Finite Group Databasebeta

← #34015576

Subgroups — #34015576

Order 18688.

1530 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 272 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
1C2×#17989533C211yesmaximal, characteristic, complemented
2C2×#17989532C211yesmaximal, characteristic, complemented
3#22625116C216yesmaximal, complemented
4#22625114C216yesmaximal
5C2×#126×D73C211yesmaximal, characteristic
6C22×C4⋊#64408C2216yes—
7C22×C22⋊#64408C2214yes—
8#17989533—24no—
9#17989532—24no—
10C2×(C2×#64408)⋊C4C2216yes—
11C2×(C2×#64408)⋊C4C22112yes—
12C2×#14556586C413yescomplemented
13C2×#14556585C411yescharacteristic, complemented
14C2×#14556584C413yescomplemented
15#126×D73—24no—
16#126×D73—24no—
17C2×C4⋊C4×D73C2213yes—
18C2×C22⋊C4×D73C2213yes—
19C2×C42×D73C2211yescharacteristic
20C2×#126×C73C411yescharacteristic, complemented
21C23×#64408C2314yes—
22C23×#64408C2314yes—
23C2×C4⋊#64408—212no—
24C2×C4⋊#64408—412no—
25C2×C22⋊#64408—412no—
26(C2×#64408)⋊C4—212no—
27(C2×#64408)⋊C4—412no—
28C24×D73C2311yescharacteristic
29C22×C4×D73C2313yes—
30C22×C4×D73C2313yes—
31#14556586—212no—
32#14556585—24no—
33#14556584—212no—
34C2×C4⋊Dic73C2×C413yes—
35C2×#11924007C2×C413yes—
36C2×#11924007C2×C413yes—
37C4⋊C4×D73—26no—
38C4⋊C4×D73—26no—
39C2×C22⋊Dic73C2×C413yes—
40C2×#11924005C2×C413yes—
41C2×#11924005C2×C413yes—
42C22⋊C4×D73—26no—
43C22⋊C4×D73—26no—
44C2×C4×Dic73C2×C413yes—
45C42×D73D414yes—
46C42×D73D414yes—
47#126×C73—24no—
48C2×C4⋊C4×C73C2×C413yes—
49C2×C22⋊C4×C73C2×C413yes—
50C2×C42×C73C2×C411yescharacteristic
51C22×#64408—24no—
52C22×#64408—24no—
53C22×#64408—212no—
54C22×#64408—212no—
55C22×#64408—212no—
56C22×#64408—212no—
57C4⋊#64408—424no—
58C22⋊#64408—416no—
59C23×D73—21no—
60C23×D73—21no—
61C23×D73—26no—
62C23×D73—26no—
63C23×D73C22×C411yescharacteristic
64C23×D73C2411yescharacteristic
65C22×Dic73C22×C413yes—
66C22×Dic73C22×C413yes—
67C2×C4×D73—26no—
68C2×C4×D73—26no—
69C2×C4×D73—23no—
70C2×C4×D73—23no—
71C2×C4×D73—26no—
72C2×C4×D73—26no—
73C2×C4×D73C2×D416yes—
74C2×C4×D73C2×D416yes—
75C4⋊Dic73—26no—
76#11924007—26no—
77#11924007—26no—
78C22⋊Dic73—26no—
79#11924005—26no—
80#11924005—26no—
81C4×Dic73C22⋊C4112yescomplemented
82C24×C73C22×C411yescharacteristic
83C22×C4×C73C22×C413yes—
84C22×C4×C73C22×C413yes—
85C4⋊C4×C73—26no—
86C22⋊C4×C73—26no—
87C42×C73C22⋊C414yescomplemented
88C23×C73—21no—
89C23×C73—26no—
90C23×C73C23×C411yescharacteristic
91C22×D73—21no—
92C22×D73—26no—
93C22×D73—43no—
94C22×D73—43no—
95C22×D73—46no—
96C22×D73—44no—
97C22×D73—412no—
98C22×D73C22×D411yescharacteristic
99C22×D73C22×D411yescharacteristic
100C22×D73H(5,𝔽2)16yes—
101C22×D732−1+416yes—
102C2×#64408—412no—
103C2×#64408—412no—
104C2×#64408—44no—
105C2×#64408—44no—
106C2×#64408—412no—
107C2×#64408—412no—
108C2×C4×C73—26no—
109C2×C4×C73—23no—
110C2×C4×C73—26no—
111C2×C4×C73C2×C22⋊C416yes—
112C2×Dic73—26no—
113C2×Dic73—23no—
114C2×Dic73—26no—
115C2×Dic73C2×C22⋊C416yes—
116C4×D73—26no—
117C4×D73—26no—
118C4×D73—212no—
119C4×D73—43no—
120C4×D73—43no—
121C4×D73—46no—
122C22×C73—43no—
123C22×C73—44no—
124C22×C73C22×C22⋊C411yescharacteristic
125C22×C73#52016yes—
126C2×D73—43no—
127C2×D73—44no—
128C2×D73—81no—
129C2×D73—83no—
130C2×D73—84no—
131C2×D73#53513yes—
132C2×D73#53613yes—
133C2×D73#53716yes—
134C2×D73#56014yes—
135C2×D73#563112yes—
136#64408—84no—
137#64408—84no—
138C4×C73—26no—
139C4×C73—43no—
140Dic73—26no—
141Dic73—43no—
142#14665—731nomaximal, Sylow
143C2×C73—81no—
144C2×C73#231113yes—
145C2×C73#238314yes—
