Finite Group Databasebeta

← #34077124

Subgroups — #34077124

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×#18012721C211yesmaximal, characteristic
2C2×#18012719C211yesmaximal, characteristic, complemented
3#23752144C212yesmaximal, complemented
4#23752143C211yesmaximal, characteristic, complemented
5#23752141C212yesmaximal, complemented
6#23752135C212yesmaximal, complemented
7#23752133C211yesmaximal, characteristic, complemented
8#23752128C212yesmaximal
9#23752124C211yesmaximal, characteristic, complemented
10#23752071C211yesmaximal, characteristic, complemented
11#397×C151C211yesmaximal, characteristic, complemented
12C22×Dic302C2211yescharacteristic
13C22×C22⋊D151C2211yescharacteristic
14C22×C4×D151C2211yescharacteristic
15#18012721—24no—
16#18012719—24no—
17C2×C4⋊Dic151C2212yes—
18C2×C4⋊Dic151C2211yescharacteristic
19C2×C4⋊Dic151C2212yes—
20C2×Dic151⋊C4C2211yescharacteristic
21C2×Dic151⋊C4C2212yes—
22C2×Dic151⋊C4C2212yes—
23C2×C22⋊Dic151C2211yescharacteristic
24C2×C22⋊Dic151C2212yes—
25C2×(C2×D151)⋊C4C2212yes—
26C2×(C2×D151)⋊C4C2211yescharacteristic
27C2×(C2×D151)⋊C4C2212yes—
28C151⋊#95C2211yescharacteristic
29C151⋊#95C2212yes—
30C151⋊#95C2212yes—
31C151⋊#95C2212yes—
32C2×C4⋊C4×C151C2211yescharacteristic
33C2×C22⋊C4×C151C2211yescharacteristic
34C2×C22⋊C4×C151C2212yes—
35#95×C151C2211yescharacteristic
36#95×C151C2212yes—
37C23×D151C2311yescharacteristic
38C2×Dic302—22no—
39C2×Dic302—42no—
40C2×C22⋊D151—22no—
41C2×C22⋊D151—42no—
42C22×Dic151C2311yescharacteristic
43C22×Dic151C2312yes—
44C22×Dic151C2312yes—
45C22×Dic151C2311yescharacteristic
46C22×Dic151C2311yescharacteristic
47C2×C4×D151—42no—
48C2×C4×D151D414yes—
49C4⋊Dic151—24no—
50C4⋊Dic151—22no—
51C4⋊Dic151—42no—
52Dic151⋊C4—22no—
53Dic151⋊C4—42no—
54Dic151⋊C4—42no—
55C22⋊Dic151—22no—
56C22⋊Dic151—42no—
57(C2×D151)⋊C4—24no—
58(C2×D151)⋊C4—22no—
59(C2×D151)⋊C4—42no—
60C24×C151C2311yescharacteristic
61C22×C4×C151C2312yes—
62C22×C4×C151C2312yes—
63C22×C4×C151C2311yescharacteristic
64C22×C4×C151C2311yescharacteristic
65C4⋊C4×C151—22no—
66C22⋊C4×C151—22no—
67C22⋊C4×C151—42no—
68C22×D151—21no—
69C22×D151—22no—
70C22×D151—22no—
71C22×D151—21no—
72C22×D151C2×D412yes—
73Dic302—44no—
74C22⋊D151—44no—
75C2×Dic151—22no—
76C2×Dic151—21no—
77C2×Dic151—22no—
78C2×Dic151—21no—
79C2×Dic151—22no—
80C2×Dic151—22no—
81C2×Dic151—22no—
82C2×Dic151—21no—
83C2×Dic151—22no—
84C2×Dic151—22no—
85C2×Dic151—21no—
86C2×Dic151—22no—
87C2×Dic151C2×D412yes—
88C4×D151—44no—
89C23×C151—21no—
90C23×C151—22no—
91C23×C151—21no—
92C23×C151—22no—
93C23×C151—21no—
94C23×C151C2411yescharacteristic
95C2×C4×C151—22no—
96C2×C4×C151—21no—
97C2×C4×C151—22no—
98C2×C4×C151—22no—
99C2×C4×C151—22no—
100C2×C4×C151—22no—
101C2×C4×C151—22no—
102C2×C4×C151—22no—
103C2×C4×C151—22no—
104C2×C4×C151C2×D412yes—
105C22×C151—41no—
106C22×C151—41no—
107C22×C151—41no—
108C22×C151—42no—
109C22×C151—42no—
110C22×C151C22×D411yescharacteristic
111C22×C151H(5,𝔽2)11yescharacteristic
112C22×C151H(5,𝔽2)12yes—
113C22×C1512−1+412yes—
114C22×C1512−1+411yescharacteristic
115C2×D151—22no—
116C2×D151—22no—
117C2×D151—22no—
118C2×D151—42no—
119C2×D151—42no—
120C4×C151—22no—
121C4×C151—41no—
122C4×C151—42no—
123C4×C151—42no—
124Dic151—22no—
125Dic151—41no—
126Dic151—42no—
127Dic151—41no—
128Dic151—42no—
