Finite Group Databasebeta

← C22×#17989491 #34029917

Subgroups — C22×#17989491 #34029917

Order 18688.

2062 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 2062 conjugacy classes of subgroups — too many to draw. Only its 240 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556650C212yesmaximal, complemented
2C2×#17989491C2112yesmaximal, direct product factor
3C22×#14556587C211yesmaximal, characteristic
4C22×C22⋊#64408—22no—
5C2×#14556650—26no—
6C2×#14556650C22112yescomplemented
7#17989491C22116yesdirect product factor
8C23×D292C2211yescharacteristic
9C23×C22⋊D73—22no—
10C2×#14556587—26no—
11C2×#14556587C416yescomplemented
12C2×#14556587C2216yes—
13C22×C4×Dic73C411yescharacteristic, complemented
14C23×D4×C73C411yescharacteristic, complemented
15C23×#64408—42no—
16C2×C22⋊#64408—212no—
17C2×C22⋊#64408—46no—
18#14556650—212no—
19#14556650C2318yescomplemented
20C24×D73D412yes—
21C22×D292—41no—
22C22×D292—43no—
23C22×D292C2×C413yes—
24C22×D292D416yes—
25C22×D292C2313yes—
26C22×C22⋊D73—212no—
27C22×C22⋊D73—212no—
28C22×C22⋊D73—416no—
29C23×Dic73—21no—
30#14556587—212no—
31#14556587—212no—
32#14556587C2×C4112yescomplemented
33#14556587C2314yes—
34C2×C4×Dic73—23no—
35C2×C4×Dic73C2×C416yescomplemented
36C25×C73—21no—
37C22×D4×C73—23no—
38C22×D4×C73—43no—
39C22×D4×C73—41no—
40C22×D4×C73C2×C416yescomplemented
41C23×C4×C73C2×C411yescharacteristic
42C22×#64408—412no—
43C22×#64408—88no—
44C22⋊#64408—28no—
45C22⋊#64408—412no—
46C23×D73—22no—
47C23×D73—26no—
48C23×D73—48no—
49C23×D73C22⋊C416yes—
50C23×D73C2×D416yes—
51C2×D292—43no—
52C2×D292—43no—
53C2×D292—46no—
54C2×D292—46no—
55C2×D292—43no—
56C2×D292—43no—
57C2×D292C22⋊C416yes—
58C2×D292C22×C413yes—
59C2×D292C2×D416yes—
60C2×D292C2411yescharacteristic
61C2×C22⋊D73—224no—
62C2×C22⋊D73—28no—
63C2×C22⋊D73—448no—
64C2×C22⋊D73—448no—
65C22×Dic73—28no—
66C22×Dic73—26no—
67C4×Dic73—26no—
68C4×Dic73C22×C414yescomplemented
69C24×C73—26no—
70C24×C73—48no—
71C24×C73—43no—
72C24×C73—41no—
73C24×C73C22⋊C411yescharacteristic
74C2×D4×C73—26no—
75C2×D4×C73—46no—
76C2×D4×C73—46no—
77C2×D4×C73—46no—
78C2×D4×C73—46no—
79C2×D4×C73C22×C414yescomplemented
80C22×C4×C73—23no—
81C22×C4×C73—21no—
82C22×C4×C73C22⋊C413yes—
83C22×C4×C73C22×C413yes—
84C23×C73—24no—
85C23×C73—24no—
86C23×C73—424no—
87C23×C73—46no—
88C23×C73—46no—
89C23×C73—812no—
90C23×C73—84no—
91C23×C73#9913yes—
92C23×C73#9911yescharacteristic
93C23×C73C2×C22⋊C413yes—
94C22×D73—26no—
95C22×D73—26no—
96C22×D73—26no—
97C22×D73—26no—
98C22×D73—48no—
99C22×D73—412no—
100C22×D73—412no—
101C22×D73—812no—
102C22×D73C2×C22⋊C416yes—
103C22×D73C22×D412yes—
104C2×#64408—48no—
105C2×#64408—824no—
106D4×C73—44no—
107D4×C73—412no—
108C2×C4×C73—23no—
109C2×C4×C73—23no—
110C2×C4×C73—23no—
111C2×C4×C73—23no—
112C2×C4×C73C2×C22⋊C413yes—
113C2×C4×C73C23×C411yescharacteristic
114C22⋊D73—448no—
115C22⋊D73—416no—
116C2×Dic73—224no—
117C2×Dic73—24no—
118D292—46no—
119D292—43no—
120D292—46no—
121D292—41no—
122C22×C73—26no—
123C22×C73—26no—
124C22×C73—48no—
125C22×C73—44no—
126C22×C73—812no—
127C22×C73—812no—
128C22×C73#35314yescomplemented
129C22×C73C2×#9913yes—
130C22×C73C2×#9913yes—
131C22×C73C22×C22⋊C411yescharacteristic
132C2×D73—22no—
133C2×D73—26no—
134C2×D73—412no—
135C2×D73—412no—
136C2×D73—46no—
137C2×D73—42no—
138C2×D73—86no—
139C2×D73—86no—
140#64408—88no—
141C4×C73—23no—
142C4×C73—21no—
143Dic73—28no—
144C22×#353—731nomaximal, Sylow
145C2×C73—21no—
146C2×C73—23no—
147C2×C73—84no—
148C2×C73C2×#35316yescomplemented
149C2×C73C22×#9911yescharacteristic
150D73—42no—
151D73—46no—
152C22×#127—731no—
153C22×#99—732no—
154C2×#353—7312no—
155C73C22×#35311yescharacteristic, complemented, Sylow
