×C412yes—
105C42×C7C22×C412yes—
106C42×C7C22×C412yes—
107C42×C7C22×C412yes—
108#1136—216no—
109C2×C7⋊C8—24no—
110C2×C7⋊C8—24no—
111C2×C7⋊C8—24no—
112C2×C7⋊C8C22×C414yes—
113C2×C7⋊C8C22×C414yes—
114C2×C7⋊C8C22×C414yes—
115C2×C7⋊C8C22×C414yes—
116C2×C7⋊C8#4118yes—
117C22×M16—72no—
118C2×#117Dic712yes—
119C22×C42C2×D711yescharacteristic
120#431—78no—
121C2×C4⋊C8—72no—
122C2×C22⋊C8—72no—
123C4×M16—78no—
124C2×#97—72no—
125C2×#97—74no—
126C2×C4×C8—71no—
127C2×C4×C8—71no—
128C2×C4×C8—71no—
129C2×C4×C8—71no—
130C2×C4×C8—74no—
131C4×C4⋊C4Dic712yes—
132C4×C4⋊C4C2×D712yes—
133C4×C22⋊C4Dic712yes—
134C4×C22⋊C4C2×D712yes—
135#376Dic712yes—
136C43C2×D711yescharacteristic
137C43C2×D711yescharacteristic
138#336—74no—
139#336—74no—
140C23×C7—22no—
141C23×C7—22no—
142C23×C7—22no—
143C23×C7C2×C4212yes—
144C23×C7C23×C411yescharacteristic
145C2×C4×C7—24no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—24no—
149C2×C4×C7—22no—
150C2×C4×C7—24no—
151C2×C4×C7—24no—
152C2×C4×C7—28no—
153C2×C4×C7—28no—
154C2×C4×C7—24no—
155C2×C4×C7—24no—
156C2×C4×C7—24no—
157C2×C4×C7C2×C4212yes—
158C2×C4×C7C2×C4212yes—
159C2×C4×C7C2×C4212yes—
160C2×C4×C7C2×C4212yes—
161C2×C4×C7C2×C4212yes—
162C2×C4×C7C2×M1618yes—
163C2×C4×C7C2×M1614yes—
164C2×C4×C7#13114yes—
165C2×C4×C7C23×C411yescharacteristic
166C2×C4×C7C23×C411yescharacteristic
167C2×C4×C7C2×#4114yes—
168C7⋊C8—24no—
169C7⋊C8—24no—
170C7⋊C8—24no—
171C7⋊C8#11718yescomplemented
172C23×C4C2×Dic711yescharacteristic
173C23×C4C2×Dic711yescharacteristic
174C23×C4C22×D711yescharacteristic
175C2×M16—78no—
176C2×M16—148no—
177C22×C8—72no—
178C22×C8—72no—
179C22×C8—72no—
180C22×C8—72no—
181#117—28no—
182C2×C4⋊C4C2×Dic714yes—
183C2×C22⋊C4C2×Dic714yes—
184C2×C42C4×D714yes—
185C2×C42C4×D714yes—
186C2×C42C2×Dic712yes—
187C2×C42C2×Dic712yes—
188C2×C42C2×Dic712yes—
189C2×C42C2×Dic712yes—
190C2×C42C2×Dic711yescharacteristic
191C2×C42C2×Dic711yescharacteristic
192C2×C42C22×D712yes—
193C2×C42C22×D712yes—
194C2×C42C22×D711yescharacteristic
195C2×C42C22×D711yescharacteristic
196C4⋊C8—78no—
197C22⋊C8—78no—
198#97—78no—
199#97—148no—
200C4×C8—74no—
201C4×C8—74no—
202C4×C8—74no—
203C4×C8—74no—
204C4×C8—148no—
205#95C2×Dic714yes—
206#95C2×Dic714yes—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7—44no—
211C22×C7C22×C4211yescharacteristic
212C22×C7C4×#4112yes—
213C22×C7C22×M1612yes—
214C22×C7C2×#13112yes—
215C4×C7—22no—
216C4×C7—22no—
217C4×C7—28no—
218C4×C7—22no—
219C4×C7—44no—
220C4×C7C4×M1614yes—
221C4×C7#40514yes—
222C4×C7C2×#11712yes—
223C4×C7C4×#4112yes—
224C24C22×Dic711yescharacteristic
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4C4×Dic714yes—
229C22×C4C4×Dic714yes—
