Finite Group Databasebeta

← #34077121

Subgroups — #34077121

Order 19328.

478 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 133 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×#18012722C211yesmaximal, characteristic
2#23752145C212yesmaximal, complemented
3#23752143C211yesmaximal, characteristic, complemented
4#23752140C212yesmaximal, complemented
5#23752138C211yesmaximal, characteristic, complemented
6#23752134C212yesmaximal, complemented
7#23752131C211yesmaximal, characteristic, complemented
8#23752127C212yesmaximal
9#23752123C211yesmaximal, characteristic, complemented
10#23752072C211yesmaximal, characteristic, complemented
11C2×C22⋊Q8×C151C211yesmaximal, characteristic, complemented
12C22×C22⋊D151C2211yescharacteristic
13#18012722—24no—
14C2×C4⋊Dic151C2212yes—
15C2×Dic151⋊C4C2211yescharacteristic
16C2×Dic151⋊C4C2211yescharacteristic
17C2×Dic151⋊C4C2212yes—
18C2×Dic151⋊C4C2212yes—
19C2×C22⋊Dic151C2211yescharacteristic
20C2×C22⋊Dic151C2212yes—
21C2×(C2×D151)⋊C4C2211yescharacteristic
22C2×(C2×D151)⋊C4C2211yescharacteristic
23C2×(C2×D151)⋊C4C2212yes—
24C2×(C2×D151)⋊C4C2212yes—
25C151⋊#95C2211yescharacteristic
26C151⋊#95C2211yescharacteristic
27C151⋊#95C2212yes—
28C151⋊#95C2212yes—
29C151⋊#95C2212yes—
30C151⋊#95C2212yes—
31C22×Q8×C151C2211yescharacteristic
32C23×C4×C151C2211yescharacteristic
33C22⋊Q8×C151—24no—
34C2×C4⋊C4×C151C2212yes—
35C2×C4⋊C4×C151C2211yescharacteristic
36C2×C22⋊C4×C151C2212yes—
37C23×D151C2311yescharacteristic
38C2×C22⋊D151—22no—
39C2×C22⋊D151—42no—
40C22×Dic151C2311yescharacteristic
41C22×Dic151C2312yes—
42C22×Dic151C2312yes—
43C22×Dic151C2311yescharacteristic
44C22×Dic151C2311yescharacteristic
45C4⋊Dic151—42no—
46Dic151⋊C4—22no—
47Dic151⋊C4—22no—
48Dic151⋊C4—42no—
49Dic151⋊C4—42no—
50C22⋊Dic151—22no—
51C22⋊Dic151—42no—
52(C2×D151)⋊C4—22no—
53(C2×D151)⋊C4—22no—
54(C2×D151)⋊C4—42no—
55(C2×D151)⋊C4—42no—
56C24×C151C2311yescharacteristic
57C2×Q8×C151—22no—
58C2×Q8×C151—42no—
59C22×C4×C151—42no—
60C22×C4×C151D414yes—
61C22×C4×C151C2312yes—
62C22×C4×C151C2312yes—
63C22×C4×C151C2311yescharacteristic
64C22×C4×C151C2311yescharacteristic
65C4⋊C4×C151—24no—
66C4⋊C4×C151—22no—
67C22⋊C4×C151—24no—
68C22×D151—21no—
69C22×D151—21no—
70C22×D151—22no—
71C22×D151—21no—
72C22×D151—22no—
73C22⋊D151—44no—
74C2×Dic151—21no—
75C2×Dic151—21no—
76C2×Dic151—21no—
77C2×Dic151—22no—
78C2×Dic151—22no—
79C2×Dic151—22no—
80C2×Dic151—21no—
81C2×Dic151—22no—
82C2×Dic151—22no—
83C2×Dic151—21no—
84C2×Dic151—22no—
85C2×Dic151—22no—
86C2×Dic151—22no—
87C23×C151—21no—
88C23×C151—22no—
89C23×C151—21no—
90C23×C151—22no—
91C23×C151C2×D412yes—
92C23×C151C2411yescharacteristic
93Q8×C151—44no—
94C2×C4×C151—22no—
95C2×C4×C151—22no—
96C2×C4×C151—22no—
97C2×C4×C151—22no—
98C2×C4×C151—22no—
99C2×C4×C151—22no—
100C2×C4×C151—22no—
101C2×C4×C151—22no—
102C2×C4×C151—44no—
103C2×C4×C151C2×D412yes—
104C2×C4×C151C2×D412yes—
105C22×C151—22no—
106C22×C151—22no—
107C22×C151—22no—
108C22×C151—42no—
109C22×C151—42no—
110C22×C151C22×D411yescharacteristic
111C22×C151H(5,𝔽2)12yes—
112C22×C151H(5,𝔽2)11yescharacteristic
113C22×C1512−1+411yescharacteristic
114C22×C1512−1+412yes—
115C2×D151—41no—
116C2×D151—41no—
117C2×D151—41no—
118C2×D151—42no—
119C2×D151—42no—
120C4×C151—22no—
121C4×C151—22no—
122C4×C151—42no—
123C4×C151—42no—
124Dic151—41no—
125Dic151—41no—
126Dic151—41no—
127Dic151—42no—
128Dic151—42no—
129C2×C151—42no—
