Finite Group Databasebeta

← #34077126

Subgroups — #34077126

Order 19328.

478 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 125 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×#18012724C211yesmaximal, characteristic
2C2×#18012719C211yesmaximal, characteristic, complemented
3#23752146C212yesmaximal, complemented
4#23752138C212yesmaximal, complemented
5#23752138C211yesmaximal, characteristic, complemented
6#23752134C212yesmaximal, complemented
7#23752132C211yesmaximal, characteristic, complemented
8#23752130C212yesmaximal
9#23752125C211yesmaximal, characteristic, complemented
10#23752071C211yesmaximal, characteristic, complemented
11#398×C151C211yesmaximal, characteristic, complemented
12C22×Dic302C2211yescharacteristic
13C22×D604C2211yescharacteristic
14C22×C4×D151C2211yescharacteristic
15#18012724—24no—
16#18012719—24no—
17C2×C4⋊Dic151C2212yes—
18C2×Dic151⋊C4C2211yescharacteristic
19C2×Dic151⋊C4C2212yes—
20C2×Dic151⋊C4C2212yes—
21C2×(C2×D151)⋊C4C2212yes—
22C2×(C2×D151)⋊C4C2211yescharacteristic
23C2×(C2×D151)⋊C4C2211yescharacteristic
24C2×(C2×D151)⋊C4C2211yescharacteristic
25C2×(C2×D151)⋊C4C2212yes—
26C2×(C2×D151)⋊C4C2212yes—
27C2×(C2×D151)⋊C4C2212yes—
28C151⋊#95C2211yescharacteristic
29C151⋊#95C2212yes—
30C151⋊#95C2212yes—
31C151⋊#95C2212yes—
32C2×C4⋊C4×C151C2211yescharacteristic
33C2×C4⋊C4×C151C2211yescharacteristic
34C2×C4⋊C4×C151C2212yes—
35#95×C151C2211yescharacteristic
36#95×C151C2212yes—
37C23×D151C2311yescharacteristic
38C23×D151C2311yescharacteristic
39C2×Dic302—22no—
40C2×Dic302—42no—
41C2×D604—22no—
42C2×D604—42no—
43C22×Dic151C2312yes—
44C22×Dic151C2312yes—
45C22×Dic151C2311yescharacteristic
46C22×Dic151C2311yescharacteristic
47C2×C4×D151—42no—
48C2×C4×D151D414yes—
49C4⋊Dic151—24no—
50Dic151⋊C4—22no—
51Dic151⋊C4—42no—
52Dic151⋊C4—42no—
53(C2×D151)⋊C4—24no—
54(C2×D151)⋊C4—22no—
55(C2×D151)⋊C4—22no—
56(C2×D151)⋊C4—22no—
57(C2×D151)⋊C4—42no—
58(C2×D151)⋊C4—42no—
59(C2×D151)⋊C4—42no—
60C22×C4×C151C2312yes—
61C22×C4×C151C2312yes—
62C22×C4×C151C2311yescharacteristic
63C22×C4×C151C2311yescharacteristic
64C22×C4×C151C2311yescharacteristic
65C4⋊C4×C151—22no—
66C4⋊C4×C151—22no—
67C4⋊C4×C151—42no—
68C22×D151—21no—
69C22×D151—22no—
70C22×D151—21no—
71C22×D151—22no—
72C22×D151—21no—
73C22×D151—21no—
74C22×D151—22no—
75C22×D151—21no—
76C22×D151—22no—
77C22×D151C2×D412yes—
78Dic302—44no—
79D604—44no—
80C2×Dic151—22no—
81C2×Dic151—21no—
82C2×Dic151—22no—
83C2×Dic151—22no—
84C2×Dic151—22no—
85C2×Dic151—22no—
86C2×Dic151—22no—
87C2×Dic151—22no—
88C2×Dic151—22no—
89C2×Dic151C2×D412yes—
90C4×D151—44no—
91C23×C151C2411yescharacteristic
92C2×C4×C151—22no—
93C2×C4×C151—21no—
94C2×C4×C151—22no—
95C2×C4×C151—21no—
96C2×C4×C151—22no—
97C2×C4×C151—22no—
98C2×C4×C151—22no—
99C2×C4×C151—22no—
100C2×C4×C151—21no—
101C2×C4×C151—22no—
102C2×C4×C151—21no—
103C2×C4×C151—22no—
104C2×C4×C151C2×D412yes—
105C22×C151C22×D411yescharacteristic
106C22×C151H(5,𝔽2)11yescharacteristic
107C22×C151H(5,𝔽2)12yes—
108C22×C1512−1+411yescharacteristic
109C22×C1512−1+412yes—
110C2×D151—22no—
111C2×D151—22no—
112C2×D151—22no—
113C2×D151—42no—
114C2×D151—42no—
115C2×D151—41no—
116C2×D151—41no—
117C2×D151—41no—
118C2×D151—42no—
119C2×D151—42no—
120C4×C151—22no—
121C4×C151—41no—
122C4×C151—41no—
123C4×C151—42no—
124C4×C151—42no—
125Dic151—22no—
126Dic151—42no—
127Dic151—41no—