230C22×C4C2×C4×D712yes—
231C22×C4C2×C4×D712yes—
232C22×C4C22×Dic714yes—
233C22×C4C22×Dic714yes—
234C22×C4C22×Dic711yescharacteristic
235C22×C4C22×Dic711yescharacteristic
236C22×C4C22×Dic711yescharacteristic
237C22×C4C22×Dic711yescharacteristic
238C22×C4C22×Dic711yescharacteristic
239C22×C4C23×D711yescharacteristic
240M16—1416no—
241C2×C8—74no—
242C2×C8—78no—
243C2×C8—74no—
244C2×C8—74no—
245C2×C8—74no—
246C2×C8—144no—
247C2×C8—144no—
248C2×C8—144no—
249C4⋊C4—28no—
250C22⋊C4—28no—
251C42—24no—
252C42—24no—
253C42—24no—
254C42—24no—
255C42—24no—
256C42#113618yescomplemented
257C42C4×Dic714yes—
258C42C4×Dic714yes—
259C42C2×C4×D712yes—
260C42C2×C4×D712yes—
261C42C2×C4×D712yes—
262C42C2×C4×D712yes—
263C2×C7—42no—
264C2×C7C4×#11711yescharacteristic
265C2×C7C2×C4×M1611yescharacteristic
266C2×C7C2×#40511yescharacteristic
267C2×C7#296514yes—
268C23—22no—
269C23—22no—
270C23—22no—
271C23C2×C4×Dic712yes—
272C23C23×Dic711yescharacteristic
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—28no—
279C2×C4—24no—
280C2×C4—24no—
281C2×C4—28no—
282C2×C4—24no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4C2×#113618yes—
286C2×C4C2×#113614yes—
287C2×C4C2×C4×Dic712yes—
288C2×C4C2×C4×Dic712yes—
289C2×C4C2×C4×Dic712yes—
290C2×C4C2×C4×Dic712yes—
291C2×C4C2×C4×Dic712yes—
292C2×C4#651114yes—
293C2×C4C22×C4×D711yescharacteristic
294C2×C4C22×C4×D711yescharacteristic
295C2×C4D7×#4114yes—
296C8—78no—
297C8—144no—
298C8—144no—
299C8—144no—
300C7#2047211yescharacteristic, complemented, Sylow
301C22—22no—
302C22—22no—
303C22—22no—
304C22—44no—
305C22C22×#113612yes—
306C22C22×C4×Dic711yescharacteristic
307C22C2×#651112yes—
308C22#41×Dic712yes—
309C4—22no—
310C4—28no—
311C4—22no—
312C4—22no—
313C4—44no—
314C4C4×#113614yes—
315C4#8976914yes—
316C4D7×#11712yes—
317C4#9021312yes—
318C2—42no—
319C2C2×C4×#113611yescharacteristic
320C2C2×#8976911yescharacteristic
321C2Dic7×#11711yescharacteristic
322C2#1173996214yes—
subgroupG/H
centerC2×C42C22×D7
commutatorC22×C7C22×C42
FrattiniC22×C4C23×D7
FittingC4×C7×#117C2
socleC23×C7C23×C4
prime radicalC24×C7C22×C4
cosocleC22×C4×C7C24
hypercenterC4×#117D7
hypocenterC7#20472
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
4. #11727232C2semidirect
6. #11727142C2semidirect
7. #11727142C2semidirect
8. C4×#6377C2semidirect
9. C42×C7⋊C8C2semidirect
21. #89697C4semidirect
23. C4×#1136C4semidirect
103. C42×C7M16semidirect
171. C7⋊C8#117semidirect
256. C42#1136semidirect
300. C7#20472semidirect

Maximal subgroups

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

subgroupindex
C4×C7×#1172
C2×#896972
C2×C4×#11362
#117272322
#117272022
#117271422
C4×#63772
C42×C7⋊C82
#204727