130C2×C151#53411yescharacteristic
131C2×C151#53611yescharacteristic
132C2×C151#53711yescharacteristic
133C2×C151#56212yes—
134C2×C151#56312yes—
135D151—81no—
136C151#281411yescharacteristic, complemented, Sylow
137#2814—1511nomaximal, Sylow
138C2×#123—1511no—
139C2×C22⋊Q8D15111yescharacteristic, complemented
140#398—1511no—
141#398—1512no—
142#397—1511no—
143#397—1512no—
144#396—1512no—
145#395—1511no—
146#394—1512no—
147#393—1511no—
148#380—1511no—
149C22×Q8C2×D15111yescharacteristic
150C22×D4—1511no—
151C23×C4C2×D15111yescharacteristic
152#123—3024no—
153C22⋊Q8—24no—
154C2×C4⋊C4—1511no—
155C2×C4⋊C4—1511no—
156C2×C4⋊C4—1512no—
157C2×C4⋊C4—1512no—
158C2×C4⋊C4—1512no—
159C2×C4⋊C4C2×D15112yes—
160C2×C4⋊C4C2×D15111yescharacteristic
161C2×C22⋊C4—1511no—
162C2×C22⋊C4—1511no—
163C2×C22⋊C4—1511no—
164C2×C22⋊C4—1512no—
165C2×C22⋊C4—1512no—
166C2×C22⋊C4—1512no—
167C2×C22⋊C4C2×D15112yes—
168#95—1511no—
169#95—1511no—
170#95—1512no—
171#95—1512no—
172#95—1512no—
173#95—1512no—
174C24—1511no—
175C24C22×D15111yescharacteristic
176C2×Q8—22no—
177C2×Q8—42no—
178C2×D4—3022no—
179C2×D4—6042no—
180C22×C4—42no—
181C22×C4—1511no—
182C22×C4—1512no—
183C22×C4—1512no—
184C22×C4—1511no—
185C22×C4—1511no—
186C22×C4C22⋊D15114yes—
187C22×C4C22×D15112yes—
188C22×C4C22×D15112yes—
189C22×C4C22×D15111yescharacteristic
190C22×C4C22×D15111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—3022no—
194C4⋊C4—3022no—
195C4⋊C4—6042no—
196C4⋊C4—6042no—
197C4⋊C4—6042no—
198C22⋊C4—24no—
199C22⋊C4—3022no—
200C22⋊C4—3022no—
201C22⋊C4—3022no—
202C22⋊C4—6042no—
203C22⋊C4—6042no—
204C22⋊C4—6042no—
205C23—22no—
206C23—21no—
207C23—21no—
208C23—22no—
209C23—3021no—
210C23—3022no—
211C23—3021no—
212C23—3021no—
213C23—3022no—
214C23C2×C22⋊D15112yes—
215C23C23×D15111yescharacteristic
216Q8—44no—
217D4—6044no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—44no—
227C2×C4—3021no—
228C2×C4—3021no—
229C2×C4—3021no—
230C2×C4—3022no—
231C2×C4—3022no—
232C2×C4—3022no—
233C2×C4—3022no—
234C2×C4—3022no—
235C2×C4—3021no—
236C2×C4—3021no—
237C2×C4—3022no—
238C2×C4—3022no—
239C2×C4—3022no—
240C2×C4C2×C22⋊D15112yes—
241C2×C4C2×C22⋊D15112yes—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—6042no—
248C22—6041no—
249C22—6041no—
250C22—6041no—
251C22—6042no—
252C22C22×C22⋊D15111yescharacteristic
253C22#1801278412yes—
254C22#1801278511yescharacteristic
255C22#1801278611yescharacteristic
256C22#1801278712yes—
257C4—22no—
258C4—22no—
259C4—42no—
260C4—42no—
261C4—6041no—
262C4—6041no—
263C4—6041no—
264C4—6042no—
265C4—6042no—
266C2—42no—
267C2—12081no—
268C2#2375277111yescharacteristic
269C2#2375278611yescharacteristic
270C2#2375279011yescharacteristic
271C2#2375295812yes—
272C2#2375296312yes—
subgroupG/H
center, FrattiniC23C23×D151
commutator, socle, cosocleC23×C151C24
FittingC2×C22⋊Q8×C151C2
prime radicalC22×C22⋊D151C22
hypercenterC2×C22⋊Q8D151
hypocenterC151#2814
LCSG ▷ C23×C151 ▷ C151
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #23752145C2semidirect
3. #23752143C2semidirect
4. #23752140C2semidirect
5. #23752138C2semidirect
6. #23752134C2semidirect
7. #23752131C2semidirect
9. #23752123C2semidirect
10. #23752072C2semidirect
11. C2×C22⋊Q8×C151C2semidirect
136. C151#2814semidirect
139. C2×C22⋊Q8D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127222
#237521452
#237521432
#237521402
#237521382
#237521342
#237521312
#237521272
#237521232
#237520722
C2×C22⋊Q8×C1512
#2814151