Finite Group Databasebeta

← #13513865

Subgroups — #13513865

Order 1792.

2390 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C7×#117C211yesmaximal, characteristic
2#11741472C212yesmaximal
3#11739685C218yesmaximal, complemented
4#11739684C218yesmaximal, complemented
5#11739601C212yesmaximal, complemented
6#11739589C212yesmaximal, complemented
7#11739538C214yesmaximal
8#11739505C214yesmaximal, complemented
9C24×C4×C7C2211yescharacteristic
10C2×C7×#117C4116yescomplemented
11C2×C7×#117C2214yes—
12C2×C7×#117C2214yes—
13C22×C7×C4⋊C4C2211yescharacteristic
14C22×C7×C4⋊C4C2211yescharacteristic
15C22×C7×C22⋊C4C2212yes—
16C22×C42×C7C2212yes—
17C2×#6489C2214yes—
18#90019C2214yes—
19#89984C2214yes—
20#89984C2214yes—
21#89984C2218yes—
22#89983C2214yes—
23C4×C22⋊Dic7C2214yes—
24C4×C22⋊Dic7C2214yes—
25C4×C22⋊Dic7C2218yes—
26#89757C22116yescomplemented
27C4⋊C4×Dic7C22116yescomplemented
28#89723C22132yes—
29#89709C2218yescomplemented
30#89708C2218yescomplemented
31C4×C4⋊Dic7C2218yescomplemented
32#89704C2218yescomplemented
33#34179—71nomaximal, Sylow
34C25×C7C2311yescharacteristic
35C23×C4×C7—28no—
36C23×C4×C7C2×C418yes—
37C23×C4×C7C2312yes—
38C23×C4×C7C2312yes—
39C23×C4×C7C2311yescharacteristic
40C23×C4×C7C2312yes—
41C23×C4×C7C2312yes—
42C23×C4×C7C2311yescharacteristic
43C7×#117C2×C4132yes—
44C7×#117C2×C4132yes—
45C2×C7×C4⋊C4C2×C418yes—
46C2×C7×C4⋊C4C2×C418yes—
47C2×C7×C4⋊C4C2314yes—
48C2×C7×C4⋊C4C2314yes—
49C2×C7×C22⋊C4C2×C4116yes—
50C2×C7×C22⋊C4C2318yes—
51C2×C42×C7C2×C4116yes—
52C2×C42×C7C2314yes—
53C2×C42×C7C2314yes—
54C2×C22⋊Dic7C2318yes—
55C2×C22⋊Dic7C2318yes—
56C2×C22⋊Dic7C2318yes—
57#6489—216no—
58C2×C4⋊Dic7C2318yes—
59C2×C4⋊Dic7C23116yes—
60C2×C4×Dic7C2318yes—
61C2×C4×Dic7C23116yes—
62C7⋊#95C2318yes—
63C7⋊#95C23116yes—
64C7⋊#95C2318yes—
65C7⋊#95C23116yes—
66C22×#117D711yescharacteristic
67#2347—78no—
68#2346—78no—
69#2316—72no—
70#2312—74no—
71#2283—74no—
72#2280—72no—
73#2274—72no—
74C24×C7—24no—
75C24×C7—42no—
76C24×C7—42no—
77C24×C7C22×C414yes—
78C24×C7C2412yes—
79C24×C7C2411yescharacteristic
80C22×C4×C7—24no—
81C22×C4×C7—232no—
82C22×C4×C7—28no—
83C22×C4×C7—28no—
84C22×C4×C7—44no—
85C22×C4×C7—44no—
86C22×C4×C7C22×C414yes—
87C22×C4×C7C22×C418yes—
88C22×C4×C7C22×C418yes—
89C22×C4×C7C22×C418yes—
90C22×C4×C7C22×C418yes—
91C22×C4×C7C22×C414yes—
92C22×C4×C7#4118yes—
93C22×C4×C7C2411yescharacteristic
94C22×C4×C7C2411yescharacteristic
95C22×C4×C7C2414yes—
96C22×C4×C7C2414yes—
97C22×C4×C7C2412yes—
98C22×Dic7C24116yes—
99C7×C4⋊C4C22×C4116yes—
100C7×C4⋊C4C22×C4116yes—
101C7×C22⋊C4C22×C4132yes—