146D73—81no—
147D73#281811yescharacteristic
148D73#282113yes—
149D73#282314yes—
150C22×#126—731no—
151C2×#384—731no—
152C2×#383—731no—
153#1441—736no—
154#1439—736no—
155C73#1466511yescharacteristic, complemented, Sylow
156C2×#126—733no—
157C2×#126—733no—
158C2×#126—731no—
159C2×#126—1464no—
160C2×#126—1464no—
161C2×#126#6440811yescharacteristic, complemented
162C22×C4⋊C4—736no—
163C22×C4⋊C4—733no—
164C22×C22⋊C4—734no—
165C22×C22⋊C4—733no—
166C22×C42—731no—
167#384—1464no—
168#383—1464no—
169C2×#95—7312no—
170C2×#95—736no—
171C25—731no—
172C23×C4—733no—
173C23×C4—733no—
174C23×C4—734no—
175C23×C4—734no—
176#126—24no—
177#126—14612no—
178#126—14612no—
179#126—1464no—
180C2×C4⋊C4—29212no—
181C2×C4⋊C4—733no—
182C2×C4⋊C4—733no—
183C2×C4⋊C4—733no—
184C2×C4⋊C4—1466no—
185C2×C4⋊C4—1466no—
186C2×C4⋊C4—14612no—
187C2×C4⋊C4C2×#6440813yes—
188C2×C22⋊C4—29212no—
189C2×C22⋊C4—733no—
190C2×C22⋊C4—733no—
191C2×C22⋊C4—733no—
192C2×C22⋊C4—1466no—
193C2×C22⋊C4—1466no—
194C2×C22⋊C4C2×#6440813yes—
195C2×C42—733no—
196C2×C42—734no—
197C2×C42—734no—
198C2×C42C2×#6440811yescharacteristic
199#95—29212no—
200#95—14612no—
201C24—731no—
202C24—731no—
203C24—1466no—
204C24—1461no—
205C24—1461no—
206C24—1466no—
207C24C22×#6440811yescharacteristic
208C22×C4—733no—
209C22×C4—736no—
210C22×C4—736no—
211C22×C4—733no—
212C22×C4—1466no—
213C22×C4—1463no—
214C22×C4—1463no—
215C22×C4—1466no—
216C22×C4—1466no—
217C22×C4—1466no—
218C22×C4—14612no—
219C22×C4—14612no—
220C22×C4—1464no—
221C22×C4—14612no—
222C22×C4—1464no—
223C22×C4—14612no—
224C22×C4C22×#6440813yes—
225C22×C4C22×#6440813yes—
226C4⋊C4—26no—
227C4⋊C4—29224no—
228C4⋊C4—1466no—
229C4⋊C4—1466no—
230C4⋊C4—1466no—
231C22⋊C4—26no—
232C22⋊C4—29216no—
233C22⋊C4—1466no—
234C22⋊C4—1466no—
235C22⋊C4—1466no—
236C42—7312no—
237C42C22⋊#6440814yescomplemented
238C23—21no—
239C23—26no—
240C23—2924no—
241C23—2923no—
242C23—2926no—
243C23—2923no—
244C23—29212no—
245C23—731no—
246C23—731no—
247C23—736no—
248C23—736no—
249C23—1461no—
250C23—1466no—
251C23C23×#6440811yescharacteristic
252C2×C4—26no—
253C2×C4—23no—
254C2×C4—26no—
255C2×C4—2923no—
256C2×C4—2926no—
257C2×C4—2923no—
258C2×C4—2924no—
259C2×C4—2924no—
260C2×C4—29212no—
261C2×C4—29212no—
262C2×C4—29212no—
263C2×C4—29212no—
264C2×C4—736no—
265C2×C4—1466no—
266C2×C4—14612no—
267C2×C4—1466no—
268C2×C4—1463no—
269C2×C4—1466no—
270C2×C4—1466no—
271C2×C4C2×C22⋊#6440816yes—
272C22—43no—
273C22—44no—
274C22—2923no—
275C22—2924no—
276C22—5841no—
277C22—5844no—
278C22—5843no—
279C22—733no—
280C22—736no—
281C22—733no—
282C22—734no—
283C22—7312no—
284C22C22×C22⋊#6440811yescharacteristic
285C22#1798965616yes—
286C4—26no—
287C4—43no—
288C4—2923no—
289C4—5844no—
290C4—5844no—
291C4—1466no—
292C2—81no—
293C2—5841no—
294C2—731no—
295C2—734no—
296C2—733no—
297C2#2262640613yes—
298C2#2262647814yes—
subgroupG/H
center, FrattiniC23C23×#64408
commutator, socleC23×C73C23×C4
FittingC2×#126×C73C4
prime radicalC24×D73C23
cosocleC23×D73C24
hypercenterC2×#126#64408
hypocenterC73#14665
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×#17989533C2semidirect
2. C2×#17989532C2semidirect
3. #22625116C2semidirect
12. C2×#14556586C4semidirect
13. C2×#14556585C4semidirect
14. C2×#14556584C4semidirect
20. C2×#126×C73C4semidirect
81. C4×Dic73C22⋊C4semidirect
87. C42×C73C22⋊C4semidirect
155. C73#14665semidirect
161. C2×#126#64408semidirect
237. C42C22⋊#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179895332
C2×#179895322
#226251162
#226251142
C2×#126×D732
#1466573