129C2×C151—81no—
130C2×C151#53411yescharacteristic
131C2×C151#53611yescharacteristic
132C2×C151#53711yescharacteristic
133C2×C151#56212yes—
134C2×C151#56312yes—
135D151—42no—
136C151#281411yescharacteristic, complemented, Sylow
137#2814—1511nomaximal, Sylow
138C2×#123—1511no—
139C2×C22⋊Q8—1511no—
140#398—1511no—
141#398—1512no—
142#397—1512no—
143#397D15111yescharacteristic, complemented
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—1511no—
157C2×C4⋊C4—1512no—
158C2×C4⋊C4—1512no—
159C2×C4⋊C4—1512no—
160C2×C4⋊C4C2×D15111yescharacteristic
161C2×C22⋊C4—1511no—
162C2×C22⋊C4—1512no—
163C2×C22⋊C4—1511no—
164C2×C22⋊C4—1512no—
165C2×C22⋊C4—1512no—
166C2×C22⋊C4C2×D15111yescharacteristic
167C2×C22⋊C4C2×D15112yes—
168#95—1511no—
169#95—1512no—
170#95—1512no—
171#95—1512no—
172#95C2×D15111yescharacteristic
173#95C2×D15112yes—
174C24—1511no—
175C24C22×D15111yescharacteristic
176C2×Q8—3022no—
177C2×Q8—6042no—
178C2×D4—3022no—
179C2×D4—6042no—
180C22×C4—6042no—
181C22×C4—1512no—
182C22×C4—1511no—
183C22×C4—1512no—
184C22×C4—1511no—
185C22×C4—1511no—
186C22×C4—1514no—
187C22×C4C22×D15111yescharacteristic
188C22×C4C22×D15112yes—
189C22×C4C22×D15112yes—
190C22×C4C22×D15111yescharacteristic
191C4⋊C4—22no—
192C4⋊C4—3024no—
193C4⋊C4—3022no—
194C4⋊C4—3022no—
195C4⋊C4—6042no—
196C4⋊C4—6042no—
197C4⋊C4—6042no—
198C22⋊C4—22no—
199C22⋊C4—42no—
200C22⋊C4—3024no—
201C22⋊C4—3022no—
202C22⋊C4—3022no—
203C22⋊C4—6042no—
204C22⋊C4—6042no—
205C23—21no—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—3021no—
211C23—3022no—
212C23—3022no—
213C23—3021no—
214C23—1512no—
215C23C23×D15111yescharacteristic
216Q8—6044no—
217D4—6044no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—3022no—
228C2×C4—3021no—
229C2×C4—3022no—
230C2×C4—3021no—
231C2×C4—3022no—
232C2×C4—3021no—
233C2×C4—3022no—
234C2×C4—3022no—
235C2×C4—3022no—
236C2×C4—3022no—
237C2×C4—3021no—
238C2×C4—3022no—
239C2×C4—6044no—
240C2×C4—1512no—
241C2×C4D4×D15112yes—
242C22—42no—
243C22—41no—
244C22—41no—
245C22—41no—
246C22—42no—
247C22—3022no—
248C22—3022no—
249C22—3022no—
250C22—6042no—
251C22—6042no—
252C22C2×D4×D15111yescharacteristic
253C22#1801278411yescharacteristic
254C22#1801278512yes—
255C22#1801278611yescharacteristic
256C22#1801278612yes—
257C4—22no—
258C4—41no—
259C4—42no—
260C4—42no—
261C4—3022no—
262C4—6041no—
263C4—6042no—
264C4—6041no—
265C4—6042no—
266C2—81no—
267C2—6042no—
268C2#2375277311yescharacteristic
269C2#2375278211yescharacteristic
270C2#2375279111yescharacteristic
271C2#2375295912yes—
272C2#2375296412yes—
subgroupG/H
center, FrattiniC23C23×D151
commutator, socle, cosocleC23×C151C24
Fitting#397×C151C2
prime radicalC22×C22⋊D151C22
hypercenter#397D151
hypocenterC151#2814
LCSG ▷ C23×C151 ▷ C151
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. C2×#18012719C2semidirect
3. #23752144C2semidirect
4. #23752143C2semidirect
5. #23752141C2semidirect
6. #23752135C2semidirect
7. #23752133C2semidirect
9. #23752124C2semidirect
10. #23752071C2semidirect
11. #397×C151C2semidirect
136. C151#2814semidirect
143. #397D151semidirect

Maximal subgroups

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

subgroupindex
C2×#180127212
C2×#180127192
#237521442
#237521432
#237521412
#237521352
#237521332
#237521282
#237521242
#237520712
#397×C1512
#2814151