102C42×C7C22×C4116yes—
103C42×C7C22×C4116yes—
104C22⋊Dic7—216no—
105C22⋊Dic7—48no—
106C22⋊Dic7—48no—
107C4⋊Dic7—216no—
108C4⋊Dic7—416no—
109C4×Dic7—216no—
110C4×Dic7—416no—
111C24×C4C2×D711yescharacteristic
112C2×#117—74no—
113C2×#117Dic7116yescomplemented
114C2×#117C2×D714yes—
115C2×#117C2×D714yes—
116C22×C4⋊C4C2×D711yescharacteristic
117C22×C4⋊C4C2×D711yescharacteristic
118C22×C22⋊C4C2×D712yes—
119C22×C42C2×D712yes—
120#389—716no—
121#388—732no—
122#383—78no—
123#382—78no—
124#381—74no—
125#380—78no—
126#380—74no—
127#380—74no—
128#379—74no—
129C4×C4⋊C4—716no—
130C4×C4⋊C4—78no—
131C4×C22⋊C4—78no—
132C4×C22⋊C4—74no—
133C4×C22⋊C4—74no—
134#376—78no—
135C23×C7—22no—
136C23×C7—22no—
137C23×C7—28no—
138C23×C7—24no—
139C23×C7—24no—
140C23×C7—24no—
141C23×C7—44no—
142C23×C7—44no—
143C23×C7—416no—
144C23×C7C23×C414yes—
145C23×C7C23×C412yes—
146C23×C7C2×#4114yes—
147C23×C7C2511yescharacteristic
148C2×C4×C7—28no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—28no—
152C2×C4×C7—28no—
153C2×C4×C7—216no—
154C2×C4×C7—216no—
155C2×C4×C7—48no—
156C2×C4×C7—48no—
157C2×C4×C7#117116yes—
158C2×C4×C7C23×C412yes—
159C2×C4×C7C23×C412yes—
160C2×C4×C7C23×C418yes—
161C2×C4×C7C23×C418yes—
162C2×C4×C7C23×C414yes—
163C2×C4×C7C2×#4118yes—
164C2×Dic7—216no—
165C2×Dic7—216no—
166C2×Dic7—216no—
167C25C22×D711yescharacteristic
168C23×C4—28no—
169C23×C4C2×Dic718yes—
170C23×C4C22×D712yes—
171C23×C4C22×D712yes—
172C23×C4C22×D711yescharacteristic
173C23×C4C22×D712yes—
174C23×C4C22×D712yes—
175C23×C4C22×D711yescharacteristic
176#117—1416no—
177#117C2×Dic7132yes—
178#117C2×Dic7132yes—
179C2×C4⋊C4—78no—
180C2×C4⋊C4—716no—
181C2×C4⋊C4C2×Dic718yes—
182C2×C4⋊C4C2×Dic718yes—
183C2×C4⋊C4C22×D714yes—
184C2×C4⋊C4C22×D714yes—
185C2×C22⋊C4—78no—
186C2×C22⋊C4—78no—
187C2×C22⋊C4—78no—
188C2×C22⋊C4C2×Dic7116yes—
189C2×C22⋊C4C22×D718yes—
190C2×C42—78no—
191C2×C42—716no—
192C2×C42C2×Dic7116yes—
193C2×C42C22×D714yes—
194C2×C42C22×D714yes—
195#95—78no—
196#95—716no—
197#95—78no—
198#95—716no—
199C22×C7—22no—
200C22×C7—22no—
201C22×C7—28no—
202C22×C7—24no—
203C22×C7—24no—
204C22×C7—24no—
205C22×C7—416no—
206C22×C7—48no—
207C22×C7C2×#11714yes—
208C22×C7C24×C411yescharacteristic
209C22×C7C22×#4112yes—
210C22×C7#58512yes—
211C22×C7#58512yes—
212C4×C7—22no—
213C4×C7—22no—
214C4×C7—28no—
215C4×C7—28no—
216C4×C7C2×#11718yes—
217Dic7—416no—
218C24—24no—
219C24—42no—
220C24—42no—
221C24C22×Dic714yes—
222C24C23×D712yes—
223C24C23×D711yescharacteristic
224C22×C4—24no—
225C22×C4—232no—
226C22×C4—28no—
