Finite Group Databasebeta

← C151×#23928 #50328853

Subgroups — C151×#23928 #50328853

Order 38656.

1696 conjugacy classes of subgroups; only the 302 normal subgroups are shown.

normal subgroups only

This group has 1696 conjugacy classes of subgroups — too many to draw. Only its 302 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×#536×C151C212yesmaximal
2C2×#451×C151C214yesmaximal, complemented
3#2001×C151C214yesmaximal, complemented
4#1873×C151C214yesmaximal, complemented
5#1780×C151C211yesmaximal, characteristic
6C22×#41×C151C2211yescharacteristic
7C23×Q8×C151C2212yes—
8C22×Q16×C151C2214yes—
9#536×C151—216no—
10C2×C4⋊Q8×C151—22no—
11C2×#124×C151—22no—
12C2×#123×C151—22no—
13C2×#123×C151—22no—
14C2×C22⋊Q8×C151C2214yes—
15C2×C22⋊Q8×C151C2214yes—
16C2×#117×C151C2212yes—
17#451×C151—216no—
18C2×#103×C151C2218yes—
19C2×C22⋊C8×C151C2214yes—
20C2×C22⋊C8×C151C2212yes—
21#340×C151C2214yes—
22C2×#41×C151—28no—
23C2×#41×C151—42no—
24C22×Q8×C151—22no—
25C22×Q8×C151—28no—
26C22×Q8×C151—28no—
27C22×Q8×C151—44no—
28C22×Q8×C151D411yescharacteristic
29C22×Q8×C151C2314yes—
30C22×D4×C151D411yescharacteristic
31C22×D4×C151D412yes—
32C23×C4×C151—22no—
33C23×C4×C151C2311yescharacteristic
34C23×C4×C151C2312yes—
35C2×Q16×C151—216no—
36C2×Q16×C151—44no—
37C22×C8×C151C2314yes—
38C4⋊Q8×C151—216no—
39#124×C151—216no—
40#123×C151—216no—
41#123×C151—216no—
42C22⋊Q8×C151—28no—
43C22⋊Q8×C151—28no—
44C22⋊Q8×C151—28no—
45C22⋊Q8×C151D4116yes—
46#117×C151—28no—
47C2×C4⋊C4×C151—24no—
48C2×C4⋊C4×C151—24no—
49C2×C4⋊C4×C151C2314yes—
50C2×C22⋊C4×C151—22no—
51C2×C22⋊C4×C151—24no—
52C2×C22⋊C4×C151—24no—
53C2×C42×C151—22no—
54#103×C151—216no—
55C22⋊C8×C151—28no—
56C22⋊C8×C151—42no—
57C24×C151C2×D411yescharacteristic
58C24×C151C2×D412yes—
59#41×C151—416no—
60C2×Q8×C151—28no—
61C2×Q8×C151—24no—
62C2×Q8×C151—24no—
63C2×Q8×C151—28no—
64C2×Q8×C151—28no—
65C2×Q8×C151—41no—
66C2×Q8×C151—432no—
67C2×Q8×C151—44no—
68C2×Q8×C151—44no—
69C2×Q8×C151C2×D418yes—
70C2×D4×C151—24no—
71C2×D4×C151—24no—
72C2×D4×C151—24no—
73C2×D4×C151—42no—
74C2×D4×C151—41no—
75C22×C4×C151—24no—
76C22×C4×C151—24no—
77C22×C4×C151—28no—
78C22×C4×C151—24no—
79C22×C4×C151—24no—
80C22×C4×C151—24no—
81C22×C4×C151—22no—
82C22×C4×C151—44no—
83C22×C4×C151—44no—
84C22×C4×C151C2×D411yescharacteristic
85C22×C4×C151C2×D412yes—
86C22×C4×C151C2×D418yes—
87C22×C4×C151C2411yescharacteristic
88Q16×C151—416no—
89C2×C8×C151—28no—
90C2×C8×C151—24no—
91C4⋊C4×C151—216no—
92C4⋊C4×C151—216no—
93C4⋊C4×C151—24no—
94C4⋊C4×C151C2×D418yes—
95C22⋊C4×C151—28no—
96C22⋊C4×C151—216no—
97C22⋊C4×C151—216no—
98C42×C151—28no—
99C23×C151—22no—
100C23×C151—22no—
101C23×C151—28no—
102C23×C151—22no—
103C23×C151—21no—
104C23×C151—24no—
105C23×C151#12014yes—
106C23×C151C22×D411yescharacteristic
107Q8×C151—216no—
108Q8×C151—44no—
109Q8×C151—416no—
110Q8×C151—48no—
111D4×C151—48no—
112D4×C151—44no—
113C2×C4×C151—22no—
114C2×C4×C151—22no—
115C2×C4×C151—216no—
116C2×C4×C151—28no—
117C2×C4×C151—22no—
118C2×C4×C151—28no—
119C2×C4×C151—21no—
120C2×C4×C151—416no—
121C2×C4×C151—44no—
122C2×C4×C151—44no—
123C2×C4×C151—48no—
124C2×C4×C151—44no—
125C2×C4×C151#12014yes—
126C2×C4×C151C22×D414yes—
127C2×C4×C151H(5,𝔽2)12yes—
128C8×C151—44no—
129C22×C151—24no—
130C22×C151—28no—
131C22×C151—44no—
132C22×C151—44no—
133C22×C151—41no—
134C22×C151—42no—
135C22×C151—44no—
136C22×C151C2×#12012yes—
137C22×C151#53411yescharacteristic
138C22×C151#57814yes—
139C4×C151—24no—
140C4×C151—41no—
141C4×C151—48no—
142C4×C151—48no—
143C4×C151D4214yes—
144C2×C151—44no—
145C2×C151—81no—