156C23×D4—731no—
157C23×D4—1462no—
158C23×D4#6440811yescharacteristic, complemented
159C2×#127—736no—
160C2×#127—736no—
161C2×#127—1466no—
162C22×C22⋊C4—1462no—
163C22×C42—731no—
164C2×#99—7312no—
165C2×#99—1466no—
166#353—7316no—
167C25—21no—
168C25—732no—
169C22×D4—23no—
170C22×D4—43no—
171C22×D4—41no—
172C22×D4—2923no—
173C22×D4—2921no—
174C22×D4—29216no—
175C22×D4—736no—
176C22×D4—733no—
177C22×D4—733no—
178C22×D4—14612no—
179C22×D4—14612no—
180C22×D4C2×#6440816yescomplemented
181C23×C4—2922no—
182C23×C4—1461no—
183C23×C4C2×#6440811yescharacteristic
184#127—734no—
185#127—7312no—
186#127—14612no—
187#127—14612no—
188C2×C22⋊C4—2926no—
189C2×C22⋊C4—14612no—
190C2×C42—736no—
191C2×C42—1463no—
192#99—738no—
193#99—14612no—
194C24—26no—
195C24—48no—
196C24—43no—
197C24—41no—
198C24—2928no—
199C24—736no—
200C24—736no—
201C24—1462no—
202C24—1466no—
203C24C22⋊#6440811yescharacteristic
204C2×D4—26no—
205C2×D4—46no—
206C2×D4—46no—
207C2×D4—46no—
208C2×D4—46no—
209C2×D4—2926no—
210C2×D4—2926no—
211C2×D4—2923no—
212C2×D4—2923no—
213C2×D4—2923no—
214C2×D4—2923no—
215C2×D4—29248no—
216C2×D4—29248no—
217C2×D4—733no—
218C2×D4—731no—
219C2×D4—736no—
220C2×D4—736no—
221C2×D4—1468no—
222C2×D4—14624no—
223C2×D4C22×#6440814yescomplemented
224C22×C4—23no—
225C22×C4—21no—
226C22×C4—29212no—
227C22×C4—5848no—
228C22×C4—1468no—
229C22×C4—1466no—
230C22×C4C22⋊#6440813yes—
231C22×C4C22×#6440813yes—
232C22⋊C4—29212no—
233C22⋊C4—1468no—
234C42—734no—
235C42—1466no—
236C23—24no—
237C23—24no—
238C23—424no—
239C23—46no—
240C23—46no—
241C23—812no—
242C23—84no—
243C23—29212no—
244C23—29212no—
245C23—2928no—
246C23—58412no—
247C23—732no—
248C23—736no—
249C23—1466no—
250C23—1466no—
251C23—1466no—
252C23—1466no—
253C23#1455664913yes—
254C23#1455664911yescharacteristic
255C23C2×C22⋊#6440813yes—
256D4—44no—
257D4—412no—
258D4—2921no—
259D4—2923no—
260D4—2926no—
261D4—2926no—
262D4—29216no—
263D4—29248no—
264C2×C4—23no—
265C2×C4—23no—
266C2×C4—23no—
267C2×C4—23no—
268C2×C4—2928no—
269C2×C4—58424no—
270C2×C4—14624no—
271C2×C4—1464no—
272C2×C4C2×C22⋊#6440813yes—
273C2×C4C23×#6440811yescharacteristic
274C22—26no—
275C22—26no—
276C22—48no—
277C22—44no—
278C22—812no—
279C22—812no—
280C22—2922no—
281C22—2926no—
282C22—29212no—
283C22—29212no—
284C22—5846no—
285C22—5846no—
286C22—1462no—
287C22—1466no—
288C22#1798949114yesdirect product factor
289C22C2×#1455664913yes—
290C22C2×#1455664913yes—
291C22C22×C22⋊#6440811yescharacteristic
292C4—21no—
293C4—23no—
294C4—5848no—
295C4—1468no—
296C2—21no—
297C2—23no—
298C2—84no—
299C2—2922no—
300C2—2926no—
301C2C2×#1798949116yesdirect product factor
302C2C22×#1455664911yescharacteristic
subgroupG/H
centerC23#14556649
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
FittingC23×D4×C73C4
socleC23×C73#99
prime radicalC22×#14556587C2
cosocleC2×D292C24
hypercenterC23×D4#64408
hypocenterC73C22×#353
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C22×#14556650C2semidirect
2. C2×#17989491C2direct (N × H)
6. C2×#14556650C22semidirect
7. #17989491C22direct (N × H)
11. C2×#14556587C4semidirect
13. C22×C4×Dic73C4semidirect
14. C23×D4×C73C4semidirect
19. #14556650C23semidirect
32. #14556587C2×C4semidirect
35. C2×C4×Dic73C2×C4semidirect
40. C22×D4×C73C2×C4semidirect
68. C4×Dic73C22×C4semidirect
79. C2×D4×C73C22×C4semidirect
128. C22×C73#353semidirect
148. C2×C73C2×#353semidirect
155. C73C22×#353semidirect
158. C23×D4#64408semidirect
180. C22×D4C2×#64408semidirect
223. C2×D4C22×#64408semidirect
288. C22#17989491direct (N × H)
301. C2C2×#17989491direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145566502
C2×#179894912
C22×#145565872
C22×#35373