227C22×C4—28no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—716no—
231C22×C4#115718yes—
232C22×C4C22×Dic714yes—
233C22×C4C22×Dic718yes—
234C22×C4C22×Dic718yes—
235C22×C4C22×Dic718yes—
236C22×C4C22×Dic718yes—
237C22×C4C22×Dic714yes—
238C22×C4C23×D711yescharacteristic
239C22×C4C23×D711yescharacteristic
240C22×C4C23×D714yes—
241C22×C4C23×D714yes—
242C22×C4C23×D712yes—
243C4⋊C4—1416no—
244C4⋊C4—2816no—
245C4⋊C4C22×Dic7116yes—
246C4⋊C4C22×Dic7116yes—
247C22⋊C4—1416no—
248C22⋊C4—288no—
249C22⋊C4—288no—
250C22⋊C4C22×Dic7132yes—
251C42—1416no—
252C42—2816no—
253C42C22×Dic7116yes—
254C42C22×Dic7116yes—
255C2×C7—24no—
256C2×C7—44no—
257C2×C7C22×#11711yescharacteristic
258C2×C7#342811yescharacteristic
259C2×C7#342811yescharacteristic
260C2×C7#349314yes—
261C23—22no—
262C23—22no—
263C23—28no—
264C23—24no—
265C23—24no—
266C23—24no—
267C23—44no—
268C23—44no—
269C23—416no—
270C23C2×#115714yes—
271C23C23×Dic714yes—
272C23C23×Dic712yes—
273C23C24×D711yescharacteristic
274C2×C4—28no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—28no—
278C2×C4—28no—
279C2×C4—216no—
280C2×C4—216no—
281C2×C4—48no—
282C2×C4—48no—
283C2×C4—1416no—
284C2×C4—1416no—
285C2×C4—1416no—
286C2×C4#6489116yes—
287C2×C4C2×#115718yes—
288C2×C4C23×Dic712yes—
289C2×C4C23×Dic712yes—
290C2×C4C23×Dic718yes—
291C2×C4C23×Dic718yes—
292C2×C4C23×Dic714yes—
293C7#3417911yescharacteristic, complemented, Sylow
294C22—22no—
295C22—22no—
296C22—28no—
297C22—24no—
298C22—24no—
299C22—24no—
300C22—416no—
301C22—48no—
302C22C2×#648914yes—
303C22C22×#115712yes—
304C22#9061612yes—
305C22#9061612yes—
306C22C24×Dic711yescharacteristic
307C4—22no—
308C4—22no—
309C4—28no—
310C4—28no—
311C4—2816no—
312C4C2×#648918yes—
313C2—24no—
314C2—44no—
315C2#1174281414yes—
316C2C22×#648911yescharacteristic
317C2#1174432811yescharacteristic
318C2#1174432811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D7
commutatorC22×C7C24×C4
FittingC22×C7×#117C2
socle, cosocleC23×C7C25
prime radicalC25×C7C23
hypercenterC22×#117D7
hypocenterC7#34179
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
3. #11739685C2semidirect
4. #11739684C2semidirect
5. #11739601C2semidirect
6. #11739589C2semidirect
8. #11739505C2semidirect
10. C2×C7×#117C4semidirect
26. #89757C22semidirect
27. C4⋊C4×Dic7C22semidirect
29. #89709C22semidirect
30. #89708C22semidirect
31. C4×C4⋊Dic7C22semidirect
32. #89704C22semidirect
113. C2×#117Dic7semidirect
293. C7#34179semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1172
#117414722
#117396852
#117396842
#117396012
#117395892
#117395382
#117395052
#341797