146C2×C151#240311yescharacteristic
147C2×C151#301114yes—
148C2×C151#320212yes—
149#23928C15111yesmaximal, characteristic, direct product factor, Sylow
150C151#2392811yescharacteristic, direct product factor, Sylow
151C2×#536C2×C15112yes—
152C2×#451C2×C15114yescomplemented
153#2001C2×C15114yescomplemented
154#1873C2×C15114yescomplemented
155#1780C2×C15111yescharacteristic
156C22×#41C22×C15111yescharacteristic
157C23×Q8C22×C15112yes—
158C22×Q16C22×C15114yes—
159#536—216no—
160C2×C4⋊Q8—22no—
161C2×#124—22no—
162C2×#123—22no—
163C2×#123—22no—
164C2×C22⋊Q8C22×C15114yes—
165C2×C22⋊Q8C22×C15114yes—
166C2×#117C22×C15112yes—
167#451—216no—
168C2×#103C22×C15118yes—
169C2×C22⋊C8C22×C15114yes—
170C2×C22⋊C8C22×C15112yes—
171#340C22×C15114yes—
172C2×#41—28no—
173C2×#41—42no—
174C22×Q8—28no—
175C22×Q8—22no—
176C22×Q8—28no—
177C22×Q8—44no—
178C22×Q8D4×C15111yescharacteristic
179C22×Q8C23×C15114yes—
180C22×D4D4×C15111yescharacteristic
181C22×D4D4×C15112yes—
182C23×C4—22no—
183C23×C4C23×C15111yescharacteristic
184C23×C4C23×C15112yes—
185C2×Q16—216no—
186C2×Q16—44no—
187C22×C8C23×C15114yes—
188C4⋊Q8—216no—
189#124—216no—
190#123—216no—
191#123—216no—
192C22⋊Q8—28no—
193C22⋊Q8—28no—
194C22⋊Q8—28no—
195C22⋊Q8D4×C151116yes—
196#117—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4C23×C15114yes—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C42—22no—
204#103—216no—
205C22⋊C8—28no—
206C22⋊C8—42no—
207C24C2×D4×C15111yescharacteristic
208C24C2×D4×C15112yes—
209#41—416no—
210C2×Q8—28no—
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—28no—
214C2×Q8—28no—
215C2×Q8—41no—
216C2×Q8—432no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8C2×D4×C15118yes—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—41no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—28no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4C2×D4×C15111yescharacteristic
235C22×C4C2×D4×C15112yes—
236C22×C4C2×D4×C15118yes—
237C22×C4C24×C15111yescharacteristic
238Q16—416no—
239C2×C8—28no—
240C2×C8—24no—
241C4⋊C4—216no—
242C4⋊C4—216no—
243C4⋊C4—24no—
244C4⋊C4C2×D4×C15118yes—
245C22⋊C4—28no—
246C22⋊C4—216no—
247C22⋊C4—216no—
248C42—28no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—22no—
253C23—21no—
254C23—24no—
255C23#120×C15114yes—
256C23C22×D4×C15111yescharacteristic
257Q8—216no—
258Q8—44no—
259Q8—416no—
260Q8—48no—
261D4—48no—
262D4—44no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—216no—
266C2×C4—28no—
267C2×C4—28no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—416no—
271C2×C4—44no—
272C2×C4—44no—
273C2×C4—48no—
274C2×C4—44no—
275C2×C4#120×C15114yes—
276C2×C4C22×D4×C15114yes—
277C2×C4H(5,𝔽2)×C15112yes—
278C8—44no—
279C22—24no—
280C22—28no—
281C22—44no—
282C22—44no—
283C22—41no—
284C22—42no—
285C22—44no—
286C22C2×#120×C15112yes—
287C22#534×C15111yescharacteristic
288C22#578×C15114yes—
289C4—24no—
290C4—41no—
291C4—48no—
292C4—48no—
293C4D42×C15114yes—
294C2—44no—
295C2—81no—
296C2#2403×C15111yescharacteristic
297C2#3011×C15114yes—
298C2#3202×C15112yes—
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×#41×C151C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C22×#41×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#451×C151C2semidirect
3. #2001×C151C2semidirect
4. #1873×C151C2semidirect
149. #23928C151direct (N × H)
150. C151#23928direct (N × H)
152. C2×#451C2×C151semidirect
153. #2001C2×C151semidirect
154. #1873C2×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#536×C1512
C2×#451×C1512
#2001×C1512
#1873×C1512
#1780×C1512
#23928151