128Dic151—42no—
129C2×C151#53411yescharacteristic
130C2×C151#53611yescharacteristic
131C2×C151#53711yescharacteristic
132C2×C151#56212yes—
133C2×C151#56312yes—
134D151—42no—
135D151—81no—
136C151#281411yescharacteristic, complemented, Sylow
137#2814—1511nomaximal, Sylow
138C2×#123—1511no—
139C2×C22⋊Q8—1511no—
140#398—1512no—
141#398D15111yescharacteristic, complemented
142#397—1511no—
143#397—1512no—
144#396—1512no—
145#395—1511no—
146#394—1512no—
147#393—1511no—
148#380—1511no—
149C22×Q8—1511no—
150C22×D4—1511no—
151C23×C4—1511no—
152#123—3024no—
153C22⋊Q8—3024no—
154C2×C4⋊C4—1512no—
155C2×C4⋊C4—1511no—
156C2×C4⋊C4—1512no—
157C2×C4⋊C4—1512no—
158C2×C4⋊C4C2×D15111yescharacteristic
159C2×C4⋊C4C2×D15111yescharacteristic
160C2×C4⋊C4C2×D15112yes—
161C2×C22⋊C4—1512no—
162C2×C22⋊C4—1511no—
163C2×C22⋊C4—1511no—
164C2×C22⋊C4—1511no—
165C2×C22⋊C4—1512no—
166C2×C22⋊C4—1512no—
167C2×C22⋊C4—1512no—
168#95—1511no—
169#95—1512no—
170#95—1512no—
171#95—1512no—
172#95C2×D15111yescharacteristic
173#95C2×D15112yes—
174C24—1511no—
175C24—1511no—
176C2×Q8—3022no—
177C2×Q8—6042no—
178C2×D4—3022no—
179C2×D4—6042no—
180C22×C4—6042no—
181C22×C4—1511no—
182C22×C4—1512no—
183C22×C4—1512no—
184C22×C4—1511no—
185C22×C4—1514no—
186C22×C4C22×D15111yescharacteristic
187C22×C4C22×D15112yes—
188C22×C4C22×D15112yes—
189C22×C4C22×D15111yescharacteristic
190C22×C4C22×D15111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—3022no—
195C4⋊C4—3024no—
196C4⋊C4—6042no—
197C4⋊C4—6042no—
198C22⋊C4—3024no—
199C22⋊C4—3022no—
200C22⋊C4—3022no—
201C22⋊C4—3022no—
202C22⋊C4—6042no—
203C22⋊C4—6042no—
204C22⋊C4—6042no—
205C23—3021no—
206C23—3022no—
207C23—3021no—
208C23—3022no—
209C23—3021no—
210C23—3021no—
211C23—3022no—
212C23—3021no—
213C23—3022no—
214C23—1512no—
215C23C23×D15111yescharacteristic
216Q8—6044no—
217D4—6044no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—3022no—
231C2×C4—3021no—
232C2×C4—3022no—
233C2×C4—3022no—
234C2×C4—3022no—
235C2×C4—3022no—
236C2×C4—3022no—
237C2×C4—3022no—
238C2×C4—3022no—
239C2×C4—6044no—
240C2×C4—1512no—
241C2×C4D4×D15112yes—
242C22—3022no—
243C22—3022no—
244C22—3022no—
245C22—6042no—
246C22—6042no—
247C22—6042no—
248C22—6041no—
249C22—6041no—
250C22—6041no—
251C22—6042no—
252C22C2×D4×D15111yescharacteristic
253C22#1801278411yescharacteristic
254C22#1801278512yes—
255C22#1801278611yescharacteristic
256C22#1801278712yes—
257C4—22no—
258C4—41no—
259C4—41no—
260C4—42no—
261C4—42no—
262C4—3022no—
263C4—6042no—
264C4—6041no—
265C4—6042no—
266C2—6042no—
267C2—12081no—
268C2#2375277311yescharacteristic
269C2#2375278711yescharacteristic
270C2#2375279111yescharacteristic
271C2#2375295812yes—
272C2#2375296612yes—
subgroupG/H
center, FrattiniC23C23×D151
commutator, socle, cosocleC23×C151C24
Fitting#398×C151C2
prime radicalC22×D604C22
hypercenter#398D151
hypocenterC151#2814
LCSG ▷ C23×C151 ▷ C151
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. C2×#18012719C2semidirect
3. #23752146C2semidirect
4. #23752138C2semidirect
5. #23752138C2semidirect
6. #23752134C2semidirect
7. #23752132C2semidirect
9. #23752125C2semidirect
10. #23752071C2semidirect
11. #398×C151C2semidirect
136. C151#2814semidirect
141. #398D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127242
C2×#180127192
#237521462
#237521382
#237521342
#237521322
#237521302
#237521252
#237520712
#398×C1